Universidad Polit ecnica de Madrid - core.ac.uk · Ingeniero Industrial 2 0 1 6. Departamento de...

T E S I S D O C T O R A L

Universidad Politecnica de Madrid

Energy-Entropy-Momentum TimeIntegration Methods for CoupledSmooth Dissipative Problems

Sergio Conde MartınIngeniero Industrial

2 0 1 6

Departamento de Mecanica de Medios continuos

y Teorıa de Estructuras

Escuela Tecnica Superior de Ingenieros de

Caminos, Canales y Puertos

Energy-Entropy-Momentum Time

Integration Methods for Coupled

Smooth Dissipative Problems

T E S I S D O C T O R A L

A U T O R

Sergio Conde Martın

Ingeniero Industrial

D I R E C T O R

Juan Carlos Garcıa Orden

Doctor Ingeniero Aeronautico

The composition of the present text was made by LATEX,C++, Python, TikZ, Gnuplot, Paraview, Octave, Gmsh.

Madrid, 2016.Author: Sergio Conde MartınEmail: [email protected]

mailto:[email protected]

Tribunal nombrado por el Mgfco. y Excmo. Sr. Rector de la Universidad Politec-nica de Madrid, el dıa 18 de Diciembre de 2015.

Presidente D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vocal D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vocal D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Vocal D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Secretario D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Realizado el acto de defensa y lectura de la Tesis el dıa . . . . . de . . . . . . . . . . . de2016 en la E.T.S. de Ingenieros de Caminos, Canales y Puertos de la U.P.M.

Calificacion: . . . . . . . . . . . . . . . . . . . . . .

EL PRESIDENTE LOS VOCALES

EL SECRETARIO

To Vicky and Laura

Abstract

This dissertation is concerned with the formulation, analysis and imple-mentation of structure-preserving time integration methods for the solutionof the initial(-boundary) value problems describing the dynamics of smoothdissipative systems, either finite- or infinite-dimensional ones. Such systemsare understood as those involving thermo-mechanical coupling and/or inter-nal dissipative effects modeled by internal state variables considered to besmooth in the sense that their evolutions follow continuos laws. The dynam-ics of such systems are ruled by the laws of thermodynamics and symmetrieswhich constitutes the structure meant to be preserved in the numerical set-ting.

For that, dissipative systems are geometrically described by metriplecticstructures which clearly identify the reversible and irreversible parts of theirdynamical evolution. In particular, the framework known by the acronymGENERIC is used to reveal the systems’ dissipative structure in the same wayas the Hamiltonian is for conserving systems. Given that, energy-preserving,entropy-producing and momentum-preserving (EEM) second-order accuratemethods are formulated using the discrete derivative operator that enabledthe formulation of Energy-Momentum methods ensuring the preservation ofthe Hamiltonian and symmetries for conservative systems.

Following these guidelines, two kind of EEM methods are formulatedin terms of entropy and temperature as a thermodynamical state variable,involving important implications discussed throughout the dissertation. Re-markably, the formulation in temperature becomes central to accommodateDirichlet boundary conditions.

EEM methods are finally validated and proved to exhibit enhanced nu-merical stability and robustness properties compared to standard ones.

ix

Resumen

Esta tesis aborda la formulacion, analisis e implementacion de metodosnumericos de integracion temporal para la solucion de sistemas disipativossuaves de dimension finita o infinita de manera que su estructura continuasea conservada. Se entiende por dichos sistemas aquellos que involucranacoplamiento termo-mecanico y/o efectos disipativos internos modelados porvariables internas que siguen leyes continuas, de modo que su evolucion esconsiderada suave. La dinamica de estos sistemas esta gobernada por lasleyes de la termodinamica y simetrıas, las cuales constituyen la estructuraque se pretende conservar de forma discreta.

Para ello, los sistemas disipativos se describen geometricamente medianteestructuras metriplecticas que identifican claramente las partes reversible eirreversible de la evolucion del sistema. Ası, usando una de estas estructurasconocida por las siglas (en ingles) de GENERIC, la estructura disipativa delos sistemas es identificada del mismo modo que lo es la Hamiltoniana parasistemas conservativos. Con esto, metodos (EEM) con precision de segundoorden que conservan la energıa, producen entropıa y conservan los impulsoslineal y angular son formulados mediante el uso del operador derivada discretaintroducido para asegurar la conservacion de la Hamiltoniana y las simetrıasde sistemas conservativos.

Siguiendo estas directrices, se formulan dos tipos de metodos EEM basa-dos en el uso de la temperatura o de la entropıa como variable de estadotermodinamica, lo que presenta importantes implicaciones que se discuten alo largo de esta tesis. Entre las cuales cabe destacar que las condiciones decontorno de Dirichlet son naturalmente impuestas con la formulacion basadaen la temperatura.

Por ultimo, se validan dichos metodos y se comprueban sus mejoresprestaciones en terminos de la estabilidad y robustez en comparacion conmetodos estandar.

xi

Acknowledgements

Quiero empezar agradeciendo a mi director de tesis, el profesor JuanCarlos Garcıa Orden, por su apoyo constante, predisposicion permanente,paciencia, entusiasmo, buen hacer e incansable trabajo. Gracias por hacerde estos anos una experiencia amena y enriquecedora. Tambien me gustarıaagradecer al catedratico Ignacio Romero Olleros por compartir conmigo susconocimientos sobre metodos de integracion temporal termodinamicamenteconsistentes y por sus sabios consejos. A los profesores Felipe GabaldonCastillo y Juanjo Arribas debo agradecerles su apoyo diario.

Quiero dar las gracias al Prof. Javier Bonet por darme la oportunidad derealizar una fructıfera estancia de investigacion en su grupo, a cuyos miem-bros tambien quiero agradecer por su acogida y buenos ratos en Swansea.

I would also like to thank Prof. Peter Betsch for his support and keyadvice during my stay in the “Kalsruhe Institute fur Technologie” and forsponsoring my attendance to the instructive course on “Structure-preservingintegration methods” held in Udine.

Esta tesis tambien es fruto del apoyo, ayuda y tozudas discusiones que endiferentes etapas me han prestado mis companeros de fatigas: Roberto Or-tega, Mustapha el Hamdaoui, Michiel Fenaux, Cesar Polindara, Pablo An-tolın, Elena Pastuschuk, Damon Afkari, Javi Oliva, Khanh Nguyen y seguroque alguno mas que me dejo en el tintero y a los que pido disculpas. A todosellos, gracias por brindarme vuestra amistad.

Este trabajo de investigacion ha sido financiado por el programa de becaspropias de la Universidad Politecnica de Madrid, al cual me gustarıa agrade-cer por permitir a muchos como yo perseguir nuestras ambiciones y nuestrossuenos.

Quiero dar las gracias tambien a mis amigos de Alcala y Sevilla porestar siempre ahı, valorarme y aguantarme, en especial, a Jose Marıa Rivera

xiii

Rubio, “el doc”, por tu impagable apoyo e infatigable entusiasmo. Finalmente,lo hemos conseguido, enhorabuena a ti tambien.

A mi familia, en especial a mi hermana, por hacer de mi lo que soy ypor ensenarme el camino del esfuerzo y el valor de la humildad.

No puedo terminar sin agradecer por todo su amor, comprension y apoyoincondicional a la persona con la que he compartido todo lo bueno que me hapasado y la que siempre ha estado a mi lado en los momentos mas duros: miesposa, Vicky. Esta tesis es tan tuya como mıa.

Sergio Conde Martın

Contents

Abstract ix

Resumen xi

Acknowledgements xiii

Contents xv

1 Introduction1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Geometric Integration . . . . . . . . . . . . . . . . . . . . . . 3

1.3 The role of dissipation . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Structure-preserving integration for dissipative systems . . . . 7

1.4.1 On the selection of the thermodynamic variable . . . . 8

1.4.2 Dissipative effects: internal variables framework . . . . 10

1.4.3 Variational Integrators . . . . . . . . . . . . . . . . . . 11

1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Nonlinear thermo-dissipative discrete dynamics 17

xv

2.1 Two thermo-spring system . . . . . . . . . . . . . . . . . . . . 18

2.1.1 Two thermo-spring system in temperature variables . . 20

2.1.2 Two thermo-spring system in entropy variables . . . . 22

2.1.3 Symmetries and laws of thermodynamics . . . . . . . . 24

2.2 Thermo-visco-elastic system . . . . . . . . . . . . . . . . . . . 26

2.2.1 Thermo-visco-elastic system in temperature variables . 28

2.2.2 Thermo-visco-elastic system in entropy variables . . . . 31

2.2.3 Symmetries and law of thermodynamics . . . . . . . . 33

2.3 Standard numerical time integration methods . . . . . . . . . 34

2.3.1 Midpoint method for the two thermo-spring system . . 38

2.3.2 Midpoint method for the thermo-viscoelastic system . . 42

2.3.3 Numerical Experiments . . . . . . . . . . . . . . . . . . 44

3 Nonlinear thermo-dissipative continuum dynamics 59

3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Generalized Standard Materials . . . . . . . . . . . . . . . . . 62

3.2.1 Physical constitutive laws . . . . . . . . . . . . . . . . 63

3.2.2 Heat conduction . . . . . . . . . . . . . . . . . . . . . 65

3.2.3 The internal energy function . . . . . . . . . . . . . . . 66

3.2.4 Constitutive behaviors for limit cases . . . . . . . . . . 68

3.3 Balance laws: strong form of the initial-boundary value problem 69

3.3.1 Balance of mass . . . . . . . . . . . . . . . . . . . . . . 70

3.3.2 Balance of linear and angular momentum . . . . . . . . 70

3.3.3 Balance of energy . . . . . . . . . . . . . . . . . . . . . 72

3.3.4 Strong form of the governing equations . . . . . . . . . 75

3.3.5 Entropy formulation of the IBVP . . . . . . . . . . . . 77

3.3.6 Temperature formulation of the IBVP . . . . . . . . . 78

3.4 Specific thermo-dissipative material models . . . . . . . . . . . 80

3.5 Symmetries and laws of thermodynamics . . . . . . . . . . . . 82

3.6 Isothermal dynamics as a limit case . . . . . . . . . . . . . . . 83

3.6.1 Viscoelasticity at finite strain . . . . . . . . . . . . . . 85

3.6.2 Nonlinear plasticity at finite strain . . . . . . . . . . . 93

3.7 Semidiscrete thermo-dissipative dynamics . . . . . . . . . . . . 94

3.7.1 Variational statement of the IBVP . . . . . . . . . . . 96

3.7.2 Bunov-Galerkin Finite Element spatial discretization . 98

3.7.3 Element implementation . . . . . . . . . . . . . . . . . 103

3.8 Standard time integration methods . . . . . . . . . . . . . . . 105

3.8.1 Midpoint time integration method . . . . . . . . . . . . 106

3.8.2 Trapezoidal time integration method . . . . . . . . . . 112

4 Metriplectic structures: GENERIC formalism 115

4.1 Metriplectic structures . . . . . . . . . . . . . . . . . . . . . . 116

4.2 Finite-dimensional smooth dissipative systems . . . . . . . . . 118

4.2.1 Finite-dimensional dissipative systems with symmetries 121

4.3 GENERIC forms of the two thermo-spring system . . . . . . . 122

4.3.1 Entropy-based GENERIC formulation . . . . . . . . . 122

4.3.2 Temperature-based GENERIC formulation . . . . . . . 124

4.3.3 Laws of thermodynamics and symmetries . . . . . . . . 127

4.4 GENERIC form of the thermo-visco-elastic system . . . . . . 129

4.4.1 Entropy-based GENERIC formulation . . . . . . . . . 130

4.4.2 Temperature-based GENERIC formulation . . . . . . . 131

4.4.3 Laws of thermodynamics and symmetries . . . . . . . . 134

4.5 Infinite-dimensional dissipative systems . . . . . . . . . . . . . 137

4.5.1 Infinite-dimensional smooth dissipative systems withsymmetries . . . . . . . . . . . . . . . . . . . . . . . . 139

4.6 GENERIC form of nonlinear thermoelasticity . . . . . . . . . 141

4.6.1 Entropy formulation . . . . . . . . . . . . . . . . . . . 141

4.6.2 Temperature formulation . . . . . . . . . . . . . . . . . 144

4.7 GENERIC form of thermo-dissipative dynamics . . . . . . . . 147

4.7.1 Entropy formulation . . . . . . . . . . . . . . . . . . . 147

4.7.2 Temperature formulation . . . . . . . . . . . . . . . . . 149

4.8 Metriplectic structure for isothermal dissipative dynamics . . . 152

4.9 Symmetries of infinite-dimensional dissipative systems . . . . . 154

5 Thermodynamically Consistent Algorithms 159

5.1 Energy-Entropy-Momentum time integration methods . . . . . 160

5.2 General formulation of EEM methods . . . . . . . . . . . . . 161

5.2.1 Discrete finite-dimensional smooth dissipative systems 163

5.2.2 Discrete infinite-dimensional dissipative systems . . . . 165

5.3 EEM methods for the two thermo-spring system . . . . . . . . 169

5.3.1 Entropy-based EEM method . . . . . . . . . . . . . . . 170

5.3.2 Temperature-based EEM method . . . . . . . . . . . . 176

5.3.3 Validation and comparison with standard methods . . 182

5.4 EEM methods for the thermo-visco-elastic system . . . . . . . 191



5.4.3 Validation and comparison with standard methods . . 203

5.5 EEM methods for nonlinear thermoelasticity . . . . . . . . . . 211



5.6 EEM methods for nonlinear thermo-dissipative dynamics . . . 224



5.7 EEM method for nonlinear isothermal dissipative dynamics . . 232

6 Simulations237

6.1 Dynamics of isothermal viscoelastic solids . . . . . . . . . . . 238

6.1.1 Vibrating cantilever beam . . . . . . . . . . . . . . . . 238

6.1.2 A tumbling L-shaped block . . . . . . . . . . . . . . . 243

6.2 Dynamics of thermoelastic solids . . . . . . . . . . . . . . . . 248

6.2.1 Twisted Block . . . . . . . . . . . . . . . . . . . . . . . 248

6.2.2 L-shaped block with Dirichlet initial-boundary conditions255

6.3 Isothermal viscoelastic applications to multibody system dy-namics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

6.3.1 Multibody framework . . . . . . . . . . . . . . . . . . . 260

6.3.2 Rigid bodies and constraints . . . . . . . . . . . . . . . 260

6.3.3 Flexible multibody system . . . . . . . . . . . . . . . . 261

6.3.4 Consistent rigid bodies and constraints . . . . . . . . . 262

6.3.5 Consistent flexible multibody system . . . . . . . . . . 264

6.3.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . 264

7 Summary, conclusions and future work 283

A Discrete Derivative Operator 289

A.1 Mean value theorem . . . . . . . . . . . . . . . . . . . . . . . 289

A.2 The discrete derivative operator . . . . . . . . . . . . . . . . . 291

A.3 Discrete derivative operator: Examples . . . . . . . . . . . . . 295

A.4 Linearization of the discrete derivative operator . . . . . . . . 299

B Linearized balance laws and spatial discretization 303

B.1 Linearization and Newton-Raphson process . . . . . . . . . . . 303

B.2 Lagrangian linearization of the IBVP weak form . . . . . . . . 306

B.2.1 Linearization of the entropy formulation . . . . . . . . 307

B.2.2 Linearization of the temperature formulation . . . . . . 311

B.2.3 Spatially discretized evolution equations . . . . . . . . 314

B.2.4 Discretization of the linearized evolution equations . . 319

List of Figures i

Bibliography vii

Introduction

Chapter

1The main goal of this dissertation is to provide robust and stable nu-

merical integration methods that make use of all the mathematically richstructure that continuous dissipative systems have so as to provide reliablyphysical solutions.

1.1 Motivation

The current technological paradigm relies inevitably on computationalsimulations not only because of their positive impact on both production timeand costs but also because of their ability to provide solutions for not eventestable real phenomena. In parallel, a crucial factor for the consolidationof this technological paradigm lies at the extraordinary and still in progressincrement of computer power, which has enabled the addressing of everymore demanding problems.

The simulation of the dynamics of nonlinear mechanic systems often liesat the formulation of numerical methods whose solution, obtained throughcomputational means, provides an discrete approximation of the exact model.The aim of discrete solutions is not only to obtain approximated forces andtrajectories, but also to extract qualitative information so that a comprehen-sive understanding of the dynamical behavior of systems can be achieved.For this reason, numerical methods must not only be numerically accuratebut also be reliable in the capture of the main features of the dynamicalbehavior.

2 1.1. Motivation

The development of numerical methods of such characteristics for thesolution of any type of nonlinear problems has significantly progressed inthe last decades. Thus, Hamiltonian systems have attracted most of theattention due to its well-known mathematical (geometric) structure and itsimportance in different areas of the engineering and physics. The success ofsuch numerical methods, known in the specialized literature as geometric, isfounded on their efficiency to provide reliable solutions as well as their stabil-ity and accuracy for long term simulations, that are outstandingly superiorto standards numerical methods.

However, there are still a bunch of appealing problems in industry thatare not Hamiltonian, such as those involving any type of dissipative mecha-nisms: viscous or frictional ones; or those involving thermo effects that mod-ify the mechanic behavior of the systems. These problems will be referredto as dissipative/non-conserving problems as opposed to Hamiltonian/con-serving ones. Provided that these scenarios are very often in materials withindustrial interest, developing geometric-like methods for the obtention ofreliable simulations has become central. Unfortunately, the mathematicalstructure of these problems is not thus far as understood as the Hamilto-nian one, although recent works have made their way to provide a generalunderstanding of them from an unified perspective, which will doubtlesslycontribute to reach these lofty objectives. In this sense, this dissertation is ahumble contribution towards that target.

Among the most representative applications in different areas of industrywould be:

• Material engineering. Simulation of vulcanize rubber materials inany of the situations that can be required in everyday industry: fromtires to conveyor belts, marine products, windshield wipers, etc. Notonly the simulation of the type of aforementioned materials is of interestbut also their analyses can contribute to better understand their phys-ical and mechanical features, thus guiding the search for new materialsthat outperform them.

• Mechanical engineering. Shock absorbers of both automotive andrailway vehicles. Dynamical behavior of machinery whose parts cansuffer high gradient of temperatures that modifies the mechanical prop-erties.

• Civil engineering. Shock absorber elements such as the Tune-mass-Damper (TMD) typically used in the construction of bridges and high-rise buildings.

1. Introduction3

• Aerospace engineering. Solar panels of aircraft exposed to the directradiation due to The Sun during their maneuvers.

1.2 Geometric Integration

For the last decades, the formulation of geometric (or structure-preser-ving) methods have played a major role in the development of numericalmethods within the domain of computational mechanics. Monographs, suchas Hairer et al. (2006), summarize the large number of recent contribution tothe field. The emergence of this sort of formulations has completely changedthe way of approaching integration methods. Whereas they were traditionallyregarded as mere approximations to continuous problems, they are now alsoconsidered as dynamic discrete systems with their own characteristics.

Such methods primarily aim to integrate numerically evolution equationsarising from deformable or rigid solid dynamics, in such way that they in-herit as many qualitative features as possible from the continuous system.This approach arose in order to obtain more physically accurate solutions,which comply with each inherited feature modeled by the continuous prob-lem. The inherited features range from geometric aspects, symmetries of theequations, to first integrals. A well-known example prior to this approach isthe Midpoint method. Even though it was not developed within this context,it preserves symmetries and symplecticity, and can therefore be consideredas a structure-preserving method.

Initially, nonlinear elastodynamics attracted most of the researchers’ in-terest. Monographs such as Leimkuhler & Reich (2005) show evidence of thematurity of the field. The evolution equations involving nonlinear elastody-namics contain Hamiltonian or Lagrangian structures with suitable propertiesto define preserving methods. Soon, this suitability enabled the formulationof methods whose solutions successfully preserved (part of) the continuousstructure of the evolution equations, obtaining the expected physical accu-racy. In addition, structure preserving methods exhibit excellent numericalaccuracy and robustness when compared with standard methods. Analyseshave shown that these methods are remarkably accurate in long term sim-ulations and allow for the use of larger time steps than traditional implicitmethods. Moreover, energy preservation significantly improves stability androbustness, as demonstrated, for example, in Stuart & Humphries (1998).These observations inspired a fast development in different fields, amongwhich the most prominent are rigid body dynamics, nonlinear hyperelastic-ity, nonlinear structural mechanics and contact.

4 1.2. Geometric Integration

Chronologically, the first binding ideas relating the energy control tothe stability are due to Haug et al. (1977), establishing that, in the nonlin-ear regime, any integration method is unconditionally stable provided thatthe energy of the system at any time step is less or equal to the energyat its predecessor one for any time step size. Based on this criteria, theauthors therein proposed an energy-bounded integration method which col-lapses to the Trapezoidal method for linear problems. Similar conclusionswere made by Hughes et al. (1978), proposing an unconditionally stableenergy-conserving method for nonlinear problems based on a modificationof the Trapezoidal method. Later, Simo & Wong (1991) developed a second-order accurate integration method for the solution of the solid rigid dy-namics without constraints that preserves both energy and angular momen-tum and is hence unconditionally stable. Following this work, the so-calledEnergy-Momentum method was formulated for general Hamiltonian systemswith symmetries, particularly, for nonlinear elastodynamics of solids, beamsand shells in Simo & Gonzalez (1993); Simo & Tarnow (1992); Simo et al.(1995, 1992). These works laid the foundation for the formulation of Energy-Momentum methods that, by design, preserve first integrals of Hamiltoniansystems: the linear and angular momentum maps and the Hamiltonian ofthe system. Later, Gonzalez (1996) systematized the formulation by fullydeveloping the concept of the discrete derivative operator or discrete gradi-ent operator, which was previously discussed, at least, in Gotusso (1985) (seealso Itoh & Abe (1988), McLachlan et al. (1999)). This operator is basedon properties that ensure both the preservation of the first integrals and thesecond order accuracy of the discrete system.

Based on this systematic procedure, other authors proposed energy-mo-mentum methods for particular applications such as N-body systems Betsch& Steinmann (2000, 2001a), mechanical systems with holonomic constrainsBetsch & Steinmann (2002) and rigid body with restrictions Betsch & Stein-mann (2001b).

In the context of flexible multi-body system (FMS), Garcıa Orden (1999);Garcıa Orden & Goicolea (2000, 2005) developed an Energy-Momentummethod for the dynamics of flexible multi-body system formulated with fullygeometrically nonlinear displacement-based FEM for deformable bodies andwith the penalty method for joints, instead of other extended approaches suchas the Floating Frame of Reference (FFR) due to Shabana (2013). Therein,the preserving evaluation of the forces involved in different types of joints areprovided. In the same way, an alternative Energy-Momentum method forFMSs based on Augmented Lagrangian was proposed in Garcıa Orden & Or-tega Aguilera (2006). Also, the stabilization properties for the formulation of

1. Introduction5

Energy-Momentum methods for problems with constraints were reviewed inGarcıa Orden & Dopico (2007). On the other hand, the Energy-Momentummethod has also resulted in a powerful tool to study other numerical difficul-ties whose treatment alters the energy of the system. That is the case of thevelocity projection technique to alleviate the drift of the constrains when theyare approached by the penalty method, discussed in Garcıa Orden (2009) andGarcıa Orden & Conde Martın (2012).

1.3 The role of dissipation

The previous energy criteria also led to the construction of time integra-tion methods that introduced numerical, viz. artificial, dissipation so that themethod’ stability can be guaranteed by design for both linear and nonlinearproblems.

When a solid, structure or body is discretized, low frequency deforma-tion modes are well captured whereas high frequencies modes are very poorlyrepresented. Traditionally, the high frequencies have been considered “spuri-ous”, as they seemed to be related with the spatial discretization performedto find numerical solutions. This consideration suggested their eliminationso that the energy of the system can be controlled and hence stable integra-tion can be achieved. In addition, as experimental observations suggestedthat high frequencies were “naturally” damped out (air, viscosity, friction,... ), it was blindly accepted that the introduced artificial dissipation some-how compensated the physical (real) dissipation, being seldom considered inmathematical models due to the lack of knowledge about it. This modifica-tion may alter the dynamics of the system and, consequently may underminethe physical certainty of the solution thus obtained.

Often, spatial discretization techniques applied to nonlinear elastody-namics result in stiff ordinary differential equations which are difficult tosolve. The main drawbacks come from the coexistence of two highly differentranges of frequencies which usually pose instabilities in the integration. Thealleviation of these difficulties were attained by introducing controllable nu-merical dissipation so that the high frequencies are damped out. First, theHHT method named after their authors in Hilber et al. (1977), then its gen-eralization called α-Generalized method due to Chung & Hulbert (1993) oreven the Wilson−θ method Wilson (1968) became very successful in commer-cial software (Abaqus HHT, ADINA Wilson−θ, etc). Nevertheless, neithernon-unconditionally stable classical time integration methods nor the sta-bilized ones due to high frequencies cut-off are able to control the energy

6 1.3. The role of dissipation

(rather damped out in long time simulations) or to preserve momenta, thuslosing most of the rich features posed by elastodynamics problems.

Energy momentum methods are able to neatly deal with stiff problems,although some cases, very stiff ones, presented instabilities in solutions orsimply stopped converging. Discrete FFT analyses on solutions suggestedthat these instabilities may appear because the existence of energy trans-fer from low to high frequency modes of the solution. The size of the dis-cretization seemed not to be crucial. In those situations, high frequenciesdissipative methods seem to be the only way to provide solutions. Theseideas motivated the development of conserving methods endowed by thecapacity of dissipating high frequencies in a controlled manner, in an at-tempt to combine the benefits of both strategies. In this sense, severalmethods were proposed as the Constraint Energy Methods due to Hugheset al. (1978), the Energy decaying schemes by Bauchau & Theron (1996),the modified Energy-Momentum methods in the context of contact intro-duced by Armero & Petocz (1998, 1999), the methods due to Crisfield et al.(1997) and Ibrahimbegovic & Mamouri (2002) for beams, or the ConstraintEnergy-Momentum Method for shells Kuhl & Ramm (1999), none of themdevoid of drawbacks. Also, using a discontinuous Galerkin approach, Lens &Cardona (2007) proposed an energy preserving/decaying method for nonlin-early constrained multi-body systems.

Finally, based on consistent perturbations of the Energy-momentum stressand velocity formulas, Armero & Romero (2001a,b) proposed the Energy-Dissipative-Momentum-Conserving (EDMC) methods which were proved tosuccessfully handle stiff problems by dissipating high frequencies while pre-serving the linear and angular momentum and relative equilibria. Later,Armero & Romero (2003) extended this approach to the dynamics of non-linear Cosserat beams. Furthermore, EDMC methods were also proposedfor the solution of exactly geometric beams and shells in Romero & Armero(2002a,b).

The spirit of this dissertation can be summarized by the idea of consis-tently incorporating physical dissipation in numerical methods to stabilizethem so that they additionally provide physically certain solutions.

1. Introduction7

1.4 Structure-preserving integration for dissipative sys-tems

The geometric integration applied to dissipative systems is usually re-ferred to as structure-preserving or thermodynamically consistent time in-tegration methods. Namely, any numerical integration method intended tosolve thermodynamical systems in such a way that the laws of thermody-namics are discretely satisfied by construction. The practical engineeringinterest that thermodynamical systems have together with the success of theEnergy-momentum time methods for Hamiltonian (conservative) systems,see Gonzalez (2000); Simo & Tarnow (1992), motivated many works towardsthis end, such as Armero & Simo (1992); Gross & Betsch (2010, 2011); Meng(2002); Meng & Laursen (2002); Ortiz et al. (2000). However, these methods-unlike the Hamiltonian case- were not developed within a uniform proce-dure. In other words, each different problem required a different formulationof the structure-preserving method. Furthermore, there was as yet evidence(within the computational mechanics community) that this was possible, dueto the lack of a general formalism that comprises every dissipative system.

In point of fact, a general formalism for dissipative systems did existin the geometry induced by metriplectic structures on smooth manifolds.These structures were first discussed in Morrison (1986) and Kaufman (1984),and are the result of coupling the Poisson structure defining conservative(Hamiltonian) systems with purely dissipative ones described by the so calledGradient structures. In this way, the derived structure is able to reproduceboth reversible and irreversible evolutions and provides an unified frameworkfor most of the systems ruled by the laws of thermodynamics.

Thanks to a particular metriplectic structure known as GENERIC, dueto Ottinger (2005) and co-workers, Romero (2009, 2010a,b) devised a generalprocedure for the formulation of Energy-Momentum-Entropy (EEM) meth-ods for any thermodynamical system. The acronym GENERIC stands for“General Equation for the Non-Equilibrium Reversible Irreversible Coupling”and, as a metriplectic structure, provides the evolution equations of any ther-modynamical system by separating its reversible and irreversible parts. Whilethe reversible part is connected to the derivative of the total energy of thesystem, the irreversible one depends on the derivative of the total entropy.

The design of the procedure to systematically attain EEM integrationmethods is applicable to both finite and infinite dimensional systems andrelies on the use of the discrete gradient operator which is a second or-der approximation of the standard gradient operator evaluated at midpoint,

8 1.4. Structure-preserving integration for dissipative systems

and satisfies two important properties: directionality and consistency, seeGonzalez (1996). This key ingredient makes EEM methods share some ofthe appealing properties derived from conservation of structure, such as theconservation of the symmetries.

Within this approach, GENERIC formalism plays a similar role to theHamiltonian formalism does in a purely mechanical context. Therefore, withthis new approach, every non-conserving continuous evolution system mightbe considered as a conserving-like, i.e consistent with the laws of thermody-namics. Thermodynamically speaking, the term consistent acts as does theterm conservative in a purely mechanical context. In fact, for the particularcase of reversibility the GENERIC formalism simplifies to the Hamiltonianone, as pointed out in Romero (2009), so that it can be interpreted as anatural generalization of the latter.

Based on this framework, Garcıa Orden & Romero (2011) proposed anEnergy-Entropy-Momentum method for the numerical solution of the dy-namics of a particular discrete system (finite dimensional), which consistsof point masses linked by thermo-viscoelastic springs. In this work, the au-thors did not use the underlying GENERIC formalism in order to derivethe structure-preserving method which satisfies the laws of thermodynamics.On the other hand, Kruger et al. (2011) performed a complete comparisonof this framework with several structure-preserving methods for the discretethermo-elastic case. Also isothermal dissipative problems can be addressedwithin this novel framework. Thus, the work of Mielke (2011) contains theGENERIC representation for the limit case of isothermal dissipative systemswhich could be of interest to derive EEM methods for isothermal viscoelas-ticity or plasticity.

1.4.1 On the selection of the thermodynamic variable

In theory, the thermodynamical state of any dissipative system can bedescribed either by the absolute temperature or by the entropy or by theinternal energy or by any other quantity that is a combination of these three.Due to its intuitive physical interpretation, temperature is often used as thevariable for the thermodynamical state. However, the use of the entropyas thermodynamical state variable has been reported to be the most suit-able choice to yield EEM methods, see for instance Garcıa Orden & Romero(2011); Romero (2009, 2010a,b). Although therein the entropy was success-fully employed, additional restrictions had to be assumed such as the neces-sity for material models to enable the analytical provision of its potentials in

1. Introduction9

terms of the entropy. In addition, many of the thermodynamical problemsof practical interest require Dirichlet boundary conditions which can onlybe defined by means of the temperature, concluding that the entropy choicedoes not allow or at least substantially hinders the solution of a wide rangeof problems.

The search for GENERIC form of the evolution equations has so far be-come cumbersome when the temperature was considered, consequently pre-venting the formulation of a corresponding EEM method in the sense ofRomero (2009). Recently, Mielke (2011) has thoroughly elaborated a fairlysystematic procedure to reach the GENERIC form departing from any ther-modynamic variable, demonstrating that a GENERIC form in terms of tem-perature may be achieved for dissipative thermo-mechanical systems. How-ever, its application to any particular system remains non-trivial at all and,therefore, the way to formulate a Temperature-based EEM method has sofar been an issue. In this dissertation this issue is successfully resolved, prov-ing that the election of the temperature as state variable does not involve acomplex GENERIC form to deal with and hence facilitating the design of anew temperature-based EEM method.

Furthermore, the election of the temperature as thermodynamical statevariable offers advantages from the analytical and numerical point of view.In this way, as the temperature can directly be measured, it is normallythe preferable variable to work with in the material modeling community,see for instance Dillon Jr. (1962, 1963); Holzapfel (1996); Holzapfel & Simo(1996b); Reese & Govindjee (1998a). The use of any thermodynamical vari-able other than the temperature would automatically involve the redefinitionof the thermo-mechanical potentials such that they are expressed in terms ofthat other variable. For realistic models this step could become cumbersomeor even impossible from an analytical point of view, so that a full implemen-tation would then require a numerical strategy which would complicate inexcess the formulation without any doubt. This factor along with the ne-cessity of imposing Dirichlet boundary conditions in continuous approaches(normally based on the FE method) motivated the search for a EEM methodin terms of the temperature.

These methods have recently reached a considerable maturity after theworks Garcıa Orden & Romero (2011), Romero (2013) or Conde Martınet al. (2014), enabling a wide variety of dissipative problems to be formu-lated within an unified procedure. As a result of the investigations carriedout for this dissertation, this procedure has currently adopted two differentapproaches based on the election of the thermodynamic variable: entropy

10 1.4. Structure-preserving integration for dissipative systems

or temperature. Thereby, the choice of the thermodynamical variable is notan issue any longer and can be made according to the particular application.Roughly speaking, the use of entropy provides a more straightforward formu-lation, assuming certain restrictions, and enables the formulation of staggered(first order accurate) methods in which the mechanical and thermal steps arethermodynamically consistent, see Romero (2010b). On the other hand, theuse of temperature successfully deals with typically temperature-based (evennon-standard) material models and is practically mandatory for consideringDirichlet boundary conditions. Apart from these technical considerations,both strategies present superior numerical stability and robustness and en-able the use of larger time steps compared to standard integration methods.

1.4.2 Dissipative effects: internal variables framework

A widespread way for the modeling of material irreversible processesis provided by the internal variables framework introduced by Coleman &Gurtin (1967). Internal or hidden variables are supposed to describe aspectsof the internal structure of materials related with dissipative effects whichcannot externally be measurable nor controllable. These variables are sup-plemented by (thermodynamically consistent) constitutive laws that macro-scopically collect all these effects, influencing on the history of the systemthermodynamic evolution. In this dissertation, the focus is put on smoothirreversible effects in the sense that they can be described by continuous laws.That is the case of viscoelastic effects in contrast with non-smooth changestypically present in plastic or damage transformations.

Internal variables naturally fit in the GENERIC formalism, contribut-ing directly to the irreversible part of the thermodynamic evolution. In thecontinuous context, this issue has been addressed at least in Mielke (2011),where also the basis for the consideration of non-smooth kinetic processeswere established. However, few works have hitherto addressed the formula-tion of Energy-Entropy-Momentum time-stepping methods involving internalvariables. Some remarkable exceptions are Garcıa Orden & Romero (2011),Conde Martın et al. (2014) and Conde Martin & Garcia Orden (2015), al-though they were not rigorously Energy-Entropy-Momentum methods sinceno use of the unified formalism was involved to reveal their thermodynamicstructure in order to formulate their structure-preserving discrete counter-parts. However, they fully correspond to those that would have been achievedwithin the Energy-Entropy-Momentum approach. In particular, Garcıa Or-den & Romero (2011) demonstrates that the choice of the entropy facilitates

1. Introduction11

the comprehension of the geometric structure of coupled thermo-mechanicdissipative problems whereas Conde Martın et al. (2014) shows that isother-mal considerations simplify the formulation of structure-preserving meth-ods, as it closely follows the guidelines provided by the Energy-Momentummethod.

Although the unified procedure has never been applied in the literatureto evolution equations defined in terms of internal variables, the preservingstructure method could nonetheless be achieved.

1.4.3 Variational Integrators

As previously commented, structure-preserving time integration algo-rithms have become an active topic of research due to their proved reliabilityand stability, crucially in long term simulations. Apart from the Energy-Entropy-Momentum approach, Variational Integrators has found their placein the solution of dissipative systems preserving some of their (quality) fea-tures.

Ultimately, Energy-Entropy-Momentum methods are formulated via thetemporal discretization of the evolution equations, previously derived fromthe GENERIC framework, that define the particular dissipative problem. Incontrast, Variational Integrators perform the temporal approximation on anearly stage, that is, directly on the variational principle of the problem. Thisapproach dates back to the seventies in the context of Hamiltonian problemsCadzow (1970), although it was fully consolidated in the nineties Moser &Veselov (1991), Marsden & Wendlandt (1997), Kane et al. (1999); Marsden &Ratiu (1999), Marsden & West (2001), Lew et al. (2004) and Romero (2008).More recently, the same idea has been applied to the particular dissipativeproblem of adiabatic thermo-mechanics Mata & Lew (2011), by introduc-ing the concept of thermal displacement. It plays the role of the mechanicaldisplacements but on the thermal evolution of the system so that the tem-perature can be recovered as the thermal velocity. Based on this concept,attempts towards the consideration of thermodynamic systems involving ir-reversible processes, such as viscoelastic, damage or plastic effects, are beingaddressed by using a discrete version of Lagrange-D’Alembert principle, seeKern et al. (2014). However, regarding heat conduction, there are still im-portant unresolved issues as the use of Fourier’s type heat conduction laws,see also Kern et al. (2014). The problem in this case is that the evolutionequations do not derive from an autonomous Lagrangian (or equivalently, anautonomous Hamiltonian).

12 1.5. Objectives

Above all these considerations, it should be mentioned that VariationalIntegrators are not strictly thermodynamically consistent methods since theyare symplectic by design and hence unlikely to be energy-preserving, see Simoet al. (1992). Nevertheless, they manage to maintain the energy boundedprovided the time step is small enough, see Hairer et al. (2006); Leimkuh-ler & Reich (2005), and, therefore, present superior numerical performancescompared to standard methods.

1.5 Objectives

The main goal of this work is to formulate, analyze and implementsecond-order accurate structure-preserving time integration methods for thesolution of general coupled smooth dissipative problems, that is, methodsthat, by design, are thermodynamically consistent and respect the symme-tries of the equations. To this end, six principal objectives have been set:

1. To select appropriated finite- and infinite-dimensional smooth dissi-pative systems so as to study their thermodynamical soundness andsymmetries, essential requisites to formulate EEM methods. Amongthem all, nonlinear thermo-dissipative continuum dynamics will be thefinal target.

2. To study and elaborate their formulations in terms of both temperatureand entropy variables, identifying their main drawbacks in what regardsto applicability and conservation properties of both the continuous anddiscrete settings, the latter being provided by standard time integrationmethods.

2. To find their descriptions within the GENERIC framework to serveas departing points to formulate abstract time integration methodsendowed by conservation properties.

3. To further develop the recently proposed unified methodology to deriveEEM methods from the GENERIC evolution equations of the dissipa-tive systems of interest.

4. To elaborate the said methodology to arrive at second-order accuratetime integration methods that inherit all the preservation propertiespreviously identified in the continuous descriptions for both tempera-ture and entropy descriptions.

1. Introduction13

5. To prove the performance of the resulting methods respect both con-servation properties, stability and robustness.

6. To provide general guidelines for the formulation of EEM methods forgeneral smooth dissipative systems.

1.6 Overview

This dissertation is organized in seven chapters, in which the first chapterhas covered the motivation, state-of-the-art and objectives. Next, an outlineof the contents of the rest of the chapters are provided:

Chapter 2 deals with the descriptions of two simple finite-dimensionalsmooth dissipative systems so as to set the scope of the problems addressedin this dissertation. They are in-depth studied regarding their conservationproperties and how they are breached when using standard time integrationmethods to find numerical solutions of the nonlinear initial value problems.

Chapter 3 contains a detailed description of the concepts involved inthe formulation of nonlinear thermo-dissipative continuum dynamics fromthe most general perspective. Special attention is paid to the entropy andtemperature forms of the underlying partial differential equations as well asthe isolated dynamics that lead to the laws of thermodynamics and symme-tries. Also, the particular case of isothermal changes is addressed. Then, theresulting infinite-dimensional system is spatially approximated by the classi-cal Galerkin FE-based approach and standard time integration methods areformulated to provide approximated solutions.

Chapter 4 stars with a brief introduction on metriplectic structures in-duced on manifolds so as to then include one of the key point of the presentdissertation. That is, the reformulation of the previously studied smooth dis-sipative systems within the GENERIC formalism (a particular metriplecticstructure) that confers an abstract description from which thermodynami-cally consistent discrete counterparts can be derived. Also, the symmetriesof the equations are used to identify the momentum maps that, according toNoether’s theorem, are first integrals of the systems evolution.

Chapter 5 provides full details of the unified methodology employed toderived EEM methods from the previously introduced GENERIC form of thesystems studied in this dissertation. Therein, the key points for successfulimplementations of these new methods are provided and reasoned.

Chapter 6 contains simulations that claim to be useful for modeling real

14 1.7. Publications

scenarios in industry.

Chapter 7 contains the main conclusions derived from this work andsuggests future works that can be addressed to further exploit the investiga-tion carried out in this dissertation.

1.7 Publications

The following scientific publications, conference presentations and semi-nars contain parts of this thesis:

Journal articles

• Sergio Conde Martın and Juan C. Garcıa Orden. “Temperature-based thermodynamically consistent integration for finite thermoelas-todynamics”, in preparation.

• Dominik Kern, Ignacio Romero, Sergio Conde Martın, Juan C. Gar-cıa Orden. “Performance Assessment of Variational Integrators forThermomechanical Problems”, submitted in October 2015.

• Sergio Conde Martın and Juan C. Garcıa Orden. “On GENERIC-based integration methods for discrete thermo-visco-elastodynamics”,Selected for C&S Special Issue: ACME-UK 2015, submitted in August2015.

• Sergio Conde Martın, Juan C. Garcıa Orden. “Energy-consistent in-tegration scheme for multi-body systems with dissipation”, Proceedingsof the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics. S.I.: New Trends in Mechanism and Machine Science,24, 413-421, 2015. DOI: 10.1177/1464419315615068 .

• Sergio Conde Martın, Peter Betsch, Juan C. Garcıa Orden. “ATemperature-based thermodynamically consistent integration schemefor discrete thermo-elastodynamics”, Commun Nonlinear Sci NumerSimulat, 32, 63-80, 2016. DOI: 10.1016/j.cnsns.2015.08.006

• Sergio Conde Martın, Juan C. Garcıa Orden, Ignacio Romero. “En-ergy-consistent time integration for nonlinear viscoelasticity”, Compu-tational Mechanics, 54(2), 473 - 488, 2014. DOI: 10.1007/s00466-014-1000-x.

http://pik.sagepub.com/content/early/2015/11/19/1464419315615068

http://link.springer.com/article/10.1007%2Fs00466-014-1000-x

http://www.sciencedirect.com/science/article/pii/S100757041500283X

http://www.sciencedirect.com/science/article/pii/S100757041500283X

1. Introduction15

• Juan C. Garcıa Orden and Sergio Conde Martın. “Controllablevelocity projection for constraint stabilization in multibody dynamics”,Nonlinear Dynamic, 68(1-2), 245-257, 2012. DOI: 10.1007/s11071-011-0224-y

Conference proceedings

• Sergio Conde Martın, Juan C. Garcıa Orden, Peter Betsch. “ATemperature-based thermodynamically consistent integration schemefor discrete thermo-elastodynamics”, ACME-UK 2015, Swansea (UK),April 2015.

• Sergio Conde Martın, Juan C. Garcıa Orden, Ignacio Romero. “En-ergy-consistent integration scheme for multi-body systems with dissipa-tion”, EUCOMES, Guimaraes (Portugal), IFToMM, September 2014.

Conference abstracts

• Sergio Conde Martın, Juan C. Garcıa Orden. “Temperature-basedthermodynamically consistent time integration for nonlinear thermoe-lasticity”, Coupled Problems, San Servolo, Venice (Italy), IACM-ECCO-MAS, May 2015.

• Sergio Conde Martın, Juan C. Garcıa Orden, Ignacio Romero. “Energy-consistent time integration for nonlinear viscoelasticity”, WCCM XI,Barcelona (Spain), IACM-ECCOMAS, July 2014.

• Juan C. Garcıa Orden and Sergio Conde Martın. “Controllablevelocity projection for constraint stabilization in multibody dynamics”,CNME, Coimbra (Portugal), SEMNI & APMTAC, June 2011.

Seminars

• Sergio Conde Martın, Juan C. Garcıa Orden. “GENERIC-basedthermodynamically consistent time integration methods for coupleddissipative problems”, Applied Mathematics Seminar (2014-15), Pavia(Italy), I.M.A.T.I-C.N.R, November 25, 2014.

http://link.springer.com/article/10.1007%2Fs11071-011-0224-y#page-1

http://link.springer.com/article/10.1007%2Fs11071-011-0224-y#page-1

16 1.7. Publications

Nonlinearthermo-dissipative discrete

dynamics Chapter

2In order to set the scope of this dissertation, two nonlinear discrete dis-

sipative systems containing every significant concept involved in the formu-lation and solution of thermo-dissipative mechanic problems are discussed.In particular, an isolated system composed of two thermo-springs exchang-ing heat and an isolated system formed by a thermo-viscoelastic elementexchanging heat with its environment are considered. The first system ad-dresses the thermo-elastic coupling and the heat conduction phenomenon,which breaks the Hamiltonian structure of the thermo-elastic problem, or, inother words, converts the system into a dissipative one. On the other hand,the second system goes further to include a dissipative mechanism of typesmooth, specifically of viscoelastic type, which leads the problem to be dissi-pative by construction. The characterization of the viscoelastic evolution ismodeled by an internal variable which introduces a common framework todeal with dissipative behaviors.

Special attention is paid to the conservation properties of these ther-modynamic systems as well as the typical breach of them due to standardnumerical integrations. Particularly, as a representative standard integrationmethod, the Midpoint method is discussed regarding the discrete conserva-tion properties.

Framed in the same context as this dissertation, the first system was atleast studied in Romero (2009) and Kruger et al. (2011) whereas the secondone can be found in Garcıa Orden & Romero (2011). Therein, the authorsaddressed the formulation of these systems in terms of entropy to proposeEnergy-Entropy-Momentum methods. In this chapter, also a formulation

18 2.1. Two thermo-spring system

based on temperature is proposed and discussed so that temperature-basedEnergy-Entropy-Momentum methods can be achieved in subsequent Chap-ters.

2.1 Two thermo-spring system

The two thermo-spring system is a thermally isolated system consistingof two point masses m1 and m2 connected with thermo-springs of naturallengths λ0

1 and λ02, as depicted in Figure 2.1. The first spring connects m1 to

a fixed point and the second spring connects m2 to m1. The position of theparticles are provided by vectors q1 and q2 relative to an inertial referenceframe eada=1 with, for simplicity, the origin located at the fixed point inan euclidean space of dimension d. The two thermo-springs exchange heataccording to a unidimensional Fourier’s law of the form

h := k (θ2 − θ1) , (2.1)

θa ∈ R+ being the absolute temperature of the spring a = 1, 2 and k ≥ 0being the coefficient of thermal conductivity.

m1

m2

λ01

λ02

e1

e2

e3

θ1

θ2

h = k (θ2 − θ1)

q1

q2

f1

f2

Figure 2.1. Thermo-spring system

Then, the thermo-springs are assumed to response according to a smoothand convex Helmholtz free-energy function Ψa : (λa, θa) ∈ R+ × R+ →

2. Nonlinear thermo-dissipative discrete dynamics 19

Ψa(λa, θa) ∈ R defined in terms of the spring temperature θa and the springelongation λa, provided by the l2-norm, denoted as ‖ · ‖, of the relative posi-tion vectors as

λ1(q1) = ‖q1‖, λ2(q1, q2) = ‖q2 − q1‖ (2.2)

The dynamics of the system is governed by the linear momentum balanceand the laws of thermodynamics. Thus, the first balance for each massestablishes that

maqa = −2∑b=1

∂Ψb

∂λb

∂λb∂qa

+ fa, (2.3)

fa being the external forces applied to the particles, see Figure 2.1, whose ex-istence, strictly speaking, makes the system be non-isolated from the mechan-ical perspective. In view of (2.2), the partial derivatives of the elongationsrespect to the position vectors result in

∂λ1

∂q1

=q1

‖q1‖,

∂λ1

∂q2

= 0,∂λ2

∂q1

=q1 − q2

‖q2 − q1‖,

∂λ2

∂q2

=q2 − q1

‖q2 − q1‖, (2.4)

enabling the spring internal force to be identified by

fa :=∂Ψa

∂λa, (2.5)

On the other hand, the first law of thermodynamics, or energy balance,establishes that the total energy of any isolated thermodynamic system mustbe preserved. Focusing on each spring it can be stated as

f1λ1 = e1 − h,f2λ2 = e2 + h,

(2.6)

ea being the internal energy of the spring a which accounts for both elasticand thermal energy stored in the spring.

Remark 2.1. In classic thermodynamics, the internal energy is intimatelyrelated to the free-energy function via the Legendre transform, see Truesdellet al. (2004), according to

e := Ψ− θ∂Ψ

∂θ, (2.7)

where the partial derivative is identified with the entropy

η := −∂Ψ

∂θ(2.8)


two important properties of the Legendre transform are unveiled by tak-ing partial differentiation with respect to temperature to give

∂e

∂θ=∂Ψ

∂θ− ∂Ψ

∂θ− θ∂

2Ψ

∂θ2= θ

∂η

∂θ, (2.9)

and partial differentiation with respect to entropy, considering that, by def-inition, the free-energy function depends on temperature and, accordingly,using the chain rule to obtain its derivative respect to entropy

∂e

∂η=∂Ψ

∂θ

∂θ

∂η− ∂θ

∂η

∂Ψ

∂θ− θ∂

2Ψ

∂θ2

∂θ

∂η= θ

∂η

∂θ

∂θ

∂η= θ (2.10)

In Chapter 3 these crucial relations will be in-depth analyzed in thecontext of continuous media.

2.1.1 Two thermo-spring system in temperature variables

Now, the two thermo-spring system is formulated in terms of spring tem-peratures as thermodynamic variables. As the free-energy function is definedto depend on temperature by construction, the relation (2.7) straightfor-wardly yields the internal energy in terms of temperature

ea(λa, θa) = Ψa(λa, θa) + θaηa(λa, θa), (2.11)

with the entropy of each spring being

ηa(λa, θa) = −∂Ψa(λa, θa)

∂θa(2.12)

Furthermore, the energy balance (2.6) is rewritten so that the rate oftemperature explicitly appears. It suffices to apply the chain rule to thefunction ea(λa, θa), expressed in terms of positions via (2.2), to give

θa =

(∂ea∂θa

)−1[

(−1)a−1h− θa∂ηa∂λa

2∑b=1

∂λa∂qb· qb]

(2.13)

Interestingly, using relation (2.9) and the chain rule yields the abovebalance in entropy form

ηa(λa, θa) = (−1)a−1 h

θa(2.14)


Remark 2.2. In the traditional calorimetry approach, the partial derivativesappearing in (2.13) are linked to the spring specific heat capacity

ca := θa∂ηa∂θa

=∂ea∂θa

> 0, (2.15)

and the spring latent heat

va := θa∂ηa∂λa

(2.16)

The first indicates the amount of energy required to produce unit increase inthe temperature while keeping the deformation fixed, and the second accountsfor the Gough-Joule heat due to the thermo-mechanical coupling, see forinstance Holzapfel (2000).

Then, by introducing the momenta pa ∈ Rd the evolution equations canbe recast in a first order PDE fashion, that is

qa =pama

pa = −2∑b=1

∂Ψb

∂λb

∂λb∂qa

+ fa

θa =

(∂ea∂θa

)−1[


2∑b=1

∂λa∂qb· pbmb

],

(2.17)

Alternatively, using the constitutive relations (2.5), (2.15) and (2.16) theyalso can be expressed as follows

qa =pama

pa = −2∑b=1

fb∂λb∂qa

+ fa

θa =1

ca

[(−1)a−1h− va

2∑b=1

∂λa∂qb· pbmb

] (2.18)

It only remains to set initial conditions for the positions q0a, the momenta

p0a and the temperatures θ0

a and the specification of the free-energy function.For example, consider the following Helmholtz free-energy function, used in


Romero (2009),

Ψa(λa, θa) =Ca(θa)

2log2 λa

λ0a

− αa(θa − θref) logλaλ0a

+ ca

(θa − θref − θa log

θaθref

),

(2.19)

where Ca(θa) is the a-th spring stiffness considered to be temperature-de-pendent, αa accounts for the thermo-mechanical coupling, ca is the specificheat capacity and θref is the reference temperature at which the unstresseddeformation is defined to be zero. Accordingly, the entropy function followsfrom differentiating once to get

ηa(λa, θa) = −C′a(θa)

2log2 λa

λ0a

+ αa logλaλ0a

+ ca logθaθref

, (2.20)

C ′a(θa) being the absolute derivative of the function Ca(θa).

Then, according to the Legendre transform (2.7), the internal energyfunction can directly be obtained as

ea(λa, θa) =Ca(θa)− θaC ′a(θa)

2log2 λa

λ0a

+ αaθref logλaλ0a

+ ca (θa − θref) (2.21)

2.1.2 Two thermo-spring system in entropy variables

If the spring entropies are chosen as thermodynamic variables the prob-lem needs to be described with thermodynamic potentials in terms of them.To do that, the Legendre transform is applied to the free-energy functionrespect to the temperature to arrive at the internal energy in terms of theentropy, which are the transformed potential and the conjugate variable, re-spectively, that is

ea(λa, ηa) = Ψa(λa, θa(λa, ηa)) + θa(λa, ηa)ηa (2.22)

In the above expression, it is assumed that the function θa(λa, ηa) canbe reached from (2.12). Moreover, the properties of the Legendre transformallow to conclude that

fa =∂ea∂λa

, θa =∂ea∂ηa

(2.23)


In view of this, applying the chain rule in (2.7) so that the rates ofentropies appear, the entropy formulation of the evolution equation yields

qa =pama

pa = −2∑b=1

∂eb∂λb

∂λb∂qa

+ fa

η1 =

(∂e1

∂η1

)−1

k

(∂e2

∂η2

− ∂e1

∂η1

)η2 =

(∂e2

∂η2

)−1

k

(∂e1

∂η1

− ∂e2

∂η2

)(2.24)

More compactly, they are expressed in terms of the constitutive relations(2.23) as

qa =pama

pa = −2∑b=1

fb∂λb∂qa

+ fa

ηa = (−1)a−1 h

θa

(2.25)

The problem is closed by setting initial conditions for the positions q0a,

the momenta p0a and the entropies η0

a. However, it is usual to know the initialtemperature θ0

a instead of the initial entropy η0a. In this case, the following

relation should be solved in order to properly set the initial conditions

η0a = −∂Ψa

∂θa(λ0

a, θ0a) (2.26)

Finally, when specifying the free energy function an important issuearises. That is, the temperature function θa(λa, θa) should be obtained byinverting (2.12) but, it is the case that this only can analytically be doneunder certain conditions, limiting the range of potential problems that canbe addressed with this formulation.

If the free-energy function (2.20) is used, the restriction falling on it isthat the dependence of the elastic parameter with temperature must be atmost linear, i.e C ′a(θa) = C, with C being a real constant. In such a case,


the temperature function results in

θa(λa, ηa) = θref exp

ηa +C

2log2 λa

λ0a

− αa logλaλ0a

ca

(2.27)

Accordingly, the particular form of internal energy in terms of the entropyis provided by

ea(λa, ηa) =Ca(θa)− θaC

2log2 λa

λ0a

+ αaθref logλaλ0a

+ caθref

exp

ηa +C

2log2 λa

λ0a

− αa logλaλ0a

ca

− 1

(2.28)

2.1.3 Symmetries and laws of thermodynamics

Associated with the dynamics of isolated systems, there exist conservedquantities throughout the motion, imposing in effect a mathematical or nat-ural constraint on it. Particularly, for discrete thermo-dissipative systems,these are the total linear and angular momentum, associated with the sym-metries of the equations, and the total energy and entropy of the system.

The proof for the linear and angular momentum is a classical result whichdeparts from the definition of the total linear and angular momentum as

L =2∑

a=1

pa, J =2∑

a=1

qa ∧ pa, (2.29)

with (·) ∧ (·) meaning cross product.

Then, it is proved that the rate of these quantities becomes zero, that is

L =2∑

a=1

pa = 0, J =2∑

a=1

(qa ∧ pa + qa ∧ pa) = 0 (2.30)

By using either (2.18)2 or (2.25)2 along with (2.4) the proof for the linearmomentum follows from

L = −(f1q1

λ1

+ f2q1 − q2

λ2

)+f1−f2

q2 − q1

λ2

+f2 = −f1q1

λ1

+f1 +f2, (2.31)


which vanishes due to the consideration of the resulting terms to be external.That is, the mechanical isolation of the system involves only the two pointmasses and the second spring. Given that, the force due to the first spring isconsidered as external and, hence, it should vanish for the linear momentumto be preserved. In other words, the inclusion of the fixed point into thesystem leads the total linear momentum (2.29)1 conservation to be breached,as in this case the system is not mechanically isolated.

The angular momentum, however, is not affected by the previous consid-eration (although strictly speaking the mechanical isolated system remainsequal) as the external force due to the first spring does not contribute to themoment respect to the fixed point, that is

J =2∑

a=1

qa ∧ pa = q1 ∧(f1q1

λ1

+ f2q1 − q2

λ2

)+ q2 ∧ f2

q2 − q1

λ2

= 0, (2.32)

where the external forces have been neglected and use has been made of thecross product property1.

On the other hand, the total energy E : S → R, S being the statespace, involves the kinetic energy K : S → R due to the dynamics of thesystem, provided by

K(‖p1‖, ‖p2‖) :=2∑

a=1

‖pa‖2

2ma

, (2.33)

and the total internal energy U : S → R of the springs provided by

U(q1, q2, s1, s2) := e1(λ1(q1), s1) + e2(λ2(q1, q2), s2), (2.34)

sa being either temperatures or entropies, depending on the formulations(2.17) or (2.24).

Then, it is easy to show that the rate of the total energy vanishes inabsence of external forces with the help of (2.18)1,2 or (2.25)1,2 and (2.6)

1a ∧ b = −b ∧ a

26 2.2. Thermo-visco-elastic system

together with the chain rule on the rate of the elongations to give

E = K + U =2∑

a=1

1

ma

pa · pa +2∑

a=1

ea(λa, sa)

=2∑

a=1

1

ma

pa ·(−

2∑b=1

fb∂λb∂qa

)+

2∑a=1

faλa

= −2∑

a=1

qa ·2∑b=1

fb∂λb∂qa

+2∑

a=1

fa

2∑b=1

∂λa∂qb· qb = 0

(2.35)

Finally, the entropy formulation of the energy balance (2.24)3 easily pro-vides the proof for the system to agree with the second law of thermody-namics. Defining the total entropy of the system S : S → R+ as the sumof the spring entropies, its rate can be demonstrated to be non-negative byadditionally using (2.1), that is

S = η1 + η2 = h

(1

θ1

− 1

θ2

)= k

(θ2 − θ1)2

θ1θ2

≥ 0 (2.36)

This result is also valid for the temperature formulation (2.17) because(2.24)3 can be recovered by (2.17)3, as demonstrated in (2.14).

2.2 Thermo-visco-elastic system

The isolated thermo-visco-elastic system is formed by a thermo-visco-elastic element and its environment, which is considered to have a variabletemperature and whose thermodynamics is defined by an unique thermalvariable. Another option for the thermodynamic isolated system is to con-sider two thermo-visco-elastic elements exchanging heat between them as theprevious example. Similarly, in the previous system, the environment couldhave been included for the exchange of heat. In any case, the thermody-namic features intended to be studied are present in both alternatives sosuch decision is not crucial at all.

A thermo-visco-elastic element consists of a generalized Maxwell elementconnecting two point masses m1 and m2, as depicted in Figure 2.2. Thegeneralized Maxwell element in turn consists of a thermo-spring arranged inparallel with a thermo-Maxwell element, i.e a thermo-spring in series with adashpot.


ν(θ)

γ

λ

m1 m2

h = k (θ − ϑ)

e2

e3

e1

q1 q2

ϑ

θ

f1

f2

Figure 2.2. Thermo-visco-elastic system

Then, the thermo-visco-elastic element is connected to the environmentby the exchange of heat which follows an unidimensional Fourier’s law of thetype

h = k (θ − ϑ) , (2.37)

where θ ∈ R+ and ϑ ∈ R+ are the absolute element and the absolute en-vironment temperatures and the scalar k ≥ 0 is the thermal conductivityparameter.

Analogously to the two thermo-spring system, the position of the pointmasses are provided by position vectors q1, q2 ∈ Rd, d being the problemdimension, relative to an inertia frame of reference eidi=1 so that the elon-gation or shortening of the element is

λ(q1, q2) = ‖q2 − q1‖ =√

(q2 − q1) · (q2 − q1), (2.38)

whose partial derivatives respect to the positions vectors fully correspond tothose of the second spring of the previous system (2.4)3,4.

The new feature that incorporates this system is the flow behavior of theMaxwell element’s dashpot, which is characterized by an internal variableγ ∈ R in the sense of Simo & Hughes (2000), and originally introduced


by Coleman & Gurtin (1967). The viscous effects are thus modeled by aNewtonian viscous fluid specified by a viscosity parameter ν > 0 which maygenerally depend on temperature. In addition, the viscosity parameter candirectly be related to the relaxation/retardation time τ > 0 through the shearmoduli of the spring in series with the dashpot µ > 0, that is, ν = 2µτ , seeHolzapfel (2000).

As a state variable, the internal variable contributes to the element con-stitutive behavior, entering the thermo-visco-elastic element free-energy func-tion Ψ(λ, θ, γ) : R+ × R+ × R→ R. Moreover, the Legendre transform (2.7)still applies, leading to the definitions of the element internal energy e andthe element entropy η.

The dynamics of this system also relies on the balance laws of linearmomentum, energy and entropy. Following Newton’s law, the first one canbe stated for each mass as

maqa = −∂Ψ

∂λ

∂λ

∂qa+ fa with a = 1, 2, (2.39)

fa being the external forces applied to the point masses. Following the firstlaw of thermodynamics, the energy balance applied to both the element andthe environment reads

∂Ψ

∂λλ = e+ h

0 = ε− h,(2.40)

ε being the internal energy of the environment.

In addition, the second law of thermodynamics bounds the rate of the in-ternal variable to be proportional to the viscous driving force via the viscosityparameter

γ = − 1

ν(θ)

∂Ψ(λ, θ, γ)

∂γ, (2.41)

defining the evolution equation for the internal variable.

2.2.1 Thermo-visco-elastic system in temperature variables

According to (2.7), the element internal energy defined in terms of theelement temperature results directly from using the free-energy functionΨ(λ, θ, γ), that is

e(λ, θ, γ) := Ψ(λ, θ, γ)− θ∂Ψ(λ, θ, γ)

∂θ, (2.42)


where the entropy function of the element is identified to be

η(λ, θ, γ) := −∂Ψ(λ, θ, γ)

∂θ(2.43)

On the other hand, the environment is thermodynamically determinedby either its internal energy ε(ϑ) or its entropy σ(ϑ) defined in terms of itstemperature so that the following identity hold

ς =dε

dϑ= ϑ

dσ

dϑ> 0, (2.44)

which is the definition of the specific heat capacity of the environment.

As in the case of the two thermo-spirng system, the evolution equations(2.39)-(2.41) can be rewritten as a first order PDE system by introducing themomenta pa ∈ Rd to give

qa =pama

pa = −∂Ψ

∂λ

∂λ

∂qa+ fa

γ = −1

ν

∂Ψ

∂γ

θ =1

θ

(∂η

∂θ

)−1[−θ

2∑a=1

∂η

∂λ

∂λ

∂qa· pama

+1

ν

∂Ψ

∂γ

(∂Ψ

∂γ+ θ

∂η

∂γ

)− h]

ϑ =

(dε

dϑ

)−1

h

(2.45)

Alternatively, the appearing partial derivatives are also linked with theelement internal force f , the dashpot viscous driving force g and the classicalcalorimetry coefficients, i.e the specific heat capacities c and ς and latentheats v and w, according to

f =∂Ψ

∂λ, g = −∂Ψ

∂γ, c =

∂e

∂θ= θ

∂η

∂θ> 0,

v = θ∂η

∂λ, w = θ

∂η

∂γ, ς =

dε

dϑ= ϑ

dσ

dϑ> 0

(2.46)

Thus, the evolution equations can be expressed in a more compact form


as

qa =pama

pa = −f ∂λ∂qa

+ fa

γ =g

ν

θ =1

c

[−v

2∑a=1

∂λ

∂qa· pama

+g

ν(g − w)− h

]ϑ =

h

ς

(2.47)

By setting initial conditions for the positions q0a, the momentum p0

a andtemperatures θ0, ϑ0 the initial value problem becomes closed, since the in-ternal variable γ0 is assumed to be zero, at the expense of the specificationof the Helmholtz free-energy function. For instance, the one used in GarcıaOrden & Romero (2011) and provided by

Ψ(λ, θ, γ) = (1 + β)Ψ∞(λ, θ) + µ(θ)γ2 − βγ∂Ψ∞(λ, θ)

∂λ, (2.48)

being Ψ∞(λ, θ) the free-energy function of the main spring, which is respon-sible for the whole response of the element when the viscous effects vanish,µ(θ) being a parameter associated to the stiffness of the Maxwell element andβ = 2µ(θref)/C(θref) relating both parameters at the reference temperature.

