Generalized Viscoelastic Material Design With Integro...

GENERALIZED VISCOELASTIC MATERIAL DESIGN WITHINTEGRO-DIFFERENTIAL EQUATIONS AND DIRECT OPTIMAL CONTROL

Lakshmi Gururaja Rao ∗Graduate Student

University of Illinois at Urbana ChampaignEmail: [email protected]

James T. AllisonAssistant Professor

University of Illinois at Urbana ChampaignEmail: [email protected]

ABSTRACTRheological material properties are examples of function-

valued quantities that depend on frequency (linear viscoelastic-ity), input amplitude (nonlinear material behavior), or both. Thisdependence complicates the process of utilizing these systems inengineering design. In this article, we present a methodology tomodel and optimize design targets for such rheological materialfunctions. We show that for linear viscoelastic systems simpleengineering design assumptions can be relaxed from a conven-tional spring-dashpot model to a more general linear viscoelasticrelaxation kernel, K(t). While this approach expands the designspace and connects system-level performance with optimal mate-rial design functions, it entails significant numerical difficulties.Namely, the associated governing equations involve a convolu-tion integral, thus forming a system of integro-differential equa-tions. This complication has two important consequences: 1)the equations representing the dynamic system cannot be writ-ten in a standard state space form as the time derivative functiondepends on the entire past state history, and 2) the dependenceon prior time-history increases time derivative function compu-tational expense. Previous studies simplified this process by in-corporating parameterizations of K(t) using viscoelastic modelssuch as Maxwell or critical gel models. While these simplifica-tions support efficient solution, they limit the type of viscoelasticmaterials that can be designed. This article introduces a moregeneral approach that can explore arbitrary K(t) designs us-ing direct optimal control methods. In this study, we analyzea nested direct optimal control approach to optimize linear vis-

∗Address all correspondence to this author.

coelastic systems with no restrictions on K(t). The study providesnew insights into efficient optimization of systems modeled us-ing integro-differential equations. The case study is based on apassive vibration isolator design problem. The resulting optimalK(t) functions can be viewed as early-stage design targets thatare material agnostic and allow for creative material design so-lutions. These targets may be used for either material-specificselection or as targets for later-stage design of novel materials.

1 INTRODUCTIONEngineers typically use hard materials or simple fluids

when designing engineering systems. Their properties are well-understood and easy to conceptualize. Soft, rheologically com-plex materials, however, have significant potential benefits, asdemonstrated by many biological systems [1–4]. These mate-rials can show dramatic transitions from elastic (solid-like) toviscous (fluid-like) behavior as a function of various parameters,including timescales (viscoelasticity), amplitude (shear-thinning,extensional thickening), or external fields. These inherent char-acteristics result in novel performance that can be important forengineering applications [5]. Soft materials have not been acomponent of conventional design as they are characterized byfunction-valued quantities that depend on frequency (linear vis-coelasticity), input amplitude (as in the case of non-linear ma-terial responses) or more generally, both. The options for di-rect mathematical-modeling with material properties (which weconsider design-driven or design-friendly modeling) is not fullyclear. In the case where soft materials are used for system design,the model is usually material-specific. While this is a useful tool

Proceedings of the ASME 2015 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

IDETC/CIE 2015 August 2-5, 2015, Boston, Massachusetts, USA

DETC2015-46768

1 Copyright © 2015 by ASME

in many design problems, it limits creative design ideation. Forexample, viscoelastic functionality may come from polymers,colloids, and many other forms of structured soft materials. Anearly stage material-agnostic approach places fewer structural re-strictions on the design space, and may help avoid design fixa-tion [6, 7]. Previous work identified need for system-level ma-terial design [8]. The focus of this kind of system-level designapproach is not the material itself, but its functionality.

Figure 1 illustrates a comprehensive framework for early-stage design that can be applied to the design of rheologicallycomplex materials. This is distinct from conventional materialdesign approaches that do not link system performance directlyto material design efforts. We envision a multi-level system de-sign process that is material-agnostic at early stages but material-specific at later stages. This multi-level hierarchical problemwill connect functional system performance down to rheologicalproperties, and finally down to the material formulation.