General smooth and convex free-energy functions Ψ∞(λ, θ) can be con-sidered, as the one used for the two thermo-spring system (2.19). Then, theentropy function follows from differentiating (2.48) once to obtain

η(λ, θ, γ) = −(1 + β)∂Ψ∞

∂θ− µ′(θ)γ2 + βγ

∂2Ψ∞

∂θ∂λ, (2.49)

µ′(θ) being the absolute derivative of the function µ(θ). Moreover, accordingto the Legendre transform (2.42), the internal energy function can be directlyobtained in terms of temperature

e(λ, θ, γ) = (1 + β)

(Ψ∞ − θ∂Ψ∞

∂θ

)+ (µ(θ)− θµ′(θ))γ2

+ βγ

(θ∂2Ψ∞

∂θ∂λ− ∂Ψ∞

∂λ

) (2.50)


Furthermore, general non-linear expressions for the stiffness C(θ), theMaxwell element parameter µ(θ) and the viscosity parameter ν(θ) could beconsidered. Particularly, the viscosity parameter ν(θ) is assumed to followan exponential law with the inverse of temperature as follows

ν(θ) = ν0 exp

[a

(1

θ− 1

θref

)]with ν0, a > 0 (2.51)

Finally, it is necessary to define a thermodynamical model for the envi-ronment. The easiest way is to consider the ideal-gas type model with aninternal energy provided by

ε(ϑ) = ς(ϑ− θref), (2.52)

and the entropy by

σ(ϑ) = ς log(ϑ/θref) (2.53)

2.2.2 Thermo-visco-elastic system in entropy variables

Alternatively, the thermo-visco-elastic system can be formulated in termsof entropies. Thus, provided e(λ, η, γ) and ε(σ), the entropy formulation ofthe evolution equations reads

qa =pama

pa = − ∂e∂λ

∂λ

∂qa+ fa

γ = −1

ν

∂e

∂γ

η =

(∂e

∂η

)−1[

1

ν

(∂e

∂γ

)2

+ k

(dε

dσ− ∂e

∂η

)]

σ =

(dε

dσ

)−1

k

(∂e

∂η− dε

dσ

)(2.54)

Identifying now the partial derivatives of the internal energies with theinternal force f , the viscous driving force g and the temperatures θ and ϑ,according to

f =∂e

∂λ, g = −∂e

∂γ, θ =

∂e

∂ηϑ =

dε

dσ, (2.55)


the evolution equations are also expressed as follows

qa =pama

pa = −f ∂λ∂qa

+ fa

γ =g

ν

η =1

θ

(g2

ν− h)

σ =h

ϑ

(2.56)

To fully bound the problem, initial conditions for the positions q0a, the

momenta p0a and the entropies η0 and σ0 must be set. Again, what it is usual

to know is the initial temperatures θ0 and ϑ0 for which the following relationsmust be considered in order to set properly the initial conditions

η0 = −∂Ψ

∂θ(λ0, θ0, γ0), σ0 = σ(ϑ0) (2.57)

Also, this formulation has the same constraints as those pointed out forthe two thermo-spring system. Namely, the relation (2.43) must be ana-lytically invertible to obtain θ(λ, η, γ). Then, the temperature-dependentstiffness C(θ), in case of assuming (2.19) for the potential Ψ∞(λ, θ), and theparameter µ(θ) must be at most linear, that is, C ′(θ) = C and µ′(θ) = µ,with C and µ being constants.

With these considerations, the element temperature function in terms ofentropy reads

θ(λ, η, γ) =

θref exp

C

2log2 λ

λ0

− α logλ

λ0

c+

η + µγ2 − βγ(C

λlog

λ

λ0

− α

λ

)(1 + β)c

(2.58)

Accordingly, the internal energy function can be directly obtained interms of entropy by using the above function in (2.50). Also, the viscosityparameter (2.51) must be evaluated via this temperature function

ν(θ(λ, η, γ)) = ν0 exp

[a

(1

θ(λ, η, γ)− 1

θref

)]with ν0, a > 0 (2.59)


In the same way, this formulation requires the inversion of the function(2.53) to obtain the environment temperature in terms of the environmententropy

ϑ(σ) = θref exp

(σ

ς

), (2.60)

leading the environment internal energy to be

ε(σ) = ςθref

[exp

(σ

ς

)− 1

](2.61)

2.2.3 Symmetries and law of thermodynamics

This subsection reviews the conservation properties of the thermo-visco-elastic system. For this case, the invariants of a free-motion are also identifiedwith the total linear and angular momentum and the total energy and en-tropy.

Then, the same definitions (2.29)1 and (2.29)2 of the linear and angularmomentums apply so, neglecting the external forces, the verification of thelinear momentum conservation is achieved by using either (2.47)2 or (2.56)2

along with (2.4) to give

L = −f q1 − q2

λ− f q2 − q1

λ= 0 (2.62)

Likewise, the angular momentum conservation is also derived from (2.47)2

and (2.56)2, yielding

J = q1 ∧ fq1 − q2

λ+ q2 ∧ f

q2 − q1

λ= 0 (2.63)

Regarding the total energy conservation, the kinetic energy answers to thesame expression as in the previous system (2.33) whereas the total internalenergy now becomes

U(q1, q2, s1, s2, γ) := e(λ(q1, q2), s1, γ) + ε(s2), (2.64)

sa being either the element and the environment temperatures or entropies,depending on the formulation.

Then, the proof for the total energy to be preserved in absence of externalforces is deduced with the help of (2.47)1,2 or (2.56)1,2 and (2.40) together

34 2.3. Standard numerical time integration methods

with the chain rule on the rate of the elongations to obtain

E = K + U =2∑

a=1

1

ma

pa · pa + e(λ, s1, γ) + ε(s2)

=2∑

a=1

1

ma

pa ·(−f ∂λ

∂qa

)+ fλ

= −2∑

a=1

qa · f∂λ

∂qa+ f

2∑a=1

∂λ

∂qa· qa = 0

(2.65)

On the other hand, the total entropy is the sum of the element and theenvironment entropies, whose rate can be proved to be non-negative via theentropy formulation of the energy balance (2.54)3 and the Fourier law (2.37),that is

S = η1 + η2 =g2

θν+ h

(1

θ− 1

ϑ

)=g2

θν+ k

(θ − ϑ)2

θϑ≥ 0, (2.66)

which also applies to the temperature formulation (2.45) for the same reasonsas those argued in the previous Section.

2.3 Standard numerical time integration methods

Each of the initial-value problems discussed so far can be recast in thefollowing simplified form

y(t) = f(y(t), t) ∀ t ∈ [0, T ], (2.67)

where the vector y ∈ S collects every state variable and f : S 7→ Rdim(y) isa nonlinear vector-valued function. Such general nonlinearity is responsiblefor an analytical integration to be unaffordable, thus requiring a numericalapproach in order to provide approximate solutions.

To this end, the continuous time interval of interest [0, T ] is decomposedinto N − 1 subintervals, such that

[0, T ] =N⋃n=0

[tn, tn+1], (2.68)

whose lengths are provided by ∆tn+1 = tn+1− tn, and are normally assumedto be constant for simplicity although it is not a necessary condition.


The time-evolution of the state variables vector is thus sampled by asequence of approximated state values at each subinterval extremities, thatis

y(t) ≈ ynNn=0, (2.69)

where yk is the approximation of y(tk). The relation between any stateand its predecessor depends on the integration method and is obtained via aparticular approximation of (2.67) inside every time subinterval, such that

yn+1 = yn +

∫ tn+1

tn

f(y(t), t)dt (2.70)

Any method thus constructed is said to be a one-step integration methodand it will be consistent provided that in the limit case of ∆tn+1 → dt thecontinuous evolution is fully recovered. Hence, the approximated solutionwill be closer to the exact one as the time subintervals are reduced.

Remark 2.3. There exist integration methods that also involve states otherthan just the previous one, referred to as multi-step methods. The mostrepresentative methods are the BDF ones, which stands for Backward Dif-ferentiation Methods, see Dong (2010).

Each one-step time integration method emerges from different approx-imations of the analytically non-accessible integral of (2.70). Thus, thestandard Forward-Euler, Backward-Euler, Trapezoidal, Midpoint or Simpsonmethods responds to the following quadratures∫ tn+1

tn

f(y(t), t)dt ≈ ∆tf(yn, tn)∫ tn+1

tn

f(y(t), t)dt ≈ ∆tf(yn+1, tn+1)∫ tn+1

tn

f(y(t), t)dt ≈ ∆t

2[f(yn+1, tn+1) + f(yn, tn)] (2.71)∫ tn+1

tn

f(y(t), t)dt ≈ ∆tf(yn+ 12, tn+ 1

2)∫ tn+1

tn

f(y(t), t)dt ≈ ∆t

6

[f(yn+1, tn+1) + 4f(yn+ 1

2, tn+ 1

2) + f(yn, tn)

],

with (·)n+ 12

= 12

[(·)n+1 + (·)n] being the evaluation at the subinterval mid-point.

A valid method needs to be stable for moderate time step sizes ∆t which,by definition, means that given two perturbations in (2.70), the difference


between the respective solutions are bounded for 0 ≤ ∆t ≤ ∆t0, with ∆t0being the maximum time step size for which the method is stable. The sta-bility issue can only be in-depth developed for partial differential equationswith the format f(y(t), t) = Ay(t), A being a real matrix of constant compo-nents whose eigenvalues’ real part are negative. In such a case, the conceptof unconditionally stable area is of importance since it defines all the possiblechoices for the time step sizes ensuring stable integrations.

For generally non-linear problems, standard methods cannot guaranteestability. However, in the literature different works, e.g. Simo et al. (1992),Stuart & Humphries (1998) or Garcıa Orden & Goicolea (2000), have re-ported that the preservation of qualitative features, particularly the en-ergy, at every time step considerably enhances the stability. This conclusionmotivated the search for non-standard integration methods with structure-preserving features, origin of the methods analyzed in this dissertation.

An important conclusion of integration methods’ theory asserts that sta-bility and consistency are necessary and sufficient conditions for a methodto be convergent, meaning that the maximal error among all time steps islimited and tends to zero as the time step is diminishing, that is

max0≤n≤N

‖y(tn)− yn‖ → 0 when ∆t→ 0 (2.72)

Another important concept is the order of accuracy of the approxima-tion. It measures the rate at which the method converge to the exact solutionwhen diminishing the time step size. That is, second-order accurate methodswill converge quadratically to the exact solution as the time step is reduced.In other words, the error can be demonstrated to be bounded by a propor-tional part of the time step to the power of the order of accuracy z, that ise( ∆t) ≤ C ∆tz. It is also intimately related to the power of the polynomialsthat can exactly be integrated. Thus, the Euler methods are just first-orderaccurate and only integrate exactly linear functions, both the Trapezoidal andMidpoint methods are second-order, exactly integrating parabolic functions,while the Simpson method is third-order accurate. The order of accuracycan be enhanced not only by incrementing the degree of the polynomial ex-actly integrated but by performing the integral approximation in successivestages within the time subinterval, each of them with low order of accuracy.One of the most prominent example of the latter approach is the family ofRunge-Kutta methods, see Butcher (1996); Kutta (1901).

In addition, if the approximation involves the value of the unknown stateat tn+1, the resulting method is said to be implicit, as an implicit equation


needs to be solved. If not, the resulting method is explicit. Generally speak-ing, the explicit methods require smaller time step sizes than implicit onesto be stable. Interestingly, there exist implicit methods that, in the linearcontext, show unconditional stability for any time step size, that is, theirunconditional stable area is infinite.

On the other hand, on-purpose methods for the solution of structuraldynamics have become widespread. The structural dynamic evolution equa-tions are dominated by second order ODE so the methods are defined interms of first and second time derivative. The family of β-Newmark (New-mark (1959)) and of α-Generalized (Chung & Hulbert (1993)) methods arethe most representative examples of these methods, usually referred to asstructural methods. An important particular case of the last family is theHHT method named after Hilber et al. (1977) and intended for the elimina-tion of non-desirable high frequencies due to the spatial discretization.

Particularly, given the typical structural dynamic second order ODE eval-uated at some instant time t = tn+1

Mn+1qn+1 + Cn+1qn+1 + Kn+1qn+1 = Fn+1, (2.73)

the β-Newmark family provides the following formulas

qn+1 = qn + ∆tqn +

(1

2− β

)∆t2qn + β∆t2qn+1

qn+1 = qn + (1− γ) ∆tqn + γ∆tqn+1

(2.74)

with β, γ being two constant limited by the method’s stability to be 0 ≤ γ ≤ 1and 0 ≤ b ≤ 1

2.

The most important methods of this family are the trapezoidal rule orthe constant acceleration method (implicit), the modified trapezoidal rule(implicit), the linear acceleration method (implicit) and the central differ-ences method (explicit). For instance, the trapezoidal rule corresponds toβ = 1

4and γ = 1

2.

Among all these choices, from now on the interest will be in the standardMidpoint and Trapezoidal methods because they are implicit and second-order accurate and entail some conservation properties. For instance, theMidpoint method is symplectic, see Simo et al. (1992), and preserves the sym-metries of dynamic equations: total linear and angular momentum. In fact,as it will be demonstrated in subsequents Chapters, the Midpoint method canbe interpreted as a first approach of the Energy-Entropy-Momentum meth-ods to be presented. Interestingly, both Trapezoidal and Midpoint methods


collapse to the same method for linear problems and preserve first integralsis such a context.

2.3.1 Midpoint method for the two thermo-spring system

According to (2.67), (2.70) and (2.71)4, the following midpoint formulascan be deduced

yn+ 12

=1

2(yn+1 + yn)

yn+ 12

=1

2(yn+1 + yn) =

yn+1 − yn∆t

(2.75)

Furthermore, it is appropriated to introduce the following notation tomake clear the midpoint-type formulations to be developed from now on

(·)n+ 12

=1

2[(·)n+1 + (·)n]

(·)|n+ 12

= evaluation at midpoint(2.76)

Then, identifying the right hand side of (2.17) with the nonlinear vector-valued function f(y(t), t) of (2.67), the midpoint temporal discretization ofthe two thermo-spring system formulated in temperature variables (2.45)results in

q1,n+1 − q1,n

∆t=

1

m1

p1,n+ 12

q2,n+1 − q2,n

∆t=

1

m2

p2,n+ 12

p1,n+1 − p1,n

∆t= − ∂Ψ1

∂λ1

∣∣∣∣n+ 1

2

q1,n+ 12

λ1|n+ 12

− ∂Ψ2

∂λ2

∣∣∣∣n+ 1

2

(q1 − q2)n+ 12

λ2|n+ 12

+ f1

p2,n+1 − p2,n

∆t= − ∂Ψ2

∂λ2

∣∣∣∣n+ 1

2

(q2 − q1)n+ 12

λ2|n+ 12

+ f2

θ1,n+1 − θ1,n

∆t=

1

c1

[−θ1,n+ 1

2

∂η1

∂λ1

∣∣∣∣n+ 1

2

q1,n+ 12

λ1|n+ 12

·p1,n+ 1

2

m1

+ h|n+ 12

]θ2,n+1 − θ2,n

∆t=

1

c2

[−θ1,n+ 1

2

∂η1

∂λ1

∣∣∣∣n+ 1

2

(q1 − q2)n+ 12

λ2|n+ 12

·p1,n+ 1

2

m1

−θ2,n+ 12

∂η2

∂λ2

∣∣∣∣n+ 1

2

(q2 − q1)n+ 12

λ2|n+ 12

·p2,n+ 1

2

m2

− h|n+ 12

],

(2.77)


where use has been made of (2.4).

In view of the resulting formulation, the key point in a midpoint-typeapproximation of (nonlinear) evolution equations is that only the state vari-ables and their time derivatives are approximated in any time interval bytheir midpoint values (2.75). Thus, all functions depending on the state vari-ables are necessarily evaluated at midpoint via them. That is, for instance,the spring elongations (2.2) are provided by

λ1|n+ 12

= λ1(q1,n+ 12) = ‖q1,n+ 1

2‖

λ2|n+ 12

= λ2(q1,n+ 12, q2,n+ 1

2) = ‖(q2 − q1)n+ 1

2‖, (2.78)

whereas the heat flux (2.1) evaluated at midpoint results in

h|n+ 12

= k(θ2,n+ 12− θ1,n+ 1

2) (2.79)

Likewise, the constitutive relations (2.5) and (2.16), for a = 1, 2, evalu-ated at midpoint are provided by

fa|n+ 12

=∂Ψa

∂λa

∣∣∣∣n+ 1

2

=∂Ψa

∂λa(λa|n+ 1

2, θa,n+ 1

2)

va|n+ 12

= θa,n+ 12

∂ηa∂λa

∣∣∣∣n+ 1

2

= θa,n+ 12

∂ηa∂λa

(λa|n+ 12, θa,n+ 1

2)

(2.80)

In the same way, the Midpoint method applied to the entropy formulation(2.24) reads

q1,n+1 − q1,n

∆t=

1

m1

p1,n+ 12

q2,n+1 − q2,n

∆t=

1

m2

p2,n+ 12

p1,n+1 − p1,n

∆t= − ∂e1

∂λ1

∣∣∣∣n+ 1

2

q1,n+ 12

λ1|n+ 12

− ∂e2

∂λ2

∣∣∣∣n+ 1

2

(q1 − q2)n+ 12

λ2|n+ 12

+ f1

p2,n+1 − p2,n

∆t= − ∂e2

∂λ2

∣∣∣∣n+ 1

2

(q2 − q1)n+ 12

λ2|n+ 12

+ f2

η1,n+1 − η1,n

∆t=

1

θ1|n+ 12

h|n+ 12

η2,n+1 − η2,n

∆t=−1

θ2|n+ 12

h|n+ 12,

(2.81)


with the temperatures being in this case a function of the state variables(2.27) whose midpoint evaluations are provided by

θa|n+ 12

=∂ea∂ηa

(λa|n+ 12, ηa,n+ 1

2) = θa(λa|n+ 1

2, ηa,n+ 1

2) (2.82)

Accordingly, the heat flux results in

h|n+ 12

= k(θ2|n+ 12− θ1|n+ 1

2), (2.83)

and the internal force constitutive relation in

fa|n+ 12

=∂ea∂λa

∣∣∣∣n+ 1

2

=∂ea∂λa

(λa|n+ 12, ηa,n+ 1

2) (2.84)

Laws of thermodynamics and symmetries. The performed approx-imation is not harmless at all regarding the conservation properties (2.31) -(2.36). In fact, the loss of energy conservation for force-free evolution is awell-known result for solutions of nonlinear Hamiltonian problems providedby the midpoint method, see Simo & Tarnow (1992). On the positive side,the midpoint method does possess the conservation of the symplectic struc-ture of the Hamiltonian flow, details can be found in Feng (1986), and of theangular momentum. The latter can be verified by computing the angularmomentum balance in any time interval as

Jn+1 − Jn =2∑

a=1

(pa,n+1 ∧ qa,n+1 − pa,n ∧ qa,n)

=2∑

a=1

[qa,n+ 1

2∧ (pa,n+1 − pa,n)− pa,n+ 1

2∧ (qa,n+1 − qa,n)

]= ∆t

[q1,n+ 1

2∧(− f1|n+ 1

2

q1,n+ 12

λ1|n+ 12

)

+ q2,n+ 12∧(− f2|n+ 1

2

(q1 − q2)n+ 12

λ2|n+ 12

− f2|n+ 12

(q2 − q1)n+ 12

λ2|n+ 12

)]= 0

(2.85)

This result applies for the both formulations (2.77) and (2.81) since it isindependent of the definition of the internal force provided by either (2.80)1

or (2.84).


On the other hand, the loss of the energy conservation can also be provedby computing its balance in any time step as

En+1 − En = Kn+1 + Un+1 −Kn − Un

=2∑

a=1

(1

2ma

‖pa,n+1‖2 − 1

2ma

‖pa,n‖2 + ea,n+1 − ea,n)

=2∑

a=1

(1

2ma

(pa,n+1 + pa,n) · (pa,n+1 − pa,n) + ea,n+1 − ea,n) (2.86)

Using either (2.77) or (2.81) and discarding the external forces, the energybalance can be further elaborated to give

En+1 − En = − f1|n+ 12

λ1|2n+1 − λ1|2nλ1|n+ 1

2

− f2|n+ 12

λ2|2n+1 − λ2|2nλ2|n+ 1

2

− f2|n+ 12

λ2|2n+1 − λ2|2nλ2|n+ 1

2

+ e1,n+1 − e1,n + e2,n+1 − e2,n 6= 0,

(2.87)

which generally holds non-zero, concluding that the energy balance is notsatisfied by the Midpoint method.

Regarding the entropy balance, the entropy formulation (2.81) readilyallows to write

Sn+1 − Sn =2∑

a=1

(ηa,n+1 − ηa,n) =

(1

θ1|n+ 12

− 1

θ2|n+ 12

)h|n+ 1

2

= k

(θ2|n+ 1

2− θ1|n+ 1

2

)2

θ1|n+ 12θ2|n+ 1

2

≥ 0,

(2.88)

demonstrating the fulfillment of the second law by this formulation. In con-trast, the temperature formulation (2.77) does not inherit this feature asthe entropy balance cannot be reached from the discrete equation written interms of temperature (2.77).


2.3.2 Midpoint method for the thermo-viscoelastic system

Similarly, the thermo-visco-elastic system evolution equations are approx-imated by the Midpoint method (2.75). The temperature formulation (2.45)thus results in

q1,n+1 − q1,n

∆t=

1

m1

p1,n+ 12

q2,n+1 − q2,n

∆t=

1

m2

p2,n+ 12

p1,n+1 − p1,n

∆t= − ∂Ψ

∂λ

∣∣∣∣n+ 1

2

(q1 − q2)n+ 12

λ|n+ 12

+ f1

p2,n+1 − p2,n

∆t= − ∂Ψ

∂λ

∣∣∣∣n+ 1

2

(q2 − q1)n+ 12

λ|n+ 12

+ f2

γn+1 − γn∆t

= − 1

ν|n+ 12

∂Ψ

∂γ

∣∣∣∣n+ 1

2

θn+1 − θn∆t

=1

c

[−θn+ 1

2

∂η

∂λ

∣∣∣∣n+ 1

2

(q2 − q1)n+ 12

λ|n+ 12

·p1,n+ 1

2

m1

+1

ν|n+ 12

∂Ψ

∂γ

∣∣∣∣n+ 1

2

(∂Ψ

∂γ

∣∣∣∣n+ 1

2

+ θn+ 12

∂η

∂γ

∣∣∣∣n+ 1

2

)− h|n+ 1

2

]ϑn+1 − ϑn

∆t=

1

ςh|n+ 1

2

(2.89)

with the midpoint evaluated functions (2.46) and (2.51) being provided by

f |n+ 12

=∂Ψ

∂λ

∣∣∣∣n+ 1

2

, g|n+ 12

= − ∂Ψ

∂γ

∣∣∣∣n+ 1

2

, v|n+ 12

= θn+ 12

∂η

∂λ

∣∣∣∣n+ 1

2

w|n+ 12

= θn+ 12

∂Ψ

∂γ

∣∣∣∣n+ 1

2

, ν|n+ 12

= ν(θn+ 12)

(2.90)


The midpoint approximation of the entropy formulation (2.54) then reads

q1,n+1 − q1,n

∆t=

1

m1

p1,n+ 12

q2,n+1 − q2,n

∆t=

1

m2

p2,n+ 12

p1,n+1 − p1,n

∆t= − ∂e

∂λ

∣∣∣∣n+ 1

2

(q1 − q2)n+ 12

λ|n+ 12

+ f1

p2,n+1 − p2,n

∆t= − ∂e

∂λ

∣∣∣∣n+ 1

2

(q2 − q1)n+ 12

λ|n+ 12

+ f2

γn+1 − γn∆t

= − 1

ν|n+ 12

∂e

∂γ

∣∣∣∣n+ 1

2

ηn+1 − ηn∆t

=1

θ|n+ 12

1

ν|n+ 12

(∂e

∂γ

∣∣∣∣n+ 1

2

)2

− h|n+ 12

σn+1 − σn

∆t=h|n+ 1

2

ϑ|n+ 12

(2.91)

In this formulation, the temperatures are functions of the state variablesso they must be computed through the midpoint evaluation of them

θ|n+ 12

=∂e

∂η(λ|n+ 1

2, ηn+ 1

2, γn+ 1

2), ϑ|n+ 1

2=

dε

dσ(σn+ 1

2) (2.92)

Accordingly, the viscosity function (2.51) must be evaluated by the ele-ment temperature function as

ν|n+ 12

= ν(θ|n+ 12) = ν(θ(λ|n+ 1

2, ηn+ 1

2, γn+ 1

2)) (2.93)

Laws of thermodynamics and symmetries. As in the two thermo-springsystem, the balances of the continuous first integrals (linear and angular mo-mentum, energy) and the entropy in any time interval lead to the preservationof both linear and and angular momentums, the breach of the energy preser-vation and the satisfaction of the entropy inequality by the entropy formula-tion (2.91) and the breach of such inequality by the temperature formulation(2.89).


2.3.3 Numerical Experiments

All the identified features of the midpoint-based time integration of thetwo thermo-spring system and the thermo-viscoelastic system are illustratednext based on five numerical experiments.

For the two thermo-spring system, three numerical examples are proposedwith the following characteristics:

• Example 1 reproduces one of the examples proposed in Romero (2009).Specifically, the first one based on a constant spring stiffnesses Ca > 0and the following constant data: m1 = 1 kg, m2 = 2 kg, C1 = 0.1 Pa,C2 = 1 Pa, λ0

1 = 2 m, λ02 = 1 m, k = 300 Js−1K−1, α1 = α2 = 0.2

JK−1, c1 = c2 = 5 JK−1 and θref = 300 K. Then, the system is set inmotion by the following initial conditions:

q01 = e1, p0

1 = 2e2, q02 = 2.2e1, p0

2 = e1, θ01 = 380, θ0

2 = 310

• Example 2 is taken from Kruger et al. (2011) and is dominated by highfrequencies giving raise to a stiff problem. It is based on constant springstiffness and uses the following constant data: m1 = 1 kg, m2 = 2 kg,C1 = 100 Pa, C2 = 100 Pa, λ0

1 = 2 m, λ02 = 1 m, k = 10 Js−1K−1,

α1 = α2 = 0.2 JK−1, c0,1 = c0,2 = 1000 JK−1 and θref = 300 K; andwith the following initial conditions

q1,0 = 1e1, p1,0 = 1e2, q2,0 = 2.2e1, p2,0 = 4.4e2, θ1,0 = 380, θ2,0 = 310

• Example 3 further explores the applicability of the temperature for-mulation in a case impossible to be solved by formulations based onentropy. It is based on a non-linear temperature-dependent functionfor the spring stiffness expressed by

Ca(θα) = C0a − C1

aθref log

(θaθref

)(2.94)

The data employed in the simulation is:

m1 = 10 kg; m2 = 20 kg; C01 = 5000 Pa; C0

2 = 10000 Pa; C11 = 50

PaK−1; C12 = 60 PaK−1; λ0

1 = 2 m; λ02 = 1 m; k = 300 Js−1K−1;

α1 = α2 = 20 JK−1; c0,1 = 5000 JK−1; c0,2 = 2000 JK−1; θref = 300 K.

The initial conditions are provided by:

q1,0 = 3e1 + 0.5e3, p1,0 = 10e2, q2,0 = 3e1 + e2 + e3, p2,0 = −20e3

θ1,0 = 380, θ2,0 = 298


The first numerical example is integrated within the time interval [0, 25] swith constant time step size ∆t = 0.3 s using the Midpoint method applied tothe temperature formulation (2.77) and the entropy formulation (2.81). Fig-ure 2.3 and Figure 2.4 show the obtained degrees of freedom and the evolutionsof total energy, total entropy and the norm of the total angular momentum.

0 5 10 15 20 25−2

0

2

4

6

t [s]

logλ λ0

0 5 10 15 20 25275

300

325

350

375

t [s]

θ[K

]

0 5 10 15 20 25418

419

420

421

422

t [s]

E[J]

0 5 10 15 20 251.24

1.26

1.28

1.3

t [s]

S[JK

−1]

0 5 10 15 20 251

1.5

2

2.5

3

t [s]

‖J‖[N·m

]

Figure 2.3. Example 1: solution obtained with the Midpoint methodformulated in temperature variables with ∆t = 0.3 s


0 5 10 15 20 25−2

0

2

4

6

t [s]

logλ λ0

0 5 10 15 20 25275

300

325

350

375

t [s]

θ[K

]

0 5 10 15 20 25419

420

421

422

t [s]

E[J]

0 5 10 15 20 251.24

1.26

1.28

1.3

t [s]

S[JK

−1]

0 5 10 15 20 251

1.5

2

2.5

3

t [s]

‖J‖[N·m

]

Figure 2.4. Example 1: solution obtained with the Midpoint methodformulated in entropy variables with ∆t = 0.3 s

Both methods are stable for the time step used as they manage to main-tain the evolution of energy bounded although, as expected, none of themprovides the correct first law of thermodynamics, confirming the theoreticalresult deduced by (2.87). Significantly, the dissatisfaction of the first law ismuch more pronounced at the beginning of the simulation when the evolu-


tions of temperatures are dramatically oscillating. Such oscillations are dueto the time step size used suggesting that it is not an appropriated one tocapture the dynamics of the problem.

0 100 200 300 400 500310

327.5

345

362.5

380

t [s]

θ[K

]

0 100 200 300 400 5000

20

40

60

80

100

t [s]λ

0 100 200 300 400 5008.95

9

9.05

9.1

9.15·104

t [s]

E[J]

0 100 200 300 400 500268

272

276

280

t [s]

S[JK

−1]

0 5 10 15 20 2510.6

10.65

10.7

10.75

10.8

t [s]

‖J‖[N·m

]


On the other hand, the total entropy evolution seems to be less problem-atic although, once again, at the beginning of the simulation, the temperature


formulation, Figure 2.3, breaks the second law. In contrast, the entropy for-mulation agrees with the second law as was dictated by (2.88). In both cases,the angular momentum preservation is satisfied according to (2.85).

Now, the second example is integrated using the temperature-based andthe entropy-based midpoint methods within the time interval [0, 500] s withconstant time step size ∆t = 0.1 s.

This moderate time step leads both methods to a non-physical solutionfor the elongations, as shown in Figure 2.5 and Figure 2.6, highlighting thata solution disagreeing with the essential physical features may be completelyspurious despite a successful integration. Thus, the first law of thermody-namics is breached by both methods although, once again, they manage tohave the total energy bounded, hence its successful integration. In contrast,the second law is satisfied by the entropy-based Midpoint method while thetemperature-based Midpoint method is clearly unable to meet it. The angu-lar momentum, on the other hand, is correctly computed by both methods,as expected.

Finally, the third example is solved just by the temperature-based Mid-point method (2.77) since the entropy-based Midpoint method cannot handlewith the non-linear dependency of spring stiffnesses. This examples addi-tionally brings into play a 3D motion of the two masses and a high thermo-mechanic coupling.

Then, the temperature-based Midpoint is used with a time step of ∆t =0.1 s to integrate the motion up to 25 s. The results are summarized in Figure2.7 and clearly show that the simulation became unstable at the half of wholetime interval. The behavior of the total energy reveals that the method failswhen it dramatically and uncontrollably decays. Before this event, both lawof thermodynamics are breached although the angular momentum adheresto its conservation law.

With these three simple examples, the limitations for the standard Mid-point method to handle a simple nonlinear dissipative problem have beenhighlighted and reasoned that are intimately related to its inability to fulfillessential physical rules, chiefly with the first law of thermodynamics. Thus,the preservation of the angular momentum does not seem to be crucial for thestability of the method, neither the agreement with the second law seems toprovide with extra stability’s features as shows the entropy-based Midpointsolutions. Of course, the Midpoint method is consistent meaning that by di-minishing the time step these limitation are alleviated although incrementingthe computational costs and times. All these problems are to be bypassed


0 100 200 300 400 500310

327.5

345

362.5

380

t [s]

θ[K

]

0 100 200 300 400 5000

20

40

60

80

100

t [s]λ

0 100 200 300 400 5008.95

9

9.05

9.1

9.15·104

t [s]

E[J]

0 100 200 300 400 500268

272

276

280

t [s]

S[JK

−1]

0 5 10 15 20 2510.6

10.65

10.7

10.75

10.8

t [s]

‖J‖[N·m

]

Figure 2.6. Example 2: solution obtained with the Midpoint methodformulated in entropy variables with ∆t = 0.1 s


0 5 10 15 20 25300

320

340

360

380

t [s]

θ[K

]

0 5 10 15 20 25−0.1

0.3

0.7

1.1

1.5

t [s]

λ

0 5 10 15 20 252

3

4

5·105

t [s]

E[J]

0 5 10 15 20 251

1.2

1.4

1.6·103

t [s]

S[JK

−1]

0 5 10 15 20 2570

80

90

100

t [s]

‖J‖[N·m

]



with methods based precisely on the conservation of all the underlying es-sential physical features.

For the thermo-viscoelastic system, consider a system composed of aparticle of mass m connected by a thermo-viscoelastic element to a fixed pointof space. In addition, the parameter µ(θ) associated with the free energyfunction of the Maxwell element is assumed to have a linear dependencewith the temperature, provided by µ(θ) = µ0 − µ1(θ − θref), with µ0, µ1 > 0.

The parameters employed in the simulation are θref = 300 K, m = 1 kg,λ0 = 1 m, α = 4 JK−1, c = 1 JK−1, µ0 = 5 Jm−2, µ1 = 0.1 Jm−2K−1,ν0 = 100 Nsm−1, a = 10 K, k = 10 Js−1K−1. The motion is integrated witha constant time step size and initial conditions:

q0 = [3, 0] m , q0 = [0, 1] m/s , γ0 = 0 m , θ0 = 380 K (2.95)

Then, two different cases are considered:

• Case I. Constant environment temperature ϑ = θref and a linear temper-ature-dependent stiffness C(θ) = C0 − C1(θ − θref), with C0 = 100 Paand C1 = 0.5 PaK−1.

This case corresponds to the model solved in Garcıa Orden & Romero(2011) with the entropy formulation and validated with a referencesolution.

• Case II. Non-constant environment temperature ϑ and non-linear tem-perature-dependent stiffness

C(θ) = C0 − C1θref log(θ/θref), (2.96)

with C0 = 100 Pa and C1 = 0.5 PaK−1.

Specific heat capacity ς = 120 JK−1. The initial environment temper-ature is set to ϑ0 = θref = 300 K.

Note that Case I is the limit case of Case II for large values of ς and smalltemperature changes with respect to the environment temperature.

Results for Case I. First, the entropy formulation is analyzed. Since noanalytical results are available, the solution obtained with the entropy-basedMidpoint method (2.91) and a time step size ∆tref = 10−2 s has been adoptedas a reference. Figure 2.8 collects the curves of the reference trajectory, tem-perature, internal variable, total energy, total entropy and total angular mo-mentum up to 20 s. These results suggest that there is a limit configuration


of relative equilibrium, where the trajectory is circular and the element tem-perature equals the environment one. The initial temperature oscillations arecaused by the viscous dissipation of the dashpot, while the smooth long-termbehavior is dominated by the heat conduction process. The whole evolutionis controlled by the laws of thermodynamics and symmetries as shows thecurves for the total energy, total entropy and total angular momentum. Thetime step size is then small enough to not spoil the preservation featuresof the system, thus demonstrating the physical consistency of the Midpointmethod.

Figure 2.9 contains the results obtained with the Midpoint method for-mulated in entropy variables and with a time step twenty times larger, that is∆t = 0.2 s. This time step size makes the method become unstable leadingto the failure of integration before the 15 s. Once again, the wrong behav-ior of the total energy evolution explains the occurring instability since thetotal entropy and total angular momentum are satisfied, without seeminglycontributing to stabilize the integration.

In contrast, the same time step size of ∆t = 0.2 s is not critical for thetemperature-based Midpoint method (2.89) to provide a stable solution asshows Figure 2.10. In this solution, the total energy does not neither adhereto the constraint of the first law, however, it does not drift significantly apartfrom it, definitely stabilizing the method’s performance with this time stepsize. Unexpectedly, the second law is satisfied even though the temperatureevolution clearly suggests the time step size to be inappropriate to capturethe initial oscillations.

Results for Case II. The reference solution has been obtained with thetemperature-based Midpoint method and a time step size of ∆tref = 10−2

s. The results are displayed in Figure 2.11 and show the expected long termconvergence of the evolution of both element and environment temperaturesdue to the heat transfer between them while agreeing with the preservationfeatures.

Then, a time step size of ∆t = 0.3 is used to obtain the results con-tained in Figure 2.12. For this time step size, the temperature-based Midpointmethod fails to satisfy the first law of thermodynamics. However, as in CaseI, the behavior of the energy is good enough to attain a stable integration.Although the evolution of the total entropy does not strictly agree with thesecond law, it is quite satisfactory whereas the angular momentum is exactlypreserved, as expected.

All these numerical experiments further corroborate the limitations of the


−2 −1 0 1 2 3−2

0

2

q2 [m]

q 1[m

]

0 5 10 15 20280

300

320

340

360

380

t [s]θ,ϑ[K

]

0 5 10 15 200

0.01

0.02

0.03

0.04

t [s]

γ[m

]

0 5 10 15 2017

17.02

17.04

17.06

17.08

17.1·102

t [s]

E[J]

0 5 10 15 205.4

5.5

5.6

5.7

t [s]

S[JK

−1]

0 5 10 15 202

2.5

3

3.5

4

t [s]

‖J‖[N·m

]

Figure 2.8. Case I: reference solution obtained with the Midpoint methodformulated in entropy variables with ∆t = 0.01 s


−2 −1 0 1 2 3−2−10

1

22

q2 [m]

q 1[m

]

0 5 10 15 20280

300

320

340

360

380

t [s]

θ ,ϑ[K

]

0 5 10 15 200

0.01

0.02

0.03

0.04

t [s]

γ[m

]

0 5 10 15 201.7

1.71

1.72

1.73·103

t [s]

E[J]

0 5 10 15 205.4

5.5

5.6

5.7

t [s]

S[JK

−1]

0 5 10 15 202

2.5

3

3.5

4

t [s]

‖J‖[N·m

]

Figure 2.9. Case I: solution obtained with the Midpoint method formulatedin entropy variables with ∆t = 0.2 s


−2 −1 0 1 2 3−2−10

1

22

q2 [m]

q 1[m

]

0 5 10 15 20280

300

320

340

360

380

t [s]θ,ϑ[K

]

0 5 10 15 20

0

0.01

0.02

0.03

0.04

t [s]

γ[m

]

0 5 10 15 201.7

1.71

1.72

1.73·103

t [s]

E[J]

0 5 10 15 205.4

5.5

5.6

5.7

t [s]

S[JK

−1]

0 5 10 15 202

2.5

3

3.5

4

t [s]

‖J‖[N·m

]

Figure 2.10. Case I: solution obtained with the Midpoint methodformulated in temperature variables with ∆t = 0.2 s


Midpoint method to deal with the first law of thermodynamics when usingmoderate time step sizes, as was also observed in the numerical experimentson the two thermo-spring system.

−2 −1 0 1 2 3−2−10

1

22

q2 [m]

q 1[m

]

0 5 10 15 20280

300

320

340

360

380

t [s]

θ ,ϑ[K

]

0 5 10 15 20

0

0.01

0.02

t [s]

γ[m

]

0 5 10 15 2016.75

16.8

16.85·102

t [s]

E[J]

0 5 10 15 205.35

5.4

5.45

5.5

5.55

t [s]

S[JK

−1]

0 5 10 15 202

2.5

3

3.5

4

t [s]

‖J‖[N·m

]

Figure 2.11. Case II: reference solution obtained with the Midpoint methodformulated in temperature variables with ∆t = 0.01 s


−2 −1 0 1 2 3−2−10

1

22

q2 [m]

q 1[m

]

0 5 10 15 20280

300

320

340

360

380

t [s]θ,ϑ[K

]

0 5 10 15 20

0

0.01

0.02

t [s]

γ[m

]

0 5 10 15 2016.8

16.85

16.9

16.95·102

t [s]

E[J]

0 5 10 15 205.35

5.4

5.45

5.5

5.55

t [s]

S[JK

−1]

0 5 10 15 202

2.5

3

3.5

4

t [s]

‖J‖[N·m

]

Figure 2.12. Case II: solution obtained with the Midpoint methodformulated in temperature variables with ∆t = 0.3 s

Nonlinearthermo-dissipative

continuum dynamics Chapter

3The concern of this Chapter is to establish a general framework that en-

compasses the dynamics of thermo-dissipative continua, understood as con-tinua that may experience geometrically nonlinear displacements and defor-mation coupled with temperature changes and transformations of the in-ternal structure of the material which macroscopically result in dissipative(irreversible) changes. First, the Chapter starts with the kinematic assump-tions for deformable continua, then, a thorough description of the generalizedstandard materials is provided to enable the formulation of general thermo-dissipative material behaviors. Along with these ingredients, balance lawsare established to arrive at the strong form of the governing equations. Fur-thermore, the conservation properties of the resulting system of equationsare deduced from the isolated dynamic equations. These developments havebeen based on some classical references such as Marsden & Hughes (1983) orHolzapfel (2000). A recent review on this topic can be found in Gurtin et al.(2010).

Once the continuous description is fully elaborated, the Galerkin FE-based spatial approximation is performed to transform the original infinite-dimensional problem into a finite-dimensional one. A brief summary on dif-ferent spatial discretization techniques is also introduced. Among them all,the Galerkin approach is used due to its simplicity and ability to not damagethe conservation properties identified in continuous systems. Finally, a subse-quent temporal discretization based on standard time integration methods isproposed and studied regarding the laws of thermodynamics and symmetries.

60 3.1. Kinematics

3.1 Kinematics

Consider a continuum occupying a set of points in the undeformed (ref-erence) configuration B0 ⊂ Rd with boundary ∂B0, d being the spatialdimension (d = 1, 2, 3), whose positions are collected in X ∈ B0 (respectto an inertial frame of reference) and undergoing a time-dependent motiondescribed by

ϕ : B0 × [0, T ] 3 (X, t) 7→ ϕ(X, t) ∈ Rd, (3.1)

which maps any material point X to the point x = ϕ(X, t) ∈ Rd in thedeformed configuration Bt = ϕ(B0, t) ⊂ Rd.

B0 ⊂ R3

•P

∂B0

ρ0(X)

X

Bt = ϕ(B0, t) ⊂ R3

•p

∂Bt

ρ(X, t)

x

X2, x2

X3, x3

X1, x1

x = ϕ(X, t)

F =∂ϕ(X, t)

∂Xvp

Figure 3.1. Continumm kinematics

The motion function (3.1) is assumed to be smooth so the tangent map,denoted as F and called the deformation gradient, can be defined at eachpoint and time as

F(X, t) :=∂ϕ(X, t)

∂X= ∇0ϕ(X, t), (3.2)

∇0 being the gradient operator in the reference configuration.

The deformation gradient transforms vectors in the tangent referencespace TXB0 to vectors in the tangent spatial space TϕBt. Thus, F : TXB0 7→TϕBt is a linear transformation for eachX ∈ B0. Given the (inertial) framesof reference for each of the reference (Lagrangian) (X1, X2, X3) and spatial

3. Nonlinear thermo-dissipative continuum dynamics 61

(Eulerian) (x1, x2, x3) spaces, the deformation gradient is a second-order two-point tensor F ∈ T2

d, with Tqd being the space of qth-order tensors of dimensiond, whose components are provided by

FaB =∂ϕa∂XB

(3.3)

An infinitesimal element volume in the reference (material) configurationis then mapped to the current (spatial) configuration by the non-negativeJacobian determinant or volume ratio J provided by

J(X, t) := det [F(X, t)] > 0, (3.4)

in accordance with the principle of the local impenetrability of the matter. Asa result, the deformation gradient tensor is positive-definite F ∈ (T2

d)+, i.e ithas positive eigenvalues.

On the other hand, the time derivative of the motion is the materialvelocity v : B0 × [0, T ]→ Rd provided in the reference configuration by

v(X, t) :=∂ϕ(X, t)

∂t= ϕ(X, t), (3.5)

˙(·) indicating time differentiation hereafter.

Accordingly, the linear momentum per unit of reference volume p : B0×[0, T ]→ Rd is defined as

p(X, t) := ρ0(X)v(X, t), (3.6)

where ρ0 : B0 → R is the density in the reference configuration. All theforegoing kinematic relations are depicted in Figure 3.1.

Interestingly, the rate of the deformation gradient is linked with the ve-locity vector (3.5) or the linear momentum (3.6) as follows

F(X, t) :=∂

∂t∇0ϕ(X, t) = ∇0v(X, t) = ∇0

[p(X, t)

ρ0(X)

], (3.7)

as the material coordinates X are time-independent, so the velocity gradienttensor measures the rate at which the deformation gradient changes.

Remark 3.1. Materials phenomenologically present distinct behavior ac-cording to different modes of deformation. Often, volumetric and isochoricdeformation modes are observed to response in clearly distinguishable man-ners, for which the assumption of the dilatational-deviatoric multiplicativesplit of the deformation gradient is normally considered. It is based on amodified deformation gradient tensor whose dilatational part vanishes

F = J13 F with det F = 1 (3.8)

62 3.2. Generalized Standard Materials

3.2 Generalized Standard Materials

The types of constitutive behaviors considered in this dissertation are en-compassed by the generalized standard materials framework Celigoj (1998);Mielke (2006). Such materials may generally undergo geometrically nonlin-ear deformations coupled with thermal effects and dissipative mechanismsderived from transformations of the internal structure. Within this generalframework, any phenomenological effect can be considered, such as plastic,viscoelastic or damage, piezoelectric effects, microstructure defects, etc. Inthis sense, the material response will generally be history-dependent sinceboth elastic or inelastic evolutions will be involved.

For the consideration of coupling thermal effects the Lagrangian absolutetemperature field is introduced as

Θ : B0 × [0, T ] 3 (X, t)→ Θ(X, t) ∈ R+ (3.9)

On the other hand, general irreversible effects are described by additionalvariables that capture the dissipative transformations which take place inthe internal structure of materials. Despite existing different approaches tochoose these variables, the concept of internal variables Coleman & Gurtin(1967) has extensively been used when facing irreversible evolutions of ma-terials. Its use is motivated by numerical considerations, as it easily suitssimulations based on the Finite Element method. That is why, this disser-tation will hereafter restrict to this framework. They are also referred to ashidden variables emphasizing their non-measurable character. For continuousmedia, they are symmetric strain-like tensors

Λα : B0 × [0, T ] 3 (X, t)→ Λα(X, t) ∈ Sym(T2d), (3.10)

where α ∈ [1,m] is the number of internal variables and Sym(T2d) ⊂ T2

d isthe space of symmetric second-order tensors. For further developments, alsothe space of symmetric and positive-definite (positive eigenvalues) second-order tensor is introduced as a subset of the space of symmetric second-ordertensors, that is Sym(T2

d)+ ⊂ Sym(T2d)

Generalized standard materials are then described by the deformationϕ, the absolute temperature Θ and the internal state variables Λα. Thematerial thermodynamic evolution is fully determined by knowing these threequantities at any instant time and at any point of the body composed by thegeneral standard material. Specifically, objectivity requirements lead to the


measure of the deformations provided by either the symmetric right Cauchy-Green deformation tensor

C := FTF ∈ Sym(T2d)+, (3.11)

in the undeformed configuration B0, or by the symmetric left Cauchy-Greendeformation tensor

b := FFT ∈ Sym(T2d)+, (3.12)

in the deformed configuration Bt. Both deformation tensors are bidirection-ally related through either the push-forward or pull-back tensor operations.

3.2.1 Physical constitutive laws

The material behavior can uniquely be defined by the Helmholtz free-energy function, measured per unit reference volume, which, in the La-grangian description, is

Ψ : B0 × [0, T ] 3 (C,Θ,Λα)→ Ψ(C,Θ,Λα) ∈ R (3.13)

It follows from the constitutive theory, see Truesdell et al. (2004), whichis based on the three general principles of determinism, local action andframe-indifference. Particularly, the frame-indifference principle is responsi-ble for the free-energy function to depend on the Cauchy-Green tensors, aspreviously indicated, see Holzapfel (2000).

Furthermore, it is assumed to be a smooth function with the followingproperties

Ψ(1,Θref ,1) = 0, Ψ(C,Θ,Λα) ≥ 0, (3.14)

which physically means that the material contains no energy in the unde-formed configuration and at a given temperature known as a reference tem-perature Θref and that the energy increases with deformation, temperatureor dissipative changes in the material.

These properties limit specific types of material behaviors, some of whichwill be discussed in subsequent chapters.

Remark 3.2. If the material behaves isotropically, the free-energy functiondepend only on the invariants of Cauchy-Green tensors, normally defined as

IC := tr(C) = tr(b) = Ib

IIC :=1

2

[I2C − tr(C2)

]=

1

2

[I2b − tr(b2)

]= IIb

IIIC := det C = det b = IIIb = J2

(3.15)


Also, it is usual to define isotropy free-energy functions in terms of theprincipal stretches, i.e the square root of the eigenvalues of the Cauchy-Greentensors, denoted as λC

i , λbi for i = 1, 2, 3, which are related to the above

invariants by

IC =(λC

1

)2+(λC

2

)2+(λC

3

)2

IIC =(λC

1

)2 (λC

2

)2+(λC

1

)2 (λC

3

)2+(λC

2

)2 (λC

3

)2

IIIC =(λC

1

)2 (λC

2

)2 (λC

3

)2

(3.16)

Remark 3.3. Often, the Helmholtz free-energy function (3.13) is assumed tobe additively decoupled into volumetric and isochoric contributions Bonet &Wood (2008). Under this assumption, the thermal effects are commonly con-sidered to contribute to the volumetric part (Holzapfel, 1996; Reese & Govin-djee, 1998a) whereas internal dissipative transformations are observed to berelated mainly with the isochoric deformations, see for instance (Holzapfel &Simo, 1996b; Reese & Govindjee, 1998b).

Remark 3.4. The described generalized standard material model may gen-erally rely on any hyperelastic model either defined in terms of the invariantsΨ(IC, IIC, IIIC) or in terms of the principal stretches Ψ(λC

1 , λC2 , λ

C3 ) of the

right Green-Lagrange tensor.

For thermodynamic consistency, the previously introduced free-energyfunction (3.13) must satisfy the Clausius-Planck inequality, see Truesdellet al. (2004), which establishes that the rate of internal dissipation mustalways be non-negative at any particle and any instant time and reads

D :=1

2S : C− Ψ− ηΘ ≥ 0, (3.17)

(·) : (·) meaning the inner product in the space of second-order tensors T2d, S :

B0× [0, T ]→ Sym(T2d) being the symmetry Piola-Kirchhoff stress tensor and

η : B0 × [0, T ] → R being the entropy function measured per unit referencevolume, which are the work conjugated quantities to the right Cauchy-Greendeformation tensor (3.12) and the absolute temperature (3.32), respectively.

The chain rule applied to the free-energy function (3.13) along with fur-ther thermodynamic arguments, see Coleman & Gurtin (1967), leads to thefollowing constitutive laws for the symmetry Piola-Kirchhoff stress tensor

S := 2∂Ψ(C,Θ,Λα)

∂C, (3.18)


and for the entropy function per unit reference volume

η := −∂Ψ(C,Θ,Λα)

∂Θ(3.19)

With these definitions the rate of internal dissipation yields

D =m∑α=0

−∂Ψ(C,Θ,Λα)

∂Λα: Λα ≥ 0, (3.20)

from where the constitutive law for the work conjugated quantities to inter-nals variables, i.e the dissipative driving stress tensors Qα : B0 × [0, T ] →Sym(T2

d), can be deduced to be

Qα := −∂Ψ(C,Θ,Λα)

∂Λα(3.21)

More importantly, the condition (3.20) determines the evolution equa-tion for each internal variables. In view of it, the rate of any internal vari-ables must generally pertain to a positive semidefinite functional K : T2

d 7→T2d | A : K(A) ≥ 0, ∀ A ∈ T2

d depending on its work conjugate quantity toyield a quadratic form that ensures the fulfillment of the inequality, that is

Λα = K(Qα) ∀ α = 1, . . . ,m (3.22)

A material thus described is said to be dissipative.

Remark 3.5. Each different dissipative effect requires a particular form ofsuch functional. For instance, linear smooth dissipative effects are providedby a linear fourth order mapping of the form

K(Qα) = Vα : Qα, (3.23)

Vα being a general positive semidefinite fourth order tensor specified by ma-terial parameters which may depend on time-dependent variables.

The constitutive laws (3.18) - (3.21) are physical in the sense that theyare functions of the physical variable temperature.

3.2.2 Heat conduction

The existence of a non-uniform temperature field in the body producesthe conduction phenomenon which tends to homogenize the temperature field


in the body by flowing heat from the warmer to the colder region of the body.The sense of the flow, naturally defined in the Eulerian description by theCauchy heat flux vector h : Bt × [0, T ] → Rd, is prescribed by the heatconduction inequality Holzapfel (2000), which reads

h · ∇[

1

Θ

]≥ 0, (3.24)

∇ being the spatial gradient, and determines the way the heat must flow,that is, opposited to the temperature gradient. A suitable constitutive lawfor the Cauchy heat flux is thus furnished by

h(ϕ, t) := Θ2κ(Θ)∇[

1

Θ

], (3.25)

which is further supported by experimental observations and known as Du-hamel’s law of heat conduction.

In this law, the symmetric second-order tensor κ : Bt×[0, T ]→ Sym(T2d)+

is the spatial thermal conductivity tensor, required to be positive semidefiniteand possibly temperature dependent. If the material is thermally isotropic,the conductivity tensor becomes κ(Θ) = κ(Θ)1, κ ≥ 0 being the thermalconductivity parameter.

The Lagrangian form of the Duhamel’s law (3.25) can be achieved byusing the relation between the spatial and the material gradient ∇(•) =F−T∇0(•) and the pull-back operation on the Cauchy heat flux to arrive atthe Piola-Kirchhoff heat flux H : B0 × [0, T ]→ Rd, see Holzapfel (2000),

H(X, t) := Θ2JF−1κ(Θ)F−T∇0

[1

Θ

](3.26)

Finally, the following simplified notation for the Lagrangian conductivitytensor K : B0 × [0, T ]→ Sym(T2

d)+ is introduced

K(X, t) := JF−1κ(Θ)F−T, (3.27)

which, in the case of thermal isotropy, simplifies to

K(X, t) := Jκ(Θ)C−1 (3.28)

3.2.3 The internal energy function

The internal energy is an important thermodynamic potential that macro-scopically collects every present microscopic form of energy in materials. The


canonical form of this potential, defined per unit reference volume, is pro-vided as a function of the entropy and other mechanical or internal variables,see Truesdell et al. (2004), that is

e : B0 × [0, T ] 3 (C, η,Λα)→ e(C, η,Λα) ∈ R, (3.29)

because it is defined through the Legendre transformation of the free-energyfunction (3.13) when replacing the temperature with its work conjugate en-tropy as the thermodynamic state variable. The Legendre transformation ofthe free-energy function with respect to the entropy thus results in

e = Ψ + Θη, (3.30)

which defines the caloric equation of state.

Remark 3.6. Although the internal energy is canonically a function of theentropy, one can readily obtain internal energy functions in terms of tempera-ture through the Legendre transform applied to free energy functions canon-ically defined in terms of temperature (3.13), see Dillon Jr. (1962, 1963).Accordingly, the internal energy defined in terms of temperature results in

e(C,Θ,Λα) = Ψ(C,Θ,Λα) + Θη(C,Θ,Λα), (3.31)

This form of the internal energy will play a central role in subsequent chap-ters.

In order for the internal energy function to be defined from canonicallytemperature-based free-energy functions (3.13), the entropy function (3.19)must be uniquely invertible respect to Θ at fixed C and Λα so that thefunction

Θ : B0 × [0, T ] 3 (C, η,Λα)→ Θ(C, η,Λα) ∈ R+, (3.32)

might be achieved. Assuming this, the canonical form of the internal energycan be obtained, following (3.30), as

e(C, η,Λα) := Ψ(C,Θ(C, η,Λα),Λα) + Θ(C, η,Λα)η (3.33)

Remark 3.7. The above requirement has to be viewed as an importanttechnical restriction to the description of specific thermoelastic models usingthe entropy as a thermodynamic state variable.

Provided that, the entropy-based description of generalized standard ma-terials can be achieved from the canonical form of the internal energy po-tential (3.29) by deriving new constitutive laws for the symmetric Piola-Kirchhoff stress tensor, the temperature function and the dissipative driving


stress tensor. To do that, the rate of internal dissipation is accordingly ex-pressed in terms of this potential using (3.30) to give

D =1

2S : C− e+ Θη ≥ 0 (3.34)

Then, applying the chain rule to the function (3.29) yields

D =1

2S : C−

(∂e

∂C: C +

∂e

∂ηη +

m∑α=1

∂e

∂Λα: Λα

)+ Θη ≥ 0 (3.35)

Thus, following the same arguments as before, the symmetric Piola-kirchhoff stress tensor is now provided by

S := 2∂e(C, η,Λα)

∂C(3.36)

Then, the temperature function (3.32) is defined as

Θ :=∂e(C, η,Λα)

∂η, (3.37)

which is consistent to the relation (3.33) and, finally, the dissipative drivingstress tensor now becomes

Qα := −∂e(C, η,Λα)

∂Λα(3.38)

Assuming these new definitions, the resulting rate of internal dissipa-tion will agree with the thermodynamics as long as the evolution equationsfor the internal variables follow the general law (3.22), now using the newconstitutive law for the dissipative driving stress tensor (3.38).

3.2.4 Constitutive behaviors for limit cases

The just discussed generalized standard materials framework is suitableto derive any kind of material behavior ranging from the purely reversiblecase to general irreversible cases both adiabatic or isothermal. It has beenintroduced in the most general way to describe general thermo-dissipativematerial models so that it encompasses every particular case.

Thus, one merely needs to neglect either the thermal or the dissipativeeffects to arrive at either isothermal models or purely elastic behaviors. To


this end, the thermodynamic potentials dependencies must be reduced ac-cordingly. That is, if the dissipative effects are ignored the general form ofthe thermodynamic potentials will be

s := s(C,Θ) or s := s(C, η), (3.39)

s representing any thermodynamic potential. This case corresponds withnonlinear thermoelasticity which can be described either in terms of tem-perature or entropy via the canonical form of the free-energy or the internalenergy potentials, respectively.

On the other hand, in the case of neglecting the thermal effects the re-sulting material behavior will be isothermal but can still dissipate energy.Therefore, the thermodynamics reduces to mechanical dynamics with dissi-pation, thereby the thermodynamic potentials collapse to the stored energyfunction that might generally depend on deformation and internal variables

Ψ := Ψ(C,Λα) = e(C,Λα) + A, (3.40)

with A being some constant of the motion.

Thus, the reversible nonlinear hyperelastic behavior can be recoveredignoring dissipative effects, leading to the well-known deformation-dependentfree-energy function

Ψ := Ψ(C) = e(C) + A (3.41)

In subsequent sections and chapters we will be considering every casepointed out before.

3.3 Balance laws: strong form of the initial-boundaryvalue problem

In this section the global balance laws, from which the strong form of thegoverning equations are derived, are briefly summarized. Also, the Cauchy’sstress theorem and its thermodynamic counterpart Stokes’ heat flux theoremare introduced.

The evolution of a continuous body composed of a generalized standardmaterial is determined by global balance laws of mass, momenta and energy.

70 3.3. Balance laws: strong form of the initial-boundary value problem

3.3.1 Balance of mass

Any continuum contains a fundamental property called mass that ac-counts for the amount of matter contained inside its boundary. A basicprinciple of non-relativistic mechanics is that mass cannot be created or de-stroyed, leading to the balance of mass expressed as

m :=

∫B0

ρ0(X)dV0 =

∫Bt

ρ(X, t)dv > 0, (3.42)

ρ0 : B0 → R+ and ρ : B0 × [0, T ] → R+ being the density field in materialand spatial description respectively. Therefore, the rate of change of the massthroughout time must vanish

m =

∫B0

ρ0(X)dV0 = 0, (3.43)

which is automatically fulfilled since the reference mass density is indepen-dent of time.

3.3.2 Balance of linear and angular momentum

Classical mechanics is grounded on the axioms of momentum balanceprinciples: linear or translational and angular or rotational. The total linearmomentum is a vector-valued function L : [0, T ]→ Rd defined by the volumeintegral of the linear momentum per unit (reference) volume (3.6) as follows

L(t) :=

∫B0

p(X, t)dV0 (3.44)

The balance of linear momentum establishes that the change of this vectormagnitude equals to the resultant of external forces, Fext : [0, T ] → Rd overthe body, that is

L(t) = Fext(t) (3.45)

In other words, the action of external forces on a body causes the totallinear momentum to change which is the classical statement of the secondNewton’s law.

The resultant of external forces is the sum of all forces acting in the bodyor on its boundary, as sketched in Figure 3.2, and can be expressed as

Fext(t) =

∫B0

B0(X, t)dV0 +

∫∂B0

T0(X, t,N )dA0, (3.46)


B0 ⊂ Rd

∂B0

dV0B0

T 0N

dS0

NdS0

Y

Z

X

Figure 3.2. Volume forces and tractions in the reference configuration.

B0 : B0 × [0, T ]→ Rd being any volumetric force expressed in the referenceconfiguration, such as the gravitational force ρ0g, T0 : ∂B0 → Rd being thetraction vector on the boundary measured per unit reference surface area andN denoting the unit outward normal field on ∂B0.

As the reference configuration remains invariable along the time thechange of the total linear momentum is the volume integral of the rate ofthe linear momentum per unit reference volume, enabling the balance prin-ciple to be expressed as∫

B0

p(X, t)dV0 =

∫B0

B0(X, t)dV0 +

∫∂B0

T0(X, t,N )dA0 (3.47)

Furthermore, the Cauchy’s stress theorem, see Holzapfel (2000), relatesthe current traction vector t : ∂Bt× [0, T ]→ Rd with the symmetric Cauchystress tensor σ : Bt× [0, T ]→ Sym(T2

d), defined in the spatial configuration,which can be translated to the reference configuration as

T0(X, t,N ) = (FS)(X, t) ·N (3.48)

The Gauss’ theorem1 thus allows to transform the resulting surface inte-gral into the volume integral through the material divergence operator∇0·(•),that is ∫

B0

pdV0 =

∫B0

B0dV0 +

∫B0

∇0 · (FS)dV0, (3.49)

1

∫A0

F ·NdA0 =

∫V0

∇0 · FdV0


where the magnitudes’ dependencies have been omitted to simplify the no-tation.

Finally, reallocating terms and arguing that the above identity must holdfor any reference volume, the integrand must identically be nil at any materialpoint and at any time, leading to the strong form or the linear momentumbalance in the Lagrangian description

p = ∇0 · (FS) +B0 (3.50)

Similarly, the total angular momentum is defined as vector-valued func-tion J : [0, T ]→ Rd resulting from the overall volume integral of the momentof the linear momentum vector per unit (reference) volume (3.6) with respectto any fixed point of space, e.g the origin of the inertial frame of reference,that is

J(t) =

∫B0

ϕ(X, t) ∧ p(X, t)dV0 (3.51)

The angular momentum balance states that the just introduced mag-nitude changes due to the presence of a resultant of external moment (ortorques) Mext : [0, T ]→ Rd acting over the body

J(t) = Mext(t) (3.52)

Accordingly, the resultant of external moments is provided by the sumof all volumetric forces and tractions moments

Mext(t) =

∫B0

ϕ(X, t) ∧B0(X, t)dV0 +

∫∂B0

ϕ(X, t) ∧ T0(X, t,N )dA0

(3.53)

Further elaboration of this principle leads to the already introduced sym-metry of the Cauchy stress tensor σ ∈ Sym(T2

d) which is a well-known re-sult whose demonstration can be found in classical references, for instanceMalvern (1969), and hence omitted.

3.3.3 Balance of energy

In pure mechanics the balance of energy is a natural consequence ofthe momentum postulations. Particularly, the total energy of a conservativemechanic system is said to be a first integral of the system evolution, see Hand& Finch (1998). However, the consideration of coupled thermo-dissipative


effects leads to the thermodynamic context in which the momentum balancesmust be supplemented by the balance of energy (first law) and the entropyinequality (second law).

Every internal form of energy in the continuum is collected by the totalinternal energy U : [0, T ]→ R, provided by

U(t) :=

∫B0

e(X, t)dV0 (3.54)

On the other hand, related to the motion, the continuum develops thekinetic energy K : [0, T ]→ R, defined as

K(t) :=

∫B0

1

2ρ0(X)‖p(X, t)‖2dV0, (3.55)

‖ · ‖ being the l2-norm in the euclidean space.

Furthermore, the continuum interacts with its surroundings through ex-ternal sources of energies coming from external forces and heats generallypresent either in the continuum or on its boundary.

The way these forms of energy interact follows the first law of thermo-dynamics: the energy balance. It postulates that the external mechanicaland thermal power supplied to a thermodynamic continuum is employed tochange both its kinetic and its total internal energy

K(t) + U(t) = Pext(t) +Q(t), (3.56)

The power of external mechanical forces or the rate of the work done bythe external forces (3.46), Pext : [0, T ]→ R, may generally be expressed as

Pext(t) :=

∫B0

B0(X, t) ·v(X, t)dV0 +

∫∂B0

T0(X, t,N ) ·v(X, t)dA0 (3.57)

On the other hand, the thermal power is supplied to the continuumthrough either heat reservoirs, R : B0 × [0, T ] → R measured per unittime and unit reference volume, or heat fluxes, HN : ∂B0 × [0, T ] → R perunit reference surface area, entering or escaping its boundary, see Figure 3.3.Therefore, the rate of thermal work Q : [0, T ]→ R may be expressed as

Q(t) :=

∫B0

R(X, t)dV0 +

∫∂B0

HN(X, t,N )dA0 (3.58)


B0 ⊂ R3

∂B0

dV0R

N

dS0

H

Y

Z

X

Figure 3.3. Volumetric source or sink of heat and heat flux escapingthrough the boundary in the reference configuration

Similarly to the Cauchy’s stress theorem (3.48) in continuum mechanics,the Stokes’ heat flux theorem postulates that the scalar function HN is alinear function of the outward unit normal

HN(X, t,N ) := −H(X, t) ·N , (3.59)

where H(X, t) follows the Duhamel’s law of heat conduction provided in(3.26).

Considering each of the above contributions the first law (3.56) can berewritten in the Lagrangian description as∫

B0

1

ρ0

p · pdV0 +

∫B0

edV0 =

∫B0

B0 · vdV0 +

∫∂B0

(FS)N · vdA0

+

∫B0

RdV0 −∫∂B0

H ·NdA0

(3.60)

The surface integral depending on the stress tensor must be elaboratedin order to properly apply the Gauss’ theorem, that is∫

∂B0

(FS)N · vdA0 =

∫∂B0

(FS)Tv ·NdA0 =

∫B0

∇0 · [(FS)Tv]dV0 (3.61)


The product rule of the divergence2 then allows to obtain∫B0

∇0 · [(FS)Tv]dV0 =

∫B0

(FS) : ∇0vdV0 +

∫B0

v · ∇0 · (FS)dV0 (3.62)

Using this last result in (3.60), the linear momentum balance (3.49) andapplying the Gauss’ theorem on the heat flux term, the energy balance resultsin ∫

B0

(e− FS : ∇0v −R+∇0 ·H) dV0 = 0 (3.63)

Following the definition of the rate of the deformation gradient (3.7) andsome tensor algebra3,4, the second term of (3.63) can be related to the rateof the right Cauchy-Green deformation tensor (3.11) as follows

FS : ∇0v = FS : F = S : FTF = S : Sym(FTF

)=

1

2S : C, (3.64)

where use has been made of the symmetry of the second Piola-Kirchhoffstress tensor.

As the reference volume is arbitrary, the local form of the energy balanceresults in

e =1

2S : C−∇0 ·H +R (3.65)

The balance (3.65) clearly couples the energy involved in the deformationprocess with the thermal processes taking place in the continuum.

Finally, the entropy inequality is ensured by the dissipation inequalitytogether with the heat conduction inequality introduced in subsections 3.2.1and 3.2.2.

3.3.4 Strong form of the governing equations

The above introduced balance laws expressed in local form (3.50), (3.65)and the evolution equations for internal variables (3.22) define the strongform of the governing equations for a thermo-dissipative continuum. Thatis, given B0 : B0 → Rd, R : B0 → R and constants T0 : ∂B0 → Rd,

2∇0 · (ATb) = A : ∇0b+ b · ∇0 ·A3AB : C = A : BTC4Sym(A) = 1

2 (A + AT)


HN : ∂B0 → R, find (ϕ,p, e,Λα) : B0 → Rd × Rd × R × Sym(T2d)m, such

that

ϕ =1

ρ0

p

p = ∇0 · (FS) +B0

e =1

2S : C−∇0 ·H +R

Λα = K(Qα) ∀ α = 1, . . . ,m

in B0 × [0, T ] (3.66)

It is a fully coupled first order partial differential equations (1st orderPDE) system whose solution provides the thermodynamics evolution of acontinuum from prescribed initial and boundary conditions.

In this way, initial conditions in B0 and at time t = 0 may generally beprovided by

ϕ(X, 0) = X

p(X, 0) = ρ0v0(X)

s(X, 0) = s0(X)

Λα(X, 0) = 1

in B0, (3.67)

where v0(X) is a prescribed field, s0(X) represents a prescribed scalar fieldfor any of the thermodynamic state variable employed to describe the sys-tem thermodynamics, in this case s = e, and the initial strain-like internalvariables are assumed to depart from an undeformed state.

The boundary ∂B0 is usually decomposed into the subsets ∂Bϕ0 , ∂Bσ

0

∂Bθ0 and ∂BH

0 in which Dirichlet or Neumann boundary conditions mightbe imposed. These subsets have the following properties ∂Bσ

0 ∪ ∂Bϕ0 = ∂B0

and ∂Bσ0 ∩ ∂Bϕ

0 = 0, ∂BH0 ∪ ∂Bθ

0 = ∂B0 and ∂BH0 ∩ ∂Bθ

0 = 0.

Thus, Dirichlet boundary conditions on ∂B0 a times t ≥ 0 might bespecified by

ϕ = ϕ on ∂Bϕ0 × [0, T ],

s = s on ∂Bθ0 × [0, T ],

(3.68)

ϕ being the prescribed motions on the boundary ∂Bϕ0 and s representing

one more time a prescribed field of any of the thermal variables.

Likewise, Neumann boundary conditions might be provided by

(FS) ·N = T0 on ∂Bσ0 × [0, T ],

H ·N = −HN on ∂BH0 × [0, T ]

(3.69)


Remark 3.8. The initial-boundary value problem (IBVP) (3.66) - (3.68)can further be particularized according to the election of the thermodynamicstate variable. The most representative formulations are based on eitherentropy or temperature, although there are no mathematical restrictions toconsider any other variable, for instance, the internal energy per unit refer-ence volume as been already presented. However, from the practical point ofview, this election is essential when Dirichlet’s boundary conditions are re-quired since, generally, variables other than temperature cannot technicallybe measured on boundaries. For this reason, the implications of both entropyand temperature formulations will become central in the rest of this disser-tation and, therefore, we will pay special attention to their particularities inthe subsequent subsections.

3.3.5 Entropy formulation of the IBVP

The use of the canonical form of the internal energy (3.33) along withits corresponding constitutive laws (3.36) - (3.38) enables the rate of internalenergy to be expressed as

e(C, η,Λα) =1

2S : C + Θη −

m∑α=1

Qα : Λα (3.70)

Identifying the last term on the right hand side of (3.70) as the rate ofinternal dissipation according to (3.34) and substituting it into (3.65) theentropy form of the local energy balance yields

η =1

Θ

(−∇0 ·H +R+ D

), (3.71)

with Θ being a function of the entropy, the deformation and the internalvariables according to (3.37).

The above form of the energy balance must be complemented with initialand boundary conditions expressed in terms of the state variable entropy.Without loss of generality, the initial condition for entropy can be assumedto be zero, that is

η(X, 0) = 0 in B0 (3.72)

However, in some applications an initial prescribed field of temperatureΘ0(X) in the body is required so an initial non-zero entropy needs to bespecified by solving

Θ(X, 0) =∂e

∂η(C(X, 0), η(X, 0),Λα(X, 0)) = Θ0(X) (3.73)


Regarding Dirichelt’s boundary conditions, the value of the entropy inthe Dirichlet boundary subset ∂Bθ

0 would then required. However, whatis normally available is the value of the temperature on some parts of theboundary, Θ on ∂Bθ

0 × [0, T ], so the solution of the identity

Θ =∂e

∂η(C, η,Λα) = Θ on ∂Bθ

0, (3.74)

is required at all instant times in order to translate the temperature valuesinto entropy ones, thereby they could be imposed as follows

η = η on ∂Bθ0 × [0, T ] (3.75)

This complicates in excess its numerical solution, practically boundingthe use of this formulation to applications with no initial conditions of tem-peratures and no Dirichlet’s boundary conditions.

3.3.6 Temperature formulation of the IBVP

The temperature-based formulation of the IBVP (3.66) - (3.68) is derivedfrom the use of the temperature form of the internal energy (3.31) provided bythe canonical form of the free-energy function (3.13) and its constitutive laws(3.18) - (3.21). Thus, the rate of change of the internal energy is computedthrough the chain rule as

e(C,Θ,Λα) =

(1

2S + Θ

∂η

∂C

): C+

∂e

∂ΘΘ−

m∑α=1

(Qα −Θ

∂η

∂Λα

): Λα (3.76)

The appearing partial derivatives of the entropy function (3.19) are linkedwith the calorimetric concepts of latent heat tensor defined as

V(C,Θ,Λα) := 2Θ∂η(C,Θ,Λα)

∂C, (3.77)

the specific heat capacity assumed to be a real positive number because of thematerial’s stability, see Silhavy (2013), measured per unit reference volumeand provided by

c := Θ∂η(C,Θ,Λα)

∂Θ=∂e(C,Θ,Λα)

∂Θ> 0, (3.78)

and the dissipative counterpart to the just introduced latent heat tensor,generally referred to as dissipative latent heat tensor, provided by

Wα(C,Θ,Λα) := Θ∂η(C,Θ,Λα)

∂Λα(3.79)


Using expressions (3.76) to (3.79) in (3.65), and the definition of the rateof the dissipation (3.20), the temperature form of the energy balance is

Θ =1

c

(−1

2V : C−

m∑α=1

Wα : Λα −∇0 ·H +R+ D)

(3.80)

where terms linked with the latent heat tensors (3.77) and (3.79) are referredto as structural thermoelastic heating and structural thermodissipative heatingwhich accounts for the Gough-Joule effects arising from the coupling thermo-mechanic response and for the same physic phenomenon but related to thedissipative transformations in the continuum, respectively.

The structural thermoelastic heating is thus a scalar-valued functionHe : B0 × [0, T ] → R that provides the heat generated due to mechani-cal deformations

He(X, t) := −1

2V : C (3.81)

Similarly, the scalar-valued function Hd : B0 × [0, T ] → R is introducedas the structural thermodissipative heating which accounts for the heat dis-sipated due to dissipative changes in the material

Hd(X, t) := −m∑α=1

Wα : Λα (3.82)

Moreover, by taking into account the symmetry of the latent heat tensor(3.77) along with the relation (3.64), the structural thermoelastic heatingmay be written as

He = −1

2V : C = −FV : ∇0

[p

ρ0

](3.83)

In this way, the temperature form of the energy balance (3.80) in termsof the state variables is

Θ =1

c

(−FV : ∇0

[p

ρ0

]−∇0 ·H +R+ D +Hd

)(3.84)

In contrast to the entropy formulation, the resulting local energy balanceinvolves more terms but easily allows to handle with both initial and Dirich-let and Neumann boundary conditions, commonly provided in temperaturevariables by

Θ(X, 0) = Θ0(X) in B0

Θ = Θ on ∂Bθ0 × [0, T ]

H ·N = −HN on ∂BH0 × [0, T ]

(3.85)

80 3.4. Specific thermo-dissipative material models

3.4 Specific thermo-dissipative material models

To close the presented dynamics of thermo-dissipative solids, specificforms of the free-energy function (3.13) must be proposed so that all derivedconstitutive quantities can be elaborated. Typically, generalized standardmaterial models are built from thermoelastic models which in turn rely onsimple hyperelastic models. Taking this into account, a particular isotropicthermoelastic model is introduced to serve also as a base for the formula-tion of generalized standard materials, particularly, for the purpose of thisdissertation, nonlinear (smooth) thermoviscoelastic material models will beconsidered.

The nonlinear thermoelastic regime accounts for the nonlinear defor-mations with significant thermal effects and without permanent structuralchanges or cracks even in the high strain ranges. Elastomeric (high-polymeric)substances such as rubber naturally behave in this regime. A representativemodel for them responds to the following free-energy function structure

Ψ(C,Θ) = f(Θ)Ψ(C) + T (J,Θ), (3.86)

with Ψ(C) being any hyperelastic free-energy function, f(Θ) is a temperature-dependent function for the variations of material parameters with tempera-ture and T (J,Θ) a function accounting for the thermal behavior and for thethermoelastic coupling effects.

As was previously remarked, the thermal effects are observed to be cou-pled with dilatational deformations. The form of the function T consequentlyagrees with so by considering two different effect, namely, a pure thermalresponse and a thermo-volumetric coupling effect via a thermal coefficientexpansion α, that is

T (J,Θ) = −3B0α(Θ−Θref)G(J) + c

(Θ−Θref −Θ log

Θ

Θref

), (3.87)

where G(J) is a potential that penalizes the change of volume, B(Θ) is thebulk modulus which may depend on temperature such that at reference tem-perature Θref its value is B(Θref) = B0.

This structure of the thermoelastic free-energy function can be foundin Holzapfel & Simo (1996a) with f(Θ) = Θ/Θref . Romero (2010a) alsoemployed it to derive entropy-dependent thermodynamic potentials for whichadditional assumptions must be made on the above free-energy function.Particularly, the function f must at most be linear, that is f ′(Θ) = C, so


that the following identity can analytically be inverted

η = −∂Ψ

∂Θ(C,Θ) = −f ′(Θ)Ψ(C) + 3B0αG(J) + c log

Θ

Θref

(3.88)

The result of this inversion thus defines the temperature function in termsof the right Cauchy-Green deformation tensor and the density entropy

Θ(C, η) = Θref exp

[1

c(η + CΨ(C)− 3B0αG(J))

](3.89)

This model can be extended to account for smooth thermoviscoelasticbehavior by attaching to (3.86) a term depending on internal variables whichtypically vanishes at mechanical equilibrium

Ψ(C,Θ,Λα) = Ψ(C,Θ) + Γ(C,Θ,Λα) (3.90)

The function Γ defines the particular thermoviscoelastic model. Thus,Holzapfel & Simo (1996b) proposed a quadratically-dependent function oninternal variables as follows

Γ(C,Θ,Λα) =m∑α=1

[µα(Θ) |Λα|2 − 2

∂Ψαiso(C,Θ)

∂C: Λα + Ψα

iso(C,Θ)

],

(3.91)where µα ∈ R+ is a non-negative parameter associated to the α-thermo-viscoelastic process and Ψα

iso : B0 × [0, T ] → R is a hyperelastic potentialrelated to the viscoelastic processes. This last potential is expressed in termsof isochoric deformations F because, according to real experiments on poly-mers, see Malvern (1969), the viscoelastic effects occur strongly in shear andweaker in dilatation. For objectivity considerations, this function dependson the modified strain Green-Lagrange tensor C defined as

C := J−23 C (3.92)

Unless stated otherwise the hyperelastic free-energy function is also as-sumed to be additively split into volumetric and isochoric parts, see Holzapfel(2000), as follows

Ψ∞(C,Θ) := Ψ∞vol(J,Θ) + Ψ∞iso(C,Θ) (3.93)

Associated with this model, the dissipative functional becomes

K(Qα) =1

να(Θ)Qα, να(Θα) = 2µα(Θ)τα > 0, (3.94)

82 3.5. Symmetries and laws of thermodynamics

where να(Θ) is a viscosity parameter and τα ∈ R+ is the relaxation/retarda-tion time, both characterizing the viscous evolution.

Alternatively, based on the multiplicative split of the deformation gra-dient tensor, Reese & Govindjee (1998a) proposed a function Γ that takesthe form of the free-energy function Ψ but evaluated by an objective tensorderiving from the multiplicative structure of the deformation gradient, whichis identified with the internal variable.

3.5 Symmetries and laws of thermodynamics

The evolution of isolated thermo-dissipative continua is determined byequation symmetries associated to conserved quantities. These quantitiesare revealed by the balances laws (3.45) - (3.56) when no external forces andheats are present, that is F (t) = M (t) = 0 and Q(t) = 0. Thus, followingthe momentum balances, the total linear and angular momentum must bepreserved

L(t) = J(t) = 0 (3.95)

According to Noether’s theorem, see Goldstein et al. (1965), these con-served quantities are related to the symmetries of the thermo-dissipative con-tinuum evolution equations, which will be in-depth discussed in subsequentChapters.

On the other hand, the first laws of thermodynamics for isolated systemslead to the conservation of the total energy E : [0, T ] → R, defined as thekinetic energy (3.55) plus the total internal energy (3.54), that is

E(t) := K(t) + U(t) (3.96)

The proof for the conservation of the total energy directly follows fromthe energy balance (3.60) for isolated continua, that is

E(t) = 0 (3.97)

Likewise, the second law of thermodynamics leads to the preservation(reversible processes) or increment (irreversible processes) of the total entropyS : [0, T ]→ R+ provided by

S(t) :=

∫B0

η(X, t)dV0 (3.98)


Then, the second law can be verified by using either the entropy formu-lation (3.71) or the temperature formulation (3.80) to express the rate ofentropy per unit reference volume as

S(t) =

∫B0

η(X, t)dV0 =

∫B0

1

Θ

[−∇0 ·H + D

]dV0 (3.99)

Further elaboration of (3.99) is performed by making use of divergenceproduct rule5 and the Gauss’ theorem to obtain

S(t) = −∫∂B0

1

ΘH ·NdA0 +

∫B0

∇0

[1

Θ

]·HdV0 +

∫B0

1

ΘDdV0 (3.100)

As the continuum is isolated the flux on its boundary must be zero sothe first term of (3.100) vanishes. Using the Duhamel’s law (3.26) alongwith (3.27), the rate of total entropy can be demonstrated to be always non-negative

S(t) =

∫B0

1

Θ2∇0ΘK∇0ΘdV0 +

∫B0

1

Θ

m∑α=1

Qα : K(Qα)dV0 ≥ 0, (3.101)

because both the conductivity tensor (3.27) and the functional K (3.22) arepositive semidefinite.

3.6 Isothermal dynamics as a limit case

In many engineering applications the thermal effects can be disregardedas observations show they do not significantly contribute to the system evo-lution, although dissipative effects might still condition it. Thus, the limitcase of isothermal changes represents a useful framework to study these typesof phenomena.

To derive this limit case, the temperature of the continuum is assumed tobe constant so that the material behavior might depend only on the deforma-tion and internal variables, as previously introduced in (3.40). The absenceof temperature changes obliges to consider the Clausius-Planck inequalityfor isothermal processes, also called internal dissipation inequality, to ensurethat the possible dissipated energy flows according to the second principle ofthermodynamics. Such inequality reads

D =1

2S : C− Ψ ≥ 0 (3.102)

5f∇0 · v = ∇0 · (fv)− v · ∇0f

84 3.6. Isothermal dynamics as a limit case

Remark 3.9. When there is no dissipation the above inequality becomes anidentity, giving raise to the reversible pure elastic deformation case in whichthe free-energy function depends only on deformations (3.41).

Following analogous arguments as those which led from (3.18) to (3.21)the ensuing constitutive laws are obtained

S = 2∂Ψ(C,Λα)

∂C, Qα = −∂Ψ(C,Λα)

∂Λα(3.103)

With this assumption, the isothermal formulation can be derived fromthe previous one (3.66) by neglecting all temperature-dependent terms. Pro-ceeding this way the strong form of isothermal dynamics reduces to

ϕ =1

ρ0

p

p = ∇0 · (FS) +B0

Λα = K(Qα) ∀ α = 1, . . . ,m

in B0 × [0, T ] (3.104)

The isothermal consideration brings the system back to the mechaniccontext in which the energy balance is no longer required to be fulfilled butit is a consequence of the momentum balances (3.45) and (3.52). However,for consistency, the total energy must include the total dissipated energyD : [0, T ]→ R provided by

D(t) :=

∫B0

D(X, t)dV0 =

∫B0

(∫ t

0

Dds

)dV0 (3.105)

Thus, the energy balance now postulates that the external mechanicalpower (3.57) supplied to the continuum is employed to change the total en-ergy (3.96) and to produce dissipated energy (3.105) due to the developmentof dissipative processes, that is

E(t) + D(t) = Pext(t) (3.106)

In this context, the total energy (3.96) is redefined to account for thedissipated energy

E(t) := E(t) + D(t) (3.107)

Then, the energy balance (3.106) applied to isolated systems (Pext(t) = 0)reveals that this new energy is preserved. As it is usually expressed using the


phase space variables, that is, coordinates and momenta, it is also referredto as Augmented Hamiltonian, see Mohr (2008).

Within the just introduced isothermal framework, classical nonlinear vis-coelasticity, plasticity or damage could be formulated by defining appropriatefunctionals K. We will hereafter focus on smooth changes, that is, viscoelas-ticity, although we will also give a brief summary of the canonical case ofnon-smooth problems: plasticity.

3.6.1 Viscoelasticity at finite strain

The most representative isothermal smooth dissipative problem is vis-coelasticity. Considering a linear law for the evolution equations of internalsvariables, it is provided by a dissipative functional K of the form (3.23),leading to the ensuing general form of the evolution equations

ϕ =1

ρ0

p

p = ∇0 · (FS) +B0

Λα = Vα : Qα ∀ α = 1, . . . ,m

in B0 × [0, T ] (3.108)

This form of the evolution equations fully reproduces the typical vis-coelastic behavior characterized by the irreversible processes known as relax-ation and creep. These time-dependent processes connect two consecutivestates of mechanic equilibrium under the specific boundary conditions of ei-ther fixed strain or fixed stress, sketched in Figure 3.4, and are determined bythe relaxation/retardation time τ ∈ R+ which measures the rate of decay/riseof the stress/strain.

In this context, the dissipative driving stress tensor Qα is referred to asnon-equilibrium stress tensor since it tends to disappear as the mechanicalequilibrium is reached.

Many different viscoelastic models at finite strain can be found in theliterature depending on how the internal variable is defined and how thefourth order tensor viscoelastic tensor is furnished, including linear or non-linear evolution equations for the internal variables. Next, some importantexamples are summarized.

Holzapfel and Simo. The thermoviscoelastic model due to Holzapfel &Simo (1996b) and provided by (3.91) can be used as isothermal viscoelastic


Stress

equilibrium

timeτ

at fixed strain

time

Strainequilibrium

τ

at fixed stress

Figure 3.4. Relaxation and creep processes

model neglecting the temperature field such that

Ψ(C,Λα) = Ψ∞(C) +m∑α=1

[µα |Λα|2 − 2

∂Ψαiso(C)

∂C: Λα + Ψα

iso(C)

], (3.109)

where Ψ∞ : B0 × [0, T ]→ R represents any hyperelastic potential, µα ∈ R+

is a non-negative parameter associated to the α-viscoelastic process andΨα

iso : B0 × [0, T ] → R is a hyperelastic potential related to the viscoelas-tic processes. Accordingly, the free-eergy function is additively split intovolumetric and isochoric parts, see Holzapfel (2000), as follows

Ψ∞(C) := Ψ∞vol(J) + Ψ∞iso(C) (3.110)

The above viscoelastic free energy function (3.109) ensures positive dissi-pation if the rate of internal variables follows a linear law which results fromthe following fourth order viscoelastic tensor

Vα =1

ναI ∀α = 1, ...,m, (3.111)

with I being the four rank identity tensor and να being the viscosity coefficientlinked to the relaxation/retardation time, inspired by linear viscoelasticity,via

να = 2µατα (3.112)

Furthermore, the viscoelastic free-energy function (3.109) depends quadrat-ically on internals variables and hence the non-equilibrium stress tensor(3.103)2 will result in a linear function of them

Qα = −2µαΛα + 2

∂Ψαiso(C)

∂C, (3.113)


so that the evolution equations become linear first order differential equations

Λα +1

ταΛα =

2

να

∂Ψαiso(C)

∂C, (3.114)

where use has been made of (3.112). Hence, it can analytically be integrated6

to give

Λα(X, t) =

∫ t

0

exp

[−(t− s)τα

]2

να

∂Ψαiso

∂C(X, s)ds+ exp

[−tτα

]Λα

0 (3.115)

The history-dependent character of such variables is reflected on the timeintegration over all previous instant times. Obviously, the above expressionneeds the partial derivative of the deformation-dependent potential Ψα

iso tobe known at each instant time and material point.

In the literature, such potential Ψαiso, associated to the α-dissipative pro-

cess, is widely assumed to be a proportional part of the hyperelastic potentialΨ∞iso by defining non-dimensional constitutive parameters Govindjee & Simo(1992), known as strain-energy factors βα ≥ 0, such that

Ψαiso(C) = βαΨ∞iso(C) with βα =

2µαE∞

∀ α = 1, . . . ,m, (3.116)

with E∞ being the Young’s modulus defining the hyperelastic potential.

This assumption is further supported by experimental tests on elastomerswith viscoelastic behavior characterized by a medium composed of identicalpolymer chains.

Remark 3.10. Regarding the numerical solution of this model, an impor-tant drawback may emerge when applying iterative strategies, i.e Newton-Rapshon method, in which the linearization of the governing equations arerequired. This step would involve the computation of a rank six tensor arisingfrom the linearization of the stress tensor, known as the linearized materialtensor, that is

C = 2∂S

∂C= 4

∂2Ψ

∂C2= 4

(1 +

m∑α=1

βα

)∂2Ψ∞iso∂C2

− 8m∑α=1

βα∂3Ψ∞iso∂C3︸︷︷︸∈T6

d

: Λα (3.117)

Nevertheless, as this sixth order tensor is scaled by the internal variables,in most of the cases this term can be neglected respect to the others suchthat they could be enough for a successful convergence process.

6y′ + P (x)y = Q(x)→ y(x) = exp

[−∫

P (x)dx

] ∫exp

[∫P (x)dx

]Q(x)dx


Remark 3.11. The described viscoelastic model can be based on anisotropicmodels as demonstrated in Holzapfel & Gasser (2001).

On the other hand, Holzapfel & Simo (1996b) introduced the followingsimple form of this potential to provide a rheological interpretation of theirmodel

Ψαiso(C) =

1

8(C− 1) : 2µαI : (C− 1) (3.118)

This function is defined through the whole right Cauchy-Green deformationtensor (3.11) instead of its isochoric part (3.92) as was presented. It is theclassical Saint-Venant hyperelastic model with the stiffness parameter ex-pressed in terms of the positive parameter µα. In this particular case, it canbe proved that the constitutive (3.103) laws are related in the following way

S = S∞ +m∑α=1

Qα (3.119)

Thus, the model follows the rheological behavior provided by the gener-alized Maxwell element, composed of a spring arranged in parallel with asmany as Maxwell-elements as required for the viscoelastic response, sketchedin Figure 3.5. That is, at the beginning of the deformation the materialresponse is distributed among the elastic (main) spring and the Maxwell-elements (Qα 6= 0). Instead, as the viscoelastic effects fully develop thewhole response is only due to the main spring (Qα = 0).

Now, it is convenient to approach the evolution equations for the internalvariables by using the non-equilibrium stress tensor, via time differentiationof (3.108)3 and (3.114) with (3.111), that is

Qα +1

ταQα = 2

d

dt

[∂Ψα

iso

∂C

], (3.120)

whose analytical solution is of the form

Qα = exp

[− Tτα

]Qα

0 +

∫ T

0

exp

[−(T − t)τα

]2

d

dt

[∂Ψα

iso

∂C

]dt (3.121)

This final expression thus enters (3.119) directly, so the specification ofthe strain-like internal variables are no longer needed to determine the vis-coelastic response.

Mandel stress tensor. A common way to construct ispotropic viscoelasticmodel relies on the concept of the Mandel stress tensor, extensively used in


Ψ∞, E∞ > 0

2µ1 > 0

2µα > 0

βα =2µα

E∞

η1 > 0

ηα > 0

•

•

•

•

γ1

γα

τα = ηα/2µα

ε

• σ

Figure 3.5. Rheological model

formulations of inelastic material, see Gross & Betsch (2010); Miehe (1995).The departure point is to consider an isotropic material behavior which de-pends only on invariants of the tensor

Γ := CΛ−1, (3.122)

Λ being the symmetric strain-like internal variable, such that

Ψ(C,Λ) := Ψ(Γ) = Ψ(IΓ, IIΓ, IIIΓ) (3.123)

This special structure of the viscoelastic free-energy function leads to thedefinition of the symmetric Mandel stress tensor Y : B0 × [0, T ] 7→ Sym(T2

d)which directly derives from the constitutive relationships (3.103) and reads

Y(X, t) :=∂Ψ

∂Γ= 2ΛQ = CS (3.124)

Then, the evolution equation for the internal variable is defined throughthis new stress-like tensor rather than directly through the non-equilibrium


stress tensor, as previously proposed in (3.104)3, observing that the dissipa-tion inequality (3.102) can be expressed by

D = Q : Λ = ΛQ : Λ−1Λ = Y :1

2Λ−1Λ ≥ 0 (3.125)

In this connection, the second factor in the last term is viewed as ameasure of the viscous deformation rate collected in a second order tensorZ : B0 × [0, T ] 7→ Sym(T2

d) as follows

Z :=1

2Λ−1Λ, (3.126)

such that the evolution equation emanates from making the above inequalitya quadratic form, that is

Z = V : Y, (3.127)

with the viscoelastic fourth order tensor V being positive semidefinite. Fur-thermore, it can naturally be expressed in terms of the internal variables andthe non-equilibrium stress tensor as follows

Λ = 2ΛV : (ΛQ), (3.128)

which is an alternative form of (3.23).

Reese and Govindjee. When facing inelastic effects at finite strains, oftenthe locally multiplicative decomposition of the deformation gradient (3.2) intoelastic Fe and inelastic (in this case viscous) Fi components, as proposed inLubliner (1985), is introduced

F := FeFi (3.129)

In this decomposition, it is implicitly assumed the existence of a generallyincompatible stress-free configuration x ∈ B, called elastic, at each neigh-borhood of every particle point p in the current configuration, as showed inFigure 3.6.

Accordingly, new frame-invariance strain measures are directly derivedsuch as

Ce := FeTFe, Ci := Fi

TFi, be := FeFeT, bi := FiFi

T, (3.130)

with the following crucial relationships among them

Ce = F−Ti CF−1

i , be = FC−1i FT, (3.131)


B0

•PdX

Bt

•pdx

F

B

• pdx

Y, y, y

Z, z, z

X, x, xFi

Fe

Figure 3.6. Multiplicative decomposition in the small neighborhood of aparticle

The inelastic right Cauchy-Green deformation tensor Ci is defined in thereference configuration, whereas its elastic counterpart Ce is defined withreference to the elastic configuration.

This decomposition can also be viewed as a rheological model such asthe one sketched in Figure 3.5 but with one Maxwell-element. Thus, theelastic deformation gradient can be interpreted as a deformation measureof the main spring whereas the inelastic one is seen as a deformation ofthe Maxwell-element spring, so that the latter plays the role of an internalvariable.

Based on this interpretation, Reese & Govindjee (1998b) proposed a vis-coelastic free-energy function expressed as

Ψ(C,F−1i ) := Ψ∞(C) + Ψi(F−T

i CF−1i ) (3.132)

The potential Ψi accounts for the viscoelastic effects and is chosen in away that it vanishes when no viscous deformations are present so that thebody responds elastically according to the hyperelastic free-energy functionΨ∞. In addition, the viscoelastic constitutive response is actually a functionof the elastic right Cauchy-Green deformation tensor (3.130)1 via (3.131)1.Accordingly, Ψi is widely assumed to be the same potential as Ψ∞ but eval-uated by Ce since it complies with the requirement of vanishing when noviscoelastic deformations occurs, i.e Ce = 1.

The above viscoelastic free-energy function (3.132) is fully consistent


with the previously introduced Mandel stress tensor approach in the caseof isotropy. With this assumption, the above viscoelastic free-energy func-tion can indifferently be expressed in terms of the invariants of the tensorsCe and CC−1

i such that

Ψ(C,F−1) = Ψ∞(C) + Ψi(Ce) = Ψ∞(C) + Ψi(CC−1i ) (3.133)

In view of that, the Mandel stress tensor approach follows directly fromidentifying the symmetric strain-like internal variable to be

Λ := Ci (3.134)

Hence, the evolution equation for the internal variable is

Ci = 2CiV : (QCi), (3.135)

where V is a positive semidefinite fourth order tensor specified by two pos-itive material parameters νD ∈ R+ and νV ∈ R+ referred to as deviatoricand volumetric viscosity, respectively, which may generally be deformation-dependent, so that

V =1

2νD

(I− 1

31⊗ 1

)+

1

9νV1⊗ 1 (3.136)

Following the classic plasticity theory, Reese & Govindjee (1998a,b) pre-sented the above evolution equations in the current configuration via therelation between the inverse of the inelastic right Cauchy-Green tensor andthe elastic left Cauchy-Green deformation tensor (3.131)2.

Bonet. Also based on the locally multiplicative split of the deformation gra-dient (3.129), Bonet (2001) proposed the same structure of the viscoelasticfree-energy function as (3.132) to arrive at the additive structure of the con-stitutive law for the symmetric Piola-Kirchhoff stress tensor

S = S∞ + Si (3.137)

Then, relying on kinematic assumptions rather than thermodynamic con-sistency the following evolution equation was proposed

Ci = −2

τ

[∂2Ψi(C,Ci)

∂C∂Ci

]−1

: Si (3.138)


The above relation ensures that the viscoelastic effects truly occur dueto deviatoric deformations. Nevertheless, unlike the previous models, it doesnot rely on the dissipation inequality (3.102) so, generally, it does not haveto agree with the thermodynamics. This does not mean that one cannotfind particular material models which matches this crucial requirement whilecomplying with (3.138), but it does mean that any material model used mustbe checked to fulfill the required thermodynamic consistency.

All presented models would require only the specification of a hyperelas-tic Helmholtz free-energy function Ψ∞(C) to provide particular viscoelasticmodels.

In the rest of this dissertation the model due to Holzapfel and Simo willbe chosen without losing of generality.

3.6.2 Nonlinear plasticity at finite strain

Although plasticity is a non-smooth dissipative effect and hence out ofthe scope of this dissertation, it is interesting to include some notes abouthow it can be described from the framework previously presented.

The non-smooth nature of the plasticity can mathematically be describedby using the concept of sub-differentials, denoted as ∂, see Studer (2009) forthe application to non-smooth dynamics. Thus, the positive semidefinitefunctional K (3.22) becomes a sub-differential of a positive convex functionalJ , key details can be found in Mielke (2011), so that the nonlinear plasticitycan be cast into the following general form

ϕ =1

ρ0

p

p = ∇0 · (FS) +B0

Λα ∈ ∂QJ (Qα) ∀ α = 1, . . . ,m

in B0 × [0, T ] (3.139)

The convex functional is thus linked with the flow rule, the consistencycondition and the loading-unloading conditions that define the plastic be-havior. These ingredients have traditionally been proposed to handle thenon-smooth plastic evolution. For example, assuming the multiplicative de-composition (3.129), any finite strain plastic model is proved to be thermo-dynamically sound if

− 1

2Lv(be)b

−1e = ζ

∂φ

∂τ, (3.140)

94 3.7. Semidiscrete thermo-dissipative dynamics

φ being the yield function and ζ being the consistency parameter (Lagrangianmultiplier) satisfying the loading-unloading conditions

ζ ≥ 0, φ ≥ 0, ζφ = 0, (3.141)

and the consistency condition

ζφ = 0 for φ = 0, (3.142)

see, for instance, Simo & Miehe (1992) or Bergstrom & Boyce (1998). Defin-ing appropriated convex functionals J these models can be generated from(3.139).

3.7 Semidiscrete thermo-dissipative dynamics

The modeling of thermo-dissipative continuum dynamics has resultedin an infinite-dimensional evolution system that is nonlinear in space andtime and hence it only allows for numerical solutions. Thereby, the infinite-dimensional problem should be transformed into an approximated finite-di-mensional one in which the number of variables are bounded. In this process,the partial differential equation system becomes a set of algebraic ones whichare recursively solved. This transformation may spoil the original continuousstructure of the problem and hence the numerical solution may be quitedifferent (the longer the simulation the worse) from the exact one in terms ofphysical behavior, not of numerical errors that will inevitably appear. Recallthat one of the main objectives of this dissertation is to propose numericaltools that do not ideally introduce physical spuriousness in the final numericalmethod.

Traditionally, the strategies to numerically address continuous problemsin space and time are to perform two subsequent discretizations: spatial andtemporal. However, space-temporal monolithic discretizations can also befound in the literature, see for instance Gross & Betsch (2011). For thepurpose of this dissertations, the first (classical) approach will be adoptedas its main contribution is intended for the time integration. In this sense,the spatial discretization performed in this dissertation will be standard,meaning that no further contribution will be made, and no degradation ofthe time-conservation properties will be involved. Next, a brief summary ofthe current state-of-the-art for spatial discretization techniques available toface general partial differential equations is carried out.


Since the beginning of numerical methods, different spatial discretizationtechniques have been proposed. Among them, the most prominent is the Fi-nite Element Method (FEM) Bathe (1982); Hughes (1987); Onate (2008),mainly because it can accommodate multi-physic problems, not to men-tion that it was the pioneer in numerical methods as they are understoodnowadays. In the last decades, the FEM has widely been developed for thesolutions of many physic, mathematical and engineering problems, havinghighlighted its shortcomings. In order to improve its performance in particu-lar applications many enhancements have been proposed, in some cases evengiving rise to new methods such as the Extended Finite Element Method(XFEM) Moes et al. (1999) for the solution of crack propagation, or B-barFEM or Mixed FEM for incompressible solid deformations Simo et al. (1985).Also, the search for better order of accuracy in the approximation yielded thefamilies of high order FEMs known as hp-FEM or hpk-FEM, see Babuska &Guo (1992).

On the other hand, the fluid dynamics particular demands made its ownway on the development of numerical solutions. Thus, the DiscountinuosGalerkin (DG) or Petrov-Galerkin (PG) FEM were developed for the so-lution of diffusion-advection problems, then extended to general first orderconservation laws. Also, the Finite Volume Method (FVM) LeVeque (2002)has been successful in this area. Interestingly, these methods have recentlybeen incorporated for the solution of fast solid dynamics Aguirre et al. (2014,2015); Bonet et al. (2015); Gil et al. (2014); Lee et al. (2013, 2014), thus cir-cumventing the locking phenomenon related to the standard displacement-based FEM applied to incompressible or cuasi-incompressible deformations.

More recently, Mesh-free methods Liu & Gu (2010) and IsogeometricAnalysis (IGA) Cottrell et al. (2009); Hughes et al. (2005) have attractedmost of the efforts of computational mechanics’ community. In the firstgroup, the SPH method Lucy (1977); Monaghan (1988) intended for thesolution of strong forms, the Meshfree Galerkin methods with Moving-Least-Square (MLS) approximation functions Belytschko et al. (1994) (named EFG)or the Max-Ent method due to Arroyo & Ortiz (2006) have been advocatedfor the solution of several types of demanding problems. For instance, aGalerkin Mesh-free method using Radial Basis functions (RBF) has beenproposed for the solution of Flexible Multibody Systems with holonomic con-strains in Iglesias Ibanez & Garcıa Orden (2011); Iglesias Ibanez et al. (2013).Also, the Max-Ent approach has been the basis for developing a new frame-work for the fluid-structure interaction using the Lagrangian description forboth solids and fluids in Urrecha (2014).


In theory, any of these spatial discretization techniques could be com-bined with structure-preserving integration methods provided that they donot spoil the conservation properties of the continuous model. An appealingproof is due to Hesch & Betsch (2012) where an Energy-Momentum methodis embedded within a spatial discretization based on IGA.

3.7.1 Variational statement of the IBVP

Most of discretization techniques rely on the variational statement of the(partial) differential equations under consideration, also referred to as theweak form of the (partial) differential equations. This choice is based mainlyon the fact that the weak form straightforwardly enables the imposition ofNeumann boundary conditions, and on the fact that the continuity of thesolution may be relaxed compared to the one required by strong forms. Fur-thermore, the fundamental lemma of the Calculus ensures that the solutionof the weak statement fully corresponds with the solution of the strong one,justifying the use of weak forms in discretization techniques.

Weak forms can be derived from strong forms by defining the trial and theweighting (or test) functions. The first ones must comply with the Dirichletboundary conditions while the second ones must be compatible with them.They are defined in the trial and weighting spaces as

V = v|v ∈M , v = v on ∂Bϕ,θ0 × [0, T ], (3.143)

W = w|w ∈ N , w = 0 on ∂Bϕ,θ0 × [0, T ], (3.144)

where v represents the prescribed Dirichlet boundary conditions and M andN are any two infinite-dimensional spaces.

The procedure consists in first multiplying the strong form by the weight-ing function and then integrating it over the volume occupied by the contin-uum. The weighting functions and the terms of strong forms thus furnish aninner product space such that U = v, u|v, u ∈ V ×W , 〈u, v〉 : V ×W → R.A more detailed explanation of this procedure can be found in Hughes (1987).

Let wϕ,wp, we,wαΛ ∈ W be a set of weighting functions for each of the

trial functions ϕ,p, e,Λα ∈ V , then the variational statement of the initial-


boundary value problem (3.66) may be stated by

〈wϕ, ϕ〉B0 − 〈wϕ,1

ρ0

p〉B0 = 0

〈wp, p〉B0 + 〈∇0wp,FS〉B0 − 〈wp,B0〉B0 − 〈wp,T0〉∂B0 = 0

〈we, e〉B0 − 〈we,1

2S : C〉B0 − 〈∇0we,H〉B0

− 〈we,R〉B0 + 〈we, HN〉∂B0 = 0

〈wαΛ, Λ

α〉B0 − 〈wαΛ,K(Qα)〉B0 = 0 ∀ α = 1, . . . ,m,

(3.145)

where use has been made of the notation 〈·, ·〉4 =∫4(·) · (·)d4.

In order to achieve the formulations based on entropy and temperature,weak forms of the energy balances (3.71) and (3.84) must be provided. Thus,let wη ∈ W be the test function for the entropy variable η ∈ V such that theweak form of the energy balance in entropy form (3.71) may be expressed as

〈wη, η〉B0 − 〈∇0

[wηΘ

],H〉B0 − 〈

wηΘ, D〉B0

+ 〈wηΘ, HN〉∂B0 − 〈

wΘ

Θ,R〉B0 = 0

(3.146)

Likewise, the weak form of the energy balance in temperature form (3.84)is defined via the test function wΘ ∈ W for the temperature variable Θ ∈ Vto give

〈wΘ, Θ〉B0 − 〈∇0

[wΘ

c

],H〉B0 + 〈wΘ

cFV,∇0

[p

ρ0

]〉B0

− 〈wΘ

c, D +Hd〉B0 + 〈wΘ

c,HN〉∂B0 − 〈

wΘ

c,R〉B0 = 0

(3.147)

The full entropy-based or temperature-based weak formulations are thenobtained by replacing (3.146) or (3.147) with (3.145)3 and accordingly con-sidering the correct constitutive laws.

On the other hand, the particular forms of (3.66)1 and (3.66)4 lead tothe pointless weak forms (3.145)1 and (3.145)4, in the sense that it remainsnecessary for them to be satisfied point-wise so the weak form of the IBVP


(3.66) may equivalently be written as

ϕ− 1

ρ0

p = 0


〈we, e〉B0 − 〈we,1

2S : C〉B0 − 〈∇0we,H〉B0

− 〈we,R〉B0 + 〈we, HN〉∂B0 = 0

Λα −K(Qα) = 0 ∀ α = 1, . . . ,m.

(3.148)

Finally, the weak statement for the isothermal dissipative dynamics (3.104)reads

ϕ− 1

ρ0

p = 0


Λα −K(Qα) = 0 ∀ α = 1, . . . ,m,

(3.149)

where the symmetry Piola-Kirchhoff stress tensor and the dissipative drivingstress tensor accordingly respond to the constitutive laws (3.103).

Remark 3.12. The weak statement of the linear momentum balance (3.148)2

becomes the classical virtual work principle by choosing the test function wpas the virtual displacement δϕ ∈ W , that is

〈p, δϕ〉B0 + 〈FS,∇0δϕ〉B0 − 〈B0, δϕ〉B0 − 〈T0, δϕ〉∂B0 = 0 (3.150)

Remark 3.13. In the pure mechanical context, unlike the weak form pre-sented for the linear momentum balance, a multi-field variational statementcan be constructed to deal with special demands, such as multi-body dynam-ics governed by Differential-Algebraic Equations (DAEs) or displacement-constrained formulations. The most representative example in solid mechan-ics is the Hu-Washizu three-field variational principle Hu (1955); Pompe &Washizu (1969) which is used to arrive at aforementioned B-bar FEM meth-ods, see Simo et al. (1985).

3.7.2 Bunov-Galerkin Finite Element spatial discretization

The Finite Element discretization is based on a partition of the continuumB0, schematically depicted in Figure 3.7, into a conforming regular mesh of


Ne finite elements, each denoted B0e , connecting N nodes, defining a discrete

configuration Bh0 as

Bh0 =

Ne⋃e=1

Be0 (3.151)

Xh

Be0

xh

Y, y

Z, z

X, x

ϕh(Xh, t)

Bh0

Bh

Be

Figure 3.7. Finite elements partition of the continuum

This partition is used to build continuous trial and weighting functionsin terms of unknowns on the nodes. The way these functions are specifieddetermines the particular FE method. Thus, the Bunov-Galerkin, or justGalerkin, method is constructed by finite-dimensional approximation of thespaces V and W , denoted by V h and W h, such that V h ⊂ V and W h ⊂W . These approximations consist of all linear combinations of global shapefunctions Na : Bh

0 → R, such that

V h = vh =N∑a=1

Nava|vh ∈H 1, vh = vh on ∂Bh,ϕ,θ0 × [0, T ], (3.152)

W h = wh =N∑a=1

Nawa|wh ∈H 1, wh = 0 on ∂Bh,ϕ,θ0 × [0, T ], (3.153)


where va, wa denote (unknown) values on the node a and H l is the Hilbertspace defined as

H m = v|v ∈ L2, Dαv ∈ L2, |α| ≤ l, (3.154)

with L2 being the (Lebesgue) space of square integrable functions.

Remark 3.14. Petrov-Galerkin methods approximate differently the weight-ing functions to deal with partial differential equations with odd order, suchas the advection-dominated diffusion-advection problem, see Fenaux (2013).Also, collocation methods Liu & Gu (2010), those that satisfy the partialdifferential equations only at certain chosen points, can be understood as aFEM in which the weighting space is approached by Dirac delta functions.

Specifically, the spatial-dependent variables that defines the thermo-dis-sipative dynamics are spatially discretized by

ϕh(X, t) =N∑a=1

Na(X)xa(t),

ph(X, t) =N∑a=1

Na(X)pa(t),

sh(X, t) =N∑a=1

Na(X)sa(t),

Λα,h(X, t) =N∑a=1

Na(X)Λα,a(t),

(3.155)

where xa, pa, sa (being either entropy ηa or temperature Θa) and Λα,a denotethe respective fields’ values on the node a. As a result, the problem boilsdown to finding this set of nodal values, known as degree of freedom ofthe system. It is precisely in this set of variables where the time-evolutionproblem remains continuous.

Thus, each spatial-dependent magnitude is spatially approximated byusing the above discretization (3.155). For instance, the fundamental defor-mation gradient tensor and the right Cauchy-Green deformation tensor resultin

Fh(X, t) =N∑a=1

xa(t)⊗∇0Na(X),

Ch(X, t) =N∑a=1

N∑b=1

xa(t) · xb(t)∇0Na(X)⊗∇0N

b(X),

(3.156)


where (·)⊗(·) means external product and the superscript h indicates spatially-discretized counterparts.

Apart form the direct discretization provided by (3.155)4, the internalvariables could also be approximated indirectly by observing that the solutionof the spatial discrete form of the dissipative evolution equation throughouttime provides a discrete form of Λα,h, that is

Λα,h = K(Qα,h) in Bh0 × [0, T ] ∀ α = 1, . . . ,m. (3.157)

For instance, in the isothermal viscoelastic case due to Simo and Holzapfel(3.114) they would turn out to be

Λα,h(t) =

∫ t

0

exp

[−(t− s)τα

]2

να

∂Ψαiso

∂C(Ch, s)ds+ exp

[−tτα

]Λα

0 (3.158)

Introducing the approximations ϕh,ph, ηh,Λα,h ∈ V h and whp, w

hη ∈ W h

into the weak form (3.146), the spatially-discrete entropy formulation of evo-lution equation is obtained

ϕh − 1

ρh0ph = 0,

〈whp, p

h〉Bh0

+ 〈∇0whp,F

hSh〉Bh0− 〈wh

p,B0〉Bh0− 〈wh

p,T0〉∂Bh0

= 0,

〈whη , ηh〉Bh0− 〈∇0

[whηΘh

],Hh〉Bh

0− 〈w

hη

Θh, Dh〉Bh

0

+ 〈whη

Θh, HN〉∂Bh

0− 〈w

hη

Θh,R〉Bh

0= 0,

Λα,h −K(Qα,h) = 0 ∀ α = 1, . . . ,m,

(3.159)

where the spatial discretization of the entropy-based constitutive relations(3.36)-(3.38) are computed as follows

Sh = 2∂e

∂C(Ch, ηh,Λα,h)

Θh =∂e

∂η(Ch, ηh,Λα,h)

Qα,h = − ∂e

∂Λα(Ch, ηh,Λα,h)

in Bh

0 × [0, T ] (3.160)

Similarly, the spatially-discretized temperature formulation derives from


the weak form (3.147) and reads

ϕh − 1

ρh0ph = 0,

〈whp, p

h〉Bh0

+ 〈∇0whp,F

hSh〉Bh0− 〈wh

p,B0〉Bh0− 〈wh

p,T0〉∂Bh0

= 0,

〈whΘ, Θh〉Bh0− 〈∇0

[whΘch

],Hh〉Bh

0+ 〈w

hΘ

ch,FhVh : ∇0

[ph

ρ0

]〉Bh

0

− 〈whΘ

ch, Dh +Hh

d〉Bh0

+ 〈whΘ

ch, HN〉∂Bh

0− 〈w

hΘ

ch,R〉Bh

0= 0,

Λα,h −K(Qα,h) = 0 ∀ α = 1, . . . ,m,

(3.161)

in where the spatial discrete constitutive laws (3.18)-(3.21) and (3.77)-(3.79)now becomes

Sh = 2∂Ψ

∂C(Ch,Θh,Λα,h)

ηh = −∂Ψ

∂Θ(Ch,Θh,Λα,h)

Qα,h = − ∂Ψ

∂Λα(Ch,Θh,Λα,h)

Vh = 2Θh ∂η

∂C(Ch,Θh,Λα,h)

ch = Θh ∂η

∂Θ(Ch,Θh,Λα,h) = c > 0

Wα,h = Θh ∂η

∂Λα(Ch,Θh,Λα,h)

in Bh0 × [0, T ] (3.162)

As was introduced in section 3.3.6, the thermodynamics obliges the spe-cific heat capacity (3.78) c to be a constant positive real number, simplifyingthe manipulation of the energy balance weak form (3.161)3.

Furthermore, in both formulations the heat flux vector (3.26) is spatiallydiscretized according to

Hh = −JhFh,−1κ(Θh)Fh,−T∇0Θh = −Kh∇0Θh, (3.163)

where the temperature field Θh is provided either by (3.155)3 or (3.160)2.The same applies to the rate of dissipation counterpart which is

Dh =m∑α=1

Qα,h : Λα,h =m∑α=1

Qα,h : K(Qα,h) ≥ 0, (3.164)


with the proper constitutive law for the dissipative driving stress tensor:(3.21) or (3.38).

To close the temperature formulation a spatially-discretized counterpartfor the thermo-dissipative heating (3.82) needs to be obtained as

Hhd =

m∑α=1

Wα,h : Λα,h =m∑α=1

Wα,h : K(Qα,h) (3.165)

On the other hand, the discretized spatial equations of the isothermaldissipative dynamics (3.104) reads

ϕh − 1

ρh0ph = 0,

〈whp, p

h〉Bh0

+ 〈∇0whp,F

hSh〉Bh0− 〈wh

p,B0〉Bh0− 〈wh

p,T0〉∂Bh0

= 0,

Λα,h −K(Qα,h) = 0 ∀ α = 1, . . . ,m,

(3.166)

with the spatial discretized constitutive laws (3.103) being

Sh = 2∂Ψ

∂C(Ch,Λα,h) and Qα,h = − ∂Ψ

∂Λα(Ch,Λα,h) (3.167)

Remark 3.15. The introduced spatial discretization does not spoil the time-evolution preserved quantities stated by the continuous formulation (3.95)-(3.101). These quantities have to do with the continuous nature in time andnot with spatial aspects. In addition, no use of artificial (numerical) stabi-lization techniques have been involved and, therefore, some drawbacks mightappear such as the widely-known phenomenon of locking related to incom-pressible or nearly incompressible deformations when the standard Galerkinapproach is carried out. Some of the methods proposed to fix this phe-nomenon are based on adding non-physically motivated terms which maycompromise the conservation properties. The reader is referred to Hughes(1987) for more details.

3.7.3 Element implementation

In order for Galerkin FEM to be convergent as the mesh is refined theshape functions must be smooth (at least C1) on each element so the globalshape functions are, at worse, continuous but not differentiable in the vicinityof the nodes, that is Na ∈ C0. The element functions are commonly definedin the isoparametric space in where a parent domain, e.g. the closed biunit


cube in R3, is denoted by , such that any point in the elementXe : → Be0

is provided, in the Lagrangian description, by

Xe(ξ) =n∑i=1

Ni(ξ)Xei , (3.168)

with n being the number of the nodes per element. The relationship betweenthe isoparametric and the Lagrangian spaces is then provided by the Jacobiantransformation

Je0 :=∂Xe

∂ξ=

n∑i=1

Xei ⊗

∂Ni(ξ)

∂ξ(3.169)

A FEM based on these functions is said to be isoparametric. As examplesof 3D-elements, Figure 3.8 shows the shape functions for the linear tetrahedraland the trilinear hexahedral elements which allow for geometrically nonlinearformulations.

+

1

3

4 2η

ζ

ξ

N1(ξ, η, ζ) = ξ

N2(ξ, η, ζ) = η

N3(ξ, η, ζ) = ζ

N4(ξ, η, ζ) = 1− ξ − η − ζ ++ +

+

++ +

+

1

2 3

4

5

6 7

8

η

ζ

ξ

Ni(ξ, η, ζ) =1

8(1 + ξiξ)(1 + ηiη)(1 + ηiζ)

Figure 3.8. Shape functions

Finally, the standard Galerkin FEM is complemented by Gaussian quadra-tures for the evaluation of the integrals appearing in weak forms. By usingthe isoparametric space, the integral of any function f : B0 → R is approxi-mated as follows∫

B0

f(X)dV0 ≈Ne∑e=1

∫Be

0

f(Xe)dV e0

=Ne∑e=1

∫f(ξ) det (Je0)d ≈

Ne∑e=1

ng∑i=1

ng∑j=1

ng∑k=1

wiwjwkf(ξg) det (Je0,g),

(3.170)


where ng is the number of Gauss points, ξg are the coordinates of the Gausspoint g and wg is the weight of the g-th Gauss point, see Chandrupatla &Belegundu (1991). Particularly, the trilinear hexahedral element has 8 Gausspoints and the weighting factors are given by wl = 1, whereas the tetrahedralelement has only a Gauss point and the weighting factor is wl = 1/ 3

√6.

With these tools, each term of the above weak forms (3.159)-(3.161) canfurther be elaborated at the element level to achieve the element implementa-tion. In addition, this implementation must be provided with the linearizedcounterparts. The linear momentum balance terms can easily be found inthe literature, e.g Bonet & Wood (2008); De Borst et al. (2012), in contrastwith the energy balance ones, at least within the thermo-dissipative dynamiccontext. For this reason, Appendix B collects the full details of the deriva-tion of both the linearized and the discretized evolution equations, being thelatter developed at element level as usual.

3.8 Standard time integration methods

Once the thermo-dissipative continuum dynamics has spatially been dis-cretized, the full discrete formulation is achieved by performing a temporaldiscretization. This step becomes crucial regarding the conservation proper-ties of the resulting discrete formulation, requiring on-purpose non-standardintegration methods that ensure their preservation.

In this section standard temporal techniques are discussed, emphasizingtheir implications in the preserved features of the system, stated in Section3.5. In addition, some benchmark simple problems will be introduced andsolved to check the methods’ properties and to serve as reference solutionsthat allow to calibrate the new structure-preserving methods to be presentedin Chapter 5.

The resulting semidiscrete thermo-dissipative dynamic equations (3.159),(3.161) or (3.166) also answers to the simplified form (2.67), that is

yh(t) = f(yh(t), t) ∀ t ∈ [0, T ], (3.171)

with yh = (ϕh,ph, sh,Λα,h).

Performing the division into subintervals [tn+1, tn] of the entire time inter-val of interest [0, T ], the same standard approximations as those discussed inSection 2.3 can be elaborated for the temporal discretization of semidiscretethermo-dissipative dynamic equations so full discrete counterparts can be

106 3.8. Standard time integration methods

achieved. Next, the standard midpoint and trapezoidal methods are studied,emphasizing their shortcomings regarding the preservation laws.

3.8.1 Midpoint time integration method

As was introduced in Section 2.3, the midpoint method is a one-step im-plicit second-order accurate integration method that additionally preservesboth the linear and the angular momentum, that is, the symmetries of thethermo-dissipative dynamics continuous evolution equations. However, it vi-olates the laws of thermodynamics for moderate time-step sizes (the largerthe worse). This characteristics doubtlessly make it interesting for the pur-pose of this dissertation. In fact, it is quite close to fulfill all requirementsfalling on the desired structure-preserving time integration algorithms.

The full midpoint-based discrete entropy formulation of thermo-dissipativecontinuum dynamics is obtained from (3.159) to be

ϕhn+1 −ϕhn∆t

− 1

ρh0phn+ 1

2= 0,

〈whp,phn+1 − phn

∆t〉Bh

0+ 〈∇0w

hp,F

hn+ 1

2Shn+ 1

2〉Bh

0

− 〈whp,B0,n+ 1

2〉Bh

0− 〈wh

p,T0,n+ 12〉∂Bh

0= 0,

〈whη ,ηhn+1 − ηhn

∆t〉Bh

0− 〈∇0

[whη

Θhn+ 1

2

],Hh

n+ 12〉Bh

0− 〈 w

hη

Θhn+ 1

2

, Dhn+ 1

2〉Bh

0

+ 〈 whη

Θhn+ 1

2

, HN,n+ 12〉∂Bh

0− 〈 w

hη

Θhn+ 1

2

,Rn+ 12〉Bh

0= 0,

Λα,hn+1 −Λα,h

n

∆t−K(Qα,h

n+ 12

) = 0 ∀ α = 1, . . . ,m,

(3.172)

where, contrary to the notation used in Subsection 2.3.1, the notation (·)hn+ 1

2

has been saturated for the sake of simplicity. The important fact to be con-sidered is that each state variable is evaluated at the subinterval midpointaccording to (·)h

n+ 12

= 12[(·)hn+1 +(·)hn] whereas all derived quantities and func-

tions are evaluated via them. Thus, the constitutive laws are correspondingly


evaluated as

Shn+ 1

2

= 2∂e

∂C(Ch

n+ 12

, ηhn+ 1

2

,Λα,h

n+ 12

)

Θhn+ 1

2

=∂e

∂η(Ch

n+ 12

, ηhn+ 1

2

,Λα,h

n+ 12

)

Qα,h

n+ 12

= − ∂e

∂Λα(Ch

n+ 12

, ηhn+ 1

2

,Λα,h

n+ 12

)

in Bh

0 × [tn, tn+1], (3.173)

with the right Cauchy-Green deformation tensor evaluated at midpoint being

Chn+ 1

2= Fh,T

n+ 12

Fhn+ 1

2=(∇0ϕ

hn+ 1

2

)T

∇0ϕhn+ 1

2(3.174)

Moreover, the heat flux vector (3.163) evaluated at midpoint results in

Hhn+ 1

2= −Jh

n+ 12Fh,−1

n+ 12

κ(Θhn+ 1

2)Fh,−T

n+ 12

∇0Θhn+ 1

2= −Kh

n+ 12∇0Θh

n+ 12, (3.175)

with Jhn+ 1

2

= det Fhn+ 1

2

. Note that the evaluation of the material gradient

requires the application of the chain rule to give

∇0Θhn+ 1

2=∂Θ

∂C: ∇0C

hn+ 1

2+∂Θ

∂η∇0η

hn+ 1

2+

∂Θ

∂Λα: ∇0Λ

α,h

n+ 12

(3.176)

The last two terms follow directly from the spatial discretization (3.155)but the first one needs further elaboration by using (3.156)2, making neces-sary the implementation of the second partial derivatives of the shape func-tions, details can be found in Appendix B.

On the other hand, the midpoint evaluation of the rate of dissipationreads

Dhn+ 1

2=

m∑α=1

Qα,h

n+ 12

: Λα,h

n+ 12

=m∑α=1

Qα,h

n+ 12

: K(Qα,h

n+ 12

) ≥ 0, (3.177)

In a similar way, the full midpoint-based discrete temperature formula-


tion is derived from (3.161) and reads

ϕhn+1 −ϕhn∆t

− 1

ρh0phn+ 1

2= 0,


∆t〉Bh

0+ 〈∇0w

hp,F

hn+ 1

2Shn+ 1

2〉Bh

0

− 〈whp,B0,n+ 1

2〉Bh

0− 〈wh

p,T0,n+ 12〉∂Bh

0= 0,

〈whΘ,Θhn+1 −Θh

n

∆t〉Bh

0+ 〈w

hΘ

c,Fh

n+ 12Vhn+ 1

2: ∇0

[phn+ 1

2

ρh0

]〉Bh

0

− 〈1c∇0w

hΘ,H

hn+ 1

2〉Bh

0− 〈w

hΘ

c, Dh

n+ 12

+Hhd,n+ 1

2〉Bh

0

+ 〈whΘ

c,HN,n+ 1

2〉∂Bh

0− 〈w

hΘ

c,Rn+ 1

2〉Bh

0= 0,

Λα,hn+1 −Λα,h

n

∆t−K(Qα,h

n+ 12

) = 0 ∀ α = 1, . . . ,m,

(3.178)

where the midpoint constitutive laws are defined in terms of the temperatureat midpoint Θh

n+ 12

= 12(Θh

n+1 + Θhn) according to

Shn+ 1

2

= 2∂Ψ

∂C(Ch

n+ 12

,Θhn+ 1

2

,Λα,h

n+ 12

)

ηhn+ 1

2

= −∂Ψ

∂Θ(Ch

n+ 12

,Θhn+ 1

2

,Λα,h

n+ 12

)

Qα,h

n+ 12

= − ∂Ψ

∂Λα(Ch

n+ 12

,Θhn+ 1

2

,Λα,h

n+ 12

)

Vhn+ 1

2

= 2Θhn+ 1

2

∂η

∂C(Ch

n+ 12

,Θhn+ 1

2

,Λα,h

n+ 12

)

chn+ 1

2

= Θhn+ 1

2

∂η

∂Θ(Ch

n+ 12

,Θhn+ 1

2

,Λα,h

n+ 12

) = c > 0

Wα,h

n+ 12

= Θhn+ 1

2

∂η

∂Λα(Ch

n+ 12

,Θhn+ 1

2

,Λα,h

n+ 12

)

in Bh0 × [tn, tn+1]

(3.179)

Accordingly, the heat vector counterpart (3.175) also becomes a functionof the temperature at midpoint, facilitating the evaluation of the involvedmaterial gradient as readily follows from (3.155). Moreover, the thermo-dissipative heating (3.82) reads

Hhd,n+ 1

2=

m∑α=1

Wα,h

n+ 12

: Λα,h

n+ 12

=m∑α=1

Wα,h

n+ 12

: K(Qα,h

n+ 12

) (3.180)


Finally, the midpoint-based discrete isothermal-dissipative continuumdynamic equations simplifies to

ϕhn+1 −ϕhn∆t

− 1

ρh0phn+ 1

2= 0,


∆t〉Bh

0+ 〈∇0w

hp,F

hn+ 1

2Shn+ 1

2〉Bh

0

− 〈whp,B0,n+ 1

2〉Bh

0− 〈wh

p,T0,n+ 12〉∂Bh

0= 0,

Λα,hn+1 −Λα,h

n

∆t−K(Qα,h

n+ 12

) = 0 ∀ α = 1, . . . ,m,

(3.181)

with the corresponding constitutive laws being evaluated at midpoint as fol-lows

Shn+ 1

2

= 2∂Ψ

∂C(Ch

n+ 12

,Λα,h

n+ 12

)

Qα,h

n+ 12

= − ∂Ψ

∂Λα(Ch

n+ 12

,Λα,h

n+ 12

)

in Bh0 × [tn, tn+1] (3.182)

Particularly, the internal variables of the Simo and Holzapfel viscoelasticmodel (3.114) result in

Λα,hn+1 = ∆t exp

[−∆t

2τα

]2

να

∂Ψαiso

∂C(Ch

n+ 12) + exp

[−∆t

τα

]Λαn (3.183)

Discrete symmetries and thermodynamic laws. The just introducedmidpoint approximation preserves the equation’s symmetries associated tothe free motion of the continuum in space. As was introduced in Section3.5, these symmetries lead to the conservation of the linear and angularmomentum. In a discrete setting, the conservation of any quantity can bechecked by balancing it inside any time subinterval. Then, the quantity isdiscretely conserved if its balance is zero for all time steps. For the globallinear (3.44) and angular (3.51) momentum, that is expressed as follows

Lhn+1 −Lhn = 0, Jhn+1 − Jhn = 0 ∀ [tn, tn+1] (3.184)

To verify the linear momentum balance it suffices to take whp = ∆tw,

w ∈ Rd being a constant vector such that ∇0whp = 0, in either (3.172)2,

(3.178)2 or (3.181)2 to obtain(Lhn+1 −Lhn

)·w = 〈phn+1 − phn,w〉Bh

0

= ∆t[〈B0,n+ 1

2,w〉Bh

0+ 〈T0,n+ 1

2,w〉∂Bh

0

],

(3.185)


which clearly vanishes in absence of external actions.