The functionality-drive material design process presentedhere involves identifying function-valued material properties thatare optimal for overall system performance. The approach is sim-ilar to the well-established methodology of analytical target cas-cading (ATC), where performance targets for lower-level designproblems are obtained via optimization [10, 11]. In this articlewe introduce a new approach for identifying optimal target ma-terial behavior functions for linear viscoelastic systems using anextension of direct optimal control. These target functions thenin turn may be used to guide the development of new materials.Earlier work utilized this hierarchical approach, but relied on re-strictive assumptions to simplify the target function optimizationproblem [8]. Here we generalize this approach in a way that liftsmaterial-specific assumptions through an extension of direct op-timal control [12]. This class of optimal control methods hasproven to be effective for problems involving physical design,including highly nonlinear systems and problems with inequal-ity constraints [13, 14].

The broader objective of this work is to create a paradigmfor creative and rational design of rheologically complex materi-als that directly connects system-level optimization and material-level design. Our approach is to develop an overall design pro-cess that involves design-driven mathematical modeling and op-timization techniques that are material-agnostic during the earlystages of optimal target setting, and that are material-specific dur-ing concept generation and material design. The article detailsthe modeling of viscoelastic materials, its challenges and appli-cation of optimal control to rheological design. The implemen-tation of the design philosophy is demonstrated using a vibrationisolator case study.

2 MODELING VISCOELASTIC MATERIALSIn this section we discuss generalized models for viscoelas-

tic models, generalized viscoelastic material design, related op-timization challenges, and a new perspective on modeling for

design of rheologically complex materials.

2.1 Generalized viscoelastic modelWhile previous studies have addressed viscoelastic material

design, many have excluded the important feature of frequencydependence, resulting in limited design design strategies that donot capitalize on the ability of viscoelastic materials to adjustproperties with changes in loading frequency [15]. Other workdoes account for time dependent behavior, but requires that thelinear viscoelastic functions are described by a superposition ofexponential functions [16, 17]. This restriction limits the designspace and requires a large number of design variables. An exam-ple of this modeling strategy is illustrated in Fig. 2. Viscoelasticbehavior can be approximated using a configuration of discretecomponents (springs and dashpots). The approach used here is toreplace the configuration of simple elements with a single gener-alized viscoelastic element characterized by the relaxation kernelK(t). Designing the function K(t) directly supports more flexi-ble design space exploration for early-stage design. The K(t) thatoptimizes system performance can then serve as a target functionfor material development. Here we focus on methods for settingthe target function. Material development based on optimal tar-gets is a subject of ongoing work outside the scope of this article.

Generalized

Viscoelastic

m

K(t)

m

FIGURE 2: A discrete topology can be generalized by the intro-duction of a viscoelastic element.

Here we consider one-dimensional deformation. The forceacross a viscoelastic connection can be modeled using a singlescalar equation:

FV E(t) =∫ t

−∞

K(t− t ′)x(t ′)dt ′ (1)

where FV E is the force due to a viscoelastic element, x is thevelocity experienced by the element (dimensions x=[LT-1]) andK(t− t ′)=[FL-1] ). With a change of variable s = t− t ′ this con-volution integral becomes

FV E(t) =∫

∞

0K(s)x(t− s)ds. (2)


Identify customer

needs

Establish target

specifications

Generate Product

Concepts

Select Product

ConceptsTest Product

ConceptsSet Final

SpecificationsPlan

DownstreamDevelopment

MissionStatement

Task 1-2Theory of modeling for design optimization with rheological

material functions

Task 3Optimization Algorithms

for function-valuedproperties

Task 4-5Case Studies that include

material-level design acrossa range of material classes

Development Plan

FIGURE 1: Early stages of the product design process and application to rheologically-complex materials (design process adapted fromUlrich and Eppinger [9])

The convolution integral in Eqn. (2) increases dynamic sys-tem optimization difficulty. In this article we focus on the casewhere K(t) is a general continuous function (i.e., no assump-tions on the structure or parameterization of K(t), as has beendone in previous studies). For general relaxation kernels, evalua-tion of FV E(t) requires numerical solution. This force will appearin the governing differential equations for a dynamic system in-volving such a viscoelastic element. Because this convolutionintegral depends on complete past state histories (assuming x is astate) it cannot be eliminated by adding a state variable to the sys-tem equations, and thus results in a system of integro-differentialequations.