On the other hand, the verification for the angular momentum conserva-tion follows from noting that

Jhn+1−Jhn =

∫Bh

0

[ϕhn+ 1

2∧ (phn+1 − phn)− ph

n+ 12∧ (ϕhn+1 −ϕhn)

]dV0, (3.186)

Then, the second term of the right hand side of (3.186) vanishes accordingto either (3.172)1, (3.178)1 or (3.181)1.

Now, consideringwhp = ∆tw∧ϕh

n+ 12

, w ∈ Rd being an arbitrary constant

vector, and furnishing the associated skew-symmetric tensor w ∈ Skw(T2d)

sucht that w × ϕhn+ 1

2

= wϕhn+ 1

2

and hence ∇0whp = ∆twFh

n+ 12

, the angular

momentum balance yields(Jhn+1 − Jhn

)·w = 〈ϕh

n+ 12∧ (phn+1 − phn),w〉Bh

0

= 〈phn+1 − phn,w ∧ϕhn+ 12〉Bh

0

= ∆t[−〈Fh

n+ 12Shn+ 1

2Fh,T

n+ 12

,w〉Bh0

+ 〈B0,n+ 12,w ∧ϕh

n+ 12〉Bh

0+ 〈T0,n+ 1

2,w ∧ϕh

n+ 12〉∂Bh

0

] (3.187)

The first term on the right hand side of (3.187) vanishes due to the sym-metry of the tensor Fh

n+ 12

Shn+ 1

2

Fh,T

n+ 12

and the skew-symmetry of w. Therefore,

the absence of external actions makes the angular momentum balance be zerosince (3.187) must hold for all w, proving that the midpoint scheme keepsthe symmetries of the continuous counterpart.

Unlike the momentum balances, the total energy balance does not resultin energy conservation since it reads

Ehn+1 − Eh

n = Khn+1 −Kh

n + Uhn+1 − Uh

n

=

∫Bh

0

1

2ρh0

(‖phn+1‖2 − ‖phn‖2

)dV0 +

∫Bh

0

(ehn+1 − ehn)dV0

=

∫Bh

0

1

ρh0phn+ 1

2·(phn+1 − phn

)dV0 +

∫Bh

0

(ehn+1 − ehn)dV0

(3.188)

The first term in the above expression can be further elaborated by con-sidering wh

p = phn+ 1

2

/ρh0 in either (3.172)2, (3.178)2 or (3.181)2 along with a


particular tensorial property7 to give

Ehn+1 − Eh

n = −〈Shn+ 1

2,1

2(Ch

n+1 −Chn)〉Bh

0+ 〈B0,n+ 1

2,∆ϕh〉Bh

0

+ 〈T0,n+ 12,∆ϕh〉∂Bh

0+

∫Bh

0

(ehn+1 − ehn)dV0,(3.189)

which in absence of external forces yields

Ehn+1 − Eh

n = −∫

Bh0

Shn+ 1

2:

1

2(Ch

n+1 −Chn)dV0 +

∫Bh

0

(ehn+1 − ehn)dV0 (3.190)

Unfortunately, this last expression cannot be further detailed unless as-suming some loss of generality, thus concluding that the midpoint-based en-ergy conservation cannot be generally ensured.

Remark 3.16. The energy balance (3.190) also applies in the reversibleelastic case in which the internal energy is identified with the Helmholtzfree-energy hyperelastic potential Ψ := Ψ(C) according to (3.41). In such acase, the energy conservation is clearly not fulfilled as the following inequalitygenerally holds

Ψ(Chn+1)−Ψ(Ch

n) 6= Shn+ 1

2:

1

2(Ch

n+1 −Chn) (3.191)

This observation reveals the key for a method to be energy-conserving in theHamiltonian context, which was devised in Simo & Tarnow (1992).

Interestingly, the entropy formulation complies with the second law ofthermodynamics, which directly derives from choosing whη = ∆t in (3.172)3

to give

Shn+1 − Shn =

∫Bh

0

(ηhn+1 − ηhn)dV0

= −〈 ∆t

(Θhn+ 1

2

)2∇0Θh

n+ 12,Hh

n+ 12〉Bh

0+ 〈 ∆t

Θhn+ 1

2

, Dhn+ 1

2〉Bh

0

− 〈 ∆t

Θhn+ 1

2

, HN,n+ 12〉∂Bh

0+ 〈 ∆t

Θhn+ 1

2

,Rn+ 12〉Bh

0

(3.192)

Assuming the isolation of the continuum, that is HN = R = 0, together

7AB : C = B : ATC


with the expressions (3.175) and (3.177) the entropy balance gives

Shn+1 − Shn = 〈 ∆t

(Θhn+ 1

2

)2∇0Θh

n+ 12,Kh

n+ 12∇0Θh

n+ 12〉Bh

0

+ 〈 ∆t

Θhn+ 1

2

, Dhn+ 1

2〉Bh

0≥ 0,

(3.193)

which proves the fulfillment of the second law in a midpoint-based discretefashion.

The temperature formulation (3.178), on the other hand, does not inheritthis important feature as it is impossible to relate the entropy balance to thetemperature balance appearing in (3.178)3.

3.8.2 Trapezoidal time integration method

The standard second-order trapezoidal method also provides a straight-forward formulation although it does not possess any preservation featuresother than the linear momentum one for general nonlinear problems as thedynamics of thermo-dissipative continua. As a representative full discreteformulation of trapezoidal-type, consider the temperature formulation whichreads

ϕhn+1 −ϕhn∆t

− 1

ρ0

phn+ 1

2= 0,


∆t〉Bh

0+ 〈∇0w

hp,F

hShn+ 12〉Bh

0

− 〈whp,B0,n+ 1

2〉Bh

0− 〈wh

p,T 0,n+ 12〉∂Bh

0= 0,


n

∆t〉Bh

0+ 〈w

hΘ

c,FhVh : ∇0[ph/ρ0]n+ 1

2〉Bh

0

− 〈∇0

[whΘc

],H

h

n+ 12〉Bh

0− 〈w

hΘ

c, D

h

n+ 12

+Hh

d,n+ 12〉Bh

0

+ 〈whΘ

c,H

h

N,n+ 12〉∂Bh

0− 〈w

hΘ

c,Rh

n+ 12〉Bh

0= 0

Λα,hn+1 −Λα,h

n

∆t−K(Qα,h)n+ 1

2= 0 ∀ α = 1, . . . ,m,

(3.194)

with (·)n+ 12

= 12

[(·)n+1 + (·)n] indicating the mean value, which coincides

with the midpoint evaluation for linear functions.


As an example of its inability to preserve the invariants of a free-motion,the angular momentum is elaborated using (3.194)2 in (3.187) to give(Jhn+1 − Jhn

)·w = ∆t

[−〈FhShn+ 1

2Fh,T

n+ 12

,w〉Bh0

+ 〈B0,n+ 12,w ∧ϕh

n+ 12〉Bh

0+ 〈T 0,n+ 1

2,w ∧ϕh

n+ 12〉∂Bh

0

]= ∆t

[−〈1

2(Fh

n+1Shn+1 + Fh

nShn)Fh,T

n+ 12

,w〉Bh0

+ 〈B0,n+ 12,w ∧ϕh

n+ 12〉Bh

0+ 〈T 0,n+ 1

2,w ∧ϕh

n+ 12〉∂Bh

0

],

(3.195)

where the term depending on the stress tensor is generally non-zero and hencethe angular momentum is not preserved in a force-free motion. The laws ofthermodynamics are also breached by this approximation.

Metriplectic structures:GENERIC formalism C

hapter

4Somehow hidden for the computational mechanics’ community but quite

apparent for mathematicians it was the concept of metriplectic structure de-veloped in the context of differential geometry’ theory, see the monograph dueto Frankel (2011). The metriplectic structure on a smooth manifold possessesall the mathematical features involved in the dynamic equations associatedwith general dissipative systems. This geometrical description of dissipativesystems dates back to the work of Morrison (1984), Kaufman (1985), Grmela(1986), Brockett (1993) and Nguyen & Turski (2001). With this structure,most of the dissipative problems of interest can be generated from a generalabstract (unifying) form that guarantees certain properties of their solutions.Not only that but also, such abstract form further contributes to the under-standing of the equations that pose thermodynamic soundness, to discardnon thermodynamically consistent terms, or even to suggest admissible can-didates for the time evolution equations in new physical theories.

A particular metriplectic structure is the framework introduced by Ot-tinger (2005) and co-workers Grmela & Ottinger (1997); Ottinger & Grmela(1997) for“General Equilibrium Non-Equilibrium Reversible-Irreversible Cou-pling”, known by the acronym GENERIC. It exploits the geometric descrip-tion provided by metriplectic structures to formulate isolated thermodynamicsystems of interest in the fields of physics or engineering. Thus, the evolu-tion equations describing any isolated thermodynamical system are derivedvia an additive decomposition into the system reversible evolution, which isassociated with a differential operator on the total energy, and its irreversibleone, which is connected with a differential operator on the total entropy.

116 4.1. Metriplectic structures

For the purposes of this dissertation, the key issue in the GENERICframework is that, for dissipative systems, it naturally plays the role of theHamiltonian framework for conserving ones, which revealed the general for-mulation of conserving time integration counterparts: Energy-Momentummethods. Furthermore, from the formal point of view, the metriplectic GE-NERIC framework shares the key features of Hamiltonian framework sothat the same reasoning which led from Hamiltonian systems to Energy-momentum methods can be applied to reveal thermodynamically consistenttime integration counterparts for general dissipative systems that, followingthe parallelism, will be referred to as Energy-Entropy-Momentum methods.These ideas were first devised in the works of Romero (2009, 2010a,b).

In this Chapter, the fundamental concepts on metriplectic structures areintroduced. Then, the GENERIC framework is used to define smooth dis-sipative systems in unified fashion for both finite and infinite dimensionalproblems. Finally, the different dissipative problems dealt with so far areformulated within this general framework so that full equivalence can easilybe checked and, accordingly, discrete thermodynamically consistent counter-parts can be derived in the following Chapter.

4.1 Metriplectic structures

In the context of differential geometry, a metriplectic structure is a struc-ture arising from the combination of a Poisson structure and a Gradient (ormetric) structure, see Fish (2005). The resulting structure is suitable tomodel the geometry of the dissipative systems which generally pose reversible(conserving) and irreversible (dissipative) qualities along trajectories. Thus,the Poisson structure assumes the generation of the reversible part of thesystem evolution while the Gradient structure controls the irreversible evo-lution.

Conservative systems on a phase space P can be described by a Poisson(or symplectic) structure of the form

f = f,H ∀f : P → R, (4.1)

where H : P → R is the Hamiltonian function and the bracket ·, · is thePoisson bracket. This bracket is a bilinear and skew-symmetric operator onthe phase space that additionally satisfies the Jacobi identity and the Leibnizrule, see Goldstein et al. (1965). These properties of the bracket ensure theconservation of the Hamiltonian, defining a conserving system, which directly

4. Metriplectic structures: GENERIC formalism 117

follows from taking f = H to give

H = H,H = 0, (4.2)

due to the skew-symmetry of the Poisson bracket.

Gradient or metric systems, however, were devised for the modeling ofpurely dissipative systems, that it, systems which evolve by increasing itsentropy. Geometrically, a metric tensor (in a Riemannian geometry) is re-sponsible for inducing such evolution on a phase space T , hence its name.Thus, the system evolution is provided by the following structure

f = [f, S] ∀f : T → R, (4.3)

S : T → R being a function on a phase space identified with a type ofentropy, and the bracket [·, ·] being the metric bracket which is a bilinear,symmetric and positive semi-definite operator. Then, the production of en-tropy is guaranteed as can be demonstrated by choosing f = S to give

S = [S, S] ≥ 0, (4.4)

due to the metric bracket being positive semi-definite.

These two mutually exclusive structures can be combined, thus defin-ing a metriplectic structure, with the idea of attaching dissipation terms toHamiltonian systems so that general dissipative system can be modeled by

f = f, E+ [f, S] ∀f : S → R, (4.5)

where E : S → R is the total energy of the system and S : S → R the totalentropy defined on a state space S .

For the metriplectic system (4.5) to exhibit trajectories in agreementwith the laws of thermodynamics, additional requirements must be imposedon the introduced brackets, specifically

S,E = [E, S] = 0 (4.6)

Thus, in view of (4.5) and (4.6) it follows that the law of thermodynamicsare satisfied, that is

E = E,E+ [E, S] = 0

S = S,E+ [S, S] ≥ 0(4.7)

More details about metriplectic structures for the modeling of dissipativeproblems can be found in Guha (2007) and the pioneering work of Morrison(1984), Kaufman (1985) and Grmela (1986).

118 4.2. Finite-dimensional smooth dissipative systems

4.2 Finite-dimensional smooth dissipative systems

As commented at the beginning of the Chapter, a metriplectic structurethat has gained popularity in the last years among the physic and compu-tational mechanic communities is the GENERIC formalism introduced byGrmela & Ottinger (1997); Ottinger & Grmela (1997), and fully revised inthe monograph Ottinger (2005). This framework intends for a physical ap-proach rather than geometric (mathematical) of metriplectic structures, thatis why, for instance, the introduced metric bracket, responsible for generat-ing the dissipative part, is accordingly named dissipative or friction bracket,making clear the effect of its contribution.

Due to any metriplectic structure (4.5) contains a Poisson structure (re-versible), the GENERIC formalism can be considered as a natural extensionof the Hamiltonian formalism. Based on this idea, the GENERIC frameworkallows for an unified definition of isolated (thermo)dissipative systems.

Definition 4.1. Let S be a finite-dimensional state space that containsthe 2k-dimensional phase space P defining a Hamiltonian system by thepair (q,p) ∈ Rk and all the internal variables (γ1, . . . , γm) ∈ Rm and thethermal variables (s1, . . . , st) ∈ Rt required to fully represent the isolatedsystem’ thermodynamics, furnishing each two finite-dimensional spaces D ,Tof dimension t and m, respectively, that is

S = z = (q,p, γ1, . . . , γm, s1, . . . , st) ∈ (P ×D ×T ) ⊂ R2k+m+t (4.8)

Then, the tangent space at some given state z and its dual are denoted asTzS and T ∗zS , respectively.

Definition 4.2. An finite-dimensional smooth dissipative system is an iso-lated system whose evolution equations are provided by the GENERIC equa-tions

z = L(z)DE(z) + M(z)DS(z), (4.9)

E : z ∈ S → R being the total energy function and S : z ∈ S → R beingthe total entropy function, D(·)1 being the gradient operator with respect tothe state space vector z and L2,M : T ∗zS 7→ TzS being the Poisson andDissipative matrices.

Remark 4.1. Following Mielke (2011), any non-smooth dissipative systemis an isolated system whose evolution equations are provided by

z = L(z)DE(z) + ∂M(z, DS), (4.10)

1not to be confused with the total dissipated energy D defined by (3.105).2not to be confused with the total linear momentum vector L.


where ∂M(z, ·) is the subdifferential of a non-negative and convex functionalM : S × T ∗zS 7→ R with M(z,0) = 0.

Proposition 4.1. The system (S ,L,M, E, S) is thermodynamically soundiff L is skew-symmetric, M is symmetric and positive semi-definite, and bothsatisfy the following degeneracy or non-interaction conditions

L(z)DS(z) = M(z)DE(z) = 0 (4.11)

Proof. The proof follows from the elaboration of the rate of the total energyand the total entropy of the system via the chain rule

E = DE · z = DE · LDE +DE ·MDS = DS ·MDE = 0, (4.12)

S = DS · z = DS · LDE +DS ·MDS

= −DE · LDS +DS ·MDS ≥ 0,(4.13)

where use has been made of (4.9) and (4.11) along with the properties pre-viously stated for the matrices.

Remark 4.2. Equations (4.9) - (4.13) constitute the GENERIC formalismfor finite-dimensional systems which are a finite-dimensional version of thegeneral metriplectic structure defined by (4.5) - (4.7).

In contrast to the Hamiltonian framework, there no exist canonical formsof the Poisson and Dissipative matrices. Their obtention in terms of the cho-sen state variables determines their components which, of course, dependon the particular thermo-dissipative problem. The search of their compo-nents could be cumbersome depending mainly on the thermodynamical statevariable. However, the introduced structure of the state space S leads thematrices to have a block structure, from which some general conclusions canbe made based on physical assumptions. Thus, due to the dissipative natureof internal variables, they do not contribute to the reversible evolution ofthe system, provided by the Poisson matrix L, so its blocks Lqγ, Lpγ mustvanish. In the same way, the phase variables (q,p) are responsible for thereversible changes, suggesting that the blocks of the Dissipative matrix Mqq,Mqp, Mpp are identically zero.

Moreover, Mielke (2011) provides a fairly systematic procedure to obtainthem for dissipative systems defined in a state space as (4.8). The systemati-zation is achieved by noticing that there are a set of thermodynamical statevariables, i.e si ∈ T , for which the matrices L and M become simpler, thusdefining the nearest thing of canonical forms of them. These simple forms

120 4.2. Finite-dimensional smooth dissipative systems

are the basis to unveil them for any choice of the thermodynamical variablesby resorting to change of variables techniques. Such canonical-like forms arerevealed by the orthogonality imposed by the degeneracy conditions (4.11).

Specifically, if the total entropy function S is identified in the state spaceS , the Poisson matrix becomes the classical symplectic matrix, see AlanJ. & Meyer (1972), denoted as Ω0, plus zero components out of the phasevariables, that is

L0 =

Ω0 02k×m 02k×t0T

2k×m 0m 0m×t0T

2k×t 0Tm×t 0t

if S ∈ T , with Ω0 =

(0k 1k−1k 0k

),

(4.14)where 0i×j and 1i×j are the null and the identity matrices of dimension ij,with a single index i meaning square matrices of dimension i.

Similarly, if the total energy function E is used to describe the systemthermodynamics, the Dissipative matrix adopts the following simple form

M0 = M

02k 02k×m 02k×t0T

2k×m gm 0m×t0T

2k×t 0Tm×t 0t

if E ∈ T , (4.15)

gm being the metric tensor in the space D and M ≥ 0 some scalar factordepending on the particular system.

With this reasoning, the matrices for any choice of the state variable canbe generated from two changes of variables, namely

z = S(z), z = E(z), (4.16)

so that the new Poisson matrix is provided by

L =

(∂S∂z

)−1

(L0 S)

(∂S∂z

)−T

, (4.17)

whereas the novel Dissipative matrix responds to

M =

(∂E∂z

)−1

(M0 E)

(∂E∂z

)−T

, (4.18)

(·) (·) meaning composition. From (4.17), it is clear that the resultingPoisson matrix always contains the canonical symplectic matrix for all vectorstate z ∈ S .


4.2.1 Finite-dimensional dissipative systems with symmetries

For dissipative systems, a symmetry is the action of a Lie group onthe state space that preserves both reversible and irreversible operators.Noether’s theorem indicates that associated with these actions are momen-tum maps that are constant along the solution. A detailed description ofthe purely Hamiltonian case can be found in Marsden & Ratiu (1999), whileits natural extension to the GENERIC framework was developed in Romero(2013). Next, these results are used to define dissipative systems with sym-metries.

Thus, let G be a Lie group with an associated algebra g and its dual g∗,endowed by an exponential map exp[·] : g 7→ G which takes elements of theLie algebra onto the group G. The group G acts on the state space S viaan action Φ: G ×S 7→ S denoted by Φg(z) = Φ(g,z). Furthermore, theone-parameter family of actions Φξ : R ×S 7→ S is defined by the formatΦξ(ε, z) = Φ(exp[εξ], z) for any ξ ∈ g. The tangent of this curve at theorigin is provided by

ξS (z) =d

dε

∣∣∣∣ε=0

Φξ(ε, z), (4.19)

which is the infinitesimal generator of the action corresponding to ξ.

Then, an action Φ is said to preserve the metriplectic structure of GE-NERIC if, for any g ∈ G, the following holds

DΦg(z) ·L ·DΦg(z)T = L Φg and DΦg(z) ·M ·DΦg(z)T = M Φg (4.20)

Definition 4.3. A smooth dissipative system (S ,L,M, E, S) has a momen-tum map if an action Φ of a Lie group G in the state space S is an invariantof the total energy of the system, that is, it is verified that

E Φg = E(Φ(g,z)) = E(z) ∀g ∈ G, ∀z ∈ S , (4.21)

then, the system (S , G,L,M, E, S) is a dissipative system with symmetriesand the total energy E is said to be a G-invariant function, see Appendix A.

Definition 4.4. A mapping J : z ∈ S 7→ g∗ is a momentum map associatedwith the action Φ iff the action preserves the GENERIC structure, i.e satisfies(4.20), and for any ξ ∈ g, the scalar function Jξ : S 7→ R, defined as

Jξ(z) = J(z) · ξ, (4.22)

satisfiesξS = LDJξ, 0 = MDJξ (4.23)

122 4.3. GENERIC forms of the two thermo-spring system

Theorem 4.2. The momentum maps of a dissipative system with symmetries(S ,L,M, E, S) are conserved quantities (first integrals) of the system.

Proof. The chain rule together with the evolution equations (4.9) lead therate of the scalar function (4.22) to be

Jξ = DJξ · z = DJξ · LDE +DJξ ·MDS (4.24)

that, by using (4.23), simplifies to

Jξ = −DE · ξS (4.25)

As the scalar function (4.22) depends on an arbitrary ξ, the momentummap J is preserved if the right hand side of (4.25) identically vanishes. Thisis proved by choosing g = exp[εξ] in (4.21) so the derivative with respect εgives

DE · ξS = 0, (4.26)

demonstrating that the mapping J defined according to Def. 4.4 is conservedalong the system evolution.

4.3 GENERIC forms of the two thermo-spring system

The two thermo-spring system presented in Section 2.1 is a finite-di-mensional dissipative system in the sense of the Def. 4.2, where the spaceD vanishes. This particularity invalidates the application of the systematicprocedure due to Mielke (2011) previously reviewed. However, its evolutionequations (2.24) or (2.17) can still be written in the GENERIC format. Forthat, two GENERIC forms need to be elaborated according to two choices ofthe thermodynamical variables in the state vector based on either entropy ortemperature variables. The entropy-based GENERIC form of this system wasfirst proposed in Romero (2009) while the temperature-based (so far thoughtto be difficult to attain) was recently revealed in Conde Martın et al. (2016).Both forms are briefly reviewed in the following subsections.

4.3.1 Entropy-based GENERIC formulation

The entropy-based GENERIC description of the two thermo-spring sys-tem departs from the definition of the finite-dimensional phase space to be


P = R2d × R2d and the thermodynamical space to be T = R2 which alto-gether define the state space as follows

S =z = (q1, q2,p1,p2, η1, η2) ∈ R2d × R2d × R2, q1 6= 0, q2 6= q1

(4.27)

With this state space, the total energy of the system is provided by thesum of the kinetic energy of the point masses (2.33) and the the internalenergies of the springs in terms of entropies (2.22) to give

E(z) = K(‖p1‖, ‖p2‖) +2∑b=1

eb(λb, ηb), (4.28)

Recall that the entropy formulation is restricted to those cases in whichthe analytical function ei(λi, ηi) can be achived.

For the state space (4.27), the total entropy of the system is provided bythe sum of the two of the state variables, namely the spring entropies, as

S(z) =2∑

a=1

ηa (4.29)

Thus, the first step to find the GENERIC equations is to derive the totalenergy and total entropy with respect to the state vector (4.27), yielding

DE =

∂e1

∂λ1

∂λ1

∂q1

+∂e2

∂λ2

∂λ2

∂q1

∂e2

∂λ2

∂λ2

∂q2

p1

m1

p2

m2

∂e1

∂η1

∂e2

∂η2

, DS =

000011

(4.30)

Due to the simplicity of the system, the Poisson and Dissipative matricesrequired to recast the evolution equations (2.24) in the GENERIC form (4.9)


can be constructed by inspection. The Poisson thus results in the canonicalone provided by (4.14), that is

L(z) =

0 0 1 0 0 0

0 0 0 1 0 0

−1 0 0 0 0 0

0 −1 0 0 0 0

0T 0T 0T 0T 0 0

0T 0T 0T 0T 0 0

(4.31)

The Dissipative matrix, however, is a bit more involved, that is

M(z) = k

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0T 0T 0T 0T ∂e2

∂η2

(∂e1

∂η1

)−1

−1

0T 0T 0T 0T −1∂e1

∂η1

(∂e2

∂η2

)−1

, (4.32)

where the partial derivatives are indetified with the spring temperatures via(2.23)2, that is

∂e1

∂η1

= θ1,∂e2

∂η2

= θ2 (4.33)

Finally, the use of these matrices and gradients in (4.9) leads to theevolution equations (2.24) except for the external forces applied to the pointmasses which are out of the GENERIC expressions due to being external.In any case, they can be inserted a posteriori taking into account that theirpresence will breach the conservation properties stated by the GENERICformulation.

4.3.2 Temperature-based GENERIC formulation

Consider a state vector which includes the temperatures of the springsas the thermodynamical variables so the state space is defined by

S =z = (q1, q2,p1,p2, θ1, θ2) ∈ R2d × R2d × R2

+, q1 6= 0, q2 6= q1

(4.34)


The total energy of the two thermo-spring system was introduced in 2.1as the sum of the kinetic energy of the point masses (2.33) and the internalenergy of the springs (2.7), that is

E(z) = K(‖p1‖, ‖p2‖) +2∑b=1

eb(λb, θb), (4.35)

The total entropy of the system is provided by the sum of the springentropy (2.12) to be

S(z) =2∑

a=1

ηa(λa, θa) =2∑

a=1

−∂Ψa(λa, θa)

∂θa(4.36)

Again, the gradient of the total energy and total entropy with respect tothe state vector need to be elaborated, that is

DE =

∂e1

∂λ1

∂λ1

∂q1

+∂e2

∂λ2

∂λ2

∂q1

∂e2

∂λ2

∂λ2

∂q2

p1

m1

p2

m2

∂e1

∂θ1

∂e2

∂θ2

, DS =

∂η1

∂λ1

∂λ1

∂q1

+∂η2

∂λ2

∂λ2

∂q1

∂η2

∂λ2

∂λ2

∂q2

0

0∂η1

∂θ1

∂η2

∂θ2

(4.37)

The associated Poisson and Dissipative matrices required to recast theevolution equations (2.17) in the GENERIC form (4.9) are provided by

L(z) =

0 0 1 0 0 0

0 0 0 1 0 0

−1 0 0 0 L11 L12

0 −1 0 0 0 L22

0T 0T L11 0T 0 0

0T 0T L21 L22 0 0

(4.38)


Interestingly, new terms taking into account the thermo-mechanical couplingappears with this choice of the state vector, which are provided by

Lab = −LTba =

(∂ηb∂θb

)−1∂ηb∂λb

∂λb∂qa

, (4.39)

Similar ones appear when the internal energies are considered in the statevector, see Romero (2009), demonstrating that the temperature election isnot much more involved than the other common choices.

The Dissipative matrix can be expressed in the following way

M(z) = kθ1θ2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0T 0T 0T 0T M11 M12

0T 0T 0T 0T M21 M22

, (4.40)

with the scalar values Mab being provided by

Mab = (−1)a+b

(∂ea∂θa

)−1(∂eb∂θb

)−1

(4.41)

Remark 4.3. The fact that the temperatures appear as factors of the ma-trix will have an impact in the formulation of Energy-Entropy-Momentummethods as will be explained in Chapter 5.

Due to the state vector including the temperatures, the GENERIC equa-tions (4.9) provides the evolution equations of the system in the followingparticular form

qa =pama

pa = −2∑b=1

(∂eb∂λb− θb

∂ηb∂λb

)∂λb∂qa

θa =

(∂ea∂θa

)−1[


2∑b=1

∂λa∂qb· pbmb

],

(4.42)

for a = 1, 2.

The form (2.17), besides the external forces, is then recast by using re-lation provided by the Legendre transform (2.7) in the linear momentumbalance (4.42)2.


4.3.3 Laws of thermodynamics and symmetries

The laws of thermodynamics are ensured by the GENERIC equations(4.9) and by the properties of the Poisson and Dissipative matrices. Thus,the energy conservation results trivial from (4.12). The second law, however,can be further elaborated by using the Dissipative matrix provided by either(4.32) or (4.40) along with the gradient (4.30)2 or (4.37)2, leading to the rateof entropy

S(z) = DS · z = DS ·MDS = k(θ2 − θ1)2

θ1θ2

≥ 0, (4.43)

that demonstrates the two thermo-spring system is thermodynamically sound.Note that for the entropy-based formulation the appearing temperatures arefunctions of the entropy via (4.33).

As was demonstrated in Section 2.1.3, the two thermo-spring system onlyhas a symmetry respect to rotational orbits. Recall that the isolation of thesystem includes the fixed point so the linear momentum (a symmetry respectto translational orbits) is breached.

Then, the preserved momentum map is identified with the classical angu-lar momentum. To reveal it within the GENERIC formalism, consider the ac-tion of the rotation group G = SO(3) = G : R3 7→ R3|GTG = 1, det (G) =1 on S , Φ: G×S 7→ S defined by

ΦG = Φ(G, z) = (Gq1,Gq2,Gp1,Gp2, s1, s2) with G ∈ SO(3), (4.44)

where si is either entropies ηi or temperatures θi.

First, it is easily proved that this action preserves the total energy of thesystem by noting that

‖Gq1‖ =√qT

1 GTGq1 =√qT

1 q1 = ‖q1‖‖G(q2 − q1)‖ =

√(q2 − q1)TGTG(q2 − q1) = ‖q2 − q1‖

‖Gpa‖ =√pTaGTGpa =

√pTapa = ‖pa‖ with a = 1, 2,

(4.45)

so that the action Φ on it gives

E ΦG = K(‖Gp1‖, ‖Gp2‖) + e1(λ1(Gq1), s1) + e2(λ2(Gq2,Gq1), s2)

= K(‖Gp1‖, ‖Gp2‖) + e1(‖Gq1‖, s1) + e2(‖G(q2 − q1)‖, s2)

= E(z),

(4.46)


where use has been made of the relations (2.2).

Similarly, the preservation of the reversible and irreversible operators(4.20) when applied this action can be demonstrated using the same argu-ments as above, the full details can be found in Romero (2013).

On the other hand, the infinitesimal generator of actions (4.19) results,for any ξ ∈ G, in

ξS (z) =d

dε

∣∣∣∣ε=0

Φξ(ε, z) = (ξ ∧ q1, ξ ∧ q2, ξ ∧ p1, ξ ∧ p2, 0, 0)T (4.47)

Given that, the angular momentum defined by (2.29)2 is a momentummap associated with the action ΦG if the scalar function Jξ : S 7→ R,provided by

Jξ(z) = ξ · J = ξ ·2∑

a=1

qa ∧ pa, (4.48)

satisfies the relations (4.23). Hence, the proof must be done separately foreach of the formulations employed. The first step, however, is to computethe gradient of the scalar function which gives

DJξ = (p1 ∧ ξ,p2 ∧ ξ, ξ ∧ q1, ξ ∧ q2, 0, 0)T (4.49)

Thus, the entropy-based GENERIC form is readily proved by

LDJξ =

0 0 1 0 0 0

0 0 0 1 0 0

−1 0 0 0 0 0

0 −1 0 0 0 0

0T 0T 0T 0T 0 0

0T 0T 0T 0T 0 0

p1 ∧ ξp2 ∧ ξξ ∧ q1

ξ ∧ q2

0

0

=

ξ ∧ q1

ξ ∧ q2

−p1 ∧ ξ−p2 ∧ ξ

0

0

,

(4.50)which coincides with ξS provided by (4.47). Furthermore, the second requi-site is also fulfilled as proves the following

MDJξ = k

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0T 0T 0T 0T θ2

θ1

−1

0T 0T 0T 0T −1θ1

θ2

p1 ∧ ξp2 ∧ ξξ ∧ q1

ξ ∧ q2

0

0

= 0 (4.51)


All this is the proof for the angular momentum (2.29) to be a first integralof the two thermo-spring system formulated in terms of entropy.

On the other hand, for the temperature-based GENERIC matrices, thefollowing is obtained

LDJξ =

0 0 1 0 0 0

0 0 0 1 0 0

−1 0 0 0 L11 L12

0 −1 0 0 0 L22

0T 0T L11 0T 0 0

0T 0T L21 L22 0 0

p1 ∧ ξp2 ∧ ξξ ∧ q1

ξ ∧ q2

0

0

=

ξ ∧ q1

ξ ∧ q2

−p1 ∧ ξ−p2 ∧ ξ

L11 · (ξ ∧ q1)

L21 · (ξ ∧ q1) + L22 · (ξ ∧ q2)

=

ξ ∧ q1

ξ ∧ q2

−p1 ∧ ξ−p2 ∧ ξ

0

0

(4.52)

The two last components of the resulting vector can be demonstratedto be zero by elaborating the terms Lab according to (4.38) together withproperties of the cross product3. Lastly, the second requirement is triviallysatisfied as follows

MDJξ = kθ1θ2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0T 0T 0T 0T M11 M12

0T 0T 0T 0T M21 M22

p1 ∧ ξp2 ∧ ξξ ∧ q1

ξ ∧ q2

0

0

= 0, (4.53)

concluding that the angular momentum (2.29) is a preserved quantity of thetwo thermospring system formulated in terms of temperatures.

4.4 GENERIC form of the thermo-visco-elastic system

The thermo-visco-elastic system introduced in Section 2.2 is also a dis-sipative system in the sense of (4.9). Then, it can be described within

3a · (b ∧ a) = 0, a · (b ∧ c) = b · (c ∧ a) = c · (a ∧ b)

130 4.4. GENERIC form of the thermo-visco-elastic system

the GENERIC formalism for both choices of the thermodynamical variable:entropy or temperature. Although Garcıa Orden & Romero (2011) dealtwith the formulation of the thermo-visco-elastic system using entropies, itsGENERIC form has never been advocated. In the following subsections bothentropy-based and temperature-based GENERIC forms are elaborated, thusfulfilling this gap.

4.4.1 Entropy-based GENERIC formulation

The entropy-based GENERIC form relies on a state space composed bya phase space P = R2d × R2d, a space D = R and a space T = R2 toaltogether be

S =z = (q1, q2,p1,p2, γ, η, σ) ∈ R2d × R2d × R× R2, q2 6= q1

(4.54)

This choice needs the internal energy of the element and of the environ-ment to be redefined in terms of their entropies. As has already been pointedout, this step may not be feasible for particular thermodynamic models, beingan important technical constraint for the formulation. Then, assuming thatthe constitutive models employed for both the element and the environmentenable their internal energies to be functions of their respective entropies,the total energy of the system is provided by

E(z) = K(‖p1‖, ‖p2‖) + e(λ, η, γ) + ε(σ), (4.55)

where K is the kinetic energy provided by (2.33).

For its part, the total entropy of the system is just the sum of two statevariables

S(z) = η + σ (4.56)

This simplicity is responsible for the simple form that acquires the Pois-son matrix, which in essence becomes the classical symplectic matrix, i.e itscanonical form (4.14), that is

L(z) =

0 0 1 0 0 0 0

0 0 0 1 0 0 0

−1 0 0 0 0 0 0

0 −1 0 0 0 0 0

0T 0T 0T 0T 0 0 0

0T 0T 0T 0T 0 0 0

0T 0T 0T 0T 0 0 0

(4.57)


Nevertheless, the Dissipative matrix requires to be carefully obtained toaccount for the both dissipative phenomena present in the system: heat con-duction and internal viscous dissipation. Either by using the ideas of Mielke(2011), summarized at the beginning of this Section, or by bearing in mindthe evolution equations (2.54) the Dissipative matrix can be demonstratedto be

M(z) =

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0T 0T 0T 0T 1

ν

∂e

∂η−1

ν

∂e

∂γ0

0T 0T 0T 0T −1

ν

∂e

∂γ

(∂e

∂η

)−1[

1

ν

(∂e

∂γ

)2

+ k∂ε

∂σ

]−k

0T 0T 0T 0T 0 −k k

(∂ε

∂σ

)−1∂e

∂η

(4.58)

In agreement with these matrices, the resulting evolutions equations de-rived from the use of (4.9) are those provided by (2.54) with the exception ofthe external forces and taking into account that in this case the temperaturesare provided by

θ(λ, η, γ) =∂e(λ, η, γ)

∂η, ϑ(σ) =

dε(σ)

dσ(4.59)

4.4.2 Temperature-based GENERIC formulation

For the temperature choice, the state vector is defined in a state spaceof the form

S =z = (q1, q2,p1,p2, γ, θ, ϑ) ∈ R2d × R2d × R× R2

+, q2 6= q1

(4.60)

Given this state space, the total energy of the system is provided by

E(z) = K(‖p1‖, ‖p2‖) + e(λ, θ, γ) + ε(ϑ), (4.61)


The total entropy is the sum of both the element and the environmententropies which can be expressed in terms of the above state vector as

S(z) = η(λ, γ, θ) + σ(ϑ) = −∂Ψ(λ, γ, θ)

∂θ+ σ(ϑ) (4.62)

Recalling that the elongation λ depends on the state vector via relation(2.38), the energy and entropy gradients results in

DE =

∂e

∂λ

∂λ

∂q1

∂e

∂λ

∂λ

∂q2

p1

m1

p2

m2

∂e

∂γ

∂e

∂θ

dε

dϑ

, DS =

∂η

∂λ

∂λ

∂q1

∂η

∂λ

∂λ

∂q2

0

0∂η

∂γ

∂η

∂θ

dσ

dϑ

(4.63)

A brief inspection of them reveals that they contain every term involvedin the evolution equations written in the form (2.45). Therefore, the Poissonand Dissipative matrices must be built such that they comply with theirrespective requirements and yield the correct set of the system evolutionequations. Accordingly, the Poisson matrix is found to be

L(z) =

0 0 1 0 0 0 0

0 0 0 1 0 0 0

−1 0 0 0 0 L1 0

0 −1 0 0 0 L2 0

0T 0T 0T 0T 0 0 0

0T 0T L1 L2 0 0 0

0T 0T 0T 0T 0 0 0

, (4.64)

with

La = −LTa =

(∂η

∂θ

)−1∂η

∂λ

∂λ

∂qa(4.65)


As expected, it comprises the symplectic Hamiltonian matrix plus termsderived from the thermo-mechanical coupling, i.e the heat generated whenchanging the elongation of the device.

Likewise, the Dissipative matrix has the following structure

M(z) = θ

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0T 0T 0T 0T M11 M12 0

0T 0T 0T 0T M12 M22 +M ′11 M ′

12

0T 0T 0T 0T 0 M ′12 M ′

22

, (4.66)

in which the appearing scalars are provided by

Mab = (−1)a+b 1

ν

(∂e

∂γ

)a(b−1)(∂e

∂θ

)−a(b−1)

, (4.67)

and

M ′ab = (−1)a+bkϑ

(∂e

∂θ

)a(b−1)−2(dε

dϑ

)−a(b−1)

(4.68)

Each of them accounts for the two dissipative mechanisms involved inthe system evolution, i.e the viscous flow taking place in the dashpot and theheat transfer, respectively.

Remark 4.4. The element temperature can be factorized out of the matrixwhich means that the structural properties stated over the Dissipative matrixhold for any value it may take. This issue will have consequences in theformulation of Energy-Entropy-Momentum methods.

Then, using (4.63) - (4.68) in (4.9) the system evolution equations are


obtained with the following format

qa =pama

pa = −(∂e

∂λ− θ ∂η

∂λ

)∂λ

∂qa

γ = −1

ν

(∂e

∂γ− θ ∂η

∂γ

)θ =

(∂e

∂θ

)−1[−θ

2∑a=1

∂η

∂λ

∂λ

∂qa· pama

+1

ν

(∂e

∂γ− θ ∂η

∂γ

)∂e

∂γ− h]

ϑ =

(dε

dϑ

)−1

h,

(4.69)

which are the same as those obtained in Chapter 2 provided by (2.45) afterusing the relation between the free energy and the internal energy provided bythe Legendre transform (2.7) along with its direct consequence (2.9), exceptfor the external forces applied in the point masses.

4.4.3 Laws of thermodynamics and symmetries

For both choices, the change of total energy is trivially demonstrated tobe zero

E(z) = DE · z = DE · LDE +DS ·MDE = 0 (4.70)

The change of the total entropy, on the other hand, needs separate elab-oration. Thus, the entropy formulation yields

S(z) = DS · z = −DE · LDS +DS ·MDS

=1

νθ

(∂e

∂γ

)2

+ k(θ − ϑ)2

θϑ≥ 0,

(4.71)

whereas the temperature-based one gives

S(z) = DS · z = −DE · LDS +DS ·MDS

=1

θν

(θ∂η

∂γ− ∂e

∂γ

)2

+ k(θ − ϑ)2

θϑ

=1

θν

(−∂Ψ

∂γ

)2

+ k(θ − ϑ)2

θϑ≥ 0

(4.72)


The difference between both results comes from the different definitionsthat adopt the viscous force g in each formulation (2.55)2 or (2.46)2

Regarding the symmetries of the system, in Section 2.2.3 two preservedquantities other than energy were identified to be the linear and the angularmomentum. The same can be concluded within the GENERIC formalism.For instance, the linear momentum is associated with the action of the addi-tive group G = Rd on S , Φ: G×S 7→ S defined by

Φa = Φ(a, z) = (q1 + a, q2 + a,p1,p2, γ, s1, s2) with a ∈ Rd, (4.73)

where si is either entropies ηi or temperatures θi.

This action preserves the total energy of the system as proved the fol-lowing

E Φa = K(‖p1‖, ‖p2‖) + e(λ(q2 + a, q1 + a), γ, s1) + ε(s2)

= K(‖p1‖, ‖p2‖) + e(‖q2 + a− q1 − a‖, γ, s1) + ε(s2)

= E(z),

(4.74)

Also, this action preserves the reversible and irreversible operators (4.20).To prove it, first the gradient of the action is computed to give

DΦa = (1,1,1,1, 1, 1, 1)T (4.75)

For this gradient, it is trivial to show that the products DΦa(z) · L ·DΦa(z)T or DΦa(z) ·M ·DΦa(z)T results in the matrices L,M themselves.On the other hand, the action composition operation on them yields the sameresults as their components depend on the action-perturbed state variables,i.e the positions coordinates, via the norm of the relative vector so the sameas shown in (4.74) applies.

Moreover, the infinitesimal generator of actions (4.19) for any ξ ∈ g yields

ξS (z) =d

dε

∣∣∣∣ε=0

Φξa(ε, z) = (ξ, ξ,0,0, 0, 0, 0)T (4.76)

To prove that the linear momentum (2.29)1 is a momentum map associ-ated with the action Φa, consider the scalar function Jξ : S 7→ R, providedby

Jξ(z) = ξ ·L = ξ ·2∑

a=1

pa, (4.77)


Then, it must comply with (4.23) for which the elaboration of the gradientof the scalar function is required, that is

DJξ = (0,0, ξ, ξ, 0, 0, 0)T (4.78)

Thus, for the entropy-based GENERIC form it follows that

LDJξ =

0 0 1 0 0 0 0

0 0 0 1 0 0 0

−1 0 0 0 0 0 0

0 −1 0 0 0 0 0

0T 0T 0T 0T 0 0 0

0T 0T 0T 0T 0 0 0

0T 0T 0T 0T 0 0 0

0

0

ξ

ξ

0

0

0

=

ξ

ξ

0

0

0

0

0

(4.79)

which coincides with ξS provided by (4.76). Now, looking at the requisitefor the Dissipative matrix the following is obtained

MDJξ =

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0T 0T 0T 0T 1

ν

∂e

∂η−1

ν

∂e

∂γ0

0T 0T 0T 0T −1

ν

∂e

∂γ

1

θν

(∂e

∂γ

)2

+ kϑ

θ−k

0T 0T 0T 0T 0 −k kθ

ϑ

0

0

ξ

ξ

0

0

0

= 0

(4.80)

Finally, for the temperature-based GENERIC matrices, the following isobtained

LDJξ =

0 0 1 0 0 0 0

0 0 0 1 0 0 0

−1 0 0 0 0 L1 0

0 −1 0 0 0 L2 0

0T 0T 0T 0T 0 0 0

0T 0T L1 L2 0 0 0

0T 0T 0T 0T 0 0 0

0

0

ξ

ξ

0

0

0

=

ξ

ξ

0

0

0

L1 · ξ + L2 · ξ0

(4.81)


The remaining term in the above vector vanishes due to the definitionof terms La which leads them to satisfy L1 = −L2. Lastly, in this case, theDissipative term MDJξ also yields zero due to the form of the scalar functiongradient (4.78).

These results are the proof for the linear momentum (2.29)1 to be afirst integral of the thermo-visco-elastic system for the both formulationsconsidered.

The thermo-visco-elastic system also has a symmetry associated witha rotation group. As with the two thermo-spring system, the correspondingmomentum map is the the angular momentum provided by (2.29). The proofclosely follows the one done for the two thermo-spring system (4.44) - (4.53).

4.5 Infinite-dimensional dissipative systems

In this Section the GENERIC formalism is discussed for infinite-dimen-sional spaces so as to provide a unified definition for infinite-dimensionalsmooth dissipative systems. From a practical perspective, such systems arebasically those defined in continua. Therefore, the resulting unified theorywill enable the formulation of the nonlinear thermo-dissipative continuum dy-namics discussed in Chapter 3. In order to make statements clear, the purecoupled finite strain thermoelastic case will be first studied and, then, themore general thermo-dissipative case will be addressed. Finally, the isother-mal limit case, described in Section 3.6, will be also formulated within thisunified framework under certain assumptions.

For the definition of infinite-dimensional dissipative systems, consider aninfinite-dimensional domain identified with a reference configuration occu-pying an open subset B0 ⊂ Rd. Then, the thermodynamic state of all theparticles contained in B0 at a given instant is defined by smooth mappingsz : B0 × [0, T ] 7→ Rk × Sym(T2

d)m, furnishing the infinite-dimensional state

space S assumed to be a subset of an infinite-dimensional vector space Xwith inner product 〈·, ·〉 : X ×X 7→ R identified with its dual X ∗.

If z is a state in S , then the set of tangent vectors at z furnishes theinfinite-dimensional vector space called tangent space and denoted by TzS .For subsequent developments, only linear manifolds are considered so eachof the tangent spaces are identified with the same vector space, denoted asXz, that is, TzS ∼= Xz, where Xz is a subset of X .

Then, given a tangent element U ∈ Xz and any functional F : S 7→ R,

138 4.5. Infinite-dimensional dissipative systems

the functional derivative at z is introduced as a mapping δF/δz : S 7→ Xdefined by ⟨

δFδz

,U

⟩= DF(z) · u+

m∑α=1

DF(z) : Uα, (4.82)

for all U = (u,Uα) ∈ Rk × Sym(T2d)m.

The gradient operator in the Euclidian space Rk and the gradient oper-ator in the symmetric second-order tensor space Sym(T2

d)m are canonically

defined4 via the directional derivative as follows

DF(z) · u =d

dε

∣∣∣∣ε=0

F(v + εu,Zα), DF(z) : Uα =d

dε

∣∣∣∣ε=0

F(v,Zα + εUα),

(4.83)for all z = (v,Zα) ∈ Rk × Sym(T2

d)m.

All the functionals involved in the derivation of infinite-dimensional dis-sipative systems are based on scalar-valued density functions f : S → Rintegrated over the volume occupied by the continuum as follows

F(z) =

∫B0

f(z)dV0 (4.84)

The previous definition of the functional derivative applied to this typeof functionals leads to the following identity⟨

δFδz

,U

⟩=

∫B0

Df(z) · udV0 +m∑α=1

∫B0

Df(z) : UαdV0

= 〈Df(z),U〉B0 ,

(4.85)

which will often appear for the derivation of the evolution equations ofinfinite-dimensional dissipative systems.

Definition 4.5. The infinite-dimensional state space S defining dissipa-tive systems consists of an infinite-dimensional phase space P, defined bythe set of smooth mappings ϕ,p : B0 7→ Rd × Rd, an infinite-dimensionalthermal space T , defined by a smooth mapping s : B0 7→ R and an infinite-dimensional dissipative space D , provided by the set of m internal variablesmooth mappings Λα : B0 7→ Sym(T2

d) with α = 1, . . . ,m, altogether col-lected in

S = z = (ϕ,p, s,Λ1, . . . ,Λm) : B0 × [0, T ] 7→ Rd × Rd × R× Sym(T2d)m

(4.86)4note that both gradients use the same notation in order to keep it simple.


Definition 4.6. A infinite-dimensional dissipative system is an isolated sys-tem ruled by the infinite-dimensional GENERIC evolution equations defined,for any functional F : S 7→ R, as

F(z) = F(z), E(z)+ [F(z), S(z)] , (4.87)

where E : S 7→ R is the total energy of the system, S : S 7→ R is the totalentropy of the system, ·, · : S ×S 7→ R is the bilinear Poisson operatorsatisfying the skew-symmetry, Jacobi and Leibniz properties, as introducedin Section 4.1, and [·, ·] : S × S 7→ R is the bilinear Dissipative (metric)operator being symmetric and positive-definite as also introduced in Section4.1.

Proposition 4.3. Any infinite-dimensional dissipative system provided by(4.87) agrees with the law of thermodynamics if the bilinear operators addi-tionally satisfy the degeneracy or non-interaction conditions

S(z),F(z) = [E(z),F(z)] = 0, (4.88)

for every functional F : S → R.

Proof. The proof relies on the brackets properties along with the just in-troduced degeneracy conditions as was reasoned in Section 4.1 for generalmetriplectic structures to arrive at (4.7).

Furthermore, the Poisson and Dissipative brackets are associated to re-versible and irreversible differential operators L,M : S ×X →X as follows

F ,G =

⟨δFδz

,L

[δGδz

]⟩, [F ,G] =

⟨δFδz

,M

[δGδz

]⟩(4.89)

Analogously to the finite-dimensional case, essential physical consider-ations allow to conclude that the reversible (Poisson) operator cannot begenerated by the dissipative space D while the irreversible (Dissipative) onecan only depend on the thermal and dissipative spaces T and D .

4.5.1 Infinite-dimensional smooth dissipative systems with symmetries

The notion of symmetries and momentum maps can be generalized forinfinite-dimensional systems. Assuming the algebra of Lie Groups introducedin the previous Section, an action Φ on the state space S of a Lie group G

140 4.5. Infinite-dimensional dissipative systems

preserves the infinite-dimensional GENERIC structure if, for any two func-tionals F ,G : S 7→ R, it is satisfied that

F Φg,G Φg = F ,GΦg and [F Φg,G Φg] = [F ,G]Φg, (4.90)

for any g ∈ G.

Definition 4.7. Let G be a Lie group on a state space with an associatedaction Φ that preserve the infinite-dimensional GENERIC structure. Then, ifthe total energy of an infinite-dimensional dissipative system is a G-invariantfunction, that is

E Φg = E ∀g ∈ G, (4.91)

the system is a infinite-dimensional dissipative with symmetries and has amomentum map associated with the action.

Definition 4.8. Given a Lie group G with associated algebra g and dual g∗,a mapping J : S → g∗ is a momentum map associated with the action Φ if,for any ξ ∈ g, the scalar functional Jξ : S → R defined as

Jξ(z) = 〈J(z), ξ〉, (4.92)

satisfies

ξS = L

[δJξδz

], 0 = M

[δJξδz

], (4.93)

where ξS is the infinitesimal generator of the action defined in (4.19).

Theorem 4.4. The momentum map of a infinite-dimensional dissipativesystem with symmetries is a conserved quantity of the system evolution.

Proof. The proof directly follows from the calculation of the rate of the scalarfunctional Jξ using (4.87) along with the properties of the brackets and (4.93),altogether yielding

Jξ = −E, Jξ = −⟨δE

δz,L

[δJξδz

]⟩= −

⟨δE

δz, ξS

⟩(4.94)

Finally, the term on the right hand side results from differentiating (4.91),considering (4.19), to give the identity⟨

δE

δz, ξS

⟩= 0, (4.95)

The above theorem corresponds with the Noether’s theorem for infinite-dimensional dissipative system with symmetries in the sense of Def. 4.7.


4.6 GENERIC form of nonlinear thermoelasticity

The previous definition of infinite-dimensional dissipative system encom-passes, for instance, the dynamics of isolated thermoelastic continua experi-encing finite strains and thermal coupled with heat conduction. The com-plete description of such a system results from defining appropriated Pois-son and Dissipative operators in terms of a given state variables with theformat (5.18), among which different pure thermodynamical variables canbe adopted, fundamentally either density entropy or temperature, althoughmany others could be used. The choice of the entropy can be found in eitherRomero (2010a,b, 2013). In contrast, the choice of temperature has so farpresented drawbacks and, therefore, it has not apparently been addresseduntil this work.

4.6.1 Entropy formulation

The description based on the density entropy relies on the followinginfinite-dimensional vector state space

S = z = (ϕ,p, η) : B0 × [0, T ] 7→ Rd × Rd × R, det (∇0ϕ) > 0, (4.96)

whose tangent space is

TS = w = (wϕ,wp, wη) : B0 7→ Rd × Rd × R (4.97)

Accordingly, the total energy function and total entropy function aredefined by (3.96) (with (3.55) and (3.54)) and (3.98). Taking into accountthat the material dissipative effects are neglected, that is the thermodynamicpotentials are of the form (3.39)2, and the above state vector (4.96), theirparticular expressions are

E(z) =

∫B0

(1

2ρ0

‖p‖2 + e(C, η)

)dV0, S(z) =

∫B0

ηdV0, (4.98)

Then, the directional derivatives of these functionals can be calculatedby (4.82) for any element on the tangent space (4.97) to give⟨

δE

δϕ,wϕ

⟩= 〈2 ∂e

∂C,∇0ϕ

T∇0wϕ〉B0⟨δE

δp,wp

⟩= 〈 p

ρ0

,wp〉B0⟨δE

δη, wη

⟩= 〈∂e

∂η, wη〉B0 = 〈Θ, wη〉B0

⟨δS

δϕ,wϕ

⟩= 0⟨

δS

δp,wp

⟩= 0⟨

δS

δη, wη

⟩= 〈1, wη〉B0 ,

(4.99)

142 4.6. GENERIC form of nonlinear thermoelasticity

where the first result can be explained by considering the following resulton real-valued functions f : S → R that depend on the motion function ϕvia the right Cauchy-Green deformation tensor C(ϕ) := ∇0ϕ

T∇0ϕ, whichresults from nothing more than applying the chain rule of the directionalderivative and using the symmetry of the said tensor to give

Df(C(ϕ), . . . ) · u = Df : (DC · u) = 2∂f

∂C: ∇0ϕ

T∇0u (4.100)

The resulting partial derivatives of the internal energy in (4.99) are identi-fied to be the symmetric Piola-Kirchhoff stress tensor S : S 7→ Sym(T2

d) andthe temperature field Θ : S 7→ R+ provided by (3.36) and (3.37) (ignoringthe dependency on internal variables), that is

S(C(ϕ), η) = 2∂e

∂C(C(ϕ), η), Θ(C(ϕ), η) =

∂e

∂η(C(ϕ), η) (4.101)

Given that, it only remains to specify proper reversible and irreversibleoperators that generate the balance laws governing the thermoelastic con-tinuum dynamics which are, following Chapter 3, the linear momentum andenergy balances.

As with the finite-dimensional case (4.14), the use of the entropy asthermodynamical state variable leads the reversible operator to be providedby the canonical one of Hamiltonian mechanics, see Marsden & Ratiu (1999),which, for any two functionals F ,G : S → R, reads

F ,G =

⟨δFδϕ

,δGδp

⟩−⟨δFδp

,δGδϕ

⟩(4.102)

For its part, the bilinear Dissipative operator is built to account for theconduction phenomenon (of dissipative nature), see Romero (2010a), for anytwo functionals F ,G : S → R, as follows

[F ,G] =

⟨∇0

[1

Θ

δFδη

],Θ2K∇0

[1

Θ

δGδη

]⟩(4.103)

In the above expression, K is the material conductivity tensor (3.27)which is symmetric and positive semidefinite, thus granting the bracket withthese properties.


The above Poisson and Dissipative brackets comply with the requireddegeneracy conditions (4.88). The proofs easily follow from (4.99) to give

S,F =

⟨δS

δϕ,δFδp

⟩−⟨δS

δp,δFδϕ

⟩= 0

[E,F ] =

⟨∇0

[1

ΘΘ

],Θ2K∇0

[1

Θ

δFδη

]⟩= 0

(4.104)

In order to derive the evolution equations of the thermoelastic continuumdynamics, the time derivative of a functional F : S 7→ R is calculated asfollows

F =

⟨δFδz

, z

⟩=

⟨δFδϕ

, ϕ

⟩+

⟨δFδp

, p

⟩+

⟨δFδη, η

⟩(4.105)

On the other hand, the reversible part generated by the Poisson brackethas the following expression

F , E = −E,F = −⟨δE

δϕ,δFδp

⟩+

⟨δE

δp,δFδϕ

⟩(4.106)

In the same way, the irreversible part is provided by

[F , S] =

⟨∇0

[1

Θ

δFδη

],Θ2K∇0

[1

Θ

δS

δη

]⟩(4.107)

Choosing the functional derivative to be a tangent vector 〈δF/δz, z〉 =〈w, z〉B0 and using the functional derivatives (4.99) along with (4.101), theweak statement of the balance laws read

〈wϕ, ϕ〉B0 = 〈wϕ,1

ρ0

p〉B0

〈wp, p〉B0 = −〈∇0wp,FS〉B0

〈wη, η〉B0 = −〈∇0

[wηΘ

],K∇0Θ〉B0 = 〈∇0

[wηΘ

],H〉B0

(4.108)

As expected, these equations only describe the isolated dynamics of athermoelastic continuum as external contributions are intrinsically excludedin the GENERIC formalism. However, a posteriori consideration of themcould be done to arrive at the weak statement provided by (3.146), consid-ering D = 0, by attaching them to the above equations.


Accordingly, the total energy of the system is preserved which can beproved by choosing F = E in the GENERIC equations (4.105), (4.106) and(4.107) to give

E(z) = E,E+ [E, S] = 〈∇0

[1

ΘΘ

],Θ2K∇0

[1

Θ

]〉B0 = 0 (4.109)

Similarly, the rate of the total entropy follows from taking F = S(z) togive

S(z) = S,E+ [S, S] = 〈∇0

[1

Θ

],Θ2K∇0

[1

Θ

]〉B0

= 〈 1

Θ2∇0Θ,K∇0Θ〉B0 ≥ 0,

(4.110)

which coincides with the first term of (3.101) due to the heat conductionphenomenon since is the only dissipative process involved in coupled ther-moelastic solids.

4.6.2 Temperature formulation

Now, the temperature formulation of thermoelasticity is addressed withinthe GENERIC formalism. To this end, the state space of the thermoelasticcontinuum is defined to be the infinite dimensional set

S = z = (ϕ,p,Θ) : B0× [0, T ] 7→ Rd×Rd×R+, det (∇0ϕ) > 0, (4.111)

whose tangent space is

TS = w = (wϕ,wp, wΘ) : B0 × [0, T ] 7→ Rd × Rd × R (4.112)

To express the total energy and total entropy of the system in termsof this state space, the thermodynamic potentials in terms of temperature(3.39)1 need to be used to obtain

E(z) =

∫B0

(1

2ρ0

‖p‖2 + e(C,Θ)

)dV0, S(z) =

∫B0

η(C,Θ)dV0 (4.113)


Accordingly, the functional derivatives now are⟨δE

δϕ,wϕ

⟩= 〈2 ∂e

∂C,∇0ϕ

T∇0wϕ〉B0⟨δE

δp,wp

⟩= 〈 p

ρ0

,wp〉B0⟨δE

δΘ, wΘ

⟩= 〈 ∂e

∂Θ, wΘ〉B0

⟨δS

δϕ,wϕ

⟩= 〈2 ∂η

∂C,∇0ϕ

T∇0wϕ〉B0⟨δS

δp,wp

⟩= 0⟨

δS

δΘ, wΘ

⟩= 〈 ∂η

∂Θ, wΘ〉B0

(4.114)

Then, the appearing partial derivatives are linked with the symmetricPiola-Kirchhoff stress tensor S, the latent heat tensor V and the specificheat capacity c by the constitutive laws (neglecting the internal variables)(3.18), (3.77) and (3.78) in the following manner

2∂e

∂C= S + 2Θ

∂η

∂C, 2

∂η

∂C=

1

ΘV,

∂e

∂Θ= Θ

∂η

∂Θ= c (4.115)

For this choice, the Poisson bracket needs to be extended to account forreversible mechanism of the thermo-deformation coupling, which for any twofunctionals F ,G : S → R are

F ,G =

⟨δFδϕ

,δGδp

⟩−⟨δFδp

,δGδϕ

⟩+

⟨∇0

[δFδp

],FV

1

c

δGδΘ

⟩−⟨

FV1

c

δFδΘ

,∇0

[δGδp

]⟩ (4.116)

The Dissipative bracket, however, is slightly modified due to the changeof the thermal variable. Thus, for any two functionals F ,G : S → R, it is

[F ,G] =

⟨∇0

[1

c

δFδΘ

],Θ2K∇0

[1

c

δGδΘ

]⟩(4.117)

Again, the material conductivity tensor K is responsible for the Dissipa-tive operator to be positive semidefinite.

The degeneracy conditions (4.88) are readily verified as follows

S,F =

⟨1

ΘFV,∇0

[δFδp

]⟩−⟨

1

ΘFV,∇0

[δFδp

]⟩= 0

[E,F ] =

⟨∇0

[1

cc

],Θ2K∇0

[1

c

δFδΘ

]⟩= 0

(4.118)


Following the same arguments as before, the evolution equations providedby the GENERIC formalism for the isolated dynamics of a thermoelasticcontinuum described in terms of the temperature are

〈wϕ, ϕ〉B0 = 〈wϕ,1

ρ0

p〉B0

〈wp, p〉B0 = −〈∇0wp,F

(2∂e

∂C−V

)〉B0 = −〈∇0wp,FS〉B0

〈wΘ, Θ〉B0 = −〈wΘ

cFV,∇0

[p

ρ0

]〉B0 − 〈∇0

[wΘ

c

],K∇0Θ〉B0 ,

(4.119)

which are the isolated version of the weak statement provided in (3.147).

Also, the laws of thermodynamics can easily be derived to prove theenergy preservation and the entropy production. Thus, the rate of the totalenergy is elaborated by choosing F = E(z), provided by (4.113)1, in theGENERIC equations (4.87) with the definition of the brackets provided by(4.116) and (4.117) to yield

E(z) = E(z), E(z)+ [E(z), S(z)]

= 〈2F∂e

∂C,∇0

[p

ρ0

]〉B0 − 〈∇0

[p

ρ0

], 2F

∂e

∂C〉B0

− 〈1ccFV,∇0

[p

ρ0

]〉B0 + 〈∇0

[p

ρ0

],FV

1

cc〉B0

+ 〈∇0

[1

cc

],Θ2K∇0

[1

c

c

Θ

]〉B0 = 0

(4.120)

Finally, the rate of the total entropy follows form choosing F = S(z),provided by (4.113)2, to give

S(z) = S(z), E(z)+ [S(z), S(z)]

= 〈 1

ΘFV,∇0

[p

ρ0

]〉B0 − 〈

1

c

c

ΘFV,∇0

[p

ρ0

]〉B0

+ 〈∇0

[1

c

c

Θ

],Θ2K∇0

[1

c

c

Θ

]〉B0 = 〈 1

Θ2∇0Θ,K∇0Θ〉B0 ≥ 0,

(4.121)

with the temperature field, in this case, being a state variable rather than afunction of the state variables as in (4.110).


4.7 GENERIC form of thermo-dissipative dynamics

Consider now the general case of the thermo-dissipative continuum dy-namics introduced in Chapter 3. This general case complements the pre-viously discussed thermoelastic case by including internal variables to de-scribe any dissipative transformation taking place in the material interior.In the same way, the entropy and the temperature formulations are elab-orated which, to the best of the author’s knowledge, have so far not beenaddressed.

4.7.1 Entropy formulation

Focusing on the entropy-based description of the problem, the infinite-dimensional state space now incorporates m strain-like internal variables ten-sors to be

S = z = (ϕ,p, η,Λα) : B0 7→ Rd × Rd × R× Sym(T2d)m, det (∇0ϕ) > 0

(4.122)

Accordingly, the tangent space becomes

TS = w = (wϕ,wp, wη,wΛα) : B0 7→ Rd ×Rd ×R× Sym(T2d)m (4.123)

Given that, the total energy of the system in terms of state vector vari-ables z is attained by using the thermodynamic potential (3.33) while thetotal entropy is provided by one of the state variables, that is

E(z) =

∫B0

(1

2ρ0

‖p‖2 + e(C, η,Λα)

)dV0, S(z) =

∫B0

ηdV0 (4.124)

As a result of the above definitions, the functional derivatives now read⟨δE

δϕ,wϕ

⟩= 〈2 ∂e

∂C,∇0ϕ

T∇0wϕ〉B0⟨δE

δp,wp

⟩= 〈 p

ρ0

,wp〉B0⟨δE

δη, wη

⟩= 〈∂e

∂η, wη〉B0⟨

δE

δΛα,wΛα

⟩= 〈 ∂e

∂Λα,wΛα〉B0

⟨δS

δϕ,wϕ

⟩= 0⟨

δS

δp,wp

⟩= 0⟨

δS

δη, wη

⟩= 〈1, wη〉B0⟨

δS

δΛα,wΛα

⟩= 0

(4.125)

148 4.7. GENERIC form of thermo-dissipative dynamics

From this result, the constitutive laws (3.36), (3.37) and (3.38) are iden-tified with

S = 2∂e

∂C, Qα = − ∂e

∂Λα, Θ =

∂e

∂η(4.126)

The GENERIC form of general thermo-dissipative continuum dynamicsbased on the entropy as a state variable is provided by the canonical Poissonbracket (4.102), related to the election of the entropy, and by the Dissipativebracket which, in addition to the conduction phenomenon, also involves thedissipation due to the presence of irreversible transformations in the contin-uum. For any two functionals F ,G : S → R, they are

F ,G =

⟨δFδϕ

,δGδp

⟩−⟨δFδp

,δGδϕ

⟩[F ,G] =

⟨∇0

[1

Θ

δFδη

],Θ2K∇0

[1

Θ

δGδη

]⟩+

⟨1

Θ

δFδη, DδG

δη

⟩+

m∑α=1

⟨δFδΛα

,K (Qα)δGδη

⟩,

(4.127)

where K is the material conductivity tensor (3.27), K is the positive semidef-inite dissipation functional and D is the dissipation provided by (3.34), alsodefined to be positive for thermodynamical soundness. As all of these quanti-ties are positive semidefinite, the Dissipative bracket itself will also be positivesemidefinite.

Remark 4.5. The last term in the Dissipative bracket breaks the symmetryof the operator. However, this fact does not seem to be crucial for it toprovide the thermodynamically consistent evolution equations.

Regarding the degeneracy conditions, the one falling on the Poissonbracket was proved in (4.104)1 whereas, using (4.125), the degeneracy condi-tion for the Dissipative bracket (4.11)2 yields

[E,F ] =

⟨∇0

[1

ΘΘ

],Θ2K∇0

[1

Θ

δFδη

]⟩+

⟨1

ΘΘ, DδF

δη

⟩+

m∑α=1

⟨−Qα,K(Qα)

δFδη

⟩,

(4.128)

which identically vanishes since the rate of dissipation is just provided by theform

D =m∑α=1

Qα : K(Qα) ≥ 0 (4.129)


The time derivative of any function defined on the sate space (4.122) nowbecomes

F =

⟨δFδϕ

, ϕ

⟩+

⟨δFδp

, p

⟩+

⟨δFδη, η

⟩+

m∑α=1

⟨δFδΛα

, Λα

⟩(4.130)

Then, the evolution equations derived from the GENERIC formalism forthe thermo-dissipative dynamics of continua described in terms of the densityentropy are

〈wϕ, ϕ〉B0 = 〈wϕ,1

ρ0

p〉B0

〈wp, p〉B0 = −〈∇0wp,FS〉B0

〈wη, η〉B0 = −〈∇0

[wηΘ

],K∇0Θ〉B0 + 〈wη

Θ, D〉B0

〈wΛα , Λα〉B0 = 〈wΛα ,K(Qα)〉B0 , α = 1, . . . ,m

(4.131)

The above set of weak equations corresponds with the one provided by(3.146) with the exception of the external actions on the continuum.

The laws of thermodynamics can easily be verified by computing the rateof the total energy system using F = E(z) in (4.87) to give

E(z) = E(z), E(z)+ [E(z), S(z)] = 0 (4.132)

and the rate of the total entropy choosing F = S(z) to give

S(z) = S(z), E(z)+ [S(z), S(z)]

= −〈∇0

[1

Θ

],K∇0Θ〉B0 + 〈 1

Θ, D〉B0

= 〈 1

Θ2∇0Θ,K∇0Θ〉B0 + 〈 1

Θ, D〉B0 ≥ 0

(4.133)

This last result fully coincides with (3.101) obtained by the classic ap-proach carried out in Chapter 3.

4.7.2 Temperature formulation

Finally, the GENERIC formalism for the thermo-dissipative continuumdynamics described by the temperature is presented. To this end, the state

150 4.7. GENERIC form of thermo-dissipative dynamics

space is defined as follows

S = z = (ϕ,p,Θ,Λα) : B0 7→ Rd×Rd×R+× Sym(T2d)m, det (∇0ϕ) > 0,

(4.134)and the tangent space is

TS = w = (wϕ,wp, wΘ,wΛα) : B0 7→ Rd×Rd×R× Sym(T2d)m (4.135)

Hence, the total energy and entropy of the system are provided by

E(z) =

∫B0

(1

2ρ0

‖p‖2 + e(C,Θ,Λα)

)dV0,

S(z) =

∫B0

η(C,Θ,Λα)dV0

(4.136)

Their functional derivatives then result in⟨δE

δϕ,wϕ

⟩= 〈2 ∂e

∂C,FT∇0wϕ〉B0⟨

δE

δp,wp

⟩= 〈 p

ρ0

,wp〉B0⟨δE

δΘ, wΘ

⟩= 〈 ∂e

∂Θ, wΘ〉B0⟨

δE

δΛα,wΛα

⟩= 〈 ∂e

∂Λα,wΛα〉B0

⟨δS

δϕ,wϕ

⟩= 〈2 ∂η


δS

δp,wp

⟩= 0⟨

δS

δΘ, wΘ

⟩= 〈 ∂η

∂Θ, wΘ〉B0⟨

δS

δΛα,wΛα

⟩= 〈 ∂η

∂Λα,wΛα〉B0

(4.137)

As usual, the above partial derivatives are linked with the symmetricPiola-Kirchhoff stress tensor S, the driving stress tensor Qα, the latent heattensor V, the visco-latent heat tensor Wα and the specific heat capacityc. These relations were revealed in Chapter 3, subsection 3.3.6, and aresummarized next

2∂e

∂C= S + 2Θ

∂η

∂C, 2

∂η

∂C=

1

ΘV,

∂e

∂Θ= Θ

∂η

∂Θ= c

∂e

∂Λα= −Qα + Θ

∂η

∂Λα,

∂η

∂Λα=

1

ΘWα

(4.138)

The Poisson bracket is not modified respect to the pure thermoelastic case(4.116), now the symmetric latent heat V tensor also depending on internal


variables. In contrast, the Dissipative bracket must add new terms to dealwith the new source of dissipation due to the internal variables. Thereby, forany two functionals F ,G : S → R, they are

F ,G =

⟨δFδϕ

,δGδp

⟩−⟨δFδp

,δGδϕ

⟩−⟨

FV1

c

δFδΘ

,∇0

[δGδp

]⟩+

⟨∇0

[δFδp

],FV

1

c

δGδΘ

⟩[F ,G] =

⟨∇0

[1

c

δFδΘ

],Θ2K∇0

[1

c

δGδΘ

]⟩+

⟨1

c

δFδΘ

, (D +Hd)Θ

c

δGδΘ

⟩+

m∑α=1

⟨δFδΛα

,K(Qα)Θ

c

δGδΘ

⟩,

(4.139)

where the term Hd is the thermodissipative heating provided by (3.82). Thesame arguments as before applies to conclude the positive semi-definitenessand the lack of symmetry of the Dissipative bracket.

Using (4.129) and (3.82), the degeneracy conditions are verified as follows

S,F =

⟨1

ΘFV,∇0

[δFδp

]⟩−⟨

1

c

c

ΘFV,∇0

[δFδp

]⟩= 0

[E,F ] =

⟨∇0

[1

cc

],Θ2K∇0

[1

c

δFδΘ

]⟩+

⟨1

cc, (D +Hd)

Θ

c

δFδΘ

⟩+

m∑α=1

⟨(−Qα + Wα),K(Qα)

Θ

c

δFδΘ

⟩= 0,

(4.140)

where use has been made of the definition of the thermodissipative heating(3.82).

Applying the GENERIC formulas, the weak form of the isolated evolutionequations for the thermo-dissipative continuum dynamics equations reads

〈wϕ, ϕ〉B0 = 〈wϕ,1

ρ0

p〉B0

〈wp, p〉B0 = −〈∇0wp,F

(2∂e

∂C−V

)〉B0 = −〈∇0wp,FS〉B0

〈wΘ, Θ〉B0 = −〈wΘ

cFV,∇0

[p

ρ0

]〉B0 − 〈∇0

[wΘ

c

],K∇0Θ〉B0

+ 〈wΘ

c, D +Hd〉B0

〈wΛα , Λα〉B0 = 〈wΛα ,K(Qα)〉B0 , α = 1, . . . ,m,

(4.141)

152 4.8. Metriplectic structure for isothermal dissipative dynamics

which are the same as those obtained in (3.147) but without consideringexternal actions.

To finish, the laws of thermodynamics are verified by taking F = E(z)in (4.87) to calculate the rate of total energy as follows

E(z) = E(z), E(z)+ [E(z), S(z)] = 0, (4.142)

and by choosing F = S(z) to arrive at the following rate of total entropy

S(z) = S(z), E(z)+ [S(z), S(z)]

= −〈∇0

[1

c

c

Θ

],K∇0Θ〉B0

+m∑α=1

〈 1

ΘWα,K(Qα)

Θ

c

c

Θ〉B0 + 〈1

c

c

Θ, (D +Hd)

Θ

c

c

Θ〉B0

= 〈 1

Θ2∇0Θ,K∇0Θ〉B0 +

m∑α=1

〈 1

Θ,Qα : K(Qα)〉B0 ≥ 0

(4.143)

4.8 Metriplectic structure for isothermal dissipative dy-namics

Isothermal dissipative systems can be approached from the metriplecticstructure perspective following Bloch et al. (1996). Therein, the authorsintroduced a metriplectic structure based on an energy function on a statespace E : S → R that decreases over time, which is what happens in thecontext of isothermal dissipative systems with the total energy resulting fromthe sum of the kinetic energy plus the total free energy function. With thispremise, the flow of a metriplectic structure is provided by

f = [f, E] = f, E+ (f, E) ∀f : S → R, (4.144)

defining the Poisson-Dissipative bracket, denoted as [·, ·], so that

E = [E,E] = E,E+ (E,E) = (E,E) ≤ 0, (4.145)

where the bracket (·, ·) : S × S 7→ R is a bilinear and negative semidefi-nite operator. The above definition is the proof for the system to developdissipation along its evolution.

In this particular metriplectic structure, the energy function E is respon-sible for generating both the reversible and irreversible flows. In fact, the


limit case of its rate E being zero corresponds with the purely Hamiltoniancase. For general positive functions E, its rate is always negative so it servesas a strong Lyapunov function. As a result, the system will surely relax to astate of equilibrium, i.e. to a surface of minimal energy.

On the other hand, by making certain physic assumptions, see Mielke(2011), isothermal dissipative dynamics can be described consistently formthe GENERIC approach, which, observe, is not a metriplectic structure ofthe format just introduced.

For the infinite-dimensional case, the above introduced metriplectic struc-ture is defined over functionals on the infinite-dimensional state space F :S → R.

Definition 4.9. An infinite-dimensional isothermal dissipative system is asystem governed by the following differential equations

F = [F , E], (4.146)

where E : S → R is the total energy of the system understood as the sumof the kinetic energy plus the total free-energy function, as was introducedin (3.96) taking into account the relation (3.40), which is to be decreasingalong the system evolution.

Thus, to derive the evolution equations of isothermal dissipative contin-uum dynamics, consider the infinite-dimensional state space

S = z = (ϕ,p,Λα) : B0 7→ Rd×Rd×Sym(T2d)m, det(∇0ϕ) > 0, (4.147)

along with its tangent space

TS = w = (wϕ,wp,wΛα) : B0 7→ Rd × Rd × Sym(T2d)m (4.148)

Then, the total energy of the system in terms of this state variables isprovided by

E(z) =

∫B0

(1

2ρ0

‖p‖2 + Ψ(C,Λα)

)dV0, (4.149)

whose functional derivative results in⟨δE

δϕ,wϕ

⟩= 〈2∂Ψ


δE

δp,wp

⟩= 〈 p

ρ0

,wp〉B0⟨δE

δΛα,wΛα

⟩= 〈 ∂Ψ

∂Λα,wΛα〉B0

(4.150)

154 4.9. Symmetries of infinite-dimensional dissipative systems

The Poisson-Dissipative bracket is then provided by

[F ,G] =

⟨δFδϕ

,δGδp

⟩−⟨δFδp

,δGδϕ

⟩+

m∑a=1

⟨δFδΛα

,K(− δGδΛα

)⟩(4.151)

The time derivative of any function defined on the sate space (4.147) nowbecomes Poisson-Dissipative

F =

⟨δFδz

, z

⟩=

⟨δFδϕ

, ϕ

⟩+

⟨δFδp

, p

⟩+

m∑α=1

⟨δFδΛα

, Λα

⟩(4.152)

Identifying the partial derivatives of (4.150) with the isothermal consti-tutive laws (3.103) and choosing δF/δz = w, the weak statement of isolatedevolution equations for the isothermal dissipative dynamics reads

〈wϕ, ϕ〉B0 = 〈wϕ,1

ρ0

p〉B0

〈wp, p〉B0 = −〈∇0wp,FS〉B0

〈wΛα , Λα〉B0 = 〈wΛα ,K(Qα)〉B0 , α = 1, . . . ,m

(4.153)

Finally, choosing F = E in (4.146) the rate of the total energy yields

E = [E,E] = −m∑α=1

〈Qα,K(Qα)〉B0 = −D ≤ 0→ E + D = 0, (4.154)

which reveals that the so called Augmented Hamiltonian (3.107) is preservedalong the isolated system evolution, as was demonstrated in Chapter 3.

Remark 4.6. The viscoelastic model due to Holzapfel & Simo (1996b) in-troduced in subsection 3.6.1 is thus attained by specifing the dissipative func-tional K according to (3.23).

4.9 Symmetries of infinite-dimensional dissipative sys-tems

As was identified in Chapter 3, the evolution equations of thermo-dissi-pative/isothermal isolated dynamics of continua pose symmetries associatedwith conserved quantities (first integrals) of the motion. That is, generalthermo-dissipative continua are dissipative system with symmetries accordingto Def. 4.7.


In Section 3.5 of Chapter 3, the first integrals of the motion were identifiedwith the total linear momentum (3.44) and the total angular momentum(3.51). Within the GENERIC approach, they are derived from the actionof certain Lie Groups that preserve the GENERIC operators according to(4.90).

Thus, the total linear momentum (3.44) is a momentum map associatedwith an additive group G = Rd whose action Φ: Rd ×S → S is defined by

Φa(z) = Φ(a, z) = (ϕ+ a,p, s,Λ1, . . . ,Λm) ∀a ∈ Rd, (4.155)

where the state space S has been chosen in the more general way. Then,the infinitesimal generator of the action is provided by

ξS =d

dε

∣∣∣∣ε=0

Φξ(ε, z) = (ξ,0, 0,0, . . . ,0), (4.156)

This action preserves all the GENERIC operators presented in previoussections. The proof for the pure thermoelastic case in terms of entropy, i.efor the operators (4.102) and (4.103), can be found in Romero (2013). Forthe rest, similar arguments can be used to verify their preservation.

Now, to demonstrate that the linear momentum is a momentum mapthat is preserved along the system evolution, a scalar functional is built asfollows

Jξ = 〈p, ξ〉B0 = L · ξ ∀ξ ∈ Rd (4.157)

whose rate can be elaborated by using the GENERIC equations (4.87) togive

Jξ = Jξ, E(z)+ [Jξ, S(z)] (4.158)

For each of the forms of the Dissipative brackets presented for each of thecases considered in previous subsections and for the last term of the Poisson-Dissipative bracket of the isothermal case (4.151), it can be concluded that

[Jξ, S(z)] = 0, (4.159)

as the scalar functional does not depend on internal variables nor on thermo-dynamical ones that are responsible for furnishing the Dissipative operators.Then, to proceed further, the Poisson brackets must be particularized foreach of the cases considered. Essentially, there are two different definitionsof them, each for the entropy formulation (the isothermal case can be in-cluded here) and for the temperature formulation. In the first case, either


(4.102) or (4.127)1 or the first two terms in (4.151), the rate of the scalarfunctional (4.157) yields

Jξ = Jξ, E(z) = −E(z), Jξ = −⟨δE

δϕ,δJξδp

⟩= −

⟨δE

δϕ, ξ

⟩(4.160)

For the temperature formulations, that is, the Poisson brackets providedby (4.116) or (4.139), the same result is obtained, that is

Jξ = −E(z), Jξ = −⟨δE

δϕ, ξ

⟩−⟨

FV1

c

δE

δΘ,∇0ξ

⟩= −

⟨δE

δϕ, ξ

⟩(4.161)

The verification is completed by noticing that the resulting term vanishesbecause the total energy of the system is invariant respect to the action Φuof the additive Group, as it depends on the motion ϕ only through the rightCauchy deformation tensor, thus concluding that the global linear momentum(3.44) is preserved for each of the systems perviously presented.

On the other hand, the total angular momentum (3.51) is a momentummap associated with the rotation group G = SO(3) = G : R3 7→ R3|GGT =1, det (G) = 1 whose action Φ: SO(3)×S → S is provided by

ΦG(z) = Φ(G, z) = (Gϕ,Gp, s,Λ1, . . . ,Λm) ∀G ∈ SO(3), (4.162)

and its infinitesimal generator of actions provided by

ξS =d

dε

∣∣∣∣ε=0

Φξ(ε, z) = (ξ ∧ϕ, ξ ∧ p, 0,0) (4.163)

This action also preserves all the GENERIC operators introduced aboveas can be seen in Romero (2013) for the pure thermoelastic case formulatedin terms of entropy (4.102) and (4.103). The proofs for the rest would closelyfollow the same guidelines.

Then, the verification for the angular momentum (3.51) to be a conservedmomentum map follows from building the following scalar functional

Jξ = 〈ϕ ∧ p, ξ〉B0 = J · ξ (4.164)

According to the GENERIC equations (4.87), its rate read

Jξ = Jξ, E(z)+ [Jξ, S(z)], (4.165)


where the second term on the right hand side vanishes for all of the casesconsidered due to the same reasons as for the linear momentum. Then, thePoisson brackets of the entropy formulations and the isothermal one, (4.102),(4.127)1 or (4.151), are used to arrive at

Jξ = Jξ, E(z) = −E(z), Jξ =

⟨δE

δϕ, ξ ∧ϕ

⟩−⟨δE

δp,p ∧ ξ

⟩(4.166)

Now, using the Poisson brackets for the temperature formulations (4.116)and (4.139), it follows that

Jξ = Jξ, E(z) = −E(z), Jξ =

⟨δE

δϕ, ξ ∧ϕ

⟩−⟨δE

δp,p ∧ ξ

⟩+

⟨FV

1

c

δE

δΘ,∇0[ξ ∧ϕ]

⟩ (4.167)

The last term of the right hand side identically vanishes because it resultsin the product of a symmetric tensor and a skew-symmetric tensor whicharises from the matrix format of the cross product ξ ∧ϕ = ξϕ, that is⟨

FV1

c

δE

δΘ,∇0

[ξϕ]⟩

=

⟨FV

1

c

δE

δΘ, ξF

⟩=

⟨1

c

δE

δΘFVFT, ξ

⟩= 0 (4.168)

Finally, due to the objectivity of the total energy functions of the systemspreviously described, the action ΦG does not modify them so the rate of thescalar function (4.164) is an invariant of any of the above systems evolutionand hence, the angular momentum (3.51) is a first integral of the motion ofthem.

ThermodynamicallyConsistent Algorithms C

hapter

5The aim of this Chapter is to present a general methodology to derive

thermodynamically consistent second-order accurate time integration meth-ods for general dissipative systems defined as in the previous Chapter. Theresulting methods not only agree with the laws of thermodynamics but alsopreserve the first integrals associated with the symmetries of the system whilebeing second-order accurate. The key ingredient of this methodology is a dis-crete derivative operator satisfying two essential properties, namely, the di-rectionality and the consistency properties. The first property is responsiblefor the preservation of the geometric structure of the systems, revealed in theprevious Chapter, whereas the second properties ensures the second-orderaccuracy.

In the same way as in the continuous modeling the metriplectic struc-ture extended the Poisson structure to include dissipative terms, this newmethodology extends the formulation to derive Energy-Momentum meth-ods for conserving systems to the formulation of Energy-Entropy-Momentummethods for dissipative systems. In fact, such extension rests on the samekey concepts that led to the formulation of Energy-Momentum methods, thussharing most of their appealing features.

160 5.1. Energy-Entropy-Momentum time integration methods

5.1 Energy-Entropy-Momentum time integration meth-ods

For concreteness, Energy-Entropy-Momentum (EEM) integration meth-ods are those derived from the fairly general methodology devised in thework of Romero (2009, 2010a,b), which provides discrete counterparts of theGENERIC equations in such a way that their intrinsic metriplectic structureis preserved. Hence, counterparts thus constructed are automatically ther-modynamically consistent and momentum maps-preserving, that is, Energy-preserving, Entropy-producing and Momentum-preserving by design, hencetheir names: Energy-Entropy-Momentum methods.

The methodology basically follows the guidelines that led to the generalformulation of Energy-Momentum methods. In short, it consists in a tem-poral approximation of the GENERIC equations defining general dissipativesystems of mid-point type (2.75) combined with a discrete derivative opera-tor satisfying two essential properties: directionality and consistency. As aresult of these two key elements in the formulation, the resulting time integra-tion methods are canonically structure-preserving and second order accurate.Among all possible candidates for the discrete derivative operator studied inRomero (2012), the one proposed in Gonzalez (1996) for the formulation ofEnergy-Momentum methods presents advantages from the implementationpoint of view, thus being the optimal candidate to develop Energy-Entropy-Momentum methods.

This general methodology can adopt several approaches mainly regard-ing the strategy to deal with thermo-coupled problems, which might affectthe order of accuracy of resulting methods, and regarding the thermody-namical state variable used to described the system’ thermodynamics. Thislast alternative has been recently open after the investigations carried outfor the composition of this dissertation that have enabled the formulationof temperature-based EEM methods as an alternative of the so far favoredentropy-based ones due to its apparent simplicity.

Thus, motivated by the natural partition of coupling problems into amechanical and a thermal problems, their solutions are commonly obtainedwith staggered strategies, so that advantages of the different time scales in-volved in each problem can be exploited, see for instance Holzapfel & Simo(1996a). This strategy is further backed by the metriplectic structure ofdissipative systems which clearly separates the reversible (isentropic) andthe irreversible (entropic) contributions to the system evolution. In fact,Romero (2009, 2010b) proved that using the entropy as state variable, stag-

5. Thermodynamically Consistent Algorithms 161

gered methods in which each step remains thermodynamically consistent canbe formulated within the proposed methodology. However, the second orderaccuracy cannot be retained.

Alternatively, the methodology can be performed by adopting a mono-lithic strategy, that this, solving the coupled equations at once. With this ap-proach, in contrast to the staggered strategy, second order accurate methodsare automatically achieved and the choice of the thermodynamical variable istheoretically not restricted. In this dissertation, the scope is restricted to thismonolithic second-order accurate approach considering the most significantthermodynamical state variables: entropy or temperature; so that the stabil-ity, the accuracy and the conservation properties of the both approaches canbe thoroughly discussed.

For methods with higher order of accuracy, sub-stepping procedures wereproposed by Tarnow & Simo (1994) to raise the order of accuracy whilepreserving the stability and conserving properties. However, the presenceof a ‘Dissipative bracket’ in the problems of interest seems to lead methodsformulated backward in time to be ill-posed, thus preventing an effectiveincrement of the order of accuracy.

5.2 General formulation of EEM methods

For any finite-dimensional system, either of the form of Def. 4.2 or re-sulting from the spatial discretization of infinite-dimensional systems in thesense of Def. 4.6, for instance, using the Galerkin approach summarizedin subsection 3.7.2, consider the classical partition of the time integrationinterval [0, T ] into constant subintervals of length ∆t = tn+1 − tn. Then,the state vector z ∈ S ,S h at certain instants of time ti is approximatedas zi ' z(ti) so that the construction of EEM counterparts rests on thefollowing definitions.

Definition 5.1. For any subinterval [tn, tn+1], the time derivative of thevector state z is approximated by the following second-order accurate formula

zn+ 12

=zn+1 − zn

∆t, ∀[tn, tn+1] (5.1)

Definition 5.2. Discrete derivative operator. This operator is a second orderapproximation of the standard derivative operator evaluated at midpoint ofthe time interval. According to Gonzalez (1996), for any smooth functiondefined in an inner product space (S , 〈·, ·〉S ), f : S → R, this second-order

162 5.2. General formulation of EEM methods

operator D : S ×S 7→ TS is provided by

Df(zn+1, zn) = Df(zn+ 12)+

f(zn+1)− f(zn)− 〈Df(zn+ 12),∆z〉S

〈∆z,∆z〉S∆z, (5.2)

where zn+1, zn ∈ S , zn+ 12

= (zn+1 + zn)/2, ∆z = zn+1 − zn and D1 beingstandard derivative.

For state spaces with structure of products S = L1 × · · · × Lk for somek ≥ 1, with every Li, i = 1, . . . , k endowed with an inner product 〈·, ·〉Li , asin the systems considered, the discrete derivative operator with respect tothe product structure of S , for any smooth function f : S → R, is providedby

〈Df(zn+1, zn),∆z〉S =k∑i=1

〈Dzif(zn+1, zn),∆zi〉Li , (5.3)

where the term Dzif(zn+1, zn) is the i-th second order accurate discrete coun-terpart to the i-th partial derivative of the function f at midpoint; for moredetails about its computation consult Appendix A.

These definitions of the discrete derivative operator satisfy the requiredproperties of directionality and consistency, that is

〈Df(zn+1, zn), zn+1 − zn〉S = f(zn+1)− f(zn), (5.4)

Df(zn+1, zn) = Df(zn+ 12) + O(‖zn+1 − zn‖2) (5.5)

The first one holds the key to preserve discretely the evolution structureidentified in continuous systems. The second one ensures the second orderaccuracy. The proofs for the operator to satisfy these properties can be alsofound in Appendix A.

Associated with functions being invariant to the action Φ of a Lie GroupG with algebra g and dual g∗, whose infinitesimal generator of actions is de-noted as ξS (z), i.e G-invariant functions, the G-equivariant discrete deriva-tive operator is introduced.

Definition 5.3. The G-equivariant discrete derivative operator is a discretederivative operator operator DG : S ×S 7→ TS in the sense of (5.2) thatsatisfies two additional properties:

1notice the different notations used for the standard derivative D and for the discretederivative D.


1. Equivariance

DGf(Φg(zn+1),Φg(zn)) =[DΦg(zn+ 1

2)]−T

DGf(zn+1, zn), (5.6)

for all g ∈ G and zn+1, zn ∈ S .

2. Orthogonality⟨DGf(zn+1, zn), ξS (zn+ 1

2)⟩

S= 0 ∀ξ ∈ g∗, (5.7)

for all zn+1, zn ∈ S .

Proposition 5.1. Let f : S → R be a smooth G-invariant function, π : S →Rr be a set of r functions πi : S → R which are invariant respect to an actionΦ of a Lie group G. Let f : π(S ) → R be the associated reduced functiondefined by f(π(z)) = f(z), for all z. If the invariant πi : S → R is at mostof degree two, then a G-equivariant discrete derivative for f is provided by

DGf(zn+1, zn) =[Dπ(zn+ 1

2)]T

Df(π(zn+1), π(zn)), (5.8)

where for any z ∈ S , Dπ(z) ∈ Rr×dim(S ) and r = dim(S )− o, o being thenumber of orbits associated with the Lie group G.

These properties become important to define discrete counterparts ofdissipative systems with symmetries and are verified in Appendix A.

5.2.1 Discrete finite-dimensional smooth dissipative systems

In this subsection, the previous definitions are employed to define dis-crete finite-dimensional smooth dissipative systems with symmetries, thusproviding the general form of EEM methods for finite-dimensional dissipa-tive systems provided by Def. 4.2.

Definition 5.4. A discrete finite-dimensional smooth dissipative system withsymmetries is a discrete system governed by the following discrete GENERICequations

zn+1 − zn∆t

= L(zn+1, zn)DGE(zn+1, zn) + M(zn+1, zn)DGS(zn+1, zn), (5.9)

where L(zn+1, zn) and M(zn+1, zn) are second order approximations of thecontinuous Poisson and Dissipative matrices evaluated at midpoint

L(zn+1, zn) = L(zn+ 12) + O(zn+1 − zn)2,

M(zn+1, zn) = M(zn+ 12) + O(zn+1 − zn)2

(5.10)


In analogy to the continuous case, the discrete versions of the Poissonand Dissipative matrices must satisfy the degeneracy or non-interaction con-ditions in the following form

L(zn+1, zn)DGS(zn+1, zn) = 0, M(zn+1, zn)DGE(zn+1, zn) = 0 (5.11)

Discrete laws of thermodynamics. The satisfaction of the laws of ther-modynamics by the discrete setting (5.9) directly follows from directionalityproperty (5.4) to give

En+1 − En = DGE (zn+1, zn) · (zn+1 − zn)

= ∆tDGE(zn+1, zn) · L(zn+1, zn)DGE(zn+1, zn)

+ ∆tDGE(zn+1, zn) ·M(zn+1, zn)DGS(zn+1, zn)

= ∆tDGS(zn+1, zn) ·M(zn+1, zn)DGE(zn+1, zn) = 0

(5.12)

This result can be directly deduced from the skew-symmetry of the dis-crete Poisson matrix together with the non-interaction conditions (5.11).Similarly, the total entropy balance results in

Sn+1 − Sn = DGS (zn+1, zn) · (zn+1 − zn)

= ∆tDGS(zn+1, zn) · L(zn+1, zn)DGE(zn+1, zn)

+ ∆tDGS(zn+1, zn) ·M(zn+1, zn)DGS(zn+1, zn)

= −∆tDGE(zn+1, zn) · L(zn+1, zn)DGS(zn+1, zn)

+ ∆tDGS(zn+1, zn) ·M(zn+1, zn)DGS(zn+1, zn) ≥ 0

(5.13)

Accordingly, the total entropy is always non-decreasing because of thesymmetry and positive semi-definiteness of the discrete Dissipative matrixtogether with the non-interaction conditions.

Discrete symmetries. As discrete dissipative systems have been definedvia the G-equivariant discrete derivative operator of the G-invariant totalenergy function E : S 7→ R and total entropy function S : S 7→ R, theirequations (5.9) pose symmetries with which discrete counterparts J of thecontinuous momentum maps J are associated. Thus, a mapping J : S 7→ g∗

is a discrete momentum maps if the scalar function Jξ : S 7→ R defined as

Jξ(z) = J(z) · ξ, (5.14)

for any ξ ∈ g, satisfies

ξS (zn+ 12) = L(zn+1, zn)DGJξ(zn+1, zn),

0 = M(zn+1, zn)DGJξ(zn+1, zn)(5.15)


Theorem 5.2. The momentum maps of a discrete dissipative system withsymmetries (S ,L,M,DG, E, S) are conserved quantities of the solution ofthe discrete system (5.9).

Proof. The directionality property along with the discrete equations (5.9)lead the balance of the scalar function (5.14) in any time subintervals to be

Jξ,n+1−Jξ,n = DGJξ ·(zn+1−zn) = ∆t(DGJξ ·LDGE+DGJξ ·MDGS), (5.16)

that, by using (5.15), simplifies to

Jξ,n+1 − Jξ,n = −∆tDGE · ξS (zn+ 12) = 0, (5.17)

which vanishes due to the orthogonality property of the G-equivariant dis-crete derivative operator (5.7), demonstrating that the mapping J is con-served along the discrete system evolution.

5.2.2 Discrete infinite-dimensional dissipative systems

The formulation of EEM methods for the infinite-dimensional dissipativesystems, defined in Chapter 4, encompasses two semi-discretizations: a firstone standard in space and, then, a non-standard in time based on the discretederivative concept.

Thus, any discretization technique commented in Section 3.7 that doesnot spoil its continuous evolution properties is considered standard and hencemight perfectly be used to derive EEM methods. So far, the EEM formalismhas been fully developed by performing a standard Galerkin FE-based spatialdiscretization and hence this approach will be assumed to propose EEMmethods for the infinite-dimensional systems discussed in previous Chapters.

Then, as in Section 3.7, the infinite-dimensional state spaces definingdissipative systems are approximated by finite-dimensional subsets S h ⊂ Sprovided by the linear combination of shape functions typically defined in aregular partition of the infinite-dimensional domain Bh

0 ⊂ B0 such that

S h = zh =N∑a=1

Naza|zh ∈H 1, zh = zh on ∂Bh,ϕ,θ0 × [0, T ], (5.18)

Similarly, the tangent space is also assumed to be approximated by thesame subset such that

TS h = wh =N∑a=1

Nawa|wh ∈H 1,wh = 0 on ∂Bh,ϕ,θ0 × [0, T ], (5.19)


Once the standard spatial discretization has been performed, the follow-ing definition leads to the formulation of EEM methods.

Definition 5.5. A discrete infinite-dimensional smooth dissipative systemwith symmetries is a discrete system governed by the consistent discreteGENERIC equations

F(zhn+1)−F(zhn)

∆t= F , E

(zhn+1, z

hn

)+ [[F , S]]

(zhn+1, z

hn

), (5.20)

for all F : S h 7→ R and E : S h 7→ R being the total energy function andS : S h 7→ R being the total entropy function.

The operator ·, · : S h×S h 7→ R is a consistent second-order discretecounterpart of the bilinear Poisson operator satisfying the skew-symmetry,Jacobi and Leibniz properties, whereas the operator [[·, ·]] : S h × S h 7→R is similarly a consistent second-order discrete counterpart to the bilinearDissipative operator being symmetric and positive-definite.

Proposition 5.3. The solution of discrete infinite-dimensional dissipativesystems (5.20) agrees with the discrete form of the laws of thermodynamicsprovided by

E(zhn+1) = E(zhn), S(zhn+1) ≥ S(zhn), (5.21)

if the discrete bilinear operators additionally satisfy the following discretedegeneracy or non-interaction conditions

S,F = [[E,F ]] = 0, (5.22)

for every functional F : S h → R.

Proof. The proof relies on the discrete brackets properties along with the justintroduced discrete degeneracy conditions. Then, choosing F(zh) = E(zh)in (5.20) leads to

E(zhn+1)− E(zhn) = ∆t[E,E

(zhn+1, z

hn

)+ [[E, S]]

(zhn+1, z

hn

)]= 0(5.23)

Similarly, choosing F(zh) = S(zh) leads to

S(zhn+1)−S(zhn) = ∆t[S,E

(zhn+1, z

hn

)+ [[S, S]]

(zhn+1, z

hn

)]≥ 0 (5.24)


The introduced discrete brackets derive from the continuous ones (4.89)

through the G-equivariant discrete functional derivative operator ∆GF∆z

: S h×S h 7→ TS h, also introduced by Gonzalez (1996) for the formulation ofEnergy-momentum methods for infinite-dimensional Hamiltonian problems,and discrete differential operators L,M : S h ×S h × TS h 7→ TS h satisfy-ing the same properties as their continuous counterparts, so that, given twofunctional F ,G : S h 7→ R, they can generally be expressed by

F ,G(zhn+1, z

hn

)=

⟨∆GF∆z

(zhn+1, z

hn

),L

[∆GG∆z

(zhn+1, z

hn

)]⟩(5.25)

[[F ,G]](zhn+1, z

hn

)=

⟨∆GF∆z

(zhn+1, z

hn

),M

[∆GG∆z

(zhn+1, z

hn

)]⟩(5.26)

The G-equivariant discrete functional derivative is then a second-orderapproximation of the functional derivative δF

δzat the midpoint that possesses

the crucial directionality property⟨∆GF∆z

(zhn+1, z

hn

), zhn+1 − zhn

⟩= F(zhn+1)−F(zhn) (5.27)

Accordingly, the left hand side of (5.20) can be expressed in terms of theDef. 5.1, that is

F(zhn+1)−F(zhn)

∆t=

⟨∆GF∆z

(zhn+1, z

hn

),zhn+1 − zhn

∆t

⟩(5.28)

On the other hand, for the types of functionals involved in the definitionof infinite-dimensional dissipative systems in the sense of Def. 4.6, that is

F(zh) =

∫Bh

0

f(zh)dV0, (5.29)

theG-equivariant discrete functional derivative is linked with theG-equivariantdiscrete derivative (5.8) by⟨

∆GF∆z

(zhn+1, z

hn

),V

⟩= 〈DGf

(zhn+1, z

hn

),V 〉Bh

0, ∀V ∈ TS h (5.30)

In short, for any element of the tangent space wh ∈ TS h, the generalform of Galerkin-based EEM methods for infinite-dimensional dissipative sys-


tems is provided by the following weak statement

⟨wh,

zhn+1 − zhn∆t

⟩=

⟨wh,L

[∆GE

∆z

(zhn+1, z

hn

)]⟩+

⟨wh,M

[∆GS

∆z

(zhn+1, z

hn

)]⟩ (5.31)

Discrete laws of thermodynamics. In Garlerkin-type formulations thepreservation of the continuous structure ultimately relies on the fact thatthe discrete derivatives in the sense of (5.2) belong to the Galerkin-basedspace TS h since the proof for the total energy and entropy to be discretelypreserved follows from replacing the weighting functions by them. Thus,choosing wh = ∆GE

∆zin (5.31) and using its directionality property, the energy

balance gives

E(zhn+1)− E(zhn)

∆t=

⟨∆GE

∆z

(zhn+1, z

hn

),L

[∆GE

∆z

(zhn+1, z

hn

)]⟩+

⟨∆GE

∆z

(zhn+1, z

hn

),M

[∆GS

∆z

(zhn+1, z

hn

)]⟩= 0,

(5.32)

which vanishes due to the skew-symmetry of the discrete Poisson operatorand the degeneracy conditions.

Similarly, choosing wh = ∆GS∆z

in (5.31) and using its directionality prop-erty, the entropy balance yields

S(zhn+1)− S(zhn)

∆t=

⟨∆GS

∆z

(zhn+1, z

hn

),L

[∆GE

∆z

(zhn+1, z

hn

)]⟩+

⟨∆GS

∆z

(zhn+1, z

hn

),M

[∆GS

∆z

(zhn+1, z

hn

)]⟩≥ 0

(5.33)

Further details on this issue can be found in Romero (2010a).

Discrete symmetries. Similarly to the finite-dimensional case, the defini-tion of the discrete dissipative systems via the G-equivariant discrete deriva-tive operator of the G-invariant total energy function E : S h 7→ R and totalentropy function S : S h 7→ R leads its discrete equations (5.20) to pose sym-metries. Hence, discrete momentum maps J : S h 7→ g∗, corresponding with


those continuous momentum maps J , are associated satisfying

ξS (zhn+ 1

2) = L(zhn+1, z

hn)

[∆GJξ∆z

(zhn+1, zhn)

],

0 = M(zhn+1, zhn)

[∆GJξ∆z

(zhn+1, zhn)

] (5.34)

for any the scalar function Jξ : S h 7→ R defined as

Jξ(zh) = J(zh) · ξ, ∀ξ ∈ g (5.35)

Theorem 5.4. The momentum maps of a discrete infinite-dimensional dis-sipative system with symmetries are conserved quantities of the solution ofthe discrete system (5.20).

Proof. The directionality property along with the discrete equations (5.9)lead the balance of the scalar function (5.14) in any time subintervals to be

Jξ,n+1 − Jξ,n =

⟨∆GJξ∆z

, zhn+1 − zhn

⟩

= ∆t

⟨∆GJξ∆z

,L

[∆GE

∆z

]⟩+ ∆t

⟨∆GJξ∆z

,M

[∆GS

∆z

]⟩,

(5.36)

that, by using (5.34) along with (5.30), simplifies to

Jξ,n+1 − Jξ,n = −∆t

⟨∆GE

∆z

(zhn+1, z

hn

), ξS (zh

n+ 12)

⟩= 0, (5.37)

which vanishes due to the orthogonality property of the G-equivariant dis-crete derivative operator (5.7), demonstrating that the mapping J is con-served along the discrete system evolution.

5.3 EEM methods for the two thermo-spring system

Based on the general format provided by (5.9), EEM methods are at-tained for the two thermo-spring system based on both the entropy and thetemperature descriptions.

170 5.3. EEM methods for the two thermo-spring system

5.3.1 Entropy-based EEM method

Departing from the finite-dimensional state space (4.27) and the totalenergy (4.28) and entropy (4.29) defined on it, the G-equivariant discretederivative operator applied to them can be elaborated by using (5.8) to give

DGE (zn+1, zn) =[Dπ(zn+ 1

2)]T

DE(π(zn+1), π(zn))

DGS (zn+1, zn) =[Dπ(zn+ 1

2)]T

DS(π(zn+1), π(zn)),

(5.38)

where π is an invariant function respect to the action of Lie groups that isat most quadratic on its argument according to Prop. 5.1.

In subsection 4.3.3, the two thermo-spring system was proved to be sym-metric respect to the rotation group SO(3) and associated with this symme-try, a momentum map, identified with the total angular momentum (3.51),was proved to be a conserved quantity of the system evolution according toNoether’s theorem. As the state space (4.27) results in the R10 space, nineinvariants respect to the said group are identified to be

π1 = λ21, π2 = λ2

2, π3 = ‖p1‖2

π4 = ‖p2‖2, π5 = η1, π6 = η2

π7 = q1 · p1, π8 = q2 · p2, π9 = q1 · q2

(5.39)

The proof for their invariability respect to the SO(3) group closely followsthe arguments used in (4.45). On the other hand, they are crucially linear,bilinear or quadratic functions of the state vector and hence invariants in thesense of Prop. 5.1.

Remark 5.1. The spring elongations λi are also invariants respect to theaction of the SO(3) as proved (4.45), however, their definitions are not linear,bilinear or quadratic functions of the state variables and hence cannot be usedas invariant functions according to Prop. 5.1.

Then, collecting all these invariant functions in a vector-valued func-tion π(z) = (π1, π2, π3, π4, π5, π6, π7, π8, π9)T, its gradient respect to the state


space reads

Dπ(z) =

∂λ21

∂q1

0 0 0 0 0

∂λ22

∂q1

∂λ22

∂q2

0 0 0 0

0 0 2p1 0 0 0

0 0 0 2p2 0 0

0 0 0 0 1 0

0 0 0 0 0 1

p1 0 q1 0 0 0

0 p2 0 q2 0 0

q2 q1 0 0 0 0

, (5.40)

where the partial derivatives of the elongations are provided by

∂λ21

∂q1

= 2q1,∂λ2

2

∂q1

= 2(q1 − q2),∂λ2

2

∂q2

= 2(q2 − q1), (5.41)

Furthermore, the total energy (4.28) can be expressed through these in-variants, thus obtaining the reduced total energy function to be

E(π(z)) = K(π3, π4) +2∑

a=1

ea(πa, πa+4) =2∑

a=1

(πa+2

2ma

+ ea(πa, πa+4)

)(5.42)

For its part, the reduced total entropy does not change as just dependon two of the invariants, that is

S(π(z)) = π5 + π6, (5.43)

Accordingly, the partitioned definition of the discrete derivative operator(5.3) is applied to obtain the discrete derivative of them

DE (πn+1, πn) =

Dλ21e1

Dλ22e2

(2m1)−1

(2m2)−1

Dη1 e1

Dη2 e2

0

0

0

, DS (πn+1, πn) =

0

0

0

0

1

1

0

0

0

, (5.44)


Each of the previous partitioned discrete derivatives operators applies toscalar-valued functions defined on a Cartesian product space. As an example,the partial discrete derivative of the internal energy function in terms of thefirst or the second invariants πa = λ2

a is provided by

Dλ2aea =

ea(λ2a,n+1, ηa,n+1)− ea(λ2

a,n, ηa,n+1) + ea(λ2a,n+1, ηa,n)− ea(λ2

a,n, ηa,n)

2(λ2a,n+1 − λ2

a,n)(5.45)

Due to the consistency property of the discrete derivative operator (5.5),the above expression is well-defined when λ2

a,n+1 → λ2a,n, particularly, it be-

comes

limλ2a,n+1→λ2a,n

Dλ2aea =

1

2

(∂ea∂λ2

a

(λ2a,n, ηa,n+1) +

∂ea∂λ2

a

(λ2a,n, ηa,n)

)(5.46)

Accordingly, the partial discrete derivative respect to the fifth or the sixthinvariants, that is, the spring entropies, is provided by

Dηa ea =ea(λ

2a,n+1, ηa,n+1)− ea(λ2

a,n+1, ηa,n) + ea(λ2a,n, ηa,n+1)− ea(λ2

a,n, ηa,n)

2(ηa,n+1 − ηa,n)(5.47)

Similarly, in the limit case of ηa,n+1 → ηa,n, the above partial derivativebecomes

limηa,n+1→ηa,n

Dηa ea =1

2

(∂ea∂ηa

(λ2a,n+1, ηa,n) +

∂ea∂ηa

(λ2a,n, ηa,n)

)(5.48)

The proofs of the limit cases and the general format of the partial discretederivatives for these types of functions can be found in Appendix A.

Furthermore, the definition of the discrete derivative operator (5.2) ap-plied to linear or quadratic functions simply coincides with the standardderivative evaluated at midpoint. This result has been used to obtain thethird and fourth components of the discrete derivative vector of the reducedtotal energy (5.42) as well as the fifth and sixth components of the discretederivative vector of the reduced total entropy (5.43).

Now, theG-equivariant discrete derivative of the total energy and entropy


are elaborated by using (5.38) with (5.40) and (5.44) to give

DGE =

2∑a=1

Dλ2aea

(∂λ2

a

∂q1

)n+ 1

2

Dλ22e2

(∂λ2

2

∂q2

)n+ 1

2

p1,n+ 12/m1

p2,n+ 12/m2

Dη1 e1

Dη2 e2

, DGS =

0

0

0

0

1

1

, (5.49)

According to the just obtained discrete derivatives, the discrete coun-terpart of the continuous Poisson matrix is just the continuous matrix itself(4.31), because it is constant, provided by

L(zn+1, zn) = L(z) =

0 0 1 0 0 0

0 0 0 1 0 0

−1 0 0 0 0 0

0 −1 0 0 0 0

0T 0T 0T 0T 0 0

0T 0T 0T 0T 0 0

(5.50)

In addition, the discrete counterpart of the continuous Dissipative matrix(4.32) are defined from the previous discrete derivatives to be

M(zn+1, zn) = k

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0T 0T 0T 0T Dη2 e2

Dη1 e1

−1

0T 0T 0T 0T −1Dη1 e1

Dη2 e2

(5.51)

In view of the continuous Dissipative matrix (4.32), the resulting par-titioned discrete derivatives in its discrete version are identified with thealgorithmic temperatures θ∗1 and θ∗2 according to

θ∗a = Dηa ea, (5.52)


which are a second-order approximation of the continuous one (4.33).

Finally, inserting relationships (5.67) through (5.52) into the discrete GE-NERIC form (5.9), the following representation of the entropy-based EEMmethod for the two thermo-spring system is achieved

qa,n+1 − qa,n∆t

=pa,n+ 1

2

ma

pa,n+1 − pa,n∆t

= −2∑b=1

Dλ2beb

(∂λ2

b

∂qa

)n+ 1

2

η1,n+1 − η1,n

∆t= k

(θ∗2θ∗1− 1

)η2,n+1 − η2,n

∆t= k

(θ∗1θ∗2− 1

)(5.53)

The set of discrete equations (5.53) is clearly recognized as a discretecounterpart of the continuous ones provided by (2.24) or (2.25).

Discrete laws of thermodynamics. The solution of the above discreteequations agrees with the laws of thermodynamics due to the the propertiesof the discrete Poisson (5.50) and Dissipative (5.51) matrices along withthe definition of discrete dissipative systems. Then, the energy conservationdirectly follows from (5.12) whereas the entropy production can be elaboratedby using (5.13) to give

Sn+1 − Sn = ∆tk(θ∗2 − θ∗1)2

θ∗1θ∗2

≥ 0, (5.54)

Discrete symmetries. Due to the use of the G-equivariant discrete deriva-tive operator definition based on the observation for the system to be sym-metric respect to the SO(3) group, the total angular momentum (3.51) isalso discretely preserved. The proof closely follows the one performed in thecontinuous case (4.44) - (4.53), now using the discrete counterparts of thePoisson and Dissipative matrices and of the standard derivatives given by(5.50), (5.51) and (5.67) following (5.14) - (5.15).

Linearization of discrete evolution equations. The resulting entropy-based EEM method is an implicit scheme and, as every implicit scheme,it requires a linearization of the equations in order to employ a Newton-Rapshon’s strategy to obtain the roots of the residue at machine precision.


Then, the classical tangent matrix of the method has the format

K =

Kqaqb Kqapb Kqaηb

Kpaqb Kpapb Kpaηb

Kηaqb Kηapb Kηaηb

a, b = 1, 2 (5.55)

Then, each of the blocks is elaborated next. Thus, the linearization ofthe first two equation respect to the positions and momenta yields

Kqaqb =1

∆t1d, Kqapb =

1

2ma

1d, if a = b otherwise Kqaqb = Kqapb = 0

(5.56)

The remaining linearization of these equations vanishes because they donot depend on the spring entropies

Kqaηb = 0 a, b = 1, 2 (5.57)

Then, the linearization of the linear momentum balance respect to posi-tions gives

Kpaqb =2∑c=1

[∂Dλ2c

ec

∂λ2c,n+1

(∂λ2

c

∂qa

)n+ 1

2

⊗(∂λ2

c

∂qb

)n+1

+1

2Dλ2c

ec

(∂2λ2

c

∂qa∂qb

)n+1

],

(5.58)

Now linearizing them respect to the momenta leads to

Kpapb =1

∆t1d if a = b otherwise Kpapb = 0, (5.59)

whereas the linearization in terms of springs entropies yields

Kpaηb =2∑c=1

∂Dλ2cec

∂ηb,n+1

(∂λ2

c

∂qa

)n+ 1

2

, a, b = 1, 2 (5.60)

Finally, the last blocks due to the linearization of the discrete energybalance equations result in

Kηaqb =∂κa

∂λb,n+1

(∂λb∂qb

)n+1

, a, b = 1, 2

Kηaηa =1

∆t+

∂κa∂ηa,n+1

, Kηaηb =∂κa

∂ηb,n+1

, if a 6= b, a, b = 1, 2

(5.61)


where the scalars κa have been introduced as

κ1 = k

(θ∗2θ∗1− 1

), κ2 = k

(θ∗1θ∗2− 1

)(5.62)

Since they do not depend on momenta, the corresponding blocks vanish,that is Kηapb = 0.

In the above expressions, the linearizations of the partial discrete deriva-tive follows from the format provided by (5.45) or can be derived from thegeneral linearized discrete derivative provided in Appendix A. Similarly, thesecond partial derivative of the elongations respect to positions vectors fol-lows from (5.41).

5.3.2 Temperature-based EEM method

Similarly, focusing on the finite-dimensional state space (4.34) and itscorresponding total energy (4.35) and entropy (4.36), their G-equivariantdiscrete derivatives are elaborated by identifying the nine invariants to be

π1 = λ21, π2 = λ2

2, π3 = ‖p1‖2

π4 = ‖p2‖2, π5 = θ1, π6 = θ2

π7 = q1 · p1, π8 = q2 · p2, π9 = q1 · q2,

(5.63)

whose standard derivative coincides with (5.40). Then, the reduced totalenergy is provided by

E(π(z)) = K(π3, π4) +2∑

a=1

ea(πa, πa+4) =2∑

a=1

(πa+2

2ma

+ ea(πa, πa+4)

),

(5.64)and the reduced total entropy by

S(π(z)) =2∑

a=1

ηa(πa, πa+4), (5.65)

Accordingly, their discrete derivatives directly follow from the partitioned


definition of the discrete derivative operator (5.3) to give

DE (πn+1, πn) =

Dλ21e1

Dλ22e2

(2m1)−1

(2m2)−1

Dθ1 e1

Dθ2 e2

0

0

0

, DS (πn+1, πn) =

Dλ21η1

Dλ22η2

0

0

Dθ1 η1

Dθ2 η2

0

0

0

(5.66)

Now performing the matrix multiplication of (5.40) by (5.66) accordingto (5.38), the G-equivariant discrete derivatives of the total energy and thetotal entropy reads

DGE =

2∑a=1

Dλ2aea

(∂λ2

a

∂q1

)n+ 1

2

Dλ22e2

(∂λ2

2

∂q2

)n+ 1

2

p1,n+ 12/m1

p2,n+ 12/m2

Dθ1 e1

Dθ2 e2

, DGS =

2∑a=1

Dλ2aηa

(∂λ2

a

∂q1

)n+ 1

2

Dλ22η2

(∂λ2

2

∂q2

)n+ 1

2

0

0

Dθ1 η1

Dθ2 η2

,

(5.67)

In agreement with these discrete gradients, the discrete counterpart ofthe Poisson matrix is provided by

L(zn+1, zn) =

0 0 1 0 0 0

0 0 0 1 0 0

−1 0 0 0 L11 L12

0 −1 0 0 0 L22

0T 0T L11 0T 0 0

0T 0T L21 L22 0 0

, (5.68)

with

Lab = −LTba = (Dθb ηb)

−1 Dλ2aηb

(∂λ2

b

∂qa

)n+ 1

2

(5.69)


Similarly, the discrete counterpart of the Dissipative matrix reads

M(zn+1, zn) = kθ∗1θ∗2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0T 0T 0T 0T M11 M12

0T 0T 0T 0T M12 M22

, (5.70)

withMab = (−1)a+b(Dθa ea)

−1(Dθb eb)−1 (5.71)

It can be easily verified that definitions (5.68) and (5.70) do satisfy thecrucial structural properties stated by (5.11). In particular, the conditions forthe discrete Dissipation matrix (5.70) hold independent of the specific choicefor the algorithmic temperatures θ∗1 and θ∗2. As a result, according to (5.9)-(5.13), any evaluation of them within the time interval would automaticallydefine a discrete dissipative system counterpart and, hence, a second-orderaccurate monolithic EEM method.

However, among all possible candidates, one choice appears to be morephysical. Unexpectedly, it is revealed by the discrete Poisson matrix (5.68)when is used to generate the right hand side of the discrete counterpart ofthe linear momentum balance. That is, multiplying its third and fourth rowsby the G-equivariant discrete derivative of the total energy (5.66) to give

6∑j=1

L(a+2)jDGEj = −

2∑b=1

(Dλ2b

eb −Dθb ebDθb ηb

Dλ2bηb

)(∂λ2

b

∂qa

)n+ 1

2

, a = 1, 2

(5.72)

Then, identifying each of the terms of this discrete counterpart with theones in the right hand side of the continuous equations (4.42)2, the appearingratios in the above expression must be counterparts of the spring tempera-tures.

With this reasoning, the algorithmic temperatures are chosen to be

θ∗a =Dθa eaDθa ηa

(5.73)

Remark 5.2. The above algorithmic temperature can be viewed as a canon-ical choice that provides an unique discrete counterpart of the specific heatcapacity (3.78), that is

ca = Dθa ea = θ∗aDθa ηa > 0, (5.74)


which could not otherwise be achieved by any other definition, provided thatuse is made of the discrete derivative in the sense of Gonzalez (1996).

Then, the partial discrete derivative operator respect to temperature ap-plied to both the reduced internal energy and reduced entropy functionsfollows the format provided in (5.47) replacing entropies with temperatures,so the algorithmic temperature is provided by

θ∗a =ea(λ

2a,n+1, θa,n+1)− ea(λ2

a,n+1, θa,n) + ea(λ2a,n, θa,n+1)− ea(λ2

a,n, θa,n)

ηa(λ2a,n+1, θa,n+1)− ηa(λ2

a,n+1, θa,n) + ηa(λ2a,n, θa,n+1)− ηa(λ2

a,n, θa,n),

(5.75)which, in the limit case of θa,n+1 → θa,n, collapses to

limθa,n+1→θa,n

θ∗a =

∂ea∂θa

(λ2a,n, θa,n) +

∂ea∂θa

(λ2a,n, θa,n)

∂ηa∂θa

(λ2a,n+1, θa,n) +

∂ηa∂θa

(λ2a,n, θa,n)

(5.76)

Finally, inserting relationships (5.66) through (5.73) into the discreteGENERIC form (5.9), the following representation of the temperature-basedEEM method is achieved

qa,n+1 − qa,n∆t

=pa,n+ 1

2

ma

pa,n+1 − pa,n∆t

= −2∑b=1

(Dλ2b

eb − θ∗bDλ2bηb

)(∂λ2b

∂qa

)n+ 1

2

θ1,n+1 − θ1,n

∆t= (Dθ1 e1)−1

[−θ∗1Dλ21

η1

(∂λ2

1

∂q1

)n+ 1

2

·p1,n+ 1

2

m1

+ h∗]

θ2,n+1 − θ2,n

∆t= (Dθ2 e2)−1

[−θ∗2

2∑b=1

Dλ22η2

(∂λ2

2

∂qb

)n+ 1

2

·pb,n+ 1

2

mb

− h∗],

(5.77)

where the consistent evaluation of the heat flux (2.1) is provided by

h∗ = k(θ∗2 − θ∗1) (5.78)

The temperature-based EEM method (5.77) clearly resembles its con-tinuous counterpart (4.42) and, hence (2.17), where the continuous partialderivatives have been replaced by its discrete counterparts and the tempera-tures have been consistently evaluated in the sense of (5.73).


What is more, the directionality property of the operator (5.4) enablesthe last two equations in (5.77) to be recast in the form

η1(λ1,n+1, θ1,n+1)− η1(λ1,n, θ1,n)

∆t= k

(θ∗2θ∗1− 1

)η2(λ2,n+1, θ2,n+1)− η2(λ2,n, θ2,n)

∆t= k

(θ∗1θ∗2− 1

) (5.79)

This form resembles the one obtained in the entropy-based EEM method(5.53)3,4 with the particularity of the entropies being function of the statevector rather than part of the state vector.

Remark 5.3.

1. Although the use of an algorithmic temperature other than (5.73) ispossible, as pointed out before, the method does not assume the easy-interpretable form (5.77), neither the discrete form (5.79) could beachieved. It should be noted, however, that this choice does not in anycase alter (5.77)2 since those algorithmic temperatures are imposedby the discrete Poisson matrix (5.68) and the G-equivariant discretederivative of the total energy (5.67)1 to be always provided by (5.73).Further insight on this issued will be provided in subsection 5.3.3 basedon a particular example.

2. The thermodynamical consistency of (5.77) algorithmically relies on theapplication of the discrete derivative to both the internal energy and theentropy functions. In contrast to that, the entropy-based EEM method(5.53) requires only to apply the discrete derivative operator to theinternal energy function expressed in terms of the entropy. Therefore,the temperature-based EEM method involves more computational costwhich is the price to be paid in order to overcome the problems pointedout along this dissertation regarding provision of the potentials in termsof the entropy and the Dirichlet’s boundary conditions for continuousproblems.

Discrete laws of thermodynamics and symmetries. Due to the con-struction of the discrete version of the Poisson and Dissipative matrices ac-cording to (5.68) and (5.70), the conservation of energy as well as the pro-duction of entropy are guaranteed. It only remains to compute the incrementof the entropy in each time step which follows from (5.13) to give

Sn+1 − Sn = ∆tk(θ∗2 − θ∗1)2

θ∗1θ∗2

≥ 0, (5.80)


and can be viewed as the discrete counterpart of (4.43).

The same reasons as those argued for the entropy-based EEM method ap-plies in this case to assert that the resulting temperature-based EEM method(5.77) preserves the total angular momentum (3.51).

Linearization of discrete evolution equations. In this case the tangentmatrix has the following format

K =

Kqaqb Kqapb Kqaθb

Kpaqb Kpapb Kpaθb

Kθaqb Kθapb Kθaθb

a, b = 1, 2 (5.81)

Then, first row blocks coincide with the one for the entropy-based method(5.56) and (5.57). This time the linearization of the linear momentum balancerespect to the positions vectors takes the same expression as (5.58) but interms of the algorithmic internal forces defined by

f ∗a = Dλ2aea − θ∗aDλ2a

ηa, (5.82)

so that the said matrix block now reads

Kpaqb =2∑c=1

[∂f ∗c

∂λ2c,n+1

(∂λ2

c

∂qa

)n+ 1

2

⊗(∂λ2

c

∂qb

)n+1

+1

2f ∗c

(∂2λ2

c

∂qa∂qb

)n+1

](5.83)

and, accordingly, the partial derivative of the algorithmic internal force re-spect to the current elongation gives

∂f ∗a∂λa,n+1

=∂Dλ2a

ea

∂λa,n+1

− ∂θ∗a∂λa,n+1

Dλ2aηa − θ∗a

∂Dλ2aηa

∂λa,n+1(5.84)

Similarly, the block due to the momenta Kpapb coincides with (5.59),whereas the block due to the variation of temperatures is provided by

Kpaθb =∂f ∗b

∂θb,n+1

(∂λ2

b

∂qa

)n+ 1

2

, a, b = 1, 2. (5.85)

In contrast, the blocks for the energy balance equations are much moreinvolved than their counterparts of the entropy-based EEM method. Toelaborate them, the scalars ρa, κa are introduced as

ρa =Dλ2a

ηa

Dθa ηa, κa = k

θ∗2 − θ∗1Dθa ea

, (5.86)


whose derivatives respect both elongations and temperatures are

∂ρa∂sa,n+1

=1

Dθa ηa

(∂Dλ2a

ηa

∂sa,n+1

− ρa∂Dθa ηa∂sa,n+1

), sa = λa, θa (5.87)

and

∂κa∂sc,n+1

=1

Dθa ea

[k

(∂θ∗2

∂sc,n+1

− ∂θ∗1∂sc,n+1

)− κa

∂Dθa ea∂sc,n+1

], sc = λc, θc (5.88)

Thus, the linearization respect to the position vectors is of the form

Kθaqb =2∑c=1

[∂ρa

∂λa,n+1

(∂λ2

a

∂qc

)n+ 1

2

⊗(∂λa∂qb

)n+1

+1

2ρa

(∂2λ2

a

∂qc∂qb

)n+1

]pc,n+ 1

2

mc

+ (−1)a∂κa

∂λb,n+1

(∂λb∂qb

)n+1

(5.89)

In this case, the linearized energy balances equations contributes to vari-ations of the momenta

Kθapb =ρa

2mb

(∂λ2

a

∂qb

)n+ 1

2

(5.90)

Finally, the last block of the tangent matrix is provided by

Kθaθb =2∑c=1

[∂ρa

∂θb,n+1

(∂λ2

a

∂qc

)n+ 1

2

·pc,n+ 1

2

mc

]+ (−1)a

∂κa∂θb,n+1

(5.91)

5.3.3 Validation and comparison with standard methods

In section 2.3.3 three methods were proposed to discuss the performanceof the Midpoint methods regarding the conservation properties. Now theseexamples are solved by the newly proposed EEM methods so that they canbe validated and the conservation properties can be verified.

In addition, for the temperature formulation, four different choices forthe algorithmic temperatures appearing in the discrete Dissipative matrix(5.70) have been implemented, giving raise to different temperature-basedEEM methods, so as to gain insight into how this selection can affect themethod’s performance. Particularly, they are chosen to be either the current,


the converged, the midpoint or the reference (constant) temperatures. Inmaking these choices, expressions (5.77) are not valid any more, specificallyequations(5.77)3,4 become slightly different, as pointed out in Remark 5.3.Remarkably, in these implementations the algorithmic temperature of thediscrete lineal momentum balance (5.77)2 is still provided by (5.73) since itis constrained by the Poisson matrix (5.68) and the G-equivariant discretederivative operator of the total energy (5.67).

First, to validate the EEM methods their order of accuracy based on theExample 1 of subsection 2.3.3 is studied by computing the relative error atdifferent time step sizes with respect to a reference solution as

ez =||z − zref ||||zref ||

, (5.92)

z being the solution at tf = 25 and zref being the reference solution atthe same instant tf obtained with the Midpoint method (5.183) and timestep size of ∆t = 0.001 s. The results obtained with the entropy-based andtemperature-based EEM methods along with the temperature-based Mid-point and Trapezoidal methods are plotted in Figure 5.1. It shows the secondorder of accuracy of the EEM methods and the error constants which arevery similar to the ones of Midpoint and Trapezoidal methods.

The same analysis is performed with the different temperature-basedEEM methods resulting from the aforementioned choices for the algorith-mic temperatures. The results are collected in Figure 5.2 and reveal thatthe choices inside the time interval yield the expected second order accuracywith the practically same exact constant error. Also expected, the constantchoice leads to a poor order of accuracy as it is unable to provide a properapproximation of temperatures in each time step. However, Figure 5.3 showsthe laws of thermodynamics provided by each of these EEM methods witha time step of ∆t = 0.3 s. All of them, even the constant-temperature one,provide the correct evolution of the total energy and entropy, confirming thatthe preservation properties of the discrete setting (5.9) relies on the crucialstructure of the Dissipative matrix (5.70) and supports the election based on(5.73) due to the reasons discussed in the previous subsection 5.3.2.

Figure 5.4 contains the solution of Example 1 of subsection 2.3.3 obtainedwith the entropy-based EEM method and a time step size of ∆t = 0.3, whichcoincides with that reported by Romero (2009). The strains and tempera-tures evolution are apparently identical to the midpoint solution’s shownin Figure 2.4, however, although unappreciable, they are as different as theymust be to satisfy the first law of thermodynamics, which is confirmed by the


10−3 10−2 10−110−7

10−6

10−5

10−4

10−3

10−2

Time step size [s]

e z

Temperature-based EEMEntropy-based EEM

MidpointTrapezoidal

1

2

Figure 5.1. State vector error vs. time step

10−3 10−2 10−110−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Time step size [s]

e z

Temperature-based EEMEEM Method with θ∗a = θa,n+ 1

2

EEM Method with θ∗a = θa,n+1

EEM Method with θ∗a = θa,nEEM Method with θ∗a = θref

1

2

Figure 5.2. Example 1: State vector error vs. time step


0 5 10 15 20 2542.13

42.15

42.18

42.2·101

t [s]

E[J]

0 5 10 15 20 251.24

1.26

1.28

1.3

t [s]

S[JK

−1]

Figure 5.3. Example 1: Total energy and total entropy obtained with eachof the temperature-based EEM methods and ∆t = 0.3. (Same legend as

Figure 5.2).

curve of the energy evolution. Similarly, the total entropy evolution agreeswith the second law of the thermodynamics and the angular momentum evo-lution is constant, confirming the theoretical results obtained in previoussubsections.