Without a special structure of K(t), there is not a closed-form solution for the convolution integral or the system differ-ential equations. Thus, solving for state trajectories will requirenumerical simulation of the differential equations, and numericalsolution of the convolution integral every time the time derivativefunction for the differential equations is evaluated. This is a com-putationally intensive process, particularly if coupled with opti-mization (i.e., for every optimization function call a simulationmust be performed, and for every derivative function evaluationmade during simulation the convolution integral must be solvednumerically). In the following subsection we review the math-ematical nature of integro-differential equations, their solutionmethods, and the physical systems that they are associated with.

2.2 Integro-Differential Equations (IDEs)Integro-differential equations are, as the name suggests,

equations that contain both an integral and derivative terms. Theytypically fall in one of the two categories [18]:

1. Volterra integro-differential equations of the form:

φ(x)u′′(x) = f (x)+λ

∫ x

ag(x, t)u(t)dt (3)

2. Fredholm integro-differential equations of the form:

φ(x)u′′(x) = f (x)+λ

∫ b

ag(x, t)u(t)dt (4)

Specifically, if the kernel function g(x, t) in Eqn. (3) is ofthe form g(x− t), then the integral is classified as a convolutionintegral (as is the case with Eqn. (2)). Please note that the deriva-tive term can exist within and/or outside of the integrand in anintegro-differential equation. The system model correspondingto a linear viscoelastic element is a Volterra IDE.

Many solution methods for IDEs have been explored in theliterature. The Laplace transform is often used when the struc-ture of the kernel function is known. Other traditional methodsinclude the series solution method where a Volterra IDE is con-verted to an initial value problem or a Volterra integral equa-tion [19]. Once the IDE is converted to an integral equation, thereexist a multitude of methods to solve the system [20, 21]. Brun-ner surveyed various numerical techniques that can be appliedto IDEs [22] and applied Runge-Kutta methods to second-orderIDEs [23]. Other work included quadrature methods to solveVolterra and Abel-Volterra equations [24]. Dixon studied multi-step methods used in the solution of Volterra integral and integro-differential equations of the second kind [25]. Other methods tosolve higher-order and non-linear IDEs that have been studiedare the Adomian Decomposition method [26, 27] and its modi-fications [28, 29]. The Variational Iteration Method (VIM) hasalso been used to solve non-linear IDEs numerically [30]. Mostof these efforts, however, do not focus on convolution kernel so-lution or using IDEs in conjunction with engineering design op-timization.

2.3 Hysteresis and Convolution Kernels in IDEsSystems characterized by convolution integral equations are

usually physical systems that exhibit the phenomenon of hys-teresis. Hysteresis in a system arises when the output responseof the system depends not only on the current but also past in-puts. A prime example of hysteresis is the dynamic behavior offerromagnets. When an external magnetic field is applied to amaterial like iron, it gets magnetized. Even after the removal ofthe field, it stays magnetized indefinitely. Viscoelastic materi-als exhibit a different type of hysteresis known as rate-dependenthysteresis [31]. A typical mechanical hysteresis loop is shown inFig. 3.


Stress

Strain

Loading

Unloading

FIGURE 3: A hysteresis loop for viscoelastic materials, whichhave both viscous and elastic components resulting in differentcurves for loading and unloading. Energy absorbed during oneloading-unloading cycle is given by the area within the loop.

Other engineering and biological systems beyond viscoelas-tic materials exhibit hysteresis such as electromagnets. Anotherexample is observed in aircraft wing aerodynamics when the an-gle of attack where the flow on top of the wing reattaches isgenerally lower than the angle of attack where the flow sepa-rates [32]. Physical adsorption is another example that exhibitsan unusual property: it is possible to scan within the hysteresisloop by reversing the direction of adsorption [33].

Several methods of convolution quadrature have been stud-ied, which is important for numerical evaluation of hystereticsystems. Lubich, a pioneer of the convolution quadrature, usesLaplace transforms to determine quadrature weights [34]. Lu-bich et al. also applied quadrature to fractional IDEs [35]. How-ever, in these cases the structure of the kernel functions g(x− t)was known a priori. Zhang et al. proposed using a combina-tion of linear methods with compound quadrature rules [36].Finite elements were also used for this purpose when the sys-tem involved modeling of flow and partial differential equations[37,38]. These efforts form a strong mathematical basis to under-stand convolution kernels and IDEs, but in general do not addressrate-dependent convolution kernels of the form given in Eqn. (2),and do not extend their use for engineering design optimization.