The solution of Example 1 obtained with the temperature-based EEMmethod for ∆t = 0.3 s is collected in Figure 5.5. Expectedly, the total energy,entropy and angular momentum curves agrees with their preservation laws.Interestingly, the final amount of entropy provided by this method is slightlydifferent compared to the one provided by the entropy-based method. Thereason is on the large time step size used. In fact, it can be verified that theyboth produce the same amount as the time step size is reduced. Remarkably,the reference solution provides the same final amount as the one provided bythe entropy-based EEM method for the time step size used of ∆t = 0.3 s.

Figure 5.6 and Figure 5.7 show the solution of Example 2 of subsec-tion 2.3.3 obtained with the entropy-based and the temperature-based EEMmethods, respectively. First, it is remarkable that the non-physical drift ofthe elongations observed in the midpoint solutions Figure 2.4 or Figure 2.3 hasdisappeared in both solutions. That is because the solution complies withthe laws of thermodynamics as demonstrates the curves of the total energyand total entropy for both methods. Therefore, it is apparent that both theentropy-based and temperature-based methods allow for larger time steps toprovide sound solutions.


0 5 10 15 20 25−2

0

2

4

6

t [s]

logλ λ0

0 5 10 15 20 25275

300

325

350

375

t [s]

θ[K

]

0 5 10 15 20 2542.13

42.15

42.18

42.2·101

t [s]

E[J]

0 5 10 15 20 251.24

1.26

1.28

1.3

t [s]

S[JK

−1]

0 5 10 15 20 251

1.5

2

2.5

3

t [s]

‖J‖[N·m

]

Figure 5.4. Example 1: solution obtained with entropy-based EEM methodwith ∆t = 0.3 s


0 5 10 15 20 25−2

0

2

4

6

t [s]

logλ λ0

0 5 10 15 20 25275

300

325

350

375

t [s]θ[K

]

0 5 10 15 20 2542.13

42.15

42.18

42.2·101

t [s]

E[J]

0 5 10 15 20 251.24

1.26

1.28

1.3

t [s]

S[JK

−1]

0 5 10 15 20 251

1.5

2

2.5

3

t [s]

‖J‖[N·m

]

Figure 5.5. Example 1: solution obtained with the temperature-based EEMmethod with ∆t = 0.3 s


The evolution of the elongations shows the high frequencies involved bythe stiff parameters of the springs. They are even appreciable in the evolutionof temperatures by a small ripple around the mean evaluation. These highfrequencies might not be correctly captured by the time step chosen, somehowexplaining the different results for the elongations provided by the two EEMmethods.

0 100 200 300 400 500310

327.5

345

362.5

380

t [s]

θ[K

]

0 100 200 300 400 5000

2

4

6

t [s]λ

0 100 200 300 400 5008.95

9

9.05

9.1

9.15·104

t [s]

E[J]

0 100 200 300 400 500268

272

276

280

t [s]

S[JK

−1]

0 5 10 15 20 2510.6

10.65

10.7

10.75

10.8

t [s]

‖J‖[N·m

]

Figure 5.6. Example 2: solution obtained with entropy-based EEM methodwith ∆t = 0.1 s


0 100 200 300 400 500310

327.5

345

362.5

380

t [s]

θ[K

]

0 100 200 300 400 5000

2

4

6

t [s]

λ

0 100 200 300 400 5008.95

9

9.05

9.1

9.15·104

t [s]

E[J]

0 100 200 300 400 500268

272

276

280

t [s]

S[JK

−1]

0 5 10 15 20 2510.6

10.65

10.7

10.75

10.8

t [s]

‖J‖[N·m

]

Figure 5.7. Example 2: solution obtained with temperature-based EEMmethod with ∆t = 0.1 s

Now, the temperature-based EEM method is used to solve Example 3,which, recall, cannot be approached by an entropy formulation because thethermodynamic potentials in terms of entropy are unreachable. In contrastto the Midpoint method, the time step size of ∆t = 0.1 s allows for thestable solution displayed in Figure 5.5. The stability is mainly attained bythe satisfaction of the first law of thermodynamics depicted in the curve of the


total energy. Besides, the temperature-based EEM method provides solutionssatisfying the entropy inequality and the preservation of angular momentum,as has theoretically been proved in previous subsections. Looking at thesolution, the coupling is clearly visible in the evolution of temperatures.

0 5 10 15 20 25−0.1

0.3

0.7

1.1

1.5

t [s]

λ

0 5 10 15 20 25300

320

340

360

380

t [s]θ[K

]

0 5 10 15 20 253

4

5·105

t [s]

E[J]

0 5 10 15 20 251.18

1.19

1.2

1.21

1.22

1.23·103

t [s]

S[JK

−1]

0 5 10 15 20 2560

65

70

75

80

t [s]

‖J‖[N·m

]

Figure 5.8. Example 3: solution obtained with temperature-based EEMmethod with ∆t = 0.1 s

In summary, the EEM methods outperforms their midpoint counterparts


due to their capacity to adhere to the essential conservation laws identifiedin the continuous equations. Among them all, the fulfillment of the first lawof thermodynamics is crucial for the stability of the method for moderatelylarge time step sizes.

5.4 EEM methods for the thermo-visco-elastic system

As in the case of the two thermo-spring system, the general discretedissipative system equations (5.9) are used to propose EEM methods for thethermo-visco-elastic system presented in Section 2.2 based on the use of theentropies and of the temperatures.


The entropy-based EEM method is directly derived from the finite-dimen-sional state space (4.54). The first step is then to apply the G-equivariantdiscrete derivative operator (5.8) to the total energy (4.55) and total entropy(4.56). For that, recall that this system poses two symmetries respect to theactions of the additive group G = Rd and the rotation group G = SO(3),leading the total linear and angular momentum to be first integrals of thesystem evolution, as demonstrated in Section 4.4. Then, nine invariantsfunctions respect to these actions are found to be

π1 = λ2, π2 = ‖p1‖2, π3 = ‖p2‖2

π4 = γ, π5 = η, π6 = σ (5.93)

π7 = (q1 − q2) · p1, π8 = (q2 − q1) · p2, π9 = (q1 − q2) · (q2 − q1)

It can be easily verified that these functions are indeed invariants of bothactions by following the same arguments as in (4.45) and (4.74). In addition,they all are linear, bilinear or quadratic functions of the state vector so theyare appropriated invariants to construct G-equivariant discrete derivativesaccording to (5.8).

The standard derivative of the vector-valued function that collects all of

192 5.4. EEM methods for the thermo-visco-elastic system

the above invariants π(z) = (π1, . . . , π9) results in the following matrix

Dπ(z) =

2(q1 − q2) 2(q2 − q1) 0 0 0 0 0

0 0 2p1 0 0 0 0

0 0 0 2p2 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

p1 −p1 q1 − q2 0 0 0 0

−p2 p2 0 q2 − q1 0 0 0

q2 − q1 q1 − q2 0 0 0 0 0

(5.94)

Based on these invariants, the reduced forms of the total energy and totalentropy functions (4.55) are

E(π(z)) =2∑

a=1

πa+1

2ma

+ e(π1, π5, π4) + ε(π6), S(π(z)) = π5 + π6, (5.95)

whose discrete derivatives are obtained via the partitioned definition (5.3) tobe

DE (πn+1, πn) =

Dλ2 e

(2m1)−1

(2m2)−1

Dγ e

Dηe

Dσ ε

0

0

0

, DS (πn+1, πn) =

0

0

0

0

1

1

0

0

0

(5.96)

In this case, the partial discrete derivative respect to the first invariantis, following the formula for three-variable functions in Appendix A, providedby

Dλ2 e =e(λ2

n+1, ηn+1, γn+1)− e(λ2n, ηn+1, γn)

2(λ2n+1 − λ2

n)

+e(λ2

n+1, ηn, γn+1)− e(λ2n, ηn, γn)

2(λ2n+1 − λ2

n)

(5.97)


Due to the consistency property of the discrete derivative operator (5.5),the above expression is well-defined when λ2

n+1 → λ2n, particularly, it becomes

lim∆λ2→0

Dλ2 e =1

2

(∂e

∂λ2(λ2

n, ηn+1, γn+1) +∂e

∂λ2(λ2

n, ηn, γn)

)(5.98)

On the other hand, the environment internal energy function depends onjust one variable so its discrete derivative is simply

Dσε =ε(σn+1)− ε(σn)

σn+1 − σn(5.99)

For the rest of the partial discrete derivatives consult Appendix A.

Then, the G-equivariant discrete derivatives derives from these discretederivatives together with (5.94) evaluated a midpoint zn+ 1

2to yield

DGE =

2(q1 − q2)n+ 12Dλ2 e

2(q2 − q1)n+ 12Dλ2 e

p1,n+ 12/m1

p2,n+ 12/m2

Dγ e

Dηe

Dσ ε

, DGS =

0

0

0

0

0

1

1

(5.100)

As a result, the discrete counterpart of the Poisson matrix (4.57) aresimply provided by

L(zn+1, zn) = L(z), (5.101)

whereas the discrete Dissipative matrix (4.58) yields

M(zn+1, zn) =

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0T 0T 0T 0T Dηe

ν∗−Dγ e

ν∗0

0T 0T 0T 0T −Dγ e

ν∗(Dγ e)

2 + ν∗kDσ ε

ν∗Dηe−k

0T 0T 0T 0T 0 −k kDηe

Dσε

(5.102)


In this case, the discrete derivatives are clearly identified with the algo-rithmic internal and viscous forces and the temperatures

f ∗ = 2Dλ2 e, g∗ = −Dγ e, θ∗ = Dηe, ϑ∗ = Dσ ε (5.103)

In principle, the algorithmic counterpart of the viscosity parameter (2.51)is not restricted by the current formulation so any evaluation of the elementtemperature inside the time interval could be used, particularly, the algorith-mic temperature to give

ν∗ = ν0 exp

[a

(1

θ∗− 1

θref

)], ν0, a > 0 (5.104)

Assuming these definitions, expressions (5.100) to (5.104) are used in(5.9) to arrive at the entropy-based EEM discrete evolution equations

qa,n+1 − qa,n∆t

=pa,n+ 1

2

ma

pa,n+1 − pa,n∆t

= (−1)a(q1 − q2)n+ 12f ∗

γn+1 − γn∆t

=g∗

ν∗

ηn+1 − ηn∆t

=1

θ∗

[(−g∗)2

ν∗+ k(ϑ∗ − θ∗)

]σn+1 − σn

∆t=

k

ϑ∗(θ∗ − ϑ∗)

(5.105)

The resulting entropy-based EEM method (5.105) is very similar to itscontinuous counterpart (2.56).

Discrete laws of thermodynamics. The discrete laws of thermodynam-ics are guaranteed by the matrices properties according to (5.12) and(5.13).Thus, the increment of the entropy in each time step can be calculated togive

Sn+1 − Sn = ∆tk(θ∗ − ϑ∗)2

θ∗ϑ∗+

(−g∗)2

θ∗ν∗≥ 0, (5.106)

which can be viewed as the discrete analogue of (4.71).

Discrete symmetries. The discrete symmetries are also inherited by theentropy-based EEM method (5.105) because the use of the G-equivariantdiscrete derivatives based on the invariability of the total energy and totalentropy functions respect to the action of the additive and rotation groups.


Thus, the discrete version of the Noether’s theorem 5.2 allows to assert thatthe total linear momentum (3.44) and the total angular momentum (3.51)are preserved quantities on the discrete solution. The proofs is equivalent tothose performed in the continuous case in Sections 4.3 and 4.4.

Linearization of discrete evolution equations. The resulting implicitmethod requires the linearization of its discrete equations (5.105) that arecollected in a tangent matrix of the form

K =

Kqaqb Kqapb Kqaγ Kqaη Kqaσ

Kpaqb Kpapb Kpaγ Kpaη Kpaσ

Kγqa Kγpa Kγγ Kγη Kγσ

Kηqb Kηpb Kηγ Kηη Kησ

Kσqb Kσpb Kσγ Kση Kσσ

a, b = 1, 2 (5.107)

Then, the first two blocks in the first row are identical to the blocks forthe EEM methods of the two thermo-spring variable (5.56), whereas the lastthrees blocks are just provided by the null matrix.

Looking now at the second row blocks, the first one corresponding withthe linearization of the discrete linear momentum balance respect to theposition vectors reads

Kpaqb =1

2

∂f ∗

∂λ2n+1

(∂λ2

∂qa

)n+ 1

2

⊗(∂λ2

∂qb

)n+1

+1

4f ∗(

∂2λ2

∂qa∂qb

)n+1

(5.108)

The second block fully coincides with (5.59), while the third and thefourth ones yield

Kpaγ =1

2

∂f ∗

∂γn+1

, Kpaη =1

2

∂f ∗

∂ηn+1

(5.109)

To end up with this row, the last block vanishes due to the linear mo-mentum balance being non-dependent on the environment entropy.

The third row provides the linearization of the evolution equation forthe internal variables. Thus, the linearization of this term respect to thepositions, the element entropy and the internal variable can be elaborated to


give

Kγqa = −(

1

ν∗∂g∗

∂λn+1

− g∗

(ν∗)2

∂ν∗

∂λn+1

)(∂λ

∂qa

)n+1

Kγη = − 1

ν∗∂g∗

∂ηn+1

+g∗

(ν∗)2

∂ν∗

∂ηn+1

Kγγ =1

∆t− 1

ν∗∂g∗

∂γn+1

+g∗

(ν∗)2

∂ν∗

∂γn+1

,

(5.110)

with the rest of the blocks being zero.

On the other hand, all the blocks of the fourth row but one are non-zero.In particular, the linearization respect to the position vectors

Kηqa =1

(θ∗)2

∂θ∗

∂λ2n+1

((−g∗)2

ν∗+ k(ϑ∗ − θ∗)

)(∂λ2

∂qa

)n+1

− 1

θ∗

(2g∗

ν∗∂g∗

∂λ2n+1

−(g∗

ν∗

)2∂ν∗

∂λ2n+1

− k ∂θ∗

∂λ2n+1

)(∂λ2

∂qa

)n+1

(5.111)

Similarly, the linearization respect to the internal variable reads

Kηγ =1

(θ∗)2

∂θ∗

∂γn+1

((−g∗)2

ν∗+ k(ϑ∗ − θ∗)

)− 1

θ∗

(2g∗

ν∗∂g∗

∂γn+1

−(g∗

ν∗

)2∂ν∗

∂γn+1

− k ∂θ∗

∂γn+1

),

(5.112)

and the linearization respect to the element entropy yields

Kηη =1

∆t+

1

(θ∗)2

∂θ∗

∂ηn+1

((−g∗)2

ν∗+ k(ϑ∗ − θ∗)

)− 1

θ∗

(2g∗

ν∗∂g∗

∂ηn+1

+

(g∗

ν∗

)2∂ν∗

∂ηn+1

+ k∂θ∗

∂ηn+1

) (5.113)

The last block of the fourth row is then provided by

Kησ = −k 1

θ∗dϑ∗

dσn+1

(5.114)


Finally, the blocks in the last row are provided by

Kσqa = − k

ϑ∗∂θ∗

∂λn+1

(∂λ

∂qa

)n+1

, a = 1, 2

Kσγ = − k

ϑ∗∂θ∗

∂γn+1

Kση = − k

ϑ∗∂θ∗

∂ηn+1

Kσσ =1

∆t+ k

θ∗

(ϑ∗)2

dϑ∗

dσn+1

(5.115)


Analogously, the temperature-based EEM method follows from the ap-plication of the G-equivariant discrete derivative operator to the total energy(4.61) and total entropy (4.62) expressed in terms of the state vector (4.60)and based on the following invariants

π1 = λ2, π2 = ‖p1‖2, π3 = ‖p2‖2

π4 = γ, π5 = θ, π6 = ϑ (5.116)

π7 = (q1 − q2) · p1, π8 = (q2 − q1) · p2, π9 = (q1 − q2) · (q2 − q1)

These new invariant functions lead the reduced total energy and totalentropy functions to be provided by

E(π(z)) =2∑

a=1

πa+1

2ma

+ e(π1, π5, π4) + ε(π6)

S(π(z)) = η(π1, π5, π4) + σ(π6),

(5.117)

Their respective discrete derivatives respect to the invariants are thenderived from the partitioned definition (5.3) to give

DE (πn+1, πn) =

Dλ2 e

(2m1)−1

(2m2)−1

Dγ e

Dθe

Dϑε

0

0

0

, DS (πn+1, πn) =

Dλ2 η

0

0

Dγ η

Dθη

Dϑσ

0

0

0

, (5.118)


The G-equivariant discrete derivative can be elaborated following (5.38)with (5.94) to give

DGE =

2(q1 − q2)n+ 12Dλ2 e

2(q2 − q1)n+ 12Dλ2 e

p1,n+ 12/m1

p2,n+ 12/m2

Dγ e

Dθe

Dϑε

, DGS =

2(q1 − q2)n+ 12Dλ2 η

2(q2 − q1)n+ 12Dλ2 η

0

0

Dγ η

Dθη

Dϑσ

(5.119)

Accordingly, the discrete counterpart of the Poisson (4.64) is constructedin the following way

L(zn+1, zn) =

0 0 1 0 0 0 0

0 0 0 1 0 0 0

−1 0 0 0 0 L1 0

0 −1 0 0 0 L2 0

0T 0T 0T 0T 0 0 0

0T 0T L1 L2 0 0 0

0T 0T 0T 0T 0 0 0

(5.120)

with

La = −LTa = (−1)a (Dθη)−1 2Dλ2 η(q2 − q1)n+ 1

2(5.121)

Similarly, the discrete counterpart of the Dissipative matrix (4.66) be-comes

M(zn+1, zn) =

θ∗

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0 0 0 0 0

0T 0T 0T 0T 1

ν∗− Dγ e

ν∗Dθe0

0T 0T 0T 0T − Dγ e

ν∗Dθe

(Dγ e)2 + ν∗kϑ∗

ν∗Dθe2− kϑ∗

DθeDϑε

0T 0T 0T 0T 0 − kϑ∗

DθeDϑε

kϑ∗

Dϑε2

(5.122)


As in the case of the temperature-based EEM method for the two thermo-spring system, provided that the algorithmic temperatures θ∗ and ϑ∗ areevaluated at any point within the time step, the resulting temperature-basedEEM method will be second-order accurate according to (5.9).

Similar reasons as those used in the mentioned case lead the algorithmictemperatures to be chosen as

θ∗ =Dθe

Dθη, ϑ∗ =

Dϑε

Dϑσ(5.123)

Also, the algorithmic viscosity parameter can be provided by (5.104) butwith the algorithmic element temperature just introduced.

The temperature-based EEM method then results from using the rela-tions (5.118) - (5.123) in (5.9) to give

qa,n+1 − qa,n∆t

=pa,n+ 1

2

ma

pa,n+1 − pa,n∆t

= (−1)a(2Dλ2 e− θ∗2Dλ2 η)(q1 − q2)n+ 12

γn+1 − γn∆t

=−1

ν∗(Dγ e− θ∗Dγ η)

θn+1 − θn∆t

=1

Dθe

[− θ∗2Dλ2 η

2∑a=1

(−1)a(q2 − q1)n+ 12·pa,n+ 1

2

ma

+Dγ e

ν∗(Dγ e− θ∗Dγ η) + k(ϑ∗ − θ∗)

]ϑn+1 − ϑn

∆t= k

θ∗ − ϑ∗Dϑε

(5.124)

The resulting temperature-based EEM method (5.124) resembles its con-tinuous counterpart (4.69). The discrete equation of the energy balance ofthe element (5.124)4 can be simplified by elaborating the first term on theright hand side as follows

2∑a=1

(−1)a(q2 − q1)n+ 12·pa,n+ 1

2

ma

=2∑

a=1

(−1)a(q2 − q1)n+ 12· qa,n+1 − qa,n

∆t

=λ2n+1 − λ2

n

2 ∆t(5.125)


Then, the discrete equation reads

θn+1 − θn∆t

=1

Dθe

[− θ∗2Dλ2 η

λ2n+1 − λ2

n

2 ∆t

+Dγ e

ν∗(Dγ e− θ∗Dγ η) + k(ϑ∗ − θ∗)

] (5.126)

Furthermore, by introducing the algorithmic viscous force as

g∗ = −(Dγ e− θ∗Dγ η), (5.127)

and using (5.124)3, the second term of the right hand side of (5.126) becomes

θn+1 − θn∆t

=1

Dθe

[− θ∗Dλ2 η

λ2n+1 − λ2

n

∆t

− (−g∗ + θ∗Dγ η)γn+1 − γn

∆t+ k(ϑ∗ − θ∗)

] (5.128)

Now, the definition of the algorithmic temperature (5.123) is introducedto give

θn+1 − θn∆t

= − 1

DθηDλ2 η

λ2n+1 − λ2

n

∆t− Dγ η

Dθη

γn+1 − γn∆t

− −g∗

Dθe

γn+1 − γn∆t

+k(ϑ∗ − θ∗)

Dθe

(5.129)

Rearranging terms it can be written as follows

Dθηθn+1 − θn

∆t+ Dλ2 η

λ2n+1 − λ2

n

∆t+ Dγ η

γn+1 − γn∆t

=−g∗θ∗

γn+1 − γn∆t

+k(ϑ∗ − θ∗)

θ∗

(5.130)

Finally, the three terms on the left hand side are identified with thedirectionality property of the discrete derivative operator so the equationtakes the following entropy form

η(λn+1, θn+1, γn+1)− η(λn, θn, γn)

∆t= k

(ϑ∗

θ∗− 1

)+

(−g∗)2

θ∗ν∗, (5.131)

where use has also been made of the introduced identity and (5.124)3.


In a similar but much more direct way, by using the directionality prop-erty the last equation in (5.124) can be simplified to

σ(ϑn+1)− σ(ϑn)

∆t= k

(θ∗

ϑ∗− 1

)(5.132)

These forms of the last two discrete equations of the temperature-basedEEM method can alternatively be used in its implementation.

Discrete laws of thermodynamics. The conservation of the total energyand the production of entropy due to the conduction phenomenon and theinternal viscous changes are readily verified by (5.12) and (5.13) using thediscrete matrices and gradients used to derive the temperature-based EEMmethod. Particularly, the increment of the entropy can also be checked bynoting that the total entropy balance is the balance of the element entropyplus the balance of environment entropy, provided by (5.131), to give

Sn+1 − Sn = ∆tk(θ∗ − ϑ∗)2

θ∗ϑ∗+

(Dγ e− θ∗Dγ η)2

θ∗ν∗≥ 0, (5.133)

which can be viewed as the discrete analogue of (4.72).

Discrete symmetries the total linear and angular momenta are first inte-grals of the solution provided by the temperature-based EEM method, sinceuse has been made of theG-equivariant discrete derivatives of the total energyand entropy. These functions are G-invariant functions respect to the addi-tive and rotation groups so, according to the discrete Noether’s theorem 5.2,the associated momentum maps are the total linear and angular momentums.Again, the proofs follow the guidelines provided for the continuous case.

Linearization of discrete evolution equations. The linearized discreteequations are assembled in this case in a tangent matrix with the followingblocks

K =

Kqaqb Kqapb Kqaγ Kqaθ Kqaϑ

Kpaqb Kpapb Kpaγ Kpaθ Kpaϑ

Kγqa Kγpa Kγγ Kγθ Kγϑ

Kθqa Kθpa Kθγ Kθθ Kθϑ

Kϑqa Kϑpa Kϑγ Kϑθ Kϑϑ

a, b = 1, 2 (5.134)

The first two blocks in the first row coincides with (5.56), while the lastthree blocks result in null matrices. In the second row, the first block isidentical to (5.108) but with the algorithmic internal force being provided by

f ∗ = Dλ2 e− θ∗Dλ2 η (5.135)


Also, the above definition applies for the second block provided by (5.59)and the third and fourth blocks by (5.109).

Similarly, the blocks in the third row are provided by (5.110) in wherethe definition of the algorithmic viscous force is provided by (5.127) and theentropy variable is replaced by the temperature variable.

For the blocks in the fourth row, the ensuing definitions are introduced

ρ =Dλ2 η

Dθη, ξ =

Dγ e

Dθe, κ = k

θ∗ − ϑ∗Dθe

, (5.136)

so that the evolution equation due to the energy balance can be rewritten asfollows

θn+1 − θn∆t

= −2ρDλ2 η2∑

a=1

(−1)a(q2 − q1)n+ 12·pa,n+ 1

2

ma

(5.137)

+ξ

ν∗(Dγ e− θ∗Dγ η) + κ (5.138)

To linearize it, the derivatives of the previous scalars respect to the elon-gation, the internal variable or the temperatures are provided by

∂ρ

∂sn+1

=1

Dθη

(∂Dλ2 η

∂sn+1

− ρ ∂Dθη

∂sn+1

), s = λ, θ, γ, (5.139)

∂ξ

∂sn+1

=1

Dθe

(∂Dγ e

∂sn+1

− ξ ∂Dθe

∂sn+1

), s = λ, θ, γ, (5.140)

and

∂κ

∂sn+1

=1

Dθe

[k

(∂θ∗

∂sn+1

− ∂ϑ∗

∂sn+1

)+ κ

∂Dθe

∂sn+1

], s = λ, θ, γ (5.141)

Then, the first block can be expressed as follows

Kθqa =2∑b=1

[∂ρ

∂λ2n+1

(∂λ2

∂qb

)n+ 1

2

⊗(∂λ2

∂qa

)n+1

+1

2ρ

(∂2λ2

∂qb∂qa

)n+1

]pb,n+ 1

2

mb

+

(g∗

ν∗∂ξ

∂λ2n+1

+ξ

ν∗∂g∗

∂λ2n+1

− ξg∗

(ν∗)2

∂ν∗

∂λ2n+1

+∂κ

∂λ2n+1

)(∂λ2

∂qa

)n+1

(5.142)

The contribution due to the momenta is then

Kθpa =ρ

2mb

(∂λ2

∂qa

)n+ 1

2

, a = 1, 2 (5.143)


The linearization respect to the element temperature yields

Kθθ =1

∆t+

2∑a=1

[∂ρ

∂θn+1

(∂λ2

∂qa

)n+ 1

2

·pa,n+ 1

2

ma

]

+g∗

ν∗∂ξ

∂θn+1

+ξ

ν∗∂g∗

∂θn+1

− ξg∗

(ν∗)2

∂ν∗

∂θn+1

− ∂κ

∂θn+1

(5.144)

The last block of the fourth row is then provided by

Kθϑ =∂κ

∂ϑn+1

= −k 1

Dθe

dϑ∗

dϑn+1

(5.145)

Finally, the blocks in the last row are provided by

Kϑqa = − k

Dϑε

∂θ∗

∂λ2n+1

(∂λ2

∂qa

)n+1

, a = 1, 2

Kϑγ = − k

Dϑε

∂θ∗

∂γn+1

Kϑθ = − k

Dϑε

∂θ∗

∂θn+1

Kϑϑ =1

∆t− k

Dϑε2

(dϑ∗

dϑn+1

Dϑε− ϑ∗dDϑε

dϑn+1

)(5.146)

5.4.3 Validation and comparison with standard methods

Next, the just devised EEM methods are validated and compared withthe results for the two cases considered in subsection 2.3.3. In addition,results obtained with the standard Trapezoidal method have been considered.The validation to be performed is based on convergence plots of relative errorscomputed by (5.92) using the reference solutions provided in subsection 2.3.3.

Results for Case I. Figure 5.9 plots the relative position error. The resultscorroborate the second order convergence of the temperature-based EEMmethod and also reveal a superior accuracy compared with both implicitMidpoint method (2.89) and Trapezoidal method applied to the temperature-based system (2.45). On the other hand, the entropy-based EEM methodresults in this case a little bit more accurate in position than the temperature-based EEM one.

The convergence of temperature is displayed in Figure 5.10. Comparingwith the range of relative errors in Figure 5.9, it becomes apparent that, for


this case, temperatures are more accurately computed than positions. Re-markably, the temperature-based EEM method is significantly more accuratethan other methods, including the entropy-based EEM one.

The second order accuracy and the smaller constant error of the tempera-ture-based EEM method, compared with the other methods, can also be seenin the convergence plot of the internal variable γ in Figure 5.11. Interestingly,the entropy-based EEM method provides poor accuracy, even compared withthe standard Midpoint and Trapezoidal methods.

The solution including the long-term behavior of the laws of thermody-namics and the angular momentum are displayed in Figure 5.12 and Figure5.13 for ∆t = 0.2 s and for the temperature-based EEM and entropy-basedEEM methods, respectively.

10−2 10−1

10−5

10−4

10−3

10−2

10−1

100

101

∆t [s]

e q


TrapezoidalMidpoint

1

2

Figure 5.9. Case I: Position (q) relative error

Results for Case II. This case can only be faced by the temperature-based EEM method due to the technical difficulties associated with the non-linear dependency of the spring stiffness with temperature that prevent theformulation of an entropy-based EEE method.

Figures 5.14, 5.15 and 5.16 display the relative errors in position (q),element temperature (θ) and internal variable (γ). Again, the quadraticconvergence is observed and the accuracy significantly improves compared


10−2 10−110−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

∆t [s]

e θ


TrapezoidalMidpoint

1

2

Figure 5.10. Case I: Temperature (θ) relative error

10−2 10−1

10−6

10−5

10−4

10−3

10−2

10−1

100

101

∆t [s]

e γ


TrapezoidalMidpoint

1

2

Figure 5.11. Case I: Internal variable (γ) relative error


−2 −1 0 1 2 3−2−10

1

22

q2 [m]

q 1[m

]

0 5 10 15 20280

300

320

340

360

380

t [s]

θ,ϑ[K

]

0 5 10 15 200

0.01

0.02

0.03

0.04

t [s]

γ[m

]

0 5 10 15 2017

17.05

17.1·102

t [s]

E[J]

0 5 10 15 205.4

5.5

5.6

5.7

t [s]

S[JK

−1]

0 5 10 15 202

2.5

3

3.5

4

t [s]

||J||[N·m

]

Figure 5.12. Case I: solution obtained with temperature-based EEMmethod with ∆t = 0.2 s


−2 −1 0 1 2 3−2−10

1

22

q2 [m]

q 1[m

]

0 5 10 15 20280

300

320

340

360

380

t [s]θ ,ϑ[K

]

0 5 10 15 20−0.02

0

0.02

0.04

t [s]

γ[m

]

0 5 10 15 2017

17.05

17.1·102

t [s]

E[J]

0 5 10 15 205.4

5.5

5.6

5.7

t [s]

S[JK

−1]

0 5 10 15 202

2.5

3

3.5

4

t [s]

||J||[N·m

]

Figure 5.13. Case I: solution obtained with the entropy-based EEMmethod with ∆t = 0.2 s


with both Trapezoidal and Midpoint methods. Convergence results for theenvironment temperature are omitted, since all methods exhibit similar ac-curacies for this particular case.

Figure 5.17 shows that, for the time step size ∆t = 0.3s that led theMidpoint method to not satisfy the law of thermodynamics, the temperature-based EEM method provides the right long-term behavior of the laws ofthermodynamics and the angular momentum.

All these results confirm the superior stability and robustness featuresbestowed by the preservation of essential physical rules.

10−2 10−1

10−6

10−5

10−4

10−3

10−2

10−1

100

∆t [s]

e q

Temperature-based EEMTrapezoidalMidpoint

1

2

Figure 5.14. Case II: Position (q) relative error


10−2 10−110−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

∆t [s]

e θ


1

2

Figure 5.15. Case II: Element temperature (θ) relative error

10−2 10−1

10−6

10−5

10−4

10−3

10−2

10−1

100

∆t [s]

e γ


1

2

Figure 5.16. Case II: Internal variable (γ) relative error


−2 −1 0 1 2 3−2−10

1

22

q2 [m]

q 1[m

]

0 5 10 15 20280

300

320

340

360

380

t [s]

θ,ϑ[K

]

0 5 10 15 20

0

0.01

0.02

t [s]

γ[m

]

0 5 10 15 2016.8

16.81

16.82·102

t [s]

E[J]

0 5 10 15 205.35

5.4

5.45

5.5

5.55

t [s]

S[JK

−1]

0 5 10 15 202

2.5

3

3.5

4

t [s]

||J||[N·m

]

Figure 5.17. Case II: solution obtained with temperature-based EEMmethod with ∆t = 0.3 s


5.5 EEM methods for nonlinear thermoelasticity

In this section, the methodology introduced in subsection 5.2.2 is usedto construct EEM time integration methods for the nonlinear dynamics ofthermoelastic solids presented in Section 4.6. Once again, two different ap-proaches based on the selection for the thermodynamical variable will beelaborated giving raise to the so called entropy-based EEM method and thetemperature-based EEM method. While the first one was fully elaboratedin Romero (2010a,b), the second one, to the best of the author’s knowledge,has never been advocated.


To construct an entropy-based EEM counterpart of nonlinear thermoe-lastodynamics, the infinite-dimensional state space (4.96) containing the den-sity entropy is projected onto an finite-dimensional one defined in a partitionof the continuum into regular elements by the linear combination of shapefunctions as was introduced in Section 3.7, that is

S h = zh =N∑a=1

Naza = (ϕh,ph, ηh) : Bh0 7→ Rd×Rd×R, det (∇0ϕ

h) > 0,

(5.147)

The tangent space is also projected onto the finite-dimensional space oflinear combinations of shape functions to give

TS h = wh =N∑a=1

Nawa = (whϕ,w

hp, w

hη ) : Bh

0 7→ Rd × Rd × R (5.148)

Then, in view of the continuous forms for the Poisson and the Dissipativebrackets (4.102) and (4.103), their respective discrete counterparts, for twoany functionals F ,G : S h 7→ R, are provided by

F ,G(zhn+1, z

hn

)=

⟨∆GF∆ϕ

(zhn+1, z

hn

),∆GG∆p

(zhn+1, z

hn

)⟩−⟨

∆GF∆p

(zhn+1, z

hn

),∆GG∆ϕ

(zhn+1, z

hn

)⟩ (5.149)

[[F ,G]](zhn+1, z

hn

)=

⟨∇0

[1

Θ∗∆GF∆η

(zhn+1, z

hn

)], (Θ∗)2 Kh

n+ 12∇0

[1

Θ∗∆GG∆η

(zhn+1, z

hn

)]⟩,

(5.150)

212 5.5. EEM methods for nonlinear thermoelasticity

where Θ∗ is the algorithmic temperature to be defined within the time in-terval [tn, tn+1] and Kh

n+ 12

is the second order approximation of the material

conductivity tensor provided by

Khn+ 1

2= Jh

n+ 12Fh,−1

n+ 12

κ(Θ∗)Fh,−T

n+ 12

, with Jhn+ 1

2= det Fh

n+ 12

(5.151)

In order to elaborate the appearing G-equivariant discrete functionalderivatives in the discrete brackets, recall that the total energy and the totalentropy (4.98) are invariant functions respect to the additive and the rotationgroups as was observed in Section 4.9. Then, the following invariant functionrespect to the actions of these group is identified

π(zh) = (∇0ϕhT∇0ϕ

h, ‖ph‖2, ηh) = (Fh,TFh, ‖ph‖2, ηh), (5.152)

whose components are additionally linear or quadratic functions of the statevariables so they can be used as invariant functions in the sense of Prop. 5.1.

Based on this invariant function, the reduced total energy and total en-tropy read

E(π(zh)) =

∫Bh

0

(π2

2ρ0

+ e(π1, π3)

)dV0, S(π(zh)) =

∫Bh

0

π3dV0, (5.153)

Now, with the help of (5.30) the G-equivariant discrete functional deriva-tive of the total energy can be elaborated by using G-equivariant discretederivative operator (5.8). To this end, the discrete derivative operator (5.2)of the above reduced functionals along with the standard derivative of theinvariant function are used to give⟨

∆GE

∆z

(zhn+1, z

hn

),wh

⟩= 〈DG

[‖ph‖2

2ρ0

+ e(Ch, ηh)

](zhn+1, z

hn),wh〉Bh

0

= 〈Dπ(zhn+ 1

2)TD

[π2

2ρ0

+ e(π1, π3)

](πn+1, πn),wh〉Bh

0

= 〈D[π2

2ρ0

+ e(π1, π3)

](πn+1, πn), Dπ(zh

n+ 12) ·wh〉Bh

0

= 〈Dπ1 e(πn+1, πn), Dπ1 ·whϕ〉Bh

0

+ 〈Dπ2

[π2

2ρ0

](πn+1, πn), Dπ2 ·wh

p〉Bh0

+ 〈Dπ3 e(πn+1, πn), Dπ3whη 〉Bh

0,

(5.154)


for any vector wh ∈ TS h.

Similarly, the G-equivariant discrete functional applied to the total en-tropy results in⟨

∆GS

∆z

(zhn+1, z

hn

),wh

⟩= 〈DGη(zhn+1, z

hn),wh〉B0

= 〈Dπ(zn+ 12)TDπ3,w

h〉Bh0

= 〈Dπ3π3, Dπ3whη 〉Bh

0

(5.155)

Then, their final expressions are revealed by performing the standardderivatives of each component of the invariant function, which are

Dπ1 · a = D(Fh,TFh) · a = Fh,T∇0a+∇0aTFh, ∀a ∈ Rd

Dπ2 · a = D(‖ph‖2) · a = 2ph · a, ∀a ∈ Rd

Dπ3w = D(ηh)w = w, ∀w ∈ R(5.156)

Using these results, the G-equivariant partial discrete functionals of thetotal energy are provided by⟨

∆GE

∆ϕ

(zhn+1, z

hn

),wh

ϕ

⟩= 〈2Fh

n+ 12Dπ1 e(πn+1, πn),∇0w

hϕ〉Bh

0⟨∆GE

∆p

(zhn+1, z

hn

),wh

p

⟩= 〈

phn+ 1

2

ρ0

,whp〉Bh

0⟨∆GE

∆η

(zhn+1, z

hn

), whη

⟩= 〈Dπ3 e(πn+1, πn), whη 〉Bh

0,

(5.157)

where use has been made of the symmetry of the partial discrete derivativeof the reduced internal energy density respect to the first invariant, identifiedwith the symmetric right Cauchy-Green deformation tensor π1 = Fh,TFh =Ch, and a second-order tensor product property2.

For its part, the G-equivariant discrete functional of the total entropy isfinally provided by ⟨

∆GS

∆η

(zhn+1, z

hn

), whη

⟩= 〈1, whη 〉Bh

0(5.158)

The resulting discrete derivatives of the reduced internal energy densityfunction are clearly identified as discrete counterparts of the symmetric Piola-kirchhoff stress tensor and the temperature field, that is

S∗ = 2Dπ1 e(πn+1, πn) = 2DCe, Θ∗ = Dπ3 e(πn+1, πn) = Dηe (5.159)

2A : BC = BTA : C


Following the partitioned definition of the discrete derivative operator,they are derived from the first and second discrete derivatives of functionsdepending on the right Cauchy-Green deformation tensor as a first argumentand on a scalar thermodynamical variable, in this case the entropy density,as a second argument. Their elaborated formulas can be found in AppendixA and are summarized next

S∗ = 2DCe = Sh

n+ 12

+∆en+1 + ∆en − S

h

n+ 12

: ∆Ch

∆Ch : ∆Ch∆Ch, (5.160)

where

∆ei = e(Chn+1, η

hi

)− e

(Chn, η

hi

), (5.161)

and the following simplified notation has been introduced

Sh

n+ 12

=1

2(Sn+1 + Sn) with Si = 2

∂e

∂C(Ch

n+ 12, ηhi ) (5.162)

The key point in the above evaluation relies on the fact that the rightCauchy-Green deformation tensor is provided by its mean value in the timeinterval Ch

n+ 12

= 12(Ch

n+1 +Chn). Recall that in a midpoint-type discretization

only the state variables are evaluated in such a way.

In addition, the above formula is well-defined in the limit case of ∆Ch →0 due to the consistency property of the discrete derivative operator, demon-strated in Appendix A, leading to

lim∆Ch→0

S∗ =∂e

∂C(Ch

n, ηhn+1) +

∂e

∂C(Ch

n, ηhn) (5.163)

The algorithmic temperature assumes a simplified form due to being ascalar field, that is

Θ∗ = Dηe =e(Chn+1, η

hn+1

)− e

(Chn+1, η

hn

)+(Chn, η

hn+1

)− e

(Chn, η

hn

)2(ηhn+1 − ηhn)

,

(5.164)which in the limit case of ηhn+1 = ηhn collapses to

lim∆ηh→0

Θ∗ =1

2

(∂e

∂η(Ch

n+1, ηhn) +

∂e

∂η(Ch

n, ηhn)

)(5.165)

Then, all the previous results are used in (5.20) to arrive at the following


representation of the entropy-based EEM method

〈whϕ,ϕhn+1 −ϕhn

∆t〉Bh

0= 〈wh

ϕ,1

ρ0

phn+ 1

2〉Bh

0


∆t〉Bh

0= −〈∇0w

hp,F

hn+ 1

2S∗〉Bh

0


∆t〉Bh

0= −〈∇0

[whηΘ∗

],Kh

n+ 12∇0Θ∗〉Bh

0

(5.166)

Laws of thermodynamics. According to the formalism that has led tothe entropy-based EEM method (5.166), its solution agrees with the laws ofthermodynamics and symmetries. In fact, using the discrete brackets (5.149)and (5.150) along with the discrete functional derivatives of the total energyand entropy (5.157) and (5.158) in (5.32) leads the energy balance to be

Ehn+1 − Eh

n = ∆t〈Fhn+ 1

2S∗,∇0

[phn+ 1

2

ρ0

]〉Bh

0

− ∆t〈∇0

[phn+ 1

2

ρ0

],Fh

n+ 12S∗〉Bh

0

+ ∆t〈∇0

[1

Θ∗Θ∗], (Θ∗)2Kh

n+ 12∇0

[1

Θ∗

]〉Bh

0= 0

(5.167)

For its part, the entropy balance yields

Shn+1 − Shn = ∆t〈∇0

[1

Θ∗

], (Θ∗)2Kh

n+ 12∇0

[1

Θ∗

]〉Bh

0

= ∆t〈 1

(Θ∗)2∇0Θ∗,Kh

n+ 12∇0Θ∗〉Bh

0≥ 0,

(5.168)

which is the discrete analogous of (3.101) for the dissipation due to the con-duction phenomenon.

Remark 5.4. Despite these results, neither the conservation of energy northe production of entropy can be ensured by the resulting discrete Galerkin-based EEM method, since, the resulting G-equivariant discrete functionalderivatives do not belong to the Galerkin space V h identified with the tan-gent space state TS h. Particularly, the resulting algorithmic temperature(5.159)2 does not belong to the finite-dimensional Galerkin space Θ∗∈V h,where the temperature field is naturally defined in this context. This issueleads the entropy-based EEM method (5.159) to the dissatisfaction of the


energy conservation. More details about this issue are provided in Romero(2010a) as well as a correction of the entropy-based EEM method consist-ing in a L2 projection technique that brings the consistent evaluation of thetemperature (5.159)2 back to the Galerkin space.

The positive side of this required projection is that the material gradientof the temperature field needed for heat conduction term is straightforwardas the resulting projected temperature will surely be provided by the linearcombination of shape functions, as the state variable (3.155). Thus, its gra-dient can directly be computed through the gradient of the shape functions.

Interestingly, this issue did not arise in the formulation of the Energy-Momentum methods of conserving problems due to the simple forms of thecanonical Poisson bracket and the kinetic energy that lead the G-equivariantdiscrete functional derivative to be defined within the Galerkin-based tangentspace.

Discrete symmetries. The devised EEM method does preserve the linearand angular momentum along its discrete solution since, by definition, theyboth are at most quadratic functions of the state variables, and hence theirG-equivariant discrete functional derivatives surely belong to V h as happenedwith the kinetic energy in Hamiltonian problems. The proof is similar to theone performed for the continuous case in Section 4.9, now using the discretecounterparts of the brackets. Also, similar arguments as those provided in(3.185) - (3.187) to demonstrate the preservation of the momentum mapsin a midpoint-type discretization can be followed up to arrive at the sameconclusion.


In order to derive the temperature-based EEM method the state space(4.111) is transformed into a finite-dimensional space using the Galerkin ap-proach presented in Section 3.7 such that

S h = zh =N∑a=1

Naza = (ϕh,ph,Θh) : Bh0 7→ Rd×Rd×R+, det (∇0ϕ

h) > 0,

(5.169)whose tangent space is

TS h = wh =N∑a=1

Nawa = (whϕ,w

hp, w

hΘ) : Bh

0 7→ Rd × Rd × R (5.170)


In view of the continuous Poisson bracket (4.116), its consistent discretecounterpart based on (5.25) is found, for any two functional F ,G : S 7→ R,to be

F ,G(zhn+1, z

hn

)=

⟨∆GF∆ϕ

(zhn+1, z

hn

),∆GG∆p

(zhn+1, z

hn

)⟩−⟨

∆GF∆p

(zhn+1, z

hn

),∆GG∆ϕ

(zhn+1, z

hn

)⟩+

⟨∇0

[∆GF∆p

(zhn+1, z

hn

)],Fh

n+ 12V∗

1

c

∆GG∆Θ

(zhn+1, z

hn

)⟩−⟨

Fhn+ 1

2V∗

1

c

∆GF∆Θ

(zhn+1, z

hn

),∇0

[∆GG∆p

(zhn+1, z

hn

)]⟩,

(5.171)

where V∗ is a discrete second order counterpart of the symmetric latent heattensor (3.77) (neglecting the internal variables) and c is the constant specificheat capacity defined in (3.78).

Now looking at the continuous Dissipative brackets (4.117), its discretecounterpart is constructed as follows

[[F ,G]](zhn+1, z

hn

)=

⟨∇0

[1

c

∆GF∆Θ

(zhn+1, z

hn

)], (Θ∗)2 Kh

n+ 12∇0

[1

c

∆GG∆Θ

(zhn+1, z

hn

)]⟩,

(5.172)

with Khn+ 1

2

being provided by (5.151) except for the algorithmic temperature

yet to be defined in this approach.

Similarly to the entropy-based case, the following invariant function re-spect to the action of the additive and rotation group is found

π(zh) = (Fh,TFh, ‖ph‖2,Θh) (5.173)

Accordingly, the reduced forms of the total energy and the total entropy(4.113) are provided in terms of the invariant function by

E(π(zh)) =

∫B0

(π2

2ρ0

+ e(π1, π3)

)dV0, S(π(zh)) =

∫B0

η(π1, π3)dV0,

(5.174)

Observing that the derivatives of the invariant function coincide withthose previously obtained (5.156), the G-equivariant partial discrete deriva-


tives of the total energy results in⟨∆GE

∆ϕ

(zhn+1, z

hn

),wh

ϕ

⟩= 〈2Fh

n+ 12Dπ1 e(πn+1, πn),∇0w

hϕ〉Bh

0⟨∆GE

∆p

(zhn+1, z

hn

),wh

p

⟩= 〈

phn+ 1

2

ρ0

,whp〉Bh

0⟨∆GE

∆Θ

(zhn+1, z

hn

), whΘ

⟩= 〈Dπ3 e(πn+1, πn), whΘ〉Bh

0,

(5.175)

Similarly, the G-equivariant partial discrete derivatives of the total en-tropy read⟨

∆GS

∆ϕ

(zhn+1, z

hn

),wh

ϕ

⟩= 〈2Fh

n+ 12Dπ1 η(πn+1, πn),∇0w

hϕ〉Bh

0⟨∆GS

∆p

(zhn+1, z

hn

),wh

p

⟩= 0⟨

∆GS

∆Θ

(zhn+1, z

hn

), whΘ

⟩= 〈Dπ3 η(πn+1, πn), whΘ〉Bh

0

(5.176)

Given that, the requirements imposed by the discrete degeneracy condi-tions (5.22) on the discrete brackets (5.171) and (5.172) leads to

S,F(zhn+1, z

hn

)=

⟨2Fh

n+ 12Dπ1 η(πn+1, πn),∇0

[∆GF∆p

(zhn+1, z

hn

)]⟩−⟨

Fhn+ 1

2V∗

Dπ3 η(πn+1, πn)

c,∇0

[∆GF∆p

(zhn+1, z

hn

)]⟩= 0,

(5.177)

and

[[E,F ]](zhn+1, z

hn

)=

⟨∇0

[Dπ3 e(πn+1, πn)

c

], (Θ∗)2 Kh

n+ 12∇0

[1

c

∆GF∆Θ

(zhn+1, z

hn

)]⟩= 0,

(5.178)

Thus, in order for these degeneracy conditions to be fulfilled, the followingrelations must hold

Dπ3 η(πn+1, πn)

cV∗ = 2Dπ1 η(πn+1, πn), Dπ3 e(πn+1, πn) = c (5.179)


Remarkably, the second relation is always satisfied due to the definitionof the specific heat capacity as a positive constant number.

Furthermore, the first relation is elaborated using the second one to give

V∗ = 2Dπ3 e(πn+1, πn)

Dπ3 η(πn+1, πn)Dπ1 η(πn+1, πn), (5.180)

in which, following the continuous definition of the latent heat tensor (3.77),the appearing rate is identified with an algorithmic temperature. What ismore, the resulting algorithmic temperature is the one that makes the specificheat capacity (3.78) be univocally defined within any time interval, that is

Θ∗ =Dπ3 e(πn+1, πn)

Dπ3 η(πn+1, πn)→ c = Dπ3 e(πn+1, πn) = Θ∗Dπ3 η(πn+1, πn), (5.181)

The fully elaborated formulas for the above discrete derivative are for-mally identical to (5.164) changing the density entropy by the temperatureand the internal energy function accordingly.

Simplifying the notation, the algorithmic latent heat tensor is expressedas follows

V∗ = 2Θ∗DCη, (5.182)

with the appearing discrete derivative being formally provided by (5.160),replacing the internal energy function by the density function and the densityentropy state variable by the temperature state variable.

Assuming these definitions and performing the operations in (5.20) usingall the above needed expressions, the temperature-based EEM method forthe nonlinear thermoelastic continuum dynamics reads


∆t〉Bh

0= 〈wh

ϕ,1

ρ0

phn+ 1

2〉Bh

0


∆t〉Bh

0= −〈∇0w

hp,F

hn+ 1

2(2DCe−V∗)〉Bh

0


n

∆t〉Bh

0= −〈w

hΘ

cFhn+ 1

2V∗,∇0

[phn+ 1

2

ρ0

]〉Bh

0

− 〈∇0

[whΘc

],Kh

n+ 12∇0Θ∗〉Bh

0

(5.183)

Finally, from the right hand side of the linear momentum balance, theconsistent evaluation of the symmetric Piola-Kirchhoff stress tensor is iden-tified as

S∗ = 2DCe−V∗ = 2(DCe−Θ∗DCη) = 2DCΨ∗, (5.184)


which has been expressed through an artificial function Ψ∗ to emphasizeits connection with its continuous counterpart (3.18), neglecting the internalvariables. Its elaborated formula responds to the format provided by (5.160).

Interestingly, as in the previously developed temperature-based EEMmethods for discrete finite-dimensional systems, the directionality propertyof the discrete derivative operator allows to recast the last equation, i.e theenergy balance, into a discrete entropy form

〈whΘ,Θ∗η(Ch

n+1,Θhn+1)− η(Ch

n,Θhn)

∆t〉Bh

0= −〈∇0w

hΘ,K

hn+ 1

2∇0Θ∗〉Bh

0(5.185)

Discrete laws of thermodynamics. In contrast to the entropy-basedmethod, the temperature-based method does ensure the preservation of en-ergy and the production of entropy within the Galerkin approximation. Theproof for the energy balance follows from choosing the weighting functionsin (5.183) according to

whϕ = phn+1 − phn, wh

p = ϕhn+1 −ϕhn, whΘ = c = DΘe, (5.186)

which, in this case, do belong to the Galerkin space V h since the specificheat capacity c has been introduced to be a constant positive number due tothe material’s stability issues. In other words, the internal energy functionin terms of temperature of the materials considered depends linearly on tem-perature so that its partial discrete derivative operator results in a constantnumber. Then, they are used in (5.183) to give

〈phn+1 − phn,ϕhn+1 −ϕhn

∆t〉Bh

0= 〈phn+1 − phn,

1

ρ0

phn+ 1

2〉Bh

0

〈ϕhn+1 −ϕhn,phn+1 − phn

∆t〉Bh

0= −〈Fh

n+1 − Fhn,F

hn+ 1

2(2DCe−V∗)〉Bh

0

〈c, Θhn+1 −Θh

n

∆t〉Bh

0= −〈c

cFhn+ 1

2V∗,∇0

[phn+ 1

2

ρ0

]〉Bh

0

− 〈∇0

[cc

],Kh

n+ 12∇0Θ∗〉Bh

0

(5.187)

In view of the two first equations, their right hand sides have to beidentical. On the other hand, using the first equation in the right hand sideof the second equation the following is obtained

〈phn+1 − phn,1

ρ0

phn+ 1

2〉Bh

0= −〈Fh

n+1 − Fhn,F

hn+ 1

2(2DCe−V∗)〉Bh

0

〈c,Θhn+1 −Θh

n〉Bh0

= −∆t〈Fhn+ 1

2V∗,∇0

[ϕhn+1 −ϕhn

∆t

]〉Bh

0

(5.188)


Finally, adding up these equations and using the directionality propertyof the discrete gradient operator leads to

〈phn+1 − phn,1

ρ0

phn+ 1

2〉Bh

0+ 〈1, e(Ch

n+1,Θhn+1)− e(Ch

n,Θhn)〉Bh

0= 0, (5.189)

in where the first term is identified with the total kinetic energy balance andthe second term is the total internal energy balance

Khn+1 −Kh

n + Uhn+1 − Uh

n = 0, (5.190)

demonstrating that the temperature-based EEM method is energy-preservingby design.

The entropy balance can then be derived directly from (5.24) to give

Shn+1 − Shn = ∆t〈∇0

[phn+ 1

2

ρ0

],Fh

n+ 12V∗

1

c

c

Θ∗〉Bh

0

− ∆t〈Fhn+ 1

2V∗

1

c

c

Θ∗,∇0

[phn+ 1

2

ρ0

]〉Bh

0

+ ∆t〈∇0

[1

c

c

Θ∗

],Θ∗,2Kh

n+ 12∇0

[1

c

c

Θ∗

]〉Bh

0

= ∆t〈 1

(Θ∗)2∇0Θ∗,Kh

n+ 12∇0Θ∗〉Bh

0≥ 0

(5.191)

Discrete symmetries. As in the case of the entropy-based method, it canbe demonstrated that the temperature-based poses symmetries respect to theadditive and rotation group and hence preserves the total linear and angularmomentum. The proof is omitted as firmly follows the steps provided inSection 4.9 for the continuous case or the one that led to the demonstrationfor the midpoint standard method to preserve the linear and angular mo-mentums (3.185) - (3.187), this time using the symmetry of the algorithmicPiola-Kirchhoff stress tensor instead of the symmetry of midpoint one.

Implementation issues. The discrete thermodynamical consistency of thetemperature-based method relies on the special evaluation of the temperaturefield inside the time interval provided by

Θ∗ =DΘe

DΘη=

c

DΘη(5.192)

This evaluation generally involves both the motion and the temperaturefields values at the interval extremities so the material gradient defining the


heat conduction must be carefully obtained. Doubtlessly, this issue compli-cates the implementation of the temperature-based EEM method comparedto standard integrations methods based on temperature in where the imple-mentation of this gradient is attained by the gradient of the shape functions,see, for instance, Hesch & Betsch (2011).

Then, the material gradient applied to the algorithmic temperatures re-sults in

∇0Θ∗ = ∇0

[DΘe

DΘη

]=−Θ∗

DΘη∇0[DΘη] (5.193)

The expression of the partial discrete derivative of the density entropyfunction respect to the temperature has the same format of (5.164) so itsgradient can be elaborated as follows

∇0[DΘη] = ∇0

[η(Ch


n+1,Θn) + η(Chn,Θ

hn+1)− η(Ch

n,Θhn)

2(Θhn+1 −Θh

n)

]=∇0

[η(Ch


n+1,Θn) + η(Chn,Θ

hn+1)− η(Ch

n,Θhn)]

2(Θhn+1 −Θh

n)

− DΘη

(Θhn+1 −Θh

n)∇0

[Θhn+1 −Θh

n

](5.194)

The full elaboration of the first term relies on the following type of iden-tities

∇0[η(Chi ,Θ

hn+1)− η(Ch

i ,Θhn)] =

(∂η

∂C(Ch

i ,Θhn+1)− ∂η

∂C(Ch

i ,Θhn)

): ∇0C

hi

+∂η

∂Θ(Ch

i ,Θhn+1)∇0Θh

n+1 −∂η

∂Θ(Ch

i ,Θhn)∇0Θh

n,

(5.195)

for i = n, n+ 1.

In view of this result, for general thermoelastic material models, a suc-cessful implementation of the temperature-based method requires the elabo-ration of the material gradient of the temperatures at each of the time intervalextremities, which follow from the FE-based discretization (3.155), and thematerial gradient of the right Cauchy-Green deformation tensor. This lastoperation is much more involved and requires the implementation of the sec-ond derivatives of shape functions. Recall that they also arise in the midpointstandard formulation of entropy-based continuous equations (3.176).


Fortunately, the modeling of most of the thermoelastic materials is basedon free-energy functions provided by (3.88) with at most linear dependencyon the elastic constants, that is f ′(Θ) = C, so that the term multiplyingthe above involved gradient identically vanishes and the gradient of the al-gorithmic temperatures simplifies to depend only on the gradients of thetemperature evaluated at the extremities of the time interval, thus avoid-ing the implementation and computation of the second derivatives of shapefunctions.

In fact, in such a case, the algorithmic temperature can be expressed justin terms of the temperature evaluated at the time interval extremities suchthat

Θ∗ =DΘe

DΘη=

Θhn+1 −Θh

n

log(Θhn+1/Θ

hn

) , (5.196)

which in the limit case of Θhn+1 = Θh

n is well-defined as

limΘhn+1→Θhn

Θ∗ = Θhn (5.197)

The gradient of the algorithmic temperature results then in

∇0Θ∗ =

(1− Θ∗

Θhn+1

)∇0Θh

n+1 −(

1− Θ∗

Θhn

)∇0Θh

n

log(Θhn+1/Θ

hn

) ,(5.198)

while, in the limit case of Θhn+1 = Θh

n, it reduces to

limΘhn+1→Θhn

∇0Θ∗ =1

2∇0[Θh

n+1 + Θhn] = ∇0Θh

n (5.199)

Then, as the EEM methods are by construction implicit, the linearizationof the equations are required for a full implementation. In the linearizationrespect to the temperature field, the linearized gradient of the algorithmictemperature will be involved so it is provided next

∂(∇0Θ∗)

∂Θn+1

=

−(

∂Θ∗

∂Θn+1

− Θ∗

Θn+1

)∇0Θn+1 +

Θn+1

Θn

∂Θ∗

∂Θn+1

∇0Θn −∇0Θ∗

Θn+1 log

(Θn+1

Θn

)(5.200)

with

∂Θ∗

∂Θn+1

=

Θ∗(

1− Θ∗

Θhn+1

)Θhn+1 −Θh

n

(5.201)

224 5.6. EEM methods for nonlinear thermo-dissipative dynamics

Finally, in the limit case Θhn+1 = Θh

n the linearized contribution vanishes

limΘhn+1→Θhn

∂(∇0Θ∗)

∂Θn+1

= 0 (5.202)

5.6 EEM methods for nonlinear thermo-dissipative dy-namics

EEM methods are derived from the methodology described in subsection5.2.2 for the dynamics of thermo-dissipative continuum presented in Chapter3 in a classical fashion and in Chapter 4, Section 4.7, within a geometricapproach. As for the dissipative systems discussed so far, two EEM methodsare to be proposed in terms of the thermodynamical variable used: entropyor temperature.


For the derivation of the entropy-based EEM method, the infinite-dimen-sional state space (4.122) is transformed into an finite-dimensional by usingthe shape functions space defined in a regular partition of the continuum ofFE-type, that is

S h = zh =N∑a=1

Naza = (ϕh,ph, ηh,Λα,h) : Bh0 7→

Rd × Rd × R× Sym(T2d)m, det (∇0ϕ

h) > 0,(5.203)

The tangent space is also identified with the finite-dimensional space oflinear combinations of shape functions to give

TS h = wh =N∑a=1

Nawa = (whϕ,w

hp, w

hη ,w

hΛα) : Bh

0 7→

Rd × Rd × R× Sym(T2d)m

(5.204)

Then, the discrete Poisson bracket is the canonical one provided by

F ,G(zhn+1, z

hn

)=

⟨∆GF∆ϕ

(zhn+1, z

hn

),∆GG∆p

(zhn+1, z

hn

)⟩−⟨

∆GF∆p

(zhn+1, z

hn

),∆GG∆ϕ

(zhn+1, z

hn

)⟩ (5.205)


The novelty of the thermo-dissipative continuum system rests on the Dis-sipative bracket which includes terms accounting for the internal dissipationdue to the development of internal variables. Then, according to (5.26), itsdiscrete counterparts is provided by

[[F ,G]](zhn+1, z

hn

)=

⟨∇0

[1

Θ∗∆GF∆η

(zhn+1, z

hn

)],Kh

n+ 12

(Θ∗)2∇0

[1

Θ∗∆GG∆η

(zhn+1, z

hn

)]⟩−⟨

1

Θ∗∆GF∆η

(zhn+1, z

hn

), D∗∆

GG∆η

(zhn+1, z

hn

)⟩+

m∑α=1

⟨∆GF∆Λα

(zhn+1, z

hn

),K(Qα,∗)

∆GG∆η

(zhn+1, z

hn

)⟩(5.206)

where Θ∗ is the algorithmic temperature, Khn+ 1

2

is the algorithmic conductiv-

ity tensor provided by (5.151), D∗ is the algorithmic dissipation rate, Qα,∗ isthe algorithmic thermo-dissipative driving stress tensor and K is the positive-definite functional defining dissipative materials.

In order to construct a entropy-based EEM method that respects thesymmetries of the continuous system, the following invariant function is con-sidered


h, ‖ph‖2, ηh,Λh,α) = (Fh,TFh, ‖ph‖2, ηh,Λh,α), (5.207)

with α = 1, . . . ,m. Then, the reduced internal energy and entropy functionsread

E(π(zh)) =

∫Bh

0

(π2

2ρ0

+ e(π1, π3, π3+α)

)dV0, α = 1, . . . ,m

S(π(zh)) =

∫Bh

0

π3dV0,

(5.208)

Given that, the G-equivariant discrete functional derivative of the total


energy (4.136)1 is elaborated to yield⟨∆GE

∆z

(zhn+1, z

hn

),wh

⟩= 〈DG

[‖ph‖2

2ρ0

+ e(Ch, ηh, ,Λh,α)

](zhn+1, z

hn),wh〉Bh

0

= 〈Dπ(zhn+ 1

2)TD

[π2

2ρ0

+ e(π1, π3, π3+α)

](πn+1, πn),wh〉Bh

0

= 〈D[π2

2ρ0

+ e(π1, π3, π3+α)

](πn+1, πn), Dπ(zh

n+ 12) ·wh〉Bh

0

= 〈Dπ1 e(π1, π3, π3+α), Dπ1 ·whϕ〉Bh

0

+ 〈Dπ2

[π2

2ρ0

], Dπ2 ·wh

p〉Bh0

+ 〈Dπ3 e(πn+1, πn), Dπ3whη 〉Bh

0

+m∑α=1

〈Dπ3+α e(πn+1, πn), Dπ3+α : whΛα〉Bh

0,

(5.209)

for any vector wh ∈ TS h. In the same way, the G-equivariant discretefunctional derivative of the total entropy (4.124)2 reads⟨

∆GS

∆z

(zhn+1, z

hn

),wh

⟩= 〈DGη(zhn+1, z

hn),wh〉B0

= 〈Dπ(zn+ 12)TDη(πn+1, πn),wh〉Bh

0

= 〈Dπ3π3, Dπ3 ·wh〉Bh0

(5.210)

In addition to the standard derivatives of the invariant function providedin (5.156), the derivative of the (3 +α)-th invariant must be considered, thatis

Dπ3+α : w = w, ∀w ∈ Sym(T2d) (5.211)

Thereby, the G-equivariant partial discrete functional derivatives gives⟨∆GE

∆ϕ

(zhn+1, z

hn

),wh

ϕ

⟩= 〈2Fh

n+ 12Dπ1 e(πn+1, πn),∇0w

hϕ〉Bh

0⟨∆GE

∆p

(zhn+1, z

hn

),wh

p

⟩= 〈

phn+ 1

2

ρ0

,whp〉Bh

0⟨∆GE

∆η

(zhn+1, z

hn

), whη

⟩= 〈Dπ3 e(πn+1, πn), whη 〉Bh

0⟨∆GE

∆Λα

(zhn+1, z

hn

),wh

Λα

⟩= 〈Dπ3+α e(πn+1, πn),wh

Λα〉Bh0,

(5.212)


whereas the only G-equivariant partial discrete functional derivative of thetotal entropy different from zero is⟨

∆GS

∆η

(zhn+1, z

hn

), whη

⟩= 〈1,wh

η 〉Bh0

(5.213)

As in the pure thermoelastic case, the following definitions for the algo-rithmic symmetry Piola-Kirchhoff stress tensor and temperature are assumed

S∗ = 2Dπ1 e(πn+1, πn) = 2DCe, Θ∗ = Dπ3 e(πn+1, πn) = Dηe (5.214)

Furthermore, the (3 +α)-th partial discrete derivative of the internal en-ergy is identified with the α-th algorithmic thermo-dissipative driving stresstensor

Qα,∗ = −Dπ3+α e(πn+1, πn) = −DΛαe (5.215)

Assuming the above definition, for the Dissipative bracket (5.206) tosatisfy the degeneracy condition (5.22), the algorithmic dissipation rate mustbe

D∗ =m∑α=1

Qα,∗ : K(Qα,∗) ≥ 0 (5.216)

For example, in the common case of one internal variable, the energyfunction becomes a three-variable function so the partitioned case of thediscrete derivative operator respect to the first variable, see Appendix A,leads to

S∗ = 2DCe = Sh

n+ 12

+∆en+1 + ∆en − S

h

n+ 12

: ∆Ch

∆Ch : ∆Ch∆Ch, (5.217)

with∆ei = e(Ch

n+1, ηhi ,Λ

1,hi )− e(Ch

n, ηhi ,Λ

1,hi ), (5.218)

and

Sh

n+ 12

=1

2(Sn+1 + Sn) with Si = 2

∂e

∂C(Ch

n+ 12, ηhi ,Λ

1,hi ) (5.219)

Then, respect to the second variable, the partitioned case of the discretederivative operator gives

Θ∗ = Dηe =e(Ch

n+1, ηhn+1,Λ

1,hn )− e

(Chn+1, η

hn,Λ

1,hn

)2(ηhn+1 − ηhn)

+e(Ch

n, ηhn+1,Λ

1,hn+1)− e(Ch

n, ηhn,Λ

1,hn+1)

2(ηhn+1 − ηhn),

(5.220)


And, finally, the third partitioned discrete derivative is provided by

Q1,∗ = −DΛ1e = Q1,h

n+ 12−

∆en+1 + ∆en + Q1,h

n+ 12

: ∆Λ1,h

∆Λ1,h : ∆Λ1,h∆Λ1,h, (5.221)

with, in this case, the increment of the internal energy being

∆ei = e(Chn+1, η

hi ,Λ

1,hn+1)− e(Ch

n, ηhi ,Λ

1,hn ), (5.222)

and

Q1,h

n+ 12

=1

2

(Q1n+1 + Q1

n

)with Q1

i = − ∂e∂Λ

(Chi , η

hi ,Λ

1,h

n+ 12

) (5.223)

Definitely, the entropy-based EEM method for the nonlinear thermo-dis-sipative dynamics of continua follows from (5.31) using (5.212) and (5.213)to give


∆t〉Bh

0= 〈wh

ϕ,1

ρ0

phn+ 1

2〉Bh

0


∆t〉Bh

0= −〈∇0w

hp,F

hn+ 1

2S∗〉Bh

0


∆t〉Bh

0= −〈∇0

[whηΘ∗

],Kh

n+ 12∇0Θ∗〉Bh

0− 〈w

hη

Θ∗, D∗〉Bh

0

〈whΛ,

Λα,hn+1 −Λα,h

n

∆t〉Bh

0= 〈wh

Λ,K(Q∗,α)〉Bh0

(5.224)

Discrete laws of thermodynamics. The resulting method presents thesame problem as the one for the pure thermoelastic case (5.166) regardingthe conservation of energy as the algorithmic temperature defined by thediscrete derivative operator is out of the Galerkin space V h. A L2-projectionstrategy would then be required to guarantee the energy conservation of thesolution provided by the entropy-based method.

Discrete symmetries. In contrast to the discrete laws of thermodynamics,the discrete symmetries are respected by the resulting entropy-based methoddue to the same arguments as those used for the EEM method (5.166).


The formulation of the temperature-based EEM method departs fromthe consideration of the Galerkin-based finite-dimensional state and tangent


spaces

S h = zh =N∑a=1

Naza = (ϕh,ph,Θh,Λα,h) : Bh0 7→

Rd × Rd × R× Sym(T2d)m, det (∇0ϕ

h) > 0,(5.225)

TS h = wh =N∑a=1

Nawa = (whϕ,w

hp, w

hΘ,w

hΛα) : Bh

0 7→

Rd × Rd × R× Sym(T2d)m

(5.226)

Then, the continuous Poisson brackets (4.139)1 is approximated using(5.25) to give

F ,G(zhn+1, z

hn

)=

⟨∆GF∆ϕ

(zhn+1, z

hn

),∆GG∆p

(zhn+1, z

hn

)⟩−⟨

∆GF∆p

(zhn+1, z

hn

),∆GG∆ϕ

(zhn+1, z

hn

)⟩+

⟨∇0

[∆GF∆p

(zhn+1, z

hn

)],Fh

n+ 12V∗

1

c

∆GG∆Θ

(zhn+1, z

hn

)⟩−⟨

Fhn+ 1

2V∗

1

c

∆GF∆Θ

(zhn+1, z

hn

),∇0

[∆GG∆p

(zhn+1, z

hn

)]⟩,

(5.227)

where V∗ is the algorithmic symmetric latent heat tensor (3.77) and c is theconstant specific heat capacity defined in (3.78).

Likewise, the continuous Dissipative brackets (4.139)2 is approximatedusing (5.26) to give

[[F ,G]](zhn+1, z

hn

)=

⟨1

c

∆GF∆Θ

(zhn+1, z

hn

), (D∗ +H∗d)

Θ∗

c

∆GG∆Θ

(zhn+1, z

hn

)⟩+

m∑α=1

⟨∆GF∆Λα

(zhn+1, z

hn

),K(Q∗,α)

Θ∗

c

∆GG∆Θ

(zhn+1, z

hn

)⟩+

⟨∇0

[1

c

∆GF∆Θ

(zhn+1, z

hn

)],Kh

n+ 12

(Θ∗)2∇0

[1

c

∆GG∆Θ

(zhn+1, z

hn

)]⟩,

(5.228)

where D∗ is the algorithmic dissipation rate, H∗d is the algorithmic thermo-dissipative heating, Θ∗ is the algorithmic temperature, Qα,∗ is the algorithmicthermo-dissipative driving stress tensor and K is the positive-definite func-tional defining dissipative materials and Kh

n+ 12

is provided by (5.151),


Next, the following invariants are identified


h, ‖ph‖2,Θh,Λh,α) = (Fh,TFh, ‖ph‖2,Θh,Λh,α),(5.229)

so that the G-equivariant partial discrete functional derivatives of the totalenergy and entropy functions (4.136) are provided by⟨

∆GE

∆ϕ

(zhn+1, z

hn

),wh

ϕ

⟩= 〈2Fh

n+ 12Dπ1 e(πn+1, πn),∇0w

hϕ〉Bh

0⟨∆GE

∆p

(zhn+1, z

hn

),wh

p

⟩= 〈

phn+ 1

2

ρ0

,whp〉Bh

0⟨∆GE

∆Θ

(zhn+1, z

hn

), whΘ

⟩= 〈Dπ3 e(πn+1, πn), whΘ〉Bh

0⟨∆GE

∆Λα

(zhn+1, z

hn

),wh

Λα

⟩= 〈Dπ3+α e(πn+1, πn),wh

Λα〉Bh0,

(5.230)

⟨∆GS

∆ϕ

(zhn+1, z

hn

),wh

ϕ

⟩= 〈2Fh

n+ 12Dπ1 η(πn+1, πn),∇0w

hϕ〉Bh

0⟨∆GS

∆p

(zhn+1, z

hn

),wh

p

⟩= 0⟨

∆GS

∆Θ

(zhn+1, z

hn

), whΘ

⟩= 〈Dπ3 η(πn+1, πn), whΘ〉Bh

0⟨∆GS

∆Λα

(zhn+1, z

hn

),wh

Λα

⟩= 〈Dπ3+α η(πn+1, πn),wh

Λα〉Bh0,

(5.231)

In view of the above and analogously to the thermoelastic case, the dis-crete degeneracy conditions of the discrete Poisson and Dissipative brackets,(5.228) and (5.227), are satisfied if the following holds

Dπ3 η(πn+1, πn)

cV∗ = 2Dπ1 η(πn+1, πn), Dπ3 e(πn+1, πn) = c, (5.232)

leading to the definitions for the algorithmic temperature and latent heattensor

c = Dπ3 e(πn+1, πn) = Θ∗Dπ3 η(πn+1, πn)→ Θ∗ =DΘe

DΘη,

V∗ = 2Θ∗Dπ1 η(πn+1, πn) = 2Θ∗DCη,

(5.233)


In addition, the same arguments for the degeneracy condition of thediscrete Dissipative bracket lead to the following definitions

Qα,∗ = −Dπ3+α e(πn+1, πn) = −DΛαe

Wα,∗ = Θ∗Dπ3+α η(πn+1, πn) = Θ∗DΛαη,(5.234)

and, in agreement with them, the following ones

D∗ =m∑α=1

Qα,∗ : K(Qα,∗) ≥ 0, H∗d =m∑α=1

Wα,∗ : K(Qα,∗) (5.235)

Then, the temperature-based EEM method directly derives from (5.31)along with (5.230) and (5.231) to give


∆t〉Bh

0= 〈wh

ϕ,1

ρ0

phn+ 1

2〉Bh

0


∆t〉Bh

0= −〈Fh

n+ 12S∗,∇0w

hp〉Bh

0


n

∆t〉Bh

0= −〈∇0

[whΘc

],Kh

n+ 12∇0Θ∗〉Bh

0

− 〈whΘ

c, D∗ +H∗d〉Bh

0

− 〈whΘ

cFhn+ 1

2V∗,∇0

[phn+ 1

2

ρ0

]〉Bh

0

〈whΛ,

Λα,hn+1 −Λα,h

n

∆t〉Bh

0= 〈wh

Λ,K(Qα,∗)〉Bh0,

(5.236)

where the algorithmic symmetric Piola-Kirchhoff stress tensor is provided by(5.184).

In the same way as in the temperature-based EEM method for nonlinearthermoelastic continuum dynamics, the directionality property of the discretederivative operator allows to rewrite the discrete equation due to the energybalance in an entropy form

〈whΘ,Θ∗η(Ch

n+1,Θhn+1,Λ

hn+1)− η(Ch

n,Θhn,Λ

hn)

∆t〉Bh

0= −〈∇0w

hΘ,K

hn+ 1

2∇0Θ∗〉Bh

0

− 〈whΘ, D∗〉Bh0

(5.237)

Discrete laws of thermodynamics. The laws of thermodynamics in theirdiscrete form (5.21) are ensured by the temperature-based EEM method

232 5.7. EEM method for nonlinear isothermal dissipative dynamics

(5.236) due to the same arguments argued in the pure thermoelastic case.Particularly, the total entropy balance in any time step can be detailed byusing (5.231) in (5.33) to give

Shn+1 − Shn = ∆t〈 1

(Θ∗)2∇0Θ∗,Kh

n+ 12∇0Θ∗〉Bh

0+ 〈 1

(Θ∗)2, D∗〉Bh

0≥ 0, (5.238)

which is the discrete counterpart of (3.101).

Discrete symmetries. Based on the same arguments as those used for eachof the EEM methods proposed so far, it can be asserted that the temperature-based EEM method for nonlinear thermo-dissipative dynamics is momentummaps-preserving, particularly the total linear and angular momentums arediscretely preserved along the system solution.

5.7 EEM method for nonlinear isothermal dissipative dy-namics

In order to derive EEM methods for the isothermal limit case studied inSection 3.6, a proper discrete counterpart that preserve the metriplectic struc-ture of such problems (4.146) must be proposed. To this end, the Galerkinformulation is assumed for the spatial discretization of both the infinite-dimensional state space and its tangent space S h ≡ V h and TS h ≡ W h.

Definition 5.6. A discrete infinite-dimensional smooth isothermal dissipa-tive system with symmetries is a discrete system governed by the consistentdiscrete equations

F(zhn+1)−F(zhn)

∆t= [[F , E]]

(zhn+1, z

hn

), (5.239)

for all F : S h 7→ R and E : S h 7→ R the total energy understood as kineticplus internal with the latter being provided by the free-energy function interms of deformation and internal variable (4.149).

The operator [·, ·] : S h × S h 7→ R is a consistent second-orderdiscrete counterpart of the bilinear Poisson-Dissipative operator introducedin (4.144).

Proposition 5.5. The solution of discrete infinite-dimensional dissipativesystems (5.239) agrees with the discrete form of the laws of thermodynamicsprovided by

E(zhn+1) ≤ E(zhn), (5.240)


if the discrete bilinear operator satisfy

[[E,E]] ≤ 0, (5.241)

for every functional F : S h → R.

Using this definition, an EEM method for the isothermal dissipative con-tinuum dynamics can be developed by first considering the Galerkin-basedfinite-dimensional state space

S h = z =N∑a=1

Naza = (ϕh,ph,Λα,h) : Bh0 7→

Rd × Rd × Sym(T2d)m, det(∇0ϕ) > 0, (5.242)

along with its tangent space

TS h = wh =N∑a=1

Nawa = (whϕ,w

hp,w

hΛα) : B0 7→

Rd × Rd × Sym(T2d)m

(5.243)

Then, the Poisson-Dissipative bracket (4.151) is approximated by usingthe G-equivariant discrete derivative operator to give

[[F ,G]] =

⟨∆GF∆ϕ

(zhn+1, z

hn

),∆GG∆p

(zhn+1, z

hn

)⟩−⟨

∆GF∆p

(zhn+1, z

hn

),∆GG∆ϕ

(zhn+1, z

hn

)⟩+

m∑α=1

⟨∆GF∆Λα

(zhn+1, z

hn

),K(−∆GG

∆Λα

(zhn+1, z

hn

))⟩,

(5.244)

with K being the positive semidefinite functional that defines dissipativematerials.

Now, the following invariants respect to the additive group G = Rd andthe rotation G = SO(3) are identified as

π(zh) = ((∇0ϕh)T∇0ϕ

h, ‖ph‖2,Λh,α) = (Fh,TFh, ‖ph‖2,Λh,α), (5.245)

so that the total energy (4.149) is expressed through them to obtain itsreduced form

E(π(zh)) =

∫Bh

0

(π2

2ρ0

+ Ψ(π1, π2+α)

)dV0, α = 1, . . . ,m (5.246)


The definition of theG-equivariant discrete functional (5.30) is elaboratedwith the help of the reduced energy function and the invariant function toarrive at⟨

∆GE

∆ϕ

(zhn+1, z

hn

),wh

ϕ

⟩= 〈2Fh

n+ 12Dπ1Ψ(πn+1, πn),∇0w

hϕ〉Bh

0⟨∆GE

∆p

(zhn+1, z

hn

),wh

p

⟩= 〈

phn+ 1

2

ρ0

,whp〉Bh

0⟨∆GE

∆Λα

(zhn+1, z

hn

),wh

Λα

⟩= 〈Dπ2+αΨ(πn+1, πn),wh

Λα〉Bh0,

(5.247)

from which the following discrete counterparts of the constitutive laws (3.103)are identified

S∗ = 2Dπ3Ψ(πn+1, πn) = 2DCΨ,

Qα,∗ = −Dπ3+αΨ(πn+1, πn) = −DΛαΨ(5.248)

Now, using (5.244), (5.247) and (5.248) in (5.239), the EEM method forisothermal dissipative continuum dynamics reads


∆t〉Bh

0= 〈wh

ϕ,1

ρ0

phn+ 1

2〉Bh

0


∆t〉Bh

0= −〈∇0w

hp,F

hn+ 1

2S∗〉Bh

0

〈whΛ,

Λα,hn+1 −Λα,h

n

∆t〉Bh

0= 〈wh

Λ,K(Qα,∗)〉Bh0

(5.249)

The resulting method is essentially the Energy-Momentum method com-bined with a consistent integration of the evolution equations for the internalvariables that ensure the dissipation.

Discrete laws of thermodynamics. As was discussed in Section 3.6,isothermal dissipative systems can only evolve by dissipating energy. There-fore, the energy balance in each of the time steps of a discrete setting mustbe non-positive if the discrete setting agrees with the thermodynamics.

Ehn+1 − Eh

n = 〈−Qα,∗,K(Qα,∗)〉Bh0

= −〈1, D∗〉Bh0≤ 0 (5.250)

In other words, the augmented Hamiltonian in the sense of (3.107) isdiscretely preserved

Ehn+1 = Eh

n (5.251)


Discrete symmetries. Likewise the continuous equations (3.166), the lin-ear and angular momentum are preserved quantities (first integrals) of thediscrete solution provided by (5.249). This can be verified by both usingthe discrete counterpart of the Poisson-discrete bracket (5.244) along withthe discrete version of the Noether’s theorem 5.4 or by resorting to the pro-cedure applied for the midpoint standard method carried out in (3.185) -(3.187).

Viscoelastic continuum dynamics. The viscoelastic continuum dynamicspresented in subsection (3.6.1) is based on the free-energy function (3.109)which leads to the following form of the dissipative functional

K(Qα) =1

ναQα (5.252)

Then, the EEM method simplifies to


∆t〉Bh

0= 〈wh

ϕ,1

ρ0

phn+ 1

2〉Bh

0


∆t〉Bh

0= −〈∇0w

hp,F

hn+ 1

2S∗〉Bh

0

〈whΛ,

Λα,hn+1 −Λα,h

n

∆t〉Bh

0= 〈wh

Λ,1

ναQα,∗〉Bh

0,

(5.253)

which can be seen as a natural generalization of the Energy-momentummethod for hyperelastic solids to account for viscoelastic effects.

On the other hand, the energy balance is specified by

Ehn+1 − Eh

n = −∆t

∫Bh

0

1

να‖Qα,∗‖2dV0 ≤ 0 (5.254)

Remark 5.5. The implementation of the EEM method for viscoelastic con-tinuum dynamics depends on the number of internal variables chosen tomodel the viscous effects. Not only as many dissipative evolution equationsas the total number of internal variables should be solved, but a re-calculationof the expression for the time-discrete constitutive laws (5.248)2, defined bythe discrete derivative operator, according to (A.7). In Appendix A, detailedexpressions are considered for models with one, two or three internal vari-ables. In the simulations of Chapter 6, just one internal variable will beemployed.

Remark 5.6. An elaboration of expression (5.253)3, taking into account thediscrete derivative operator detailed in the Appendix A and the free energy


function (3.109), reveals that it coincides with the standard trapezoidal ap-proximation of the evolution equations for the internal variables. Note thatthis coincidence is due to the specific dependency of the free energy functionon the internal variables.

Simulations

Chapter

6This Chapter contains a series of simulations to illustrate the poten-

tial scope of applications that can be addressed by EEM methods. Theselected simulations have been divided into three different areas: isothermalviscoelastic solids, thermoelastic solids and multi-body systems composed ofisothermal viscoelastic solids or thermo-viscoelastic stiffeners.

All the simulations make use of trilinear hexahedral eight-nodes mesheswith fully nonlinear geometrical formulation, as introduced in Chapter 3 anddetailed in Appendix B. EEM methods’ solutions will be compared withboth Trapezoidal and Midpoint methods in order to highlight their superiorperformance for the selected time step sizes.

Most of the work of this dissertation has consisted in the implementa-tion of the proposed EEM methods into an C++ open (upon request) codedeveloped within a small division of the Computational Mechanics Group atTechnical University of Madrid and led by Dr. Juan Carlos Garcıa Orden.The code, called Eppi, was primarily intended for the statics and dynamicsof multi-body systems composed by rigid and deformable bodies. After thework done for this dissertation, the code has been extended to deal withthermo-dissipative coupled deformable bodies.

Eppi is an object-oriented and MPI-based parallel computing softwarewith the following main features:

• General purpose: statics and dynamics of multi-body systems, com-posed by rigid and thermal-dissipative coupled deformable bodies.

https://bitbucket.org/scondemartin/eppi

238 6.1. Dynamics of isothermal viscoelastic solids

• Object-oriented software:

• Parallel computing: Mpi.

• Solvers and linear algebra: SuperLU, UMFPACK, Eigen.

• Graphics output:

• Version control system:

• Repositories [access upon request]:

1. https://bitbucket.org/scondemartin/eppi

2. https://code.google.com/p/eppi/

6.1 Dynamics of isothermal viscoelastic solids

In this section, two representative simulations are carried out in order toillustrate the properties and the performance of the isothermal EEM methodprovided by (5.253). For the first example, the vibration of a cantileverbeam is simulated, demonstrating the strength of accurately dissipating theenergy according to the laws of thermodynamics. Finally, the simulation ofa tumbling L-shaped block is performed to verify the preservation laws andthe corresponding effect on the stability and robustness in a problem withnotorious geometrical non-linearities (large motions and deformations).

For the viscoelastic behavior in isothermal conditions, the model due toHolzapfel and Simo described in subsection 3.6.1 is considered with only oneinternal variable. For the hyperelastic part, the Neo-Hookean material modelprovided in Bonet & Wood (2008) is used, but it would also be possible toconsider other common rubber-like materials, such as Ogden-type models,etc.

6.1.1 Vibrating cantilever beam

The vibration of a cantilever beam, depicted in Figure 6.1, under its ownweight is analyzed. The beam is 4 m long, has a 0.5 × 0.5 m cross section,and the finite element mesh has 160 elements. The material behavior is basedon a decoupled Neo-Hookean model, see for instance Bonet & Wood (2008),

https://bitbucket.org/scondemartin/eppi

https://code.google.com/p/eppi/

6. Simulations239

x

zy

g

f(t)

Figure 6.1. Cantilever beam and external actions

with constants summarized in Table 6.1. The beam is set in motion by theaction of a point load applied at the extreme cross section centroid, describedin (6.1), which vanishes at 2 s and g is set to 9.81 ms−2.

f(t) = f(t)

0

1000

1000

N, f(t) =

t for t ≤ 2 s

0 for t > 2 s(6.1)

The evolution of the energy of the system (including the dissipated en-ergy) must remain constant in the free-force phase. In other words, the energyintroduced by the force will be partially consumed by the development of theviscoelastic deformations such that the beam vibrations will be damped out.

Table 6.1. Viscoelastic decoupled Neo-hookean model properties

Shear moduli µ 422.5 kPa

Bulk modulus B 104 kPa

Retardation/Relaxation time τ1 1.0 s

Dissipative process parameter µ1 542.5 kPa

Density ρ0 600 kg/m3

The motion of the beam is integrated for the time interval [0, 50] s withtime step size of ∆t = 0.1 s using the Trapezoidal, Midpoint and the isother-mal EEM method. Some snapshots of the simulation are plotted in Figure


6.2, where the large deflection of the beam can be observed. Figure 6.3 showsthe vertical displacement of the extreme cross section centroid in time, andFigure 6.4 shows the horizontal projection of the three-dimensional trajectoryof the same point.

x

z yg

f(t)

Figure 6.2. Snapshot of the motion of the beam (from left to right, top tobottom)

Finally, Figure 6.5 depicts the evolution of the energy obtained with theclassic methods and illustrates that they are unable to comply with the first

6. Simulations241

0 10 20 30 40 50

3

2

1

0

t [s]

uz

[m]

Consistentisothermal EEM

Figure 6.3. Vertical displacement of the extreme cross section centroid

2 1 0 1 22

1

0

1

2

ux [m]

uy

[m]

Consistentisothermal EEM

Figure 6.4. XY Trajectory of the extreme cross section centroid


law of thermodynamics (they lead to non-decreasing evolution of energy atcertain time-steps in the free-force phase of the motion), and ultimately failsbefore the end of the simulation.

On the other hand, the isothermal EEM method, Figure 6.6, succeedsby assuring the fulfillment of the laws of thermodynamics, providing withstability to the simulation.

0 1 2 3 4 5 6

500

0

−500

−1000

−1500

t [s]

E[J]

Trapezoidal

Midpoint

Figure 6.5. Beam - Energy: E = K + U

6. Simulations243

0 10 20 30 40 50

1000

0

1000

2000

3000

4000

t [s]

E[J

] Consistentisothermal EEM

Figure 6.6. Beam - Energy: E = K + U

6.1.2 A tumbling L-shaped block

In this example the dynamics of an L-shaped block with viscoelastic re-sponse in isothermal conditions is considered. The L-shaped geometry is pro-vided by the mesh used, among many other works, in Armero & Zambrana-Rojas (2007); Meng & Laursen (2002); Simo & Tarnow (1992) which is de-picted in Figure 6.7. The viscoelastic material behavior is based on a decou-pled Neo-Hookean model, whose main characteristics are collected in Table6.2.

Table 6.2. Viscoelastic decoupled Neo-Hooke model properties

Bulk modulus B0 5243 Pa

Shear modulus µ 1004 Pa

Retardation/Relaxation time τ 20 s

Dissipative process parameter µ1 50.2 Pa

Density ρ0 100 kgm−3


FB(t)

FA(t)

FA(t) = −FB(t) = f(t)

256

512

768

N

f(t) =

t for t ≤ 2.5 s

5− t for 2.5 ≤ t ≤ 5 s

0 for t > 5 s

Figure 6.7. L-shaped block mesh and external forces

During an initial phase the block is subjected to two resultant loads atthe top and left end surfaces, faces A and B, respectively, according to Figure6.7, where the resultant values are also defined. These resultants arise froma nodal forces distribution, which is symmetric, consisting in the applicationof the sixteenth part of the resultant in each node in the face. At the endof this phase, these forces vanish and, as a result, the block tumbles free ofexternal forces in the space undergoing finite deformation accompanied bylarge overall rotations and translations.

This example is integrated for a time interval [0, 200] s with a time stepsize ∆t = 0.5 s using Trapezoidal, Midpoint and the isothermal EEM method.In accordance with the simulation, the energy evolution of the system mustincrease during the loaded phase and then must be unconditionally decreasingin the free-force phase.

First of all, a sequence of deformed snapshots calculated with the isother-mal EEM method is provided in Figure 6.8. Interestingly, as the simulationprogresses, the L-block resembles more to the undeformed shape. That hap-pens because the dilational-derivatoric material model concentrates the vis-coelastic effects on the derivatoric deformations.

Finaly, Figure 6.9 depicts the evolution of the energy of the system ob-tained with each of the compared methods and clearly shows that the classicalones provide a non-physical evolution of the energy in the free-force phase ofthe motion, which eventually cause these methods to collapse. Obviously, thefinal growth of the energy results in polluted solutions, distorted shapes ofthe block and non-physical viscoelastic stresses. Instead, the isothermal EEMmethod reproduces the accurate physical behavior, enhancing the stability

6. Simulations245

Figure 6.8. Snapshots of the motion of the L-shaped block (from left toright, top to bottom)


0 20 40 60 80 100 120 140 160 180 2000.8

0.9

1

1.1

1.2·104

E[J]

isothermal EEMTrapezoidalMidpoint

Figure 6.9. L-Shaped block - Energy E = K + U

and robustness of the method and allowing to calculate long term simulationswith the same time step as the one which makes classical methods explode.In order to illustrate this assertion, the evolution of the total energy of thebody (kinetic plus internal) is computed in a long term simulation of 2000 s,with the same time step as before, in Figure 6.10. Therein, the loss of energyproduced in the unloaded phase of the motion due to the viscoelastic dissi-pation is clearly visible. Both plots demonstrate that the classical methodsfail to satisfy the laws of thermodynamics for moderately large time steps,resulting either in physically inaccurate solutions or in the methods’ failure.

In addition to this characteristics, the isothermal EEM method also pre-serves the angular momentum as demonstrates Figure 6.11. It is also provideda zoom to verify that Midpoint method preserves the angular momentum too,as expected, whereas Trapezoidal method does not.

6. Simulations247

0 200 400 600 800 1000 1200 1400 1600 1800 20000.8

0.85

0.9

0.95

1

1.05

1.1

1.15·104

t [s]

E[J]

isothermal EEM

Figure 6.10. L-Shaped block - Long Term Simulation

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

·104

t [s]

‖J‖[N·m

]

5 25 50 75 100 125 15040500

41417

41800

isothermal EEMTrapezoidalMidpoint

Figure 6.11. L-Shaped block - Angular Momentum Norm

248 6.2. Dynamics of thermoelastic solids

6.2 Dynamics of thermoelastic solids

In this section, the performance of the EEM methods for the dynamicsof thermoelastic solids devised in Section 5.5 is analyzed. The temperature-based EEM method will attract most of the attention due to its novelty.Two simulations are studied based on a fairly complex mesh resulting fromtwisting a block and on the L-shaped block mesh previously introduced forthe analysis of the isothermal viscoelastic dynamics. The first mesh is usedin a simulation with no Dirichlet boundary conditions and, therefore, it canbe approached by both the entropy- and temperature-based EEM methods.In contrast, the L-shaped block mesh is employed in a example includingnatural Dirichlet boundary conditions so only the temperature-based EEMcan handle with it.

In both simulations, the thermoelastic behavior is modeled by a freeenergy function of the form provided in (3.86)-(3.87), introduced in Section3.4, and based on the decoupled Neo-Hookean hyperelastic model previouslyintroduced.

6.2.1 Twisted Block

The simulation consists in the free flight of a twisted block caused byan initial velocity applied on its top. The twisted block was introducedin Romero (2010b) and is constructed from a block of dimensions 4 × 4 ×10π whose top face is twisted 360 with respect to the opposite one. Afinite element mesh of non-regular hexahedra is constructed by partitioningthe short edges and long edges, respectively, in 3 and 19 identical divisions.Figure 6.12 depicts the twisted block mesh along with the selected frame ofreference in which the initial velocity of all the points of the top surface canbe expressed as v0 = 10ex + 10ez. The constitutive law of the thermoelasticis based on the material constants collected in Table 6.3.

Then, the long term motion of the free-flying twisted block is integratedup to 500 s with a relatively large time step size of ∆t = 2.0 s using thestandard Midpoint method formulated in entropy and temperature variablesand the temperature-based EEM method (5.183). The performance of theentropy-based EEM method is thoroughly analyzed in Romero (2010b).

Some snapshots of the motion obtained with the EEM method are shownin Figure 6.20. Also, contour plots of temperatures in the deformed meshesare displayed. The initial velocity induces two main effects; on the one hand,

6. Simulations249

X

Z

Y

v0 = 10√2

Figure 6.12. Twisted block and initial conditions

a global rotation of the block which leads it from its initial vertical positionto an horizontal one, and, on the other hand, a slight torsion of the block,causing deformations that heat up the block due to the coupling effects. Theglobal variation of temperature is moderate although the temperature dra-matically increases at the beginning of the motion, where the deformation aremore pronounced. After that, the temperature of the block is slowing homog-enizing due to the heat conduction process dominated by a low conductivityparameter.

Table 6.3. Thermoelastic decoupled Neo-Hooke model properties

Lame parameters λ, µ 1000 Pa

Specific heat capacity c 1 JK−1

Dilatation coefficient α 10−4 JK−1

Conductivity k 0.1 JK−1s−1

Ref. Temperature Θref 273 K



Apparently, the just described dynamics does not pose special challenges,however, none of the standard methods is able to provide an stable solutionfor the selected time step size. The total energy evolution, depicted in Figure6.14, reveals that the temperature-based Midpoint method outperforms theentropy-based one, although none of them agrees with the first law of thermo-dynamics. In Figure 6.15, the curves of the total entropy evolution providedby the two standard methods are depicted, confirming that the entropy-basedMidpoint method provides the correct non-decreasing behavior dictated bythe second law of thermodynamics (3.193), while the temperature-based Mid-point method fails to adhere to such principle. This observation corroboratesthe general conclusion drawn along this dissertation asserting that the capac-ity for integration methods to agree with the first law is much more criticalregarding the methods’ stability than the capacity to meet the second prin-ciple of thermodynamics.

In contrast, the temperature-based EEM method manages to meet thetwo laws of thermodynamics for such a moderately large time step size, asshown in Figure 6.16 and Figure 6.17, allowing for stable solutions in longterm simulations. Additionally, the total angular momentum is preserved ascan be seen in Figure 6.18.

6. Simulations251

273.2

273.5

273.8

273

274

Temperature

Figure 6.13. Snapshots and contour plots of temperature of the Twistedblock (from top to bottom, left to right). Temperature-based EEM

integration, ∆t = 2 s


0 50 100 150 200 250 300 350 400 450 50080

100

120

140

160

t [s]

E[J]

Temperature-based MidpointEntropy-based Midpoint

Figure 6.14. Standard methods: Total energy evolution

0 50 100 150 200 250 300 350 400 450 500−0.2

−0.15

−0.1

−0.05

0

0.05

t [s]

S[JK

−1]

Temperature-based MidpointEntropy-based Midpoint

Figure 6.15. Standard methods: Total Entropy evolution

6. Simulations253

0 50 100 150 200 250 300 350 400 450 50080

82

84

86

88

90

t [s]

E[J]

Temperature-based EEM

Figure 6.16. Total energy evolution

0 50 100 150 200 250 300 350 400 450 5000

1

2

3

4

5

6

7·10−4

t [s]

S[JK

−1]


Figure 6.17. Total Entropy evolution


0 50 100 150 200 250 300 350 400 450 500

1426

1428

1430

1432

1434

1436

1438

1440

t [s]

‖J‖[N·m

]


Figure 6.18. Total Angular momentum evolution

6. Simulations255

6.2.2 L-shaped block with Dirichlet initial-boundary conditions

In order to further exploit the performance of the temperature-basedEEM method, consider the L-shaped block of subsection 6.1.2, now made ofthermoelastic material and, again, subject to two external forces that vanishat certain time. In addition, in faces A and B, Dirichlet initial-boundary con-ditions are imposed by setting temperatures other than the reference temper-ature, as indicated in Figure 6.19. After the initial instant, the temperatureof the faces are not imposed any longer so that they will evolve accordingto the balance laws. This consideration prevents the entropy-based EEMmethod (5.166) to be directly applied, that is, its application would requirean involved preprocess to infer the values of entropy at every node.

Θ0,A = 300K

Θ0,B = 250K

X

ZY

FA(t)

FB(t)

FA(t) = −FB(t) = f(t)

256

512

768

N

f(t) =

t for t ≤ 2.5 s

5− t for 2.5 ≤ t ≤ 5 s

0 for t > 5 s

Figure 6.19. L-shaped block with Dirichlet initial-boundary conditions

The example is integrated using the standard Midpoint and Trapezoidalmethods and the temperature-based EEM up to 100 s and time step sizeof ∆t = 0.2 s. The rest of the data required to perform the simulation isprovided by the thermoelastic material parameters are in Table 6.4

Then, the result provided by the temperature-based EEM method isplotted in Figure 6.20, which contains the contour plots of temperatures fordifferent snapshots of the motion described by the block. The initial externalactions induce both large deformations and rotations in the block while thethermal evolution is controlled by the heat conduction process which slowly


Table 6.4. Thermoelastic decoupled Neo-Hooke model properties

Shear modulus µ 997.5 Pa

Bulk modulus B0 5209 Pa

Specific heat capacity c 100 JK−1

Dilatation coefficient α 2.233 · 10−4 JK−1

Conductivity k 10 JK−1

Ref. Temperature Θref 293.15 K


compensates the gradient of temperature caused by the initial-boundary con-ditions.

For the selected time step size, only the temperature-based EEM methodis able to integrate the whole time interval. The standard methods, formu-lated in temperature variable to handle with the initial-boundary conditions,become unstable before the end of the integration. The instabilities are di-rectly related to the incapacity to fulfill the first law of thermodynamics in theforce-free phase of the evolution as shows Figure 6.21. Neither is the secondlaw satisfied by the standard methods as can be observed in Figure 6.22. Incontrast, both figures clearly show that the temperature-based EEM methodprovides evolutions of the total energy and total entropy in agreement withthe laws of thermodynamics, this way successfully managing to complete thelong term simulation. In addition, Figure 6.23 demonstrates that the solutionprovided by the EEM method respects the symmetries of the isolated evolu-tion equations by plotting the total angular momentum which is preservedduring the force-free phase of the motion of the block.

6. Simulations257

Figure 6.20. Snapshots and contour plots of temperature of the L-shapedblock (from top to bottom, left to right). Temperature-based EEM

integration, ∆t = 0.2 s


0 10 20 30 40 50 60 70 80 90 100

−16

−14

−12

−10

−8

−6

−4

·103

t [s]

E[J]

Temperature-based EEMTemperature-based Midpoint

Temperature-based Trapezoidal

Figure 6.21. Total energy E = K + U

0 10 20 30 40 50 60 70 80 90 100−62

−61

−60

−59

−58

−57

−56

−55

t [s]

S[JK

−1]



Figure 6.22. Total Entropy

6. Simulations259

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

·104

t [s]

‖J‖[N·m

]



Figure 6.23. Total Angular Momentum

260 6.3. Isothermal viscoelastic applications to multibody system dynamics

6.3 Isothermal viscoelastic applications to multibody sys-tem dynamics

Before presenting particular applications, the multibody framework isbriefly introduced, focusing on how to treat the deformable isothermal dissi-pative solids so that the laws of thermodynamics are satisfied by design.

6.3.1 Multibody framework

All nodal coordinates xa and nodal variations δxa are collected in vectorsqD and δqD respectively. As a result, the discrete weak formulation (3.166)can be written in matrix format as

(δqD)T(

∫Bh

ρNTN dV︸︷︷︸MD

qhD −∫

Bh0

BSh dV0︸︷︷︸f intD

−f extD ) = 0 (6.2)

with (δqD) ∈ Vn defined in a finite-dimensional space. Accordingly, theisothermal EEM method (5.249) can be written in matrix form in terms ofthe nodal positions qD(t) and velocities qD(t) as follows

MD(qDn+1 − qDn)−∆t(f int

?

D + f ext?

D

)= 0

qDn+1 =2(qDn+1 − qDn)

∆t− qDn , (6.3)

where f int?

D and f ext?

D are second-order algorithmic approximations of the inter-nal and external forces respectively in the time interval [tn+1, tn], that ensureexact discrete energy consistency based on the discrete derivative operator.Note that internal variables Λα,h do not explicitly appear in (6.3) since theyare condensed at element level.

A MBS formed by several deformable bodies (Bi, i = 1, .., N) is achievedby simply collecting in qD all the variables of the different bodies, assemblingthe matrices and forces and adding the constraint terms associated with thekinematic pairs. These constraint terms will be explained with some moredetail in the following section.

6.3.2 Rigid bodies and constraints

A set of dependent coordinates will be used to define the position of arigid body BR, collected in vector qR : [0, T ] → R3m. Deliberatly, these

6. Simulations261

coordinates will always be the spatial (inertial) coordinates of m selectedmaterial points in BR. In terms of them, the kinetic energy (KR) can alwaysbe expressed as

KR =1

2qTR · (MRqR) (6.4)

MR being the mass matrix, which will be constant provided that m ≥ 4and the selected points are non-coplanar and non-aligned in groups of threeGarcıa Orden & Goicolea (2000).

As long as qR collects coordinates of material points within the rigid body,constant-distance constraints are necessary. These constraints are holonomicand scleronomic Φ : R3m 3 qR 7−→ Φ(qR) ∈ Rq; thus, they do not dissipateenergy. The weak form of the constrained problem, based on the introductionof Lagrange multipliers λ ∈ Rq is

δqTR ·MRqR + δqT

R ·[(DΦ)Tλ

]+ δλT ·Φ−

− δqTR · f extR = 0

∀δqR ∈ V3m,∀δλ ∈ X q, (6.5)

where f extR is the vector of external forces and fΦR = −(DΦ)Tλ is the con-straint force vector with DΦ = ∂Φ/∂qR. The enforcement of the constraintswith the penalty method is equivalent to consider λ = αΦ in the formerexpression , where α is a (q × q) constant penalty matrix. Generalising theabove approach, a MBS formed by several rigid bodies (BRi , i = 1, .., N) isachieved by simply collecting in qR all the variables of the different bodiesand adding the constraint terms associated with the kinematic pairs.

6.3.3 Flexible multibody system

Consider now a multibody system composed of sets of both rigid (BR)and deformable (BD) bodies. Rigid bodies are defined by coordinates qR :[0, T ] → Rm and the deformable ones by qD : [0, T ] → Rn. Constraints areclassified into three categories: a) constraints ΦR involving rigid body pointsonly; b) constraints ΦD involving deformable body points only and, finally,c) constraints ΦRD involving rigid and deformable body points.

The weak form of the dynamic equation with Lagrange multipliers may


be written as

δqTR ·MRqR + δqT

D ·MDqD + δqTD · f intD +

+ δqTR ·[(DΦR)TλR + (DRΦRD)TλRD

]+

+ δqTD ·[(DΦD)TλD + (DDΦRD)TλRD

]−

− δλTR ·ΦR − δλT

D ·ΦD − δλTRD ·ΦRD

− δqTR · f extR − δqT

D · f extD = 0

∀δqR ∈ Vm, ∀δqD ∈ Vn,∀δλR ∈ X qR , ∀δλD ∈ X qD , ∀δλRD ∈ X qRD , (6.6)

where DR and DD are respectively the derivative operators for ΦRD withrespect to qR and qD.

Finally, defining a new global vector of unknowns q : [0, T ] → Rm+n

merging qR and qD, being δq, δλR, δλD and δλRD arbitrary, global dynamicswill be expressed by the set of index-3 algebraic-differential equations inmatrix form: (

MR 0

0 MD

)︸︷︷︸

M

qRqD

︸︷︷︸

q

=

fΦR

fΦD

+ fΦRD︸︷︷︸

fΦ

+

0

f intD

+

f extR

f extD

︸︷︷︸

f

(6.7a)

ΦR = 0 , ΦD = 0 , ΦRD = 0 , (6.7b)

where fΦRD is the coupling term between the rigid and the deformable sub-systems. If a penalty method is employed, equations (6.7b) disappear and(6.7a) transforms into a purely differential system.

6.3.4 Consistent rigid bodies and constraints

Based on the parametrization for rigid bodies presented in the previoussection, with inertial Cartesian coordinates qR of selected material points, itbecomes apparent that the consistent formulation relies just on the adequatetreatment of the constant-distance constraints. This is due to the fact thatthe mass matrix MR is constant, and therefore the conserving formulationof the inertial terms, following (6.3), is given simply by MR(qRn+1 − qRn).

6. Simulations263

Note that the constant-distance constraints are holonomic and sclero-nomic; thus they do not dissipate energy and the consistent formulation is inthis case an energy-conserving one.

A conserving formulation of a constant-distance constraint can be ob-tained with a Lagrange multiplier method Gonzalez (1999), a penalty methodGarcıa Orden (1999); Garcıa Orden & Goicolea (2000) or an augmentedLagrangian method Garcıa Orden & Ortega Aguilera (2006). With the La-grange multipliers and augmented Lagrange methods, constraints are exactlysatisfied with exact discrete energy conservation. With the penalty method,constraints are violated but the stored energy is exactly released if they areexactly fulfilled. This means that, with conserving penalty, the summationof the kinetic energy, the potential energy of the conserving applied forcesand the constraint energy is constant.

Constraint forces, specifically formulated to satisfy exact energy conser-vation, are introduced into the scheme (6.3). Among all available options,the conserving penalty formulation is specially appealing due to its simplic-ity, as it can be obtained from a constraint potential. For example, considertwo points with positions defined by vectors x1 and x2. The relative positionvector is denoted by r = x2 − x1 and the absolute distance by r = |r|. Theconstant-distance constraint is expressed as Φ(r) = r−r0, being r0 the initialdistance. With this notation, the conserving constraint force is given by:

f∗ΦR = −α(

1− r0

rn+ 12

)(rn+ 1

2

−rn+ 12

), (6.8)

where rn+ 12

= (rn+1+rn)/2 and α ∈ R+ is the penalty parameter. Full details

about the linearization of (6.8) and other important practical issues, as thebehaviour in the limit ∆r → 0, can be found in Garcıa Orden & Goicolea(2000); Goicolea & Garcıa Orden (2002).

Joints connecting different bodies are taken into account with additionalconstraints, not necessarily of the constant-distance type. These constraintswill depend, in general, on the position vectors of many points of the bodies,and will be fully non-linear. If a conserving penalty approach is employed, dif-ferent strategies can be employed; one option is to directly apply the discretederivative operator (5.2) to the constraint potential in order to obtain theconstraint force f∗Φ, as in Leyendecker et al. (2004). Other option is to enforcethe directionality condition (5.4) with a standard derivative evaluated at anintermediate value in [tn+1, tn], as proposed in the original energy-momentummethod by Simo & Tarnow (1992). Details of this latter approach can befound in Garcıa Orden & Goicolea (2000).


6.3.5 Consistent flexible multibody system

In the previous sections a energy-consistent formulation of deformableviscoelastic bodies, rigid bodies and constraints has been obtained. Thus,the energy-consistent global formulation of a flexible multibody system isobtained in a straightforward way, just putting together both the rigid andthe deformable subsystems and adding the terms related with the constraints,representing the kinematical joints between them. Following the format givenin (6.3) and using the conserving penalty method for the constraints, the fol-lowing scheme is obtained to be solved for the global positions and velocitiesqn+1 and qn+1: (

MR 0

0 MD

)︸︷︷︸

M

[qRn+1

qDn+1

︸︷︷︸

qn+1

−

qRnqDn

︸︷︷︸

qn

]−

−

f∗ΦRf∗ΦD

− f∗ΦRD︸︷︷︸

f∗Φ

−

0

f int∗

D

−

f ext∗

R

f ext∗

D

︸︷︷︸

f∗

= 0 (6.9a)

qn+1 =2(qn+1 − qn)

∆t− qn , (6.9b)

where the symbol (∗) represents the EEM algorithmic approximation of theaffected magnitude.

6.3.6 Applications

In this section three examples inspired by real mechanisms have been con-sidered to be representative of complex multibody systems with deformableisothermal dissipative bodies, where stable and accurate long-term computa-tional solutions are usually hard to obtain.

It is necessary to remark that these two examples are used just to illus-trate the performance and the applicability of the proposed energy-consistentglobal method for flexible multibody systems; a detailed state-of-the-art sur-vey on computational models for the dynamics of these types of specificsystems are out of the scope of this dissertation and, accordingly, no detailedcritical comparisons with other methods traditionally applied in these fieldsare included. Nevertheless, a very brief outline of the key issues affectingtheir numerical solution are presented for each one of them.

6. Simulations265

Satellite with viscoelastic solar panels’ maneuver. Many references inthe literature study the dynamics of spacecrafts with flexible appendages, likesolar panels or antennas; e.g. Gao et al. (2008); Lomas (2001); Modi (1974);Sakamoto et al. (2011); Wallrapp & Wiedemann (2002); Zakrzhevskii (2008).Banerjee (2003) presents an interesting overview of the computational diffi-culties associated to very flexible space structures; namely the stiff characterof the equations of motion, the coupling between axial and bending effectsand the problems associated with large deformations. Stiffness is tackled us-ing implicit integration schemes, that usually improve stability introducingnumerical damping in the high-frequency range. The axial-bending couplingis considered by introducing additional terms to the standard floating frameof reference approach, or by a fully nonlinear formulation for the flexibleparts, that has also the advantage of being capable of representing largedeformations. The presented approach, based on a full nonlinear finite ele-ment formulation combined with the energy-consistent (implicit) integrationscheme, is thus a valid framework for dealing with these problems. Theadvantage of using the energy-consistent time-integration scheme is that itallows to obtain stable long-term solutions with time step sizes larger thanthose associated to standard methods, while being more physically accurate,as it will be illustrated next.

Consider the maneuver of a satellite with flexible solar panels made ofviscoelastic material and depicted in Figure 6.24. The body is a 1× 1× 1 mcube with a mass of 500 kg, and each panel is 4 m long, 1 m wide and 0.1 mthick. Each panel is hinged to the satellite body by three smooth sphericaljoints (see Figure 6.25) and connected by four rigid and massless bars, actingas stiffeners of the panels. The satellite body is equipped with two rocketsR1 and R2 that provide the necessary thrust, given by (6.10) expressed inkN, to perform the maneuver aimed to change of attitude of the satellite in90. The motion is illustrated in Figure 6.26.

R1

R2

X

Z

Y

Figure 6.24. Satellite: mesh and boundary condition definition.


•••

X

Z

Y

Figure 6.25. Satellite: spherical joints’ position.

R1(t) = −R2(t) =

−t for t ≤ 0.5 s

t− 1 for 0.5 < t ≤ 1 s

0 for 1 < t ≤ 12.8 s

1.15(12.8− t) for 12.8 < t ≤ 13.3 s

1.15(t− 13.8) for 13.3 < t ≤ 13.8 s

0 for t > 13.8 s

(6.10)

The satellite body is modeled with a homogeneous Saint Venant-Kirchhoffmaterial with density ρ0,b = 500 kg/m3, a Young modulus Eb = 1GPaand Poisson coefficient νb = 0.3. The panels are modeled by the viscoelas-tic material due to Holzapfel & Simo (1996b) with free-energy provided by(3.109) based on Neo-hookean hyperelasticity and one internal variable, withνp = 0.4, Ep = 100 kPa, τp = 2 s, µp = 5 kPa and ρ0,p = 100 kg/m3.

The example is integrated for a time interval [0, 50] s with a time step size∆t = 0.2 s using trapezoidal and energy-consistent methods. In addition, anintegration using the Midpoint method is performed with a smaller time stepsize, i.e ∆t = 0.02 s, in order to stress that conserving the angular-momentumdoes not help to stabilize solutions as conserving the laws of thermodynamicsdoes.

Figure 6.27 depicts the evolution of the energy of the system obtainedwith each scheme. It clearly shows that the Trapezoidal method provides anon-physical evolution of the energy in the free-force phases of the motion,which eventually cause this method to collapse. The same can be concludedfor the midpoint method which is unable to provide a full integration evenfor a time step size ten times smaller.

On the other hand, the isothermal EEM method successes in simulating

6. Simulations267

Figure 6.26. Snapshots of the maneuver (from top to bottom, left to right).Energy-consistent integration, ∆t = 0.2 s

the whole time interval, showing a decreasing total energy during the free-force periods of the motion. It can be concluded that the isothermal EEMmethod enhances the stability and robustness of the numerical integration ofthis problem, allowing to calculate long term simulations with the same timestep as the one which makes standard methods fail.

Figure 6.28 shows that the energy-consistent method exactly conservesthe angular momentum in the free-force periods of the movement, while thetrapezoidal rule fails to accomplish it even when it remains stable, duringthe first seconds of integration.

Finally, Figure 6.29 shows that the dissipation increases as the relaxationtimes decreases. Note that the energy-consistent integration provides positivedissipation in all cases.


0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

t [s]

E[J

]

isothermal EEM (∆t = 0.2 s)Trapezoidal (∆t = 0.2 s)Midpoint (∆t = 0.02 s)

Figure 6.27. Satellite. Evolution of the total energy of the system, kineticplus potential, obtained with the Trapezoidal, Midpoint and isothermal

EEM method. Only the isothermal EEM method provides a full solutionwhich, in addition, is in accordance with the laws of thermodynamics

6. Simulations269

0 5 10 15 20 25 30 35 40 45 50−40

0

80

160

240

320

t [s]

‖J‖

[N·m

]


0.5 3 5 7 9 11 13250

260

270

Figure 6.28. Satellite.Evolution of the angular momentum of the systemobtained with the trapezoidal and isothermal EEM methods.

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

t [s]

E[J]

τ = 10 sτ = 2 sτ = 0.1 s

Figure 6.29. Satellite, Total energy (kinetic plus potential), isothermalEEM integration with ∆t = 0.2 s


Satellite with thermoviscoelastic stiffeners’ maneuver. This examplesimulates a maneuver of a satellite with flexible solar panels, as depicted inFigure 6.30. The model is very similar to the one presented in Garcıa Orden& Romero (2011) , which was solved with an entropy-based EEM method aswell as standard Midpoint and Trapezoidal rules.

The body is a 1× 1× 1 m cube with a mass of 500 kg, and each panel is4 m long, 1 m wide and 0.1 m thick. Each panel is hinged at the lower sideto the satellite body by three smooth spherical joints and connected by fourdiscrete thermo-visco-elastic elements, acting as stiffeners.

(a) Upper view, showing the rocketsR1 and R2

(b) Lower view, showing the hingesof the solar panels with the satellite

body.

Figure 6.30. Satellite model, showing the solar panels’ hinges and therockets

The satellite body is equipped with two rockets R1 and R2 that providethe necessary thrust to perform the maneuver, aimed to change of attitudeof the satellite in 90. The maneuver starts firing both rockets during 1 s,initiating a rotation of the whole satellite. After one and a quarter turn, themovement of the panels stabilizes and the rockets are fired again; this secondfiring takes place in the precise moment of time required to stop the rotationleaving the satellite with the new orientation. The time-dependent forces ofthe rockets are shown in Figure 6.31, and the motion is illustrated in Figure6.26.

Both the satellite body and the panels are modeled with an homogeneousSaint Venant-Kirchhoff material. The material of the body has a densityρb = 500 kg/m3, a Young modulus Eb = 1 GPa and Poisson coefficientνb = 0.3. The panels have a density ρp = 10 kg/m3, a Young modulusEp = 5000 Pa and a Poisson coefficient νp = 0.3. Eight-noded hexahedralfinite elements with a fully nonlinear geometrical formulation are employedfor both the body and the panels.

6. Simulations271

0 5 10 15 20 25 30 35 40 45 50

−100

−50

0

50

100

t [s]

f 1,f 2

[N]

Rocket 1Rocket 2

Figure 6.31. Rocket forces vs. time

The parameters of the thermo-visco-elastic stiffeners are λ0 = 1 m, θr =300 K, c10 = 100 Nm, c11 = 0.5 NmK−1, βc = 4 NmK−1, c0 = 1 NmK−1, µ0 =5 Nm−1, µ1 = 0.1 Nm−1K−1, η0 = 100 Nsm−1, a = 10 K. The environmenttemperature is constant θref = 10 K and the thermal conductivity is k = 2Nms−1K−1.

The motion obtained with the thermodynamically consistent integrationof the dynamical equations is shown in Figure 6.26, being the consistent for-mulation of the elastic parts of the system provided by the energy-momentumscheme (Gonzalez, 2000; Simo & Tarnow, 1992; Simo et al., 1992), and thedissipative thermo-visco-elastic stiffeners implement the temperature-basedTC formulation presented in this paper.

The differences between this model and the one presented in Garcıa Or-den & Romero (2011) are:

• The number of panels. The increment of the number of panels fromtwo to four has several effects that add some numerical difficulties.Basically, it produces a kinematically richer movement, with two ofthe panels deforming with a dominant bending mode and the othertwo with a torsional mode. As a consequence higher frequencies areintroduced, because panel torsional frequencies are a little bit higher


than the bending ones.

• The conductivity (k) between the stiffeners and the environment. Thedecrement of the conductivity will produce higher temperatures in theelements, because the generated heat dissipates at a lower rate.

• The dependency of the main spring stiffness with temperature. Sincehigher temperatures are expected, the logarithmic expression (2.96)have been adopted, while in Garcıa Orden & Romero (2011) is linear.This choice poses additional technical difficulties, as it has already beenmentioned in different sections of the paper, when entropy is used toformulate the EEM method. Nevertheless, no special difficulties appearwhen temperature is used.

Figures 6.33, 6.34 and 6.35 display the angular momentum, the energy andthe entropy respectively, obtained with the temperature-based EEM methodwith ∆t = 0.2 s, which are qualitatively similar to the ones reported inGarcıa Orden & Romero (2011). Exact discrete satisfaction of the two lawsof thermodynamics and the symmetries (rotational in this case, associatedwith the angular momentum during the free-force motion) is observed asexpected.

Figures 6.36 and 6.37 show the evolution of the representative tempera-ture of the different groups of stiffeners. Specifically, Figure 6.36 displays thetemperature of the stiffeners constraining the two panels that have a distinctbending motion (parallel to the plane perpendicular to the angular veloc-ity), while Figure 6.37 shows the temperature of the rest of the stiffeners,associated to the panels that have a distinct torsional motion.

Finally, Figures 6.38, 6.39 and 6.40 have been included to emphasizethe robustness of the proposed temperature-based EEM method comparedwith standard methods, such as Midpoint and Trapezoidal methods. Thesemethods are convergent and can successfully obtain a solution provided thetime-step size is reduced. However, the numerical experiment shows that thetemperature-based EEM method may obtain a stable solution with time-stepsizes larger than those limited by stability in standard methods, with exactsatisfaction of the two laws of thermodynamics and the present symmetries.

6. Simulations273

Figure 6.32. Snapshots of the maneuver (from top to bottom, left to right).Consistent integration, ∆t = 0.2 s


0 5 10 15 20 25 30 35 40 45 50−10

0

10

20

30

40

50

60

t [s]

||J||[N·m

]JxJyJz

Total

Figure 6.33. Total angular momentum, consistent integration, ∆t = 0.2 s

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

t [s]

E[J]

SatelliteEnvironment

Total

Figure 6.34. Energy, consistent integration, ∆t = 0.2 s

6. Simulations275

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

t [s]

S[JK

−1]

SatelliteEnvironment

Total

Figure 6.35. Entropy, consistent integration, ∆t = 0.2 s

0 5 10 15 20 25 30 35 40 45 508.5

9

9.5

10

10.5

11

t [s]

θ[K

]

Stiffeners 1,2,7,8Stiffeners 3,4,5,6

Figure 6.36. Stiffeners’ temperature, EEM method, ∆t = 0.2 s


0 5 10 15 20 25 30 35 40 45 50

9.8

9.9

10

10.1

10.2

t [s]

θ[K

]

Stiffeners 9,12,13,16Stiffeners 10,11,14,15

Figure 6.37. Stiffeners’ temperature, consistent integration, ∆t = 0.2 s

0 5 10 15 20 25 30 35 40 45 50

−20

0

20

40

60

t [s]

||J||[N·m

]

TrapezoidalMidpoint


Figure 6.38. Total angular momentum, ∆t = 0.2 s

6. Simulations277

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

t [s]

E[J]

TrapezoidalMidpoint


Figure 6.39. Total energy ∆t = 0.2 s

0 5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1

t [s]

S[J/K

]

TrapezoidalMidpoint


Figure 6.40. Total entropy, ∆t = 0.2 s


Rotor blade with elastomeric bearing. For the last example, considera multibody model of a helicopter rotor blade incorporating an elastomericbearing.

There are many references in the specialized literature on multibodydynamics devoted to the dynamics of rotorcraft systems; see for instanceBauchau (2009); Chun et al. (2013); Kang et al. (2014); Maqueda et al.(2008) and references therein. Likewise the satellite example of the previoussection, the coupling between axial and bending effects along with the stiffcharacter of the related initial value problems are among the main numericalchallenges posed by these types of models. The model proposed in this paperis simple but, nevertheless, incorporates these difficulties and, therefore, wehave considered it to be representative of this type of engineering systems.Descriptions of suitable numerical approaches described in the literature aswell as the advantages of the proposed approach have already been mentionedregarding the satellite example and are omitted here.

The bearing is a 0.3 × 0.3 × 0.2 m block made of viscoelastic materialconnecting the solid rigid blade and the rotor. The blade is a rigid prism 2.7m long with a density of 2500 kg/m3. Figure 6.41 shows the whole model.The material parameters used for the viscoelastic bearing are: ν = 0.49,E = 2.1 MPa, τ = 10 s, µ = 1.05 MPa and ρ0 = 100 kg/m3.

The internal face of the prismatic bearing is rigid and is rigidly attachedto the hub (not shown, located at the center of the reference system XYZ inFigure 6.41. The two lower vertices of the bearing internal face are constrainedto frictionless move along the fixed circular path depicted also in Figure 6.41.

X

Z

Y g

ω0

Figure 6.41. Rotor blade with elastomeric bearing: mesh and boundarycondition definition.

6. Simulations279

t = 0.7 s

t = 2.5 s

t = 7.5 s

t = 4.9 s

t = 23.0 s

Figure 6.42. Snapshots of the motion of the rotor blade with elastomericbearing

The system is initially in an horizontal position and at rest, and it is setto motion by the hub with an angular velocity ω0(t) around the fixed verticalaxis which acts during the first 2 s and is defined by the law:

ω0(t) =ωs2

[t− 1

πsin (πt)

]if 0 ≤ t ≤ 2 s (6.11)

with ωs = 0.6124 rad/s. The time step size is chosen as ∆t = 0.1 s andthe motion is integrated up to 30 s with trapezoidal and energy-consistentmethods. An extra simulation is carried out by midpoint method and timestep size of ∆t = 0.01 s.

Figure 6.42 shows some snapshots of the motion, which is a composition ofthe main rotation around the vertical axis imposed by the rotor and rotationsaround other spatial axis due to the deformation of the bearing under theaction of gravity and the inertia effects. Figure 6.43 shows the vertical posi-tion of the center of mass of the blade, oscillating due to the aforementioneddeformation of the bearing under to the weight of the blade.

Figure 6.44 shows that the total energy obtained with the isothermalEEM method is monotonically decreasing during the free-force motion, whilethe trapezoidal rule shows a non-physical behavior and eventually becomesunstable. As in the previous example, the midpoint method with a smallertime step size is able to integrate further time steps but ends up failing dueto an unreasonable growth of the energy.


Figure 6.45 displays the dependency of the energy with the relaxation-time parameter, computed with the isothermal EEM method. As expected,we obtain a positive dissipation that increases as the relaxation time of theviscoelastic model decreases.

0 5 10 15 20 25 30−2

0

2

4

6

8

10·10−2

t [s]

uG,z[m

]

Figure 6.43. Vertical position of the blade center of mass

6. Simulations281

0 5 10 15 20 25 30−100

0

100

200

300

400

t [s]

E[J

]


Figure 6.44. Evolution of the total energy of the blade system.

0 5 10 15 20 25 300

100

200

300

400

t [s]

E[J]

τ = 20 sτ = 10 sτ = 5 s

Figure 6.45. Total energy (kinetic plus potential), isothermal EEM with∆t = 0.1 s

Summary, conclusions andfuture work C

hapter

7This dissertation has addressed the formulation, analysis and implemen-

tation of structure-preserving time integration methods for nonlinear smoothcoupled dissipative systems. These time integration methods are able toprovide solutions which agree with the fundamental conservation propertiesidentified in continuous models derived from the laws of thermodynamics andsymmetries. They are constructed within an uniform methodology consist-ing of a temporal discretization of mid-point type combined with a specialsecond order accurate discrete operator responsible for preservation of thethermodynamical and symmetrical structure of the continuous equations.Thus, the resulting methods are Energy-preserving Entropy-producing andMomentum-preserving (EEM) by design while rendering second-order accu-racy.

The uniform methodology conceptually relies on the underlying metriplec-tic structure of the initial-boundary value problems defining the evolutionof coupled smooth dissipative systems. Such structure provides an unifiedframework to deal with any dissipative system governed by the laws of ther-modynamics. Formally, a metriplectic structure is a generalization of the(reversible) Poisson structure that possesses Hamiltonian (conservative) sys-tems by attaching a (irreversible) Gradient structure that defines pure dissi-pative systems. Then, the evolution of dissipative systems is generated fromthe addition of two reversible and irreversible operators. Using a particularmetriplectic framework, known by the acronym GENERIC, the methodologyhas basically consisted in extending the concepts involved in the formulationof Energy-Momentum integration methods to the integration of dissipative

284

problems defined by its metriplectic structure. Thus, the concept of the dis-crete derivative operator has enabled the derivation of discrete counterpartsthat agree with the laws of thermodynamics and respect the symmetries.

Alternative strategies have been adopted deriving from the use of eitherentropies or temperatures for the thermodynamic state variable. Specialattention has been paid to the temperature-based formulation since it hasnot thus far been developed. With these strategies, most of the real problemsof interest can successfully be faced.

The formulation of EEM methods for finite-dimensional dissipative sys-tems directly follows from the introduced methodology with the only re-quirement of devising its reversible and irreversible operators in the sense ofthe metriplectic structures in terms of the thermodynamical variable chosen.Thus, the discrete preservation features are completely guaranteed.

For infinite-dimensional dissipative systems, however, the methodologymust be supplemented by a spatial discretization that converts the systeminto a finite-dimensional one in such a way that the preservation features arenot altered. The classical Galerkin FE-based technique has been proved tobe appropriated for the successful formulations of EEM methods. The finaldiscrete conservation features, however, might not be guaranteed for anyformulation based on this methodology. Specifically, Galerkin formulationsrequires that the resulting discrete derivative operators belong to the finite-dimensional Galerkin space. Unfortunately, this cannot be ensured for anyformulation of the system. Particularly, that is the case of the entropy-basedapproach whose resulting EEM method needs to be corrected with an L2-projection technique.

Several representative smooth dissipative systems have been consideredfrom finite- to infinite-dimensional ones addressing the most common sourcesof dissipation: heat conduction and internal dissipative effects. In all of them,dissipative effects have been considered smooth, in the sense that they may bedescribed by continuous laws, and have been approached by internal variableswhich naturally contributes to the irreversible operator of the metriplecticstructure.

Each of the smooth dissipative systems has been thoroughly introduceddiscussing its conservation features and reasoning the loss of some of suchfeatures when standard integration methods are used to obtain approximatesolutions. The continuous formulation of these systems has been approachedfollowing classical thermodynamical assumptions from different perspectivesaccording to the thermodynamical variable chosen for the description of the

7. Summary, conclusions and future work 285

system thermodynamics. Also, the continuous formulation has been ad-dressed by the unified GENERIC framework, carefully developing the re-versible and irreversible operators in terms of both entropy and temperatures,guaranteeing the thermodynamical soundness of the formulated systems.