In this article we introduce the use of optimal control in theoptimization of viscoelastic systems (characterized by convolu-tion IDEs), specifically for the case of general relaxation kernelsK(t). Optimal control methods have successfully been used inthe optimization of similar systems, such as wave energy convert-ers [39, 40] where the cost functional term has a rate-dependentconvolution structure. Yu and Falnes approached the problemby using a state space approximation of the convolution term;this approximation results in additional system model states [41].

The objective of this study is to identify a generalized form of theshear relaxation modulus K(t) found in Eqn. (2) to support moreflexible design strategies for early-stage design exploration in-volving viscoelastic materials. This flexibility may help reducedesign fixation and enhance design innovation using rheologi-cally complex materials.

2.4 Relaxation Modulus ParameterizationsIn the most general case the relaxation kernel, K(t), should

be treated as a function of arbitrary structure. Passive materialsor systems, however, always result in a monotonically decreasingrelaxation kernel [42,43]. This constraint is simple to implementif only passive systems are to be considered, and does not restrictdesign space exploration significantly. Actively controlled sys-tems, however, may have a relaxation kernel that is not mono-tonically decreasing. Usually K(t) is parameterized by modelsdefined in Table 1. K′(ω) and K′′(ω) are the dynamic storageand loss moduli defined by [8]:

K′(ω) = ω

∫∞

0K(s)sin(ωs)ds (5)

K′′(ω) = ω

∫∞

0K(s)cos(ωs)ds. (6)

Using these forms of K(t), we can identify optimal param-eters for each of the models. This is an extremely important ex-ercise, which explores the design space and determines the func-tionalities defined by viscoelastic models. Previous studies havedemonstrated the possibilities of expanding the design space byevaluating optimal target functions [44], but are limited to pa-rameterized relaxation modulus treatment. Here we use directoptimal control methods to allow the treatment of more generalrelaxation kernels that do not require assuming the use of a spe-cific material class or material model.

2.5 Modeling for DesignIn this article we aim to extend the utility of previous work

in viscoelastic material design by lifting design exploration re-strictions imposed by specific K(t) parameterizations, enablingmore general treatment of viscoelastic material design. An iter-ative process may be used where an optimal K(t) is identifiedvia direct optimal control methods, insights are derived from theresulting relaxation modulus, and efforts in material-level designare then made to identify material classes that may produce theideal behavior expressed by the target K(t) function identifiedvia optimization. These insights may then allow reformulationof the optimization problem (e.g., model adjustments, design pa-rameterization, or addition of constraints) to gradually guide thedesign process toward a physically realizable material design thatproduces optimal performance for the overall system. This is a


TABLE 1: Typical parameterizations for the shear relaxation modulus K(t) [44]

VE Model Parameters K(t) K′(ω) K′′(ω)

Dashpot c .= [F/(L/T )] c ·δ (t) c ·ω 0

Maxwell Elementwith m modes

Km.= [F/L]

λm.= [T ]

M∑

m=1Kme−t/λm

M∑

m=1Km

(λmω)2

1+(λmω)2

M∑

m=1Km

λmω

1+(λmω)2

Critical Gel Sk.=[F

L ·T n]

0 < n < 1Skt−n K′′(ω)

tan(n π2 )

K′(ω)

tan(n π2 )

vision for connecting material-level design directly with designfor overall system performance.

Identifying an optimal K(t) target is qualitatively similar el-ements of ATC [11, 45]. The ATC paradigm is based on hier-archical decomposition of an engineering system where high-level system analysis and design is supported by subsystem andcomponent-level analysis and design. The system-level providestarget behaviors (via optimization) for subsystems to meet, andsubsystems aim to minimize the difference between these tar-gets and their actual behavior. The iterative process convergesto system and lower-level designs that are optimal for the overallsystem. One objective of ATC is to determine appropriate andconsistent targets for subsystem and component design groupsthat will lead to system-optimal designs and minimize redesignand iteration in the design process.

Consider a linear viscoelastic system with a target functionK(t) and an objective function φ(·). In the example case of avibration isolation system, the objective could be to minimizethe acceleration of the mass that is subject to vibration. Figure 4depicts the analysis dependencies in this type of design problem.The independent material design variables x influence K(t), andK(t) influences overall system performance.

xSystem 1:GeneralizedVE Design

K(t) �(·)System 2:IDE Problem

FIGURE 4: An ATC representation of the viscoelastic designproblem

This analysis structure is similar to problems solved previ-ously using ATC where a top-level system target can be trans-lated to a subsystem-level target using optimization and system-level analysis. The bottom-level problem is to match this targetby optimizing bottom-level design variables. For example, in theelectric water pump design problem of Ref. [11] the top-levelproblem is similar to the problem of identifying K(t) that op-

timizes system performance φ(·). The bottom-level problem issimilar to designing a material with the objective of matching atarget K(t) function that optimizes system-level performance. Ifthe target cannot be matched, the top-level problem is modifiedand the process is repeated.

Utilizing direct optimal control to identify optimal K(t)functions may provide great insight into what types of materialbehavior are best for overall system utility. This article focuseson solution methods for this top-level problem. We acknowl-edge that designing a particular material that approaches the de-sired target behavior is also a significant challenge. It may notbe possible, for example, to map independent material designparameters to K(t), which is required to use an optimization-based approach to solve the bottom-level problem. Analysisof the optimal K(t) functions, however, may still provide greatinsights that are valuable for material design experts in deter-mining what material classes and formulation strategies to use.This approach supports functionality-driven design as opposedto material-specific creations that are disconnected from systemperformance.

3 OPTIMAL CONTROL SYSTEM DESIGNThe desired early-stage design approach that is not restricted

by a priori material class choices (i.e., material agnostic) requiresa generalized treatment of K(t). Instead of parameterizing K(t)and tuning parameters to improve system performance, we pro-pose to design K(t) directly. In other words, instead of specify-ing a finite set of parameters as design variables, we choose todesign with respect to an infinite-dimensional (function-valued)quantity. Optimal control methods provide the ability to solveinfinite-dimensional optimization problems.

Optimal control is a mature field of study [46]. An opti-mal control system design (OCSD) problem aims to find a statetrajectory ξξξ (t) and a control trajectory u(t) that satisfy systemdynamics relationships and maximize system performance. In


general, an optimal control problem can be formulated as fol-lows [47]:

minx

∫ t f

0L(t,ξξξ (t),x)dt +M

(t, t0,ξξξ (t0), t f ,ξξξ (t f ),x

)

subject to ξξξ = fd(t,ξξξ (t),x),C(t,ξξξ (t),x)≤ 0φφφ(t0,ξξξ (t0), t f ,ξξξ (t f ),x

)≤ 0

(7)where x is the set of optimization variables, which can includestate variable trajectories (ξξξ (·), control input trajectories u(t),and initial (t0) or final time (t f ). fd(·) is the time derivative func-tion for the state equations that describe system dynamics. C(·) isthe path constraint function, and φφφ(·) is the boundary constraintfunction. L(·) is the Lagrangian that quantifies the running costportion of the objective function, and M(·) is the objective func-tion term that depends on boundary values.

There are two classes of methods employed to solve OCSDproblems: (1) indirect and (2) direct optimal control methods(Fig. 5). Indirect methods are based on the calculus of varia-tions, and work by applying optimality conditions to the optimalcontrol problem to form a boundary value problems (BVP). Insimple cases the BVP may be solved analytically, but in moregeneral cases it must be discretized and solved numerically [12].

Direct optimal control methods take the opposite approach,where the optimal control problem is discretized in time first.The result is a nonlinear program (NLP) that can be solved us-ing standard large-scale optimization algorithms. Direct meth-ods are particularly effective for highly nonlinear systems, prob-lems with inequality constraints, or other situations where indi-rect methods fall short [13, 48]. The state space equations arediscretized to form a system of algebraic equations (known asdefect equations or defect constraints, similar to residual func-tions). This discretization makes use of a collocation method(e.g., implicit Runge-Kutta [49]).

The most straightforward of the direct methods is single-shooting; state trajectories are obtained for every NLP functionevaluation by solving the defect equations using forward simu-lation. The control is parameterized using either a polynomialapproximation or another appropriate method. Given a set of ini-tial conditions and a control parameterization, the optimizationis then performed with respect to the control parameters (e.g.,polynomial coefficients).