Then, the methodology to construct EEM methods has been fully ex-ploited so that entropy-based and temperature-based EEM methods havebeen proposed and checked their discrete conservation features. Also, bothapproaches has been compared regarding conservation features, relative er-rors, applicability and implementation issues.

Thus, from the formulation point of view, it can be asserted that the tem-perature-based EEM methods pose three main advantages respect to theirentropy-based counterparts:

• Common constitutive formulations in terms of the temperature candirectly be applied.

• Dirichlet boundary conditions are naturally imposed.

• In Galerkin formulations, no need for a L2-projection technique to en-sure the preservation properties.

However, from the computational cost perspective, the thermodynamicalconsistency of the temperature-based EEM methods relies on the notion ofdiscrete derivative operator applied to the internal energy and entropy func-tions so they requires about twice as much implementation effort and com-putational cost compared to the entropy-based EEM ones that only needsthe elaboration of the discrete derivative operator for the internal energyexpressed in terms of entropy.

Furthermore, the temperature-based EEM methods for infinite-dimen-sional dissipative systems rest on a special evaluation of the temperaturefiled within the time interval to ensure the preservation of the structurethat complicates the implementation of the heat conduction contribution.Fortunately, the particular form of classic thermo(visco)elastic models helpsto simplify the said implementation.

For the finite-dimensional dissipative systems approached with the tem-perature-based formulation, the issue of the evaluation of the temperaturesinvolved in the discrete evolution equations corresponding to the energy bal-ance ones remains open. That is because any evaluation of them within thetime interval ensures both discrete preservation features and second-order

286

accuracy. However, there exist a candidate that stands out among all be-cause it leads to an unique discrete definition of the specific heat capacitywhich contributes to the desired discrete thermodynamical consistency. Sig-nificantly, such evaluation coincides with the evaluation that univocally arosein the infinite-dimensional case.

All numerical examples proposed throughout this dissertation suggestthat EEM methods not only conserves angular and linear momentum, and thelaws of thermodynamics, but considerably improves the numerical stabilityand enables the use of larger time steps compared to standard integrationschemes.

In particular, the numerical examples of Chapter 5 have revealed that thetemperature-based EEM method for finite-dimensional dissipative systemswith internal variables are significantly more accurate for the positions, theinternal variable and the temperatures at a fixed time step than the standardimplicit methods and the EEM entropy-based method.

Another important aspect to consider when facing dissipative problems isthat the inclusion of dissipative effects naturally contributes to the stabilityof standard methods in the same was as the inclusion of artificial dissipationhelps to the stability of standard integration methods for elastodynamicsproblems. Thereby, solutions can be obtained for moderately large time stepsizes; sizes that would produce unstable solutions when no dissipation ispresent. In fact, as the dissipative effects become important, it is more likelyfor standard methods to provide stable solutions. However, these solutionsmight have accumulated an important error (the longer the simulation theworse) since neither the laws of thermodynamics nor the symmetries complywith their essential physical rules. Instead, the EEM method complies withevery evolution property inherited from the continuum, providing physicalcertainty over the solution obtained.

Finally, the main conclusions of the research carried out in this disserta-tion are summarized in the following six points:

1. Just as in the conserving case, preserving the structure of dissipa-tive problems provides solutions with physical certainty and posesnumerical advantages.

2. The basic structure of coupled smooth dissipative systems is providedby the laws of thermodynamics and the symmetries, and can beformulated within an unified framework provided by metriplecticstructures induced on manifolds.

7. Summary, conclusions and future work 287

3. General energy-preserving, entropy-producing, momentum-pre-serving methods (EEM) for dissipative problems can be formulatedby using the same ideas as those that led to the formulation of con-serving methods.

4. Temperature can be used as a thermodynamical state variable al-lowing to handle with Dirichlet’s boundary conditions and withnaturally temperature-based free-energy functions.

5. The internal variables framework naturally contributes to the irre-versible operator of the metriplectic structure. Although they canbreak its symmetry, this does not seem to be an impediment forthe derivation of thermodynamically sound evolution equations and,accordingly, the discrete setting provided by EEM methods preservesall the preservation features.

6. The resulting EEM integration methods not only conserve angular,linear momentum and the laws of thermodynamics, but considerablyimproves the stability and robustness for large time steps.

Future works

Beyond the results contained in this dissertation, future lines of researchare proposed next:

• For infinite-dimensional dissipative systems, exploring the use of multi-field variational for the treatment of the volumetric constraint of nearlyor fully incompressible materials in such a way that the preservationfeatures identified in the continuous equations remain ensured. This isof vital importance since, as has been introduced in this dissertation,the dissipative effects are usually associated with isochoric deforma-tions while the thermal coupling is naturally associated with volumet-ric deformation via an thermal expansion coefficient. A departing pointcould be the work done in Gonzalez (2000) for the formulation of anEnergy-Momentum method in such context.

• Extending the methodology to deal with the formulation of non-smoothdissipative systems, among which (thermo)plastic and (thermo)damagechanges in the material are of great interest.

• Related to the above, systematize the obtention of the reversible andirreversible operators of the GENERIC formalism for any set of state

288

variables and any source of dissipation in the line of the work due toMielke (2011).

• An exploration of the performances of these methods for anisotropymaterial behavior in which the matrix and the fibers have different me-chanical and thermal behaviors. In the aerospace industry there exista bunch of demanding problems associated with composite materialsthat could be faced with this approach.

• Comparison with the performances provided by variational integratorsregarding time step sizes, order of accuracy, relative errors, computa-tional cost, implementation issues, applicability, etc.

• Despite the good performances of structure-preserving methods basedon the discrete derivative operators, the fact that its implementationrequires a full redefinition of the elements technology has so far pre-vented them to be embedded into commercial codes. In the same waythat Energy-Momentum methods are not available in these codes, it isexpectable that the EEM methods proposed in this dissertation havethe same ‘succeed’. Because of this, the search for alternatives for thediscrete derivative operator that somehow makes the resulting EEMmethods less intrusive, that is, they do not ideally require to changethe standard element implementation, becomes a crucial issue. ForEnergy-Momentum methods, there have been several attempts towardsthis target, see for instance Romero (2001) or Krenk (2014), that couldserve as a departing point for the here proposed EEM methods.

Discrete DerivativeOperator A

ppendix

A

The design of the presented EEM integration methods relies cruciallyon a discrete derivative operator (D). Such operator applies to scalar-valuedfunctions defined in inner product spaces (U , 〈·, ·〉U ) and satisfies two essen-tial properties:

1. Directionality: the ability for the operator to relate univocally the func-tion balance in an arbitrary set [x,y] with the difference of such set,that is

f(x)− f(y) = 〈Df(x,y),x− y〉U (A.1)

2. Consistency: the ability for the operator to render second order accu-racy of the standard derivative at the set midpoint, that is

Df(x,y) = Df

(x+ y

2

)+O(‖x− y‖2). (A.2)

The concept of the discrete derivative operator dates at least back tothe mid-seventies in LaBudde & Greenspan (1976a,b) (see also Maeda (1980,1981), Gotusso (1985), Itoh & Abe (1988), McLachlan et al. (1999)). Later,Gonzalez (1996) fully developed it in the context of geometric integration ofHamiltonian problems.

A.1 Mean value theorem

As a first approach to the discrete derivative operator, consider the the-orem of the mean value which asserts that, given a scalar-valued function

290 A.1. Mean value theorem

f(x) : C → R, defined in C ⊂ Rd, continuous on [a, b] and differentiable on(a, b), where a, b ∈ C, then there exists at least a c ∈ (a, b) such that

f(b)− f(a) = ∇f(c) · (b− a) (A.3)

In view of this theorem, the existence of operators satisfying the direc-tionality property is proved for scalar-valued functions under the conditionsof continuity and derivability within the intervals of interest.

The mean value theorem for unidimensional scalar-valued functions, thatis C ⊂ R, is appealing because it is easily geometrically interpretable, seeFigure A.1. Thus, given a smooth function f : R→ R in an interval [a, b] thesecant joining the endpoints of the interval is parallel to the tangent at somepoint c within the interval.

x

f(x)

f ′(c) =f(b)− f(a)

b− a

a bc

Figure A.1. Geometrical interpretation of the mean value theorem

This geometrical interpretation readily provides an expression for theoperator satisfying the directionality property of unidimensional functions,that is

Df(a, b) = f ′(c) =f(b)− f(a)

b− a (A.4)

Moreover, the above formula turns out to be the only definition for thisspecific types of functions that furnishes such property. For more generaltypes of function, however, there exists infinite formulae endowed with thisproperty, as will be discussed in the next section.

On the other hand, the above definition of the operator collapses to the

A. Discrete Derivative Operator 291

continuos derivative as the interval becomes infinitesimal

lima→b

Df(a, b) = f ′(b) = f ′(a), (A.5)

being hence well-defined.

Furthermore, this formula also results in a second order approximation tothe standard derivative evaluated at interval midpoint so (A.4) is a valid dis-crete derivative operator for the formulation of structure-preserving methodsaccording to the requirements introduced at the beginning of this Appendix.

A.2 The discrete derivative operator

For general scalar-valued functions there exist several definitions for thediscrete derivative operator matching the directionality and consistency prop-erties, as studied in Romero (2012). Therein, it is concluded that the optimaldefinition is the one introduced by Gonzalez (1996) whose main features arediscussed next.

Definition A.1. Let f : U → R be a smooth function, where (U , 〈·, ·〉U )is an inner product space, then the second-order discrete derivative operatoris provided by

Df(x,y) = Df (v) +f(x)− f(y)− 〈Df(v),u〉U

〈u,u〉Uu, (A.6)

where x,y ∈ U , v = (x+y)/2, u = x−y and D being standard derivative1.

Remark A.1. As pointed out by Romero (2012), there exist as many second-order accurate operators as forms of defining the inner product appearing in(A.6) that verifies the directionality condition (A.1).

Definition A.2. If the inner product space has a partitioned structure U =U1 × · · · × Uk for some k > 1, and every Ui, i = 1, . . . , k is endowed withan inner product 〈·, ·〉Ui , the partitioned discrete derivative operator for anysmooth function f : U → R, is provided by

〈Df(x,y),u〉U =k∑i=1

〈Dif(xi, yi), ui〉Ui , (A.7)

1Note that the different notations used for the standard derivative operator D and thediscrete derivative operator D.

292 A.2. The discrete derivative operator

for all u = (u1, u2, . . . , uk) ∈ U , and with x = (x1, x2, . . . , xk) ∈ U andy = (y1, y2, . . . , yk) ∈ U . The term Dif(xi, yi) might be interpreted as thei-th second order accurate discrete counterpart to the i-th partial derivativeof the function f at midpoint and is provided by

Dif(xi, yi) =1

2

[Df ixy(xi, yi) + Df iyx(xi, yi)

], (A.8)

where f ixy, fiyx : Ui → R are defined by the relations

f ixy(w) = f(x1,x2, . . . ,xi−1,w,yi+1, . . . ,yk)

f iyx(w) = f(y1,y2, . . . ,yi−1,w,xi+1, . . . ,xk)(A.9)

Properties. The above definitions automatically satisfy the two essentialproperties (A.1) and (A.2) required to discrete derivatives operators for theformulation of structure-preserving methods.

Proof. The verification of the directionality property is readily checked ap-plying (A.6) to u, yielding

〈Df(x,y),u〉U = 〈Df(v),u〉U +f(x)− f(y)− 〈Df(v),u〉U

〈u,u〉U〈u,u〉U

= f(x)− f(y)

(A.10)

Remark A.2. This proof would be valid for any v in (A.6), not necessarilyat the midpoint, leading to as many discrete derivatives operator definitionsas values of v, although the second-order consistency condition (A.2) may becompromised.

Proof. The consistency property is verified by making x → y or u → 0in (A.6). For that, consider the followings Taylor’s expansions around themidpoint of both endpoints of the interval

f(x) = f(v) + 〈12Df(v),u〉U

+ 〈14D2f(v), (u,u)〉U + 〈1

8D3f(v), (u,u,u)〉U

+ 〈 1

16D4f(v), (u,u,u,u)〉U +O(||u||5)

(A.11)


f(y) = f(v)− 〈12Df(v),u〉U

+ 〈14D2f(v), (u,u)〉U − 〈

1

8D3f(v), (u,u,u)〉U

+ 〈 1

16D4f(v), (u,u,u,u)〉U +O(||u||5)

(A.12)

Combining now them yields the following crucial expansion

f(x) = f(y) + 〈Df(v),u〉U + 〈14D3f(v), (u,u,u)〉U +O(||u||5) (A.13)

This result can be used in the expression (A.6) to arrive at

Df(x,y) = Df (v) +〈14D3f(v), (u,u,u)〉U +O(||u||5)

〈u,u〉Uu, (A.14)

Considering u = αw with α > 0 and ||w|| = 1, the last expression canbe written as

Df(x,y) = Df (v) + 〈14α3D3f(v), (w,w,w)〉U +O(α4), (A.15)

proving the second order accuracy of the discrete derivative operator. Thisresult also serves to verify that the discrete gradient operator is well-definedin the limit case of u→ 0.

Definition A.3. A G-invariant function f : U → R is a function f : U → Rthat, given an action Φ of a Lie Group G on the space U , satisfies

f(Φg(z)) = f(z) ∀g ∈ G, (A.16)

Definition A.4. A G-equivariant discrete derivative for a smooth G-invariantfunction f : U → R is a mapping DGf : U × U → TU that satisfies twoadditional properties:

1. Equivariance:

DGf(Φg(x),Φg(y)) =

[DΦg

(x+ y

2

)]−T

· DGf(x,y), (A.17)

for all g ∈ G and x,y ∈ U .

294 A.2. The discrete derivative operator

2. Orthogonality:

DGf(x,y) · ξU

(x+ y

2

)= 0 ∀ξ ∈ g∗ and x,y ∈ U , (A.18)

where g∗ is the associated dual algebra of the Lie group G.

Proposition A.1. Let f : U → R be a smooth G-invariant function,π : U → Rr be a set of invariant functions πi : U → R which are invari-ant respect to an action Φ of a Lie group G. Let f : π(U ) → R be theassociated reduced function defined by f(π(z)) = f(z), for all z. If the in-variant πi : U → R is at most of degree two, then a G-equivariant discretederivative for f is provided by

DGf(x,y) = Df(π(x), π(y)) Dπ(z) = [Dπ(v)]T Df(π(x), π(y)), (A.19)

where for any z ∈ U , Dπ(z) ∈ Rr×dim(U ) and r = dim(U )− o, o being thenumber of orbits associated with the Lie group G.

Proof. The result is proved in Gonzalez (1996) by direct verification of theproperties which are summarized next.

1. The directionality property of G-equivariant discrete derivative opera-tor establishes that

〈DGf(x,y),v〉U = 〈[Dπ(z)]TDf(π(x), π(y)),v〉U= 〈Df(π(x), π(y)), Dπ(z) · v〉U

(A.20)

As, by assumption, the invariant function π is at most of degree two,it follows that

Dπ(z) · v = π(y)− π(x) (A.21)

Then, the directionality property yields

〈DGf(x,y) · v = Df(π(x), π(y)) · (π(y)− π(x))

= f(π(x))− f(π(y)) = f(y)− f(x)(A.22)

2. The consistency property follows from the consistency of the standarddiscrete derivative operator (A.15).


3. The equivariance property is proved by using the invariability of the πfunction respect to the action Φg of a Lie group G, that is π(Φg(z) =π(z), for all z ∈ U and g = G, which leads to

Dπ(Φg(z)) = Dπ(z) [DΦg(z)]−1 (A.23)

If the action satisfies 12[Φg(x) + Φg(y)] = Φg(z), that is the Lie Group

is affine, then

DGf(Φg(x),Φg(y)) = Df(π(Φg(x)), π(Φg(y))) Dπ(Φg(z))

= Df(π(x), π(y)) Dπ(Φg(z))

= Df(π(x), π(y)) Dπ(z) [DΦg(z)]−1

= [DΦg(z)]−T ·(

Df(π(x), π(y)) Dπ(z))

= [DΦg(z)]−T · DGf(x,y)

(A.24)

4. The orthogonality property can also be proved by exploiting the invari-ability of the function π which allows to conclude that

Dπ(z) · ξU (z) = 0, ∀ξ ∈ g∗ (A.25)

and, hence,

DGf(x,y) · ξU (z) = Df(π(x), π(y)) · (Dπ(z) · ξU (z)) = 0 (A.26)

A.3 Discrete derivative operator: Examples

• Functions defined on a k-product-Cartesian space R×· · ·×R. Givena function f :∈ R × · · · × R → R, the discrete derivative operator must beelaborated using the partitioned definition (5.3) in terms of k.

Thus, the case of k = 1 coincides with the mean value theorem (A.4).

For two-variable functions, k = 2, the first and the second partial discretederivatives follow from (A.8) and (A.9) and are provided by

D1f(x,y) =f(x1, x2)− f(y1, x2) + f(x1, y2)− f(y1, y2)

2(x1 − y1)(A.27)

296 A.3. Discrete derivative operator: Examples

D2f(x,y) =f(x1, x2)− f(x1, y2) + f(y1, x2)− f(y1, y2)

2(x2 − y2)(A.28)

with x = (x1, x2) and y = (y1, y2).

The resulting expressions are required in the formulation of EEM for theTwo thermo-spring system discussed in Section 5.3, as it is defined throughpotentials depending on two real variables.

For three-variable functions, k = 3, following (A.8) and (A.9), the partialdiscrete derivatives are provided by

D1f(x,y) =f(x1, x2, x3)− f(y1, x2, x3) + f(x1, y2, y3)− f(y1, y2, y3)

2(x1 − y1)(A.29)

D2f(x,y) =f(x1, x2, y3)− f(x1, y2, y3) + f(y1, x2, x3)− f(y1, y2, x3)

2(x2 − y2)(A.30)

D3f(x,y) =f(x1, x2, x3)− f(x1, x2, y3) + f(y1, y2, x3)− f(y1, y2, y3)

2(x3 − y3)(A.31)

with x = (x1, x2, x3) and y = (y1, y2, y3).

These discrete derivatives are involved in the formulation of the EEMmethods for the Thermo-visco-elastic system presented in Section 5.4.

• Functions defined on Sym(T2d). Given a function f : Sym(T2

d)→ R, thediscrete derivative operator

Df(X,Y) = Df (V) +f(X)− f(Y)−Df(V) : U

U : UU, (A.32)

for all X,Y ∈ Sym(T2d), U = X−Y and V = 1

2(X + Y).

The free-energy functions defining nonlinear elastodynamics are of thistype. Then, the stress formula that ensures the preservation of the Hamil-tonian of such as systems is provided by the above formula for the discretederivative operator. The interesting point in the above expression is on thefact that the standard derivatives are evaluated at the mean value of thetensor.

If f is at most quadratic, that is, the Saint-Vennant Kirchhoff materialmodel, the discrete gradient collapses to the standard derivative evaluated atmidpoint so the midpoint methods becomes energy-preserving.

• Functions defined on a k-product space of second-order tensors.Given a function f : (T2

d)k → R, the partitioned discrete derivative operator

is elaborated in terms of k.


The case of k = 1 has just been introduced in (A.32).

For k = 2, two partial discrete derivative follows as

D1f(X,Y) =1

2[D1f(V1,X2) +D1f(V1,Y2)]

+f(X1,X2)− f(Y1,X2) + f(X1,Y2)− f(Y1,Y2)

2U1 : U1

U1

− [D1f(V1,X2) +D1f(V1,Y2)] : U1

2U1 : U1

U1

(A.33)

D2f(X,Y) =1

2[D2f(X1,V2) +D2f(X1,V2)]

+f(X1,X2)− f(X1,Y2) + f(Y1,X2)− f(Y1,Y2)

2U2 : U2

U2

− [D2f(X1,V2) +D2f(X1,V2)] : U2

2U2 : U2

U2,

(A.34)

for all X = (X1,X2),Y = (Y1,Y2), U = X−Y and V = 12(X + Y).

These formulas are symmetric respect to the first and the second variablesso they can be exchangeable leading to the same expression for the discretederivative operator. They are involved in the formulation of the EEM methodfor isothermal dissipative dynamics defined in (5.249) and based on an inter-nal variable. The first variable is then identified with the right Cauchy-Greendeformation tensor and the second with the internal variable.

For k = 3, the corresponding three partial discrete derivatives are pro-vided by

D1f(X,Y) =1

2[D1f(V1,X2,X3) +D1f(V1,Y2,Y3)]

+f(X1,X2,X3)− f(Y1,X2,X3) + f(X1,Y2,Y3)− f(Y1,Y2,Y3)

2U1 : U1

U1

− [D1f(V1,X2,X3) +D1f(V1,Y2,Y3)] : U1

2U1 : U1

U1

(A.35)

D2f(X,Y) =1

2[D2f(X1,V2,X3) +D2f(Y1,V2,Y3)]

+f(X1,X2,Y3)− f(X1,Y2,Y3) + f(Y1,X2,X3)− f(Y1,Y2,X3)

2U2 : U2

U2

− [D2f(X1,V2,X3) +D2f(Y1,V2,Y3)] : U2

2U2 : U2

U2

(A.36)

298 A.3. Discrete derivative operator: Examples

D3f(X,Y) =1

2[D3f(X1,X2,V3) +D3f(Y1,Y2,V3)]

+f(X1,X2,X3)− f(X1,X2,Y3) + f(Y1,Y2,X3)− f(Y1,Y2,Y3)

2U3 : U3

U3

− [D3f(X1,X2,V3) +D3f(Y1,Y2,V3)] : U3

2U3 : U3

U3,

(A.37)

for all X = (X1,X2,X3),Y = (Y1,Y2,Y3), U = X−Y and V = 12(X+Y).

In this case, the formulas is not symmetric respect to three variables,although the crucial properties of the operator is ensured for any combinationof them. Then, the EEM method for isothermal dissipative dynamics basedon two internal variables requires the use of these expressions.

The appealing differences between the elaborated discrete derivatives fordifferent value of k lead the formulation of EEM methods to depend sub-stantially on this issue, as was remarked in 5.5 for the number of internalvariables chosen to model the viscoelastic effects.

• Functions defined on a k-product space of real numbers and sec-ond order tensors. Among all possible combinations, two choices are ofinterest since they are involved in the formulation of EEM methods for cou-pled thermoelastic and thermo dissipative dynamics.

Thus, consider a function defined on a product space of a real numberand a symmetric second order tensor, that is f : Sym(T2

d) × R → R, then,the partial discrete derivative result in

D1f(X, Y ) =1

2[D1f(V, x) +D1f(V, y)]

+f(X, x)− f(Y, x) + f(X, y)− f(Y, y)

2U : UU

− [D1f(V, x) +D1f(V, y)] : U

2U : UU

(A.38)

D2f(X, Y ) =f(X, x)− f(X, y) + f(Y, x)− f(Y, y)

2u(A.39)

with X = (X, x) ∈ Sym(T2d) × R, Y = (Y, y) ∈ Sym(T2

d) × R, U = X − Yand V = 1

2(X + Y ).

The above formulas are necessary for the formulation of EEM methodsfor thermo-elastic continuum dynamics. In particular, the above partial dis-crete derivatives are directly identified with the algorithmic stress tensor andthe algorithmic temperature of the entropy-based EEM method. For the


temperature-based EEM method, however, these expressions are involved inits formulation but are not directly identified with algorithmic quantities.

Finally, consider a function defined on a product space of a real numberand more than one second order tensors, that is f : Sym(T2

d)×R×Sym(T2d)×

· · ·×Sym(T2d)→ R. These types of functions are involved in thermo dissipa-

tive potential so its partial discrete derivatives are needed for the formulationof EEM method for such a problem.

For the particular case of just one internal variable, the partial discretederivative arising from the definitions (A.8) and (A.9) are

D1f(X, Y ) =1

2[D1f(V1, x,X3) +D1f(V1, y,Y3)]

+f(X1, x,X3)− f(Y1, x,X3) + f(X1, y2,Y3)− f(Y1, y,Y3)

2U1 : U1

U1

− [D1f(V1, x,X3) +D1f(V1, y,Y3)] : U1

2U1 : U1

U1

(A.40)

D2f(X, Y ) =f(X1, x,Y3)− f(X1, y,Y3) + f(Y1, x,X3)− f(Y1, y,X3)

2u(A.41)

D3f(X, Y ) =1

2[D3f(X1, x,V3) +D3f(Y1, y2,V3)]

+f(X1, x,X3)− f(X1, x,Y3) + f(Y1, y,X3)− f(Y1, y,Y3)

2U3 : U3

U3

− [D3f(X1, x,V3) +D3f(Y1, y,V3)] : U3

2U3 : U3

U3,

(A.42)

with X = (X1, x,X3) ∈ Sym(T2d) × R × Sym(T2

d), Y = (Y1, y,Y3) ∈Sym(T2

d)× R× Sym(T2d), U = X − Y and V = 1

2(X + Y ).

The resulting expressions enable the formulation of EEM methods forthe thermo dissipative continuum dynamics presented in Section 5.6 basedon just one internal variables, which is besides the most common choice.

A.4 Linearization of the discrete derivative operator

All the proposed EEM methods are implicit and lie at the discrete deriva-tive operator. Therefore, in the required linearization process the linearized

300 A.4. Linearization of the discrete derivative operator

discrete derivative operator will be involved. For this reason, it is elaboratednext with the help of the directional (Frechet) derivative defined as follows

d

dε[Df(x+ εw,y)]

∣∣∣∣ε=0

= Dx(Df(x,y))[w] =∂Df(x,y)

∂x·w (A.43)

Using the definition (A.6) together with the associative property andthe chain product rule of the directional derivative the linearized discretederivative can be demonstrated to be

Dx(Df(x,y))[w] =

=1

2D2f (v) ·w +Dx

(f(x)− f(y)− 〈Df(v),u〉U

〈u,u〉U

)[w]u

+f(x)− f(y)− 〈Df(v),u〉U

〈u,u〉UI ·w

(A.44)

with I being the identity in the tangent space TU .

Applying the chain rule to the second term on the right hand side of theabove equation leads to

Dx

(f(x)− f(y)− 〈Df(v),u〉U

〈u,u〉U

)[w]

=Dx (f(x)− f(y)− 〈Df(v),u〉U ) [w]

〈u,u〉U− (f(x)− f(y)− 〈Df(v),u〉U )Dx (〈u,u〉U ) [w]

(〈u,u〉U )2

=〈Df(x),w〉U − 〈〈D2f(v),

1

2w〉U ,u〉U − 〈Df(v),w〉U

〈u,u〉U− 2 (f(x)− f(y)− 〈Df(v),u〉U ) 〈u,w〉U

(〈u,u〉U )2

(A.45)

Finally, all these terms can be rearranged so that the linearization canbe expressed as follows

∂Df(x,y)

∂x=

1

2D2f (v)

+u⊗Df(x)− u⊗ (

1

2D2f(v) · u)− u⊗Df(v)

〈u,u〉U+ 2

f(x)− f(y)− 〈Df(v),u〉U〈u,u〉U

(1

2I − u⊗ u〈u,u〉U

) (A.46)


This result leads to non-symmetric tangents which tends to disappear asthe time step size is reduced or u→ 0

limu→0

∂Df(x,y)

∂x=

1

2D2f (x) =

1

2D2f (y) (A.47)

302 A.4. Linearization of the discrete derivative operator

Linearized balance laws andspatial discretization A

ppendix

B

B.1 Linearization and Newton-Raphson process

The weak formulation of the thermo-dissipative continuum dynamics pre-sented in Chapter 3 and recovered by the GENERIC framework in Chapter 4is nonlinear respect to both the geometry and the material. Therefore, for agiven instant time, there exist a state of thermodynamic equilibrium given bya deformed configuration and a thermodynamic variable: temperature, en-tropy or others. As a result, this nonlinear formulation must be linearized inorder to obtain such thermodynamic equilibrium states by using the Newton-Raphson iterative process.

This linearization can be performed in either the Lagrangian or Euleriandescriptions, leading both to the same quantities. For convenience, all thebalance laws have been elaborated in the Lagrangian description where everyvolume integral is performed over the same fixed initial volume: V0.

Unlike other approaches found in the literature, all the methods dealtwith in this dissertation are monolithic in the sense that the mechanic andthermal coupling is solved at once. Thereby, each term in the weak formu-lation must be linearized respect to both the mechanic and thermal configu-rations.

Based on the mechanical context, the notation of the virtual work prin-ciple for the linear momentum weak form (3.148)2 is assummed by choosingwp = δp/ρ0 = δv, and accordingly define the inertial, internal and external

304 B.1. Linearization and Newton-Raphson process

virtual work terms

δW (ϕ,p, s,Λα, δv) = δWine(p, δv) + δWint(ϕ, s,Λα, δv)

− δWext(ϕ, δv)(B.1)

with s representing the thermodynamic variable and

δWine(p, δv) =

∫B0

δv · pdV0

δWint(ϕ, s,Λα, δv) =

∫B0

∇0δv : FSdV0

δWext(ϕ, δv) =

∫B0

δv ·B0dV0 +

∫∂B0

δv · T0dA0

(B.2)

In analogy with the virtual work principle, the weak form of the energybalance, formulated either in entropy form (3.146), with wη = δη, or intemperature form (3.147), with wΘ = δΘ, is expressed as follows

δQs(ϕ,p, s,Λα, δs) = δQsine(s, δs) + δQs

int(ϕ,p, s,Λα, δs)

+ δQsdis(ϕ, s,Λ

α, δs)− δQsext(s, δs),

(B.3)

where the inertial-like term is

δQsine(s, δs) =

∫B0

δssdV0, (B.4)

with s playing the role of either entropy or temperature.

The rest of the terms needs differentiated definitions for the entropy orthe temperature formulation. Thus, in the entropy formulation the termcalled internal considers the heat conduction phenomenon, that is

δQηint(ϕ, η,Λ

α, δη) = −∫

B0

∇0

[δη

Θ

]·HdV0

= −∫

B0

∇0δηΘ− δη∇0Θ

Θ2·HdV0

(B.5)

whereas in the temperature formulation, in addition to it, the term account-ing for the thermoelastic heating is also encompassed to be

δQΘint(ϕ,p,Θ,Λ

α, δΘ) = −∫

B0

1

c∇0δΘ ·HdV0

+

∫B0

1

cδΘFV : ∇0

[p

ρ0

]dV0

(B.6)

B. Linearized balance laws and spatial discretization 305

The same happens to the dissipation term, which, in the entropy formu-lation, only accounts for the rate of dissipation

δQηdis(ϕ, η,Λ

α, δη) =

∫B0

δη

ΘDdV0 =

∫B0

δη

Θ

m∑α=1

Qα : K(Qα)dV0, (B.7)

whereas, in the temperature formulation, it also collects the thermodissipa-tive heating to be

δQΘdis(ϕ,Θ,Λ

α, δΘ) =

∫B0

δΘ

cDdV0 +

∫B0

δΘ

cHddV0 (B.8)

Finally, the external term in the entropy case is provided by

δQηext(η, δη) =

∫B0

δη

ΘRdV0 +

∫∂B0

δη

ΘHNdA0, (B.9)

whereas, in the temperature one, is provided by

δQΘext(Θ, δΘ) =

∫B0

δΘ

cRdV0 +

∫∂B0

δΘ

cHNdA0 (B.10)

So far, the thermo-dissipative continuum evolution equations have beenpresented as a first order partial differential equation (3.148). However, inorder to simplify the linearization process, the density linear momentum andinternal variables will be eliminated by using (3.148)1 and (3.148)4, thustransforming the original 1st order PDE system into a 2nd order PDE one,typically handled in continuum dynamics. The second elimination agreeswith a condensation strategy for the internal variables. Thereby, the ther-modynamic state is fully defined by the configuration ϕ, which somehowretains the information stored by the density linear momentum and the in-ternal variables, and the chosen thermodynamic state variable s.

Then, the Newton-Rapshon departs from a trial solution (ϕk, sk) so thatthe thermodynamic equilibrium can be achieved by the linearization of theabove equations in the direction of an increment u in ϕk, henceforth referredto as configurational linearization, by using the directional derivative D(·)[·],as

δW (ϕk,p, s,Λα, δv) +DδW (ϕk,p, s,Λ

α, δv)[u] = 0

δQ(ϕk,p, s,Λα, δs) +DδQ(ϕk,p, s,Λ

α, δs)[u] = 0(B.11)

and in the direction of an increment t in sk, hereafter referred to as thermallinearization, as

δW (ϕ,p, sk,Λα, δv) +DδW (ϕ,p, sk,Λ

α, δv)[t] = 0

δQ(ϕ,p, sk,Λα, δs) +DδQ(ϕ,p, sk,Λ

α, δs)[t] = 0(B.12)

306 B.2. Lagrangian linearization of the IBVP weak form

B.2 Lagrangian linearization of the IBVP weak form

The directional derivative is applied to each term defined in the previoussection to obtain the linearized form of the problem’s weak form. Accordingto the Newton-Rapshon process (B.11) - (B.12), each term requires the elab-oration of two types of directional derivatives, one that accounts for changesin the configuration and another that measures the changes due to a pertur-bation on the thermodynamical state variable.

Linearization of the inertial virtual work. Considering the relation be-tween the motion and the density linear momentun (3.148)1 the linearizationwith respect to the configuration of the inertial term results in

DδWine(p, δv)[u] =

∫B0

δv ·Dp[u]dV0 =

∫B0

δv · ρ0udV0 (B.13)

This term does not contribute to the linearization with respect to thethermodynamical variable.

Linearization of the internal virtual work. Using the product rule fordirectional derivatives and the definition of the material elasticity tensor, thedirectional derivative in the direction of an increment of u is obtained as

DδWint(ϕ, s,Λα, δv)[u] =

∫B0

D(∇0δv : FS)[u]dV0

=

∫B0

1

2DδC[u] : SdV0 +

∫B0

1

2δC : DS[u]dV0

=

∫B0

S :(∇0δv

T∇0u)

dV0 +

∫B0

1

4DC[δv] : C : DC[u]dV0

=

∫B0

S :(∇0δv

T∇0u)

dV0 +

∫B0

1

4DC[δv] : C : DC[u]dV0,

(B.14)

where use has been made of the crucial relation

DC[δv] = ∇0δvTF + FT∇0δv = δFTF + FTδF = δC, (B.15)

and the linearized material fourth order tensor C : B0 × [0, T ] 7→ Sym(T4d)

has been introduced as

C(X, t) := 2∂S(C, s,Λα)

∂C, (B.16)

with S(C, s,Λα) being provided either by (3.36) or (3.18) according to theelection of the thermodynamical variable.


The resulting linearized term (B.14) is a classical result in nonlinear con-tinuum mechanics. In fact, its terms are normally referred to as geometricaland material tangents.

On the other hand, the thermodynamically-linearized term is elaboratedas follows

DδWint(ϕ, s,Λα, δv)[t] =

∫B0

D(∇0δv : FS)[t]dV0

=

∫B0

1

2δC : DS[t]dV0 =

∫B0

1

2δC :

∂S

∂stdV0

=

∫B0

1

2DC[δv] : TtdV0,

(B.17)

where T : B0 × [0, T ] 7→ Sym(T2d) the referential stress-temperature tensor.

Linearization of the external virtual work. In some applications the ex-ternal actions may be motion-dependent, i.e B0(ϕ) or T0(ϕ), so that theywould contribute to the linearized equations as follows

DδWext(ϕ, δv)[u] =

∫B0

δv ·DB0[u]dV0 +

∫∂B0

δv ·DT0[u]dA0

=

∫B0

δv · ∂B0

∂ϕudV0 +

∫∂B0

δv · ∂T0

∂ϕudA0

(B.18)

For the energy balance weak form, the distinction for the entropy andthe temperature formulations must be made.

B.2.1 Linearization of the entropy formulation

Linearization of the inertial-like virtual energy. The application of thedirectional derivative with respect the thermodynamical variable in the di-rection of s to (B.4), s = η, yields

DδQηine(η, δη)[s] =

∫B0

δηDη[s]dV0 =

∫B0

δηsdV0 (B.19)

As this term does not depend on the configuration ϕ, it has no contri-bution to the configurational linearized equations.

Linearization of the internal virtual energy. This term contributes toboth the configurational and thermal linearized equations.


Then, the linearization respect to the thermal variable entropy is elabo-rated by using the chain rule of the directional derivative in (B.5) to yield

DδQηint(ϕ, η,Λ

α, δη)[s] = −∫

B0

∇0

[D

(δη

Θ

)[s]

]·HdV0

−∫

B0

∇0

[δη

Θ

]·DH [s]dV0

(B.20)

The first directional derivative is developed by the product rule to be

D

(δη

Θ

)[s] = − δη

Θ2DΘ[s] = − δη

Θ2

∂Θ

∂ηs, (B.21)

while the second one results in

DH [s] = D(−JF−1κF−T∇0Θ)[s]

= −JF−1Dκ[s]F−T∇0Θ− JF−1κF−T∇0DΘ[s]

= −JF−1 ∂κ

∂ΘDΘ[s]F−T∇0Θ− JF−1κF−T∇0

[∂Θ

∂η

]s

(B.22)

In the common case of thermal isotropy the above term can be simplifiedto be

DH [s] = −k′(Θ)∂Θ

∂ηsJC−1∇0Θ− k(Θ)JC−1∇0

[∂Θ

∂η

]s (B.23)

Thereby, the full thermally-linearized internal term can be rearranged as

DδQηint(ϕ, η,Λ

α, δη)[s] =

∫B0

∇0

[δη

Θ2

∂Θ

∂ηs

]·HdV0

+

∫B0

∇0

[δη

Θ

]· k′(Θ)

∂Θ

∂ηsJC−1∇0ΘdV0

+

∫B0

∇0

[δη

Θ

]· k(Θ)JC−1∇0

[∂Θ

∂η

]sdV0

(B.24)

The linearization respect to the configuration is provided by the following


directional derivative

DδQηint(ϕ, η, δη)[u] = −

∫B0

∇0

[D

(δη

Θ

)[u]

]·HdV0

−∫

B0

∇0

[δη

Θ

]·DH [u]dV0

=

∫B0

∇0

[δη

Θ2DΘ[u]

]·HdV0

+

∫B0

∇0

[δη

Θ

]·D(JF−1κF−T)[u]∇0ΘdV0

+

∫B0

∇0

[δη

Θ

]· JF−1κF−T∇0DΘ[u]dV0

(B.25)

To proceed further, the ensuing directional derivative needs to be elabo-rated

D(JF−1κF−T)[u] = DJ [u]F−1κF−T + JF−1Dκ[u]F−T

+ JDF−1[u]κF−T + JF−1κDF−T[u],(B.26)

where the resulting directional derivatives are classic in nonlinear mechanics,see Bonet & Wood (2008), and are provided by

DJ [u] = D det F[DF[u]] = JF−T : ∇0u

DF−1[u] = DF−1[DF[u]] = −F−1∇0uF−1

DF−T [u] = DF−T[DF[u]] = −F−T∇0uTF−T

(B.27)

In addition, the spatial conductivity tensor is linearized as

Dκ[u] = Dκ[DΘ[u]] =∂κ

∂ΘDΘ[u] (B.28)

Finally, both (B.25) and (B.28) can be fully elaborated by using

DΘ[u] = DΘ[DC[u]] = 2∂Θ

∂C: FT∇0u, (B.29)

where use has been made of (B.15) and the symmetry of the tensor ∂Θ∂C

.

Linearization of the dissipated virtual energy. As the above term, thedissipated virtual energy one has contributions to both the configurationaland thermal linearized equations.


On the one hand, the thermal linearization yields

DδQηdis[s] =

∫B0

D

(δη

Θ

)[s]

m∑α=1

Qα : K(Qα)dV0

+

∫B0

δη

Θ

m∑α=1

DQα[s] : K(Qα)dV0

+

∫B0

δη

Θ

m∑α=1

Qα : DK(Qα)[s]dV0

= −∫

B0

δη

Θ2

∂Θ

∂ηsDdV0 +

∫B0

δη

Θ

m∑α=1

∂Qα

∂ηs : K(Qα)dV0

+

∫B0

δη

Θ

m∑α=1

Qα :

(dK

dQα:∂Qα

∂η

)sdV0

(B.30)

On the other hand, the configurational linearized contribution results in

DδQηdis[u] =

∫B0

D

(δη

Θ

)[u]

m∑α=1

Qα : K(Qα)dV0

+

∫B0

δη

Θ

m∑α=1

DQα[u] : K(Qα)dV0

+

∫B0

δη

Θ

m∑α=1

Qα : DK[DQα[u]]dV0

(B.31)

where the first term is fully elaborated using the chain rule together with(B.29) while the rest ones lie at the following directional derivative

DQα[u] = DQα[DC[u]] = Qα : FT∇0u, (B.32)

with Qα : B0 × [0, T ] 7→ Sym(T4d) being the linearized material-dissipative

fourth order tensor provided by

Qα := 2∂Qα(C, η,Λα)

∂C(B.33)

Rearranging terms the contribution to the configurational linearized equa-


tions yields

DδQηdis[u] = −

∫B0

2D δηΘ2

∂Θ

∂C: FT∇0udV0

+

∫B0

δη

Θ

m∑α=1

Qα : FT∇0u : ΛαdV0

+

∫B0

δη

Θ

m∑α=1

Qα :

(dK

dQα: Qα : FT∇0u

)dV0

(B.34)

B.2.2 Linearization of the temperature formulation

Linearization of the inertial-like virtual energy. This term only con-tributes to the thermal linearized equations, that is

DδQΘine(Θ, δΘ)[t] =

∫B0

δΘDΘ[t]dV0 =

∫B0

δΘtdV0 (B.35)

Linearization of the internal virtual energy. As was defined in (B.6),it has two terms that will hereafter be indicated by the subscript 1 and 2,respectively.

Thus, the first term is thermally linearized as follows

DδQΘint,1(ϕ,Θ, δΘ)[t] = −

∫B0

1

c∇0δΘ ·DH [t]dV0 (B.36)

where the directional derivative applied to the heat vectorH nearly resemblesthe expression (B.22), this time considering the temperature as state variableto give

DH [t] = −JF−1 ∂κ

∂ΘtF−T∇0Θ− JFκ(Θ)F−T∇0t (B.37)

The second term is conveniently expressed in terms of the rate of theCauchy-Green deformation tensor, using (3.83) to be linearized as follows

DδQΘint,2(ϕ,Θ, δΘ)[t] =

∫B0

D

(δΘ

c

1

2C : V

)[t]dV0

=

∫B0

δΘ

c

1

2C : DV[t]dV0 =

∫B0

δΘ

c

1

2C :

∂V

∂ΘtdV0

=

∫B0

δΘ

c

1

2C : DV[t]dV0 =

∫B0

δΘ

c

1

2DC[v] :

∂V

∂ΘtdV0

(B.38)


On the other hand, the first term contributes to the configurational lin-earized equations in the following way

DδQΘint,1(ϕ,Θ, δΘ)[u] = −

∫B0

1

c∇0δΘ ·DH [u]dV0 (B.39)

The linearization of the heat vector then follows the expression (B.26)except for the term derived from the conductivity tensor as, in the temper-ature formulation, it does not depend any longer on the configuration, thatis

DH [u] = −(DJ [u]F−1κF−T + JDF−1[u]κF−T + JF−1κDF−T[u])∇0Θ,

(B.40)

with the appearing directional derivatives being again provided by (B.27).

In analogy with (B.14), the second term can be demonstrated to con-tribute to the configurational linearization as follows

DδQΘint,2(ϕ,Θ, δΘ)[u] =

∫B0

δΘ

cD

(1

2C : V

)[u]dV0

=

∫B0

1

cδΘ

(1

2DC[u] : V +

1

2C : DV[u]

)dV0

=

∫B0

1

cδΘ

(1

2DC[u] : V +

1

4DC[v] : D : DC[u]

)dV0

=

∫B0

1

cδΘ

(V : (∇0v

T∇0u) +1

4DC[v] : D : DC[u]

)dV0,

(B.41)

where following the definition of the material tensor (B.16), the fourth ordertensor D : B0 × [0, T ] 7→ Sym(T4

d) has been introduced as

D := 2∂V(C,Θ,Λα)

∂C(B.42)

Linearization of the dissipated virtual energy. This term was also in-troduced as sum of two contributions, the first accounting for the rate ofdissipation and the second for the thermo-dissipative heating, which will behereafter referred to 1 and 2, respectively.

Then, the linearization of the first term respect to the temperature is

DδQΘdis,1[t] =

∫B0

δΘ

c

m∑α=1

DQα[t] : K(Qα)dV0

+

∫B0

δΘ

c

m∑α=1

Qα : DK[DQα[t]]dV0

(B.43)


whereas the second term counterpart similarly yields

DδQΘdis,2[t] =

∫B0

δΘ

cDHd[t]dV0

=

∫B0

δΘ

c

m∑α=1

DWα[t] : K(Qα)dV0

+

∫B0

δΘ

c

m∑α=1

Wα : DK[DQα[t]]dV0

(B.44)

These last expressions are completed by the following directional deriva-tives

DQα[t] =∂Qα

∂Θt

DWα[t] =∂Wα

∂Θt

DK[DQα[t]] =dK

dQα:∂Qα

∂Θt

(B.45)

Likewise, the resulting linearization of the first term is alike (B.31), thatis

DδQΘdis,1[u] =

∫B0

1

cδΘ

m∑α=1

DQα[DC[u]] : K(Qα)dV0

+

∫B0

1

cδΘ

m∑α=1

Qα : DK[DQα[DC[u]]dV0

=

∫B0

1

cδΘ

m∑α=1

Qα : FT∇0u : K(Qα)dV0

+

∫B0

1

cδΘ

m∑α=1

Qα :dK

dQα: Qα : FT∇0udV0,

(B.46)

now Qα : B0 7→ Sym(T4d) being provided by

Qα := 2∂Qα(C,Θ,Λα)

∂C(B.47)

Finally, similarly proceeding with the second term, its configurationally


linearized counterpart results in

DδQΘdis,2[u] =

∫B0

δΘ

cDHd[u]dV0

=

∫B0

δΘ

c

m∑α=1

DWα[DC[u]] : K(Qα)dV0

+

∫B0

δΘ

c

m∑α=1

Wα : DK[DQα[DC[u]]]dV0

=

∫B0

δΘ

c

m∑α=1

Wα : FT∇0u : K(Qα)dV0

+

∫B0

δΘ

c

m∑α=1

Wα :dK

dQα: Qα : FT∇0udV0,

(B.48)

where, in analogy with (B.33), the resulting symmetric fourth order tensorWα : B0 7→ Sym(T4

d) has been introduced as

Wα := 2∂Wα(C,Θ,Λα)

∂C(B.49)

B.2.3 Spatially discretized evolution equations

The discretization is performed in the undeformed configuration usingisoparametric elements defined by nodal particles Xa

Xe =

p∑a=1

Na(ξ, η, ζ)Xa, (B.50)

with Na being the a-th standard element shape function and p being thenumber of nodes of the element.

Discretized state variables. In the isoparametric approach, the statevariables are approximated using the same standard shape functions, suchthat

ϕe =

p∑a=1

Naxa(t), se =

p∑a=1

Nasa(t), Λα,e =

p∑a=1

NaΛαa (t), (B.51)

with xa being the current position of the nodal particles.


Discretized kinematics. All the kinematic quantities are discretized basedon the discretization of the motion function. Thus, the discretized defor-mation gradient tensor results from applying the material gradient to it asfollows

Fe =

p∑a=1

xa(t)⊗∇0Na, (B.52)

and, consequently, the discretized right Cauchy-Green deformation tensor is

Ce =

p∑a=1

p∑b=1

xa(t) · xb(t)∇0Na ⊗∇0Nb, (B.53)

The material gradient of the shape functions can be obtained from thegradient in the isoparametric space (∇ξ), where they are naturally defined,as follows

∇0Na =

(∂X

∂ξ

)−T

∇ξNa, with∂X

∂ξ=

p∑a=1

Xa ⊗∇ξNa (B.54)

On the other hand, the discretized material velocity, or the linear mo-mentum density, results from time differentiating the discretized motion togive

ve = pe/ρe0 =

p∑a=1

Naxa(t), (B.55)

Discretized linear momentum equation. Consider the contribution toδW caused by a single virtual nodal velocity δva occurring at a typical nodei of element e. Following the format provided by (B.1), the contribution dueto the inertial term is elaborated as follows

δW eine(p, Naδva) = δva ·

∫Be

0

NapdV e0 (B.56)

Similarly, the contribution due to the internal forces results in

δW eint(ϕ

e, se,Λα,e, Naδva) =

∫Be

0

(δva ⊗∇0Na) : FeSedV e0

= δva ·∫

Be0

FeSe∇0NadVe

0

(B.57)


And, finally, the contribution due to the external forces yields

δW eext(ϕ

e, Naδva) = δva ·∫

Be0

NaB0dV e0 + δva ·

∫∂Be

0

NaT0dAe0 (B.58)

Arranging each of the above contribution, the searched contribution causedby a single virtual velocity at a node e is provided by

δW e(ϕ, s,Λα, Naδva) = δva ·(∫

Be0

NapdV e0 +

∫Be

0

FeSe∇0NadVe

0

−∫

Be0

NaB0dV e0 −

∫∂Be

0

NaT0dAe0

) (B.59)

It only remains to assemble each of the element contribution concurringin a node to obtain the discretized linear momentum equation

δW (ϕ, s,Λα, Naδva) =ma∑e=1e3a

δW e(ϕ, s,Λα, Naδva) =ma∑e=1e3a

δva · (Iea + T ea − F e

a )

(B.60)

Discretized energy equation. Analogously, consider the contribution toδQ caused by a single variation of the thermal variable δsa occurring at atypical node i of element e. Then, the contribution of each of the term in(B.1) is elaborated next according to the choice for the thermal variable.First, the inertial-like term contributes as follows

δQs,eine(s

e, Naδsa) = δsa

∫Be

0

NasedV e

0 , (B.61)

with s playing the role of either entropy or temperature.

The rest of the terms needs differentiated elaborations for the entropy orthe temperature formulation. Thus, in the entropy formulation the contri-bution due to the internal term (B.5) results in

δQη,eint(ϕ

e, ηe,Λα,e, Naδηa) = −∫

Be0

∇0

[Naδηa

Θe

]·HedV e

0

= δηa

(−∫

Be0

1

Θe∇0Na ·HedV e

0 +

∫Be

0

Na

(Θe)2∇0Θe ·HedV e

0

) (B.62)


with

He = −Ke∇0Θe (B.63)

The entropy formulation requires further elaboration of the gradient oftemperature to express it in terms of the state variables. To this end, thechain rule is applied to give

∇0Θe =∂Θ

∂C: ∇0C

e +∂Θ

∂η∇0η

e +∂Θ

∂Λα: ∇0Λ

α,e, (B.64)

where the gradients applied to state variables are easily computed through thegradient of the shape functions. However, the gradient of the right Cauchy-Green deformation tensor needs further elaboration to relate it with the dis-cretized motion state variable. The directional derivative is used to arrive atthe identity

DCe[U ] = ∇0C : U = FT∇0FU +∇0FUTF

= FT∇0F : U +∇0FTF : U

= (FT∇0F +∇0FTF) : U

(B.65)

Therefore, the problem is translated to compute the gradient of the dis-cretized deformation gradient (B.52). The application of the material gradi-ent to it leads to

∇0Fe =

p∑a=1

xa ⊗∇20Na

=

p∑a=1

xa ⊗[∂2Na

∂ξ2:

(∂ξ

∂X⊗ ∂ξ

∂X

)+∂Na

∂ξ:∂2ξ

∂X2

]

=

p∑a=1

xa ⊗[(

∂X

∂ξ⊗ ∂X

∂ξ

)−T

:∂2Na

∂ξ2+∂Na

∂ξ:∂2ξ

∂X2

],

(B.66)

where use has been made of (B.54)2.

The main implication of the above expression is that second derivativesof the standard shape functions are required, imposing an extra requirementover the definition of shape functions, that is, they must have non-trivialsecond derivatives. This issue was pointed out in Section 3.8.1.

Summarizing, the gradient of the discretized right Cauchy-Green defor-


mation tensor results in

∇0C = (Fe)T∇0Fe + Fe(∇0F

e)T =

p∑a=1

(∇0Na ⊗ xa)(xa ⊗∇2

0Na

)+

p∑a=1

(xa ⊗∇0Na)(∇2

0Na ⊗ xa)

= 2

p∑a=1

(xa · xa)(∇0Na ⊗∇2

0Na

)(B.67)

On the other hand, for the temperature formulation, the contribution ofthe internal term (B.6) is

δQΘ,eint (ϕe,pe,Θe,Λα,e, NaδΘa) = −δΘa

∫Be

0

1

c∇0Na ·HedV e

0

+ δΘa

∫Be

0

1

cFeVe : ∇0

[pe

ρe0

]NadV

e0

(B.68)

The dissipation term also adopts different definitions for each of the for-mulations considered. In the entropy one provided by (B.7), its contributionis

δQη,edis(ϕ

e, ηe,Λα,e, Naδηa) = δηa

∫Be

0

1

ΘeDeNadV

e0

= δηa

∫Be

0

1

Θe

m∑α=1

Qα,e : K(Qα,e)NadVe

0 ,

(B.69)

whereas, in the temperature formulation (B.8), it becomes

δQΘ,edis (ϕe,Θe,Λα,e, NaδΘa) = δΘa

∫Be

0

1

cDeNadV

e0

+ δΘa

∫Be

0

1

cHedNadV

e0

(B.70)

with

Hed =

m∑α=1

Wα,e : K(Qα,e) (B.71)

Finally, the external term in the entropy case (B.9) is discretized as fol-lows

δQη,eext(η

e, Naδηa) = δηa

∫Be

0

1

ΘeRNadV

e0 + δηa

∫∂Be

0

1

ΘeHNNadA

e0, (B.72)


whereas, in the temperature one (B.10), is

δQΘ,eext (Θe, NaδΘa) = δΘa

∫Be

0

1

cRNadV

e0 + δΘa

∫∂Be

0

1

cHNNadA

e0 (B.73)

Then, each of the above contributions are regrouped and assembled in thesame way as was performed for the linear momentum balance to arrive at thediscretized energy balance formulated either in entropy or in temperaturesvariables

δQs(ϕ,p, η,Λα, Naδηa) =ma∑e=1e3a

δQs(ϕe,pe, se,Λα,e, Naδηa)

=ma∑e=1e3a

δsa (J ea +Qea +De

a −Gea) ,

(B.74)

where J ea is the inertial-like equivalent nodal heat, Qea is the equivalent nodal

heat, Dea is the dissipated equivalent nodal heat and Ge

a is the externalequivalent nodal heat.

B.2.4 Discretization of the linearized evolution equations

The linearization of discretized evolution equation of linear momentumand energy can be derived from the discretized form of the evolution equa-tions presented in the previous subsection and following (B.11) and (B.12),such that

DδW e(ϕ,p, s,Λα, Naδva)[Nbub] = D(δva · (Iea + T ea − F e

a ))[Nbub]

= δva ·D(Iea + T ea − F e

a )[Nbub]

= δva ·Meabub + δva ·Ke

abub

(B.75)

DδW e(ϕ,p, s,Λα, Naδsa)[Nbtb] = D(δsa(Iea + T e

a − F ea ))[Nbtb]

= δsaD(T ea )[Nbtb]

= δsaSeabtb

(B.76)

DδQe(ϕ,p, s,Λα, Naδsa)[Nbtb] = D(δsa(Jea +Qe

a +Dea −Ge

a))[Nbtb]

= δsaD(J ea +Qea +De

a −Gea)[Nbtb]

= δsaCeabtb + δsaQ

eabtb + δsaD

eabtb

(B.77)


DδQe(ϕ,p, s,Λα, Naδsa)[Nbub] = D(δsa(Jea +Qe

a +Dea −Ge

a))[Nbub]

= δsaD(J ea +Qea +De

a −Gea)[Nbub]

= δsaCeab · ub + δsaP

e · ub + δsaLe · ub

(B.78)

Similar procedures as those carried out in the previous subsection allow toobtain the expressions for the arising quantities. Then, the classical exampleof the linearization of the inertial terms leads to the definition of the massmatrix according to

DδW eine(p, Naδva)[Nbub] =

∫B0

Naδva · ρ0NbubdV0

= δva ·∫

B0

ρ0NaNbdV0ub

(B.79)

Similarly, the expression for the stiffness matrix can be obtained. For theinertial-like term of the energy balance, a similar matrix arose, that is

DδQs,eine(Naδsa)[Nbsb] =

∫B0

NaδsaNbsbdVe

0 = δsa

∫B0

NaNbdVe

0 sb (B.80)

Finally, as an example, the matrix Qe is derived for the entropy formu-lation

DδQη,eint(ϕ

e, ηe, Naδηa)[Nbsb] = δηa

∫B0

∇0Na ·Ke∇0

[∂Θ

∂η

]NbdV0sb

+ δηa

∫B0

k′(Θe)∂Θe

∂ηe∇0Na ·

[JCe,−1∇0Θe

]dNbV0sb

(B.81)

where, as in the discretized energy balance, the appearing material gradientsmust be computed according to (B.65) - (B.67) with

∇0

[∂Θ

∂η

]=

∂2Θ

∂η∂C: ∇0C +

∂2Θ

∂η2∇0η +

∂2Θ

∂η∂Λα: ∇0Λ

α (B.82)

Once each of the contributions has been computed, the global matricesare obtained by performing the classical assembly.

List of Figures

2.1 Thermo-spring system . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Thermo-visco-elastic system . . . . . . . . . . . . . . . . . . . 27

2.3 Example 1: solution obtained with the Midpoint method for-mulated in temperature variables with ∆t = 0.3 s . . . . . . . 45

2.4 Example 1: solution obtained with the Midpoint method for-mulated in entropy variables with ∆t = 0.3 s . . . . . . . . . . 46


2.6 Example 2: solution obtained with the Midpoint method for-mulated in entropy variables with ∆t = 0.1 s . . . . . . . . . . 49


2.8 Case I: reference solution obtained with the Midpoint methodformulated in entropy variables with ∆t = 0.01 s . . . . . . . 53

2.9 Case I: solution obtained with the Midpoint method formu-lated in entropy variables with ∆t = 0.2 s . . . . . . . . . . . 54

2.10 Case I: solution obtained with the Midpoint method formu-lated in temperature variables with ∆t = 0.2 s . . . . . . . . . 55

2.11 Case II: reference solution obtained with the Midpoint methodformulated in temperature variables with ∆t = 0.01 s . . . . . 56

2.12 Case II: solution obtained with the Midpoint method formu-lated in temperature variables with ∆t = 0.3 s . . . . . . . . . 57

i

3.1 Continumm kinematics . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Volume forces and tractions in the reference configuration. . . 71

3.3 Volumetric source or sink of heat and heat flux escaping throughthe boundary in the reference configuration . . . . . . . . . . . 74

3.4 Relaxation and creep processes . . . . . . . . . . . . . . . . . 86

3.5 Rheological model . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.6 Multiplicative decomposition in the small neighborhood of aparticle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.7 Finite elements partition of the continuum . . . . . . . . . . . 99

3.8 Shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.1 State vector error vs. time step . . . . . . . . . . . . . . . . . 184

5.2 Example 1: State vector error vs. time step . . . . . . . . . . 184

5.3 Example 1: Total energy and total entropy obtained with eachof the temperature-based EEM methods and ∆t = 0.3. (Samelegend as Figure 5.2). . . . . . . . . . . . . . . . . . . . . . . . 185

5.4 Example 1: solution obtained with entropy-based EEM methodwith ∆t = 0.3 s . . . . . . . . . . . . . . . . . . . . . . . . . . 186

5.5 Example 1: solution obtained with the temperature-basedEEM method with ∆t = 0.3 s . . . . . . . . . . . . . . . . . . 187

5.6 Example 2: solution obtained with entropy-based EEM methodwith ∆t = 0.1 s . . . . . . . . . . . . . . . . . . . . . . . . . . 188

5.7 Example 2: solution obtained with temperature-based EEMmethod with ∆t = 0.1 s . . . . . . . . . . . . . . . . . . . . . 189

5.8 Example 3: solution obtained with temperature-based EEMmethod with ∆t = 0.1 s . . . . . . . . . . . . . . . . . . . . . 190

5.9 Case I: Position (q) relative error . . . . . . . . . . . . . . . . 204

5.10 Case I: Temperature (θ) relative error . . . . . . . . . . . . . . 205

5.11 Case I: Internal variable (γ) relative error . . . . . . . . . . . . 205

5.12 Case I: solution obtained with temperature-based EEM methodwith ∆t = 0.2 s . . . . . . . . . . . . . . . . . . . . . . . . . . 206

5.13 Case I: solution obtained with the entropy-based EEM methodwith ∆t = 0.2 s . . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.14 Case II: Position (q) relative error . . . . . . . . . . . . . . . . 208

5.15 Case II: Element temperature (θ) relative error . . . . . . . . 209

5.16 Case II: Internal variable (γ) relative error . . . . . . . . . . . 209

5.17 Case II: solution obtained with temperature-based EEM methodwith ∆t = 0.3 s . . . . . . . . . . . . . . . . . . . . . . . . . . 210

6.1 Cantilever beam and external actions . . . . . . . . . . . . . . 239

6.2 Snapshot of the motion of the beam (from left to right, top tobottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

6.3 Vertical displacement of the extreme cross section centroid . . 241

6.4 XY Trajectory of the extreme cross section centroid . . . . . . 241

6.5 Beam - Energy: E = K + U . . . . . . . . . . . . . . . . . . . 242

6.6 Beam - Energy: E = K + U . . . . . . . . . . . . . . . . . . . 243

6.7 L-shaped block mesh and external forces . . . . . . . . . . . . 244

6.8 Snapshots of the motion of the L-shaped block (from left toright, top to bottom) . . . . . . . . . . . . . . . . . . . . . . . 245

6.9 L-Shaped block - Energy E = K + U . . . . . . . . . . . . . . 246

6.10 L-Shaped block - Long Term Simulation . . . . . . . . . . . . 247

6.11 L-Shaped block - Angular Momentum Norm . . . . . . . . . . 247

6.12 Twisted block and initial conditions . . . . . . . . . . . . . . . 249

6.13 Snapshots and contour plots of temperature of the Twistedblock (from top to bottom, left to right). Temperature-basedEEM integration, ∆t = 2 s . . . . . . . . . . . . . . . . . . . . 251

6.14 Standard methods: Total energy evolution . . . . . . . . . . . 252

6.15 Standard methods: Total Entropy evolution . . . . . . . . . . 252

6.16 Total energy evolution . . . . . . . . . . . . . . . . . . . . . . 253

6.17 Total Entropy evolution . . . . . . . . . . . . . . . . . . . . . 253

6.18 Total Angular momentum evolution . . . . . . . . . . . . . . . 254

6.19 L-shaped block with Dirichlet initial-boundary conditions . . . 255

6.20 Snapshots and contour plots of temperature of the L-shapedblock (from top to bottom, left to right). Temperature-basedEEM integration, ∆t = 0.2 s . . . . . . . . . . . . . . . . . . . 257

6.21 Total energy E = K + U . . . . . . . . . . . . . . . . . . . . . 258

6.22 Total Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

6.23 Total Angular Momentum . . . . . . . . . . . . . . . . . . . . 259

6.24 Satellite: mesh and boundary condition definition. . . . . . . . 265

6.25 Satellite: spherical joints’ position. . . . . . . . . . . . . . . . 266

6.26 Snapshots of the maneuver (from top to bottom, left to right).Energy-consistent integration, ∆t = 0.2 s . . . . . . . . . . . . 267

6.27 Satellite. Evolution of the total energy of the system, kineticplus potential, obtained with the Trapezoidal, Midpoint andisothermal EEM method. Only the isothermal EEM methodprovides a full solution which, in addition, is in accordancewith the laws of thermodynamics . . . . . . . . . . . . . . . . 268

6.28 Satellite.Evolution of the angular momentum of the systemobtained with the trapezoidal and isothermal EEM methods. . 269

6.29 Satellite, Total energy (kinetic plus potential), isothermal EEMintegration with ∆t = 0.2 s . . . . . . . . . . . . . . . . . . . . 269

6.30 Satellite model, showing the solar panels’ hinges and the rockets270

6.31 Rocket forces vs. time . . . . . . . . . . . . . . . . . . . . . . 271

6.32 Snapshots of the maneuver (from top to bottom, left to right).Consistent integration, ∆t = 0.2 s . . . . . . . . . . . . . . . . 273

6.33 Total angular momentum, consistent integration, ∆t = 0.2 s . 274

6.34 Energy, consistent integration, ∆t = 0.2 s . . . . . . . . . . . . 274

6.35 Entropy, consistent integration, ∆t = 0.2 s . . . . . . . . . . . 275

6.36 Stiffeners’ temperature, EEM method, ∆t = 0.2 s . . . . . . . 275

6.37 Stiffeners’ temperature, consistent integration, ∆t = 0.2 s . . . 276

6.38 Total angular momentum, ∆t = 0.2 s . . . . . . . . . . . . . . 276

6.39 Total energy ∆t = 0.2 s . . . . . . . . . . . . . . . . . . . . . . 277

Bibliography v

6.40 Total entropy, ∆t = 0.2 s . . . . . . . . . . . . . . . . . . . . . 277

6.41 Rotor blade with elastomeric bearing: mesh and boundarycondition definition. . . . . . . . . . . . . . . . . . . . . . . . . 278

6.42 Snapshots of the motion of the rotor blade with elastomericbearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

6.43 Vertical position of the blade center of mass . . . . . . . . . . 280

6.44 Evolution of the total energy of the blade system. . . . . . . . 281

6.45 Total energy (kinetic plus potential), isothermal EEM with∆t = 0.1 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

A.1 Geometrical interpretation of the mean value theorem . . . . . 290

vi Bibliography

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