Another class of direct optimal control, known as DirectTranscription (DT), parameterizes both control and state trajecto-ries. The dynamic equations are also discretized and are writtenas defect constraints that are solved by the optimization algo-rithm instead of with simulation. To formulate the DT problem,let time be discretized into nt steps: t1, t2, . . . , tnt , where the stepsize at ti is hi = ti+1− ti. Discretizing the control input forms U,where row i is the control input vector at time step i: u(ti). Dis-

cretizing state trajectories forms the matrix ΞΞΞ, where row i is thestate vector at time step i: ξξξ (ti). If the objective includes onlythe Lagrangian term, the resulting DT formulation is:

minU,ΞΞΞ

nt−1

∑i=1

L(ξξξ i,ui)hi, subject to: ζζζ (ΞΞΞ,U) = 0 (8)

where ζζζ (U,ΞΞΞ) = 0 are the defect constraints that arise fromdiscretization of state equations. If these constraints are satis-fied, then the original state-space equations are satisfied approxi-mately. DT allows addition of other constraints and optimizationvariables, enhancing its utility for design. DT avoids the needfor nested simulations, and often has better numerical behaviorthan single shooting. The DT NLP usually a sparse structurethat can be exploited to reduce total computational expense. Theextension of DT for solution of problems based on IDEs has asignificant impact on the sparsity structure since the derivativefunction depends on previous states. Ongoing work is address-ing techniques for managing this new problem structure.

SolveR t0 K(s)x(t � s)ds

Forward Simulation

Optimization Algorithm Optimization Algorithm

SolveR t0 K(s)x(t � s)ds

Defect ConstraintEvaluation

�(K) K,⌅�(K,⌅)⇣(K,⌅)

K

(a) Single Shooting (b) Direct Transcription

FIGURE 5: The two direct optimal control methods applied tothe viscoelastic design problem. The relaxation modulus K(t)is treated as the control input. (a) Single shooting involves anested simulation of the dynamic system and is an example of asequential approach. (b) Direct Transcription is a simultaneousapproach that requires the evaluation of the dynamic equationsas defect constraints and solves a large nonlinear program.

4 CASE STUDYThe application of direct optimal control to solve the optimal

kernel target setting problem for a vibration isolator problem ispresented here. In the initial case a mass m is connected to a basewith a spring of stiffness k. The base is subjected to a prescribeddisplacement of y(t) (Fig. 6a). The objective is to isolate thetop mass from the base displacement. One approach to improvevibration isolation is to add a linear dashpot in parallel with thespring (Fig. 6b). Here we study the implications and benefitsof using a generalized viscoelastic element instead of a lineardashpot (Fig. 6c).


m

y(t)

x(t)

Generalized

Viscoelastick K(t)

(c)

m

y(t)

x(t)

ck

(b)(a)

m

k

y(t)

x(t)

FIGURE 6: The introduction of a generalized viscoelastic element aims to enhance design freedom while improving design performance.

4.1 Problem FormulationGiven an initial condition Fve(t = 0) = 0, Eqn. (2) has limits

of integration from 0 to t. For the particular system in Fig. 5cwith a generalized viscoelastic element and the initial conditionFve(0) = 0, the governing equation for conservation of linear mo-mentum is most generally written as:

−k(x− y)−∫ t f

t0K(s) [x(t− s)− y(t− s)]ds = mx(t) (9)

where s = t− t ′ as defined in Eqn. (2).The convolution integral structure has two important con-

sequences. First, the equations cannot be written in linear ma-trix form. Second, the numerical simulation of this model incursincreased computational expense as simulation time increases,since each time derivative function evaluation requires an in-tegration of the entire prior time-history of velocities, and thelength of this history increases as the simulation proceeds. In pre-vious studies that used specific parameterizations this was not anissue because a closed-form integral solution was possible [44].Solving this problem based on an arbitrary K(t) is a more signif-icant challenge.

While treating K(t) as a system control input is an intu-itive solution strategy, the convolution structure of the problempresents some challenges. The integral corresponds to the areaof overlap between two curves: K(s) and x(t− s)− y(t− s). Inthis sense K(t) is subtly different from the control input used ina standard optimal control problem. The objective function ofthe system is the area under the position trajectory of the massm. The states of the system are the mass position and velocity:[x, x]T = [ξ1,ξ2]

T . The kernel K(t) is treated as a control inputto be optimized. The excitation y(t) is defined as a harmonicfunction y(t) = Y0 sin(ωt). If K(t) = 0, the system is unstabledynamics (Fig. 7). Two methods are used to solve this kerneltarget setting problem: single shooting and DT.

Time(s)0 50 100 150 200

x(t)

-4000

-3000

-2000

-1000

0

1000

2000

3000

FIGURE 7: Unstable system dynamics without the use of a vis-coelastic element (K(t) = 0)

4.1.1 Single Shooting The optimal control problemfor the single shooting approach is formulated as follows:

minK

∫ t f

t0|ξ1|

subject to ξξξ (t0) = 0dK(t)

dt≤ 0

(10)

where ξξξ = fd(ξξξ (t),K(t), t) is satisfied via simulation, and thetime derivative function is defined as:

fd(·) =[

ξ2k(ξ1− y)− ∫ t

0 K(s)(ξ2(t− s)− y(t− s))ds

]

The constraint on dK(t)/dt is added ensures a time-decaying K(t), which is a typical characteristic reported in lit-erature for passive materials [42, 43]. In this case K(t), isparameterized at uniformly discretized points in time: K =[K(t1),K(t2), . . . ,K(tnt )]

T, and the optimization is performedwith respect to these nt discrete values. As a result the derivativeconstraint is transformed into a set of linear algebraic inequalityconstraints.


4.1.2 Direct Transcription In the DT formulationboth the state and control trajectories are discretized in time. Theoverall size of the optimization problem is larger but the needfor nested simulation is now eliminated and numerical behavioris improved. The OCSD problem for the DT implementation isformulated as:

minΞΞΞ,K

∫ t f

t0|ξ1|

subject to ξξξ (t0) = 0dK(t)

dt≤ 0

ζζζ (ΞΞΞ(t),K) = 0

(11)

where ΞΞΞ is the matrix of discretized state trajectories (the ith rowis the state vector values for time ti), ζζζ (·) are the defect con-straints. Trapezoidal quadrature was used to formulate both thedefect constraints and to evaluate the objective function integral.For nt discretized points in time, the objective function and de-fect constraints are:

∫ t f

t0|ξ1|=

12

nt

∑k=1

hk (|ξ1[tk]|+ |ξ1[tk−1]|) (12)

ζζζ [k] = ξξξ [tk]−ξξξ [tk−1]−∫ t f

t0fd(ξξξ (t),K(t), t) = 0 (13)

ζζζ [k] = ξξξ [tk]−ξξξ [tk−1]−hk

2(fd[tk]+ fd[tk−1]) = 0 (14)

Since K(t) is discretized at all points in time , the constraint onthe derivative of K(t) becomes a set of linear inequality con-straints:

K[tk]−K[tk−1]≥ 0 (15)

4.1.3 Convolution sum The value of the convolutionintegral in Eqn. (9) at discrete points in time is denoted I =[I1, I2, . . . , Int ]

T, where Ik is the value of the convolution integralat tk. Here this is evaluated using a convolution sum:

Ik =k

∑j=0

K( j)(ξ2(k− j+1)− y(k− j+1)) (16)

4.2 ResultsDirect Transcription achieves better attenuation of the vibra-

tion amplitude than with single shooting (as shown in Fig. 8), butrequires more function evaluations to solve the discretized NLP.

The objective function values and function evaluations are re-ported in Table 2. These results are also compared in Fig. 8 tothe optimal design based on a Maxwell model parameterization.The DT implementation here is a basic approach; several stepscould be taken to take advantage of problem structure and im-prove efficiency, including polynomial representations of trajec-tories that reduce the number of time steps required and supportclosed-form integral solutions.

The optimal K(t) trajectories are also shown in Fig. 9. Thesetrajectories do not strictly fall into any of the parameterizationsidentified in Table 1. This is an exciting result that goes be-yond what might be achieved using conventional material designmethods. A physical realization of K(t) could require one ormany combinations of various sub-classes of materials that areavailable to a material designer today. This type of target func-tion also presents an opportunity for significant innovation in ma-terial design with potential for impact on system performance.The optimal target provides new insights into what kind of func-tionality is needed to maximize overall system performance.

TABLE 2: The results from using DT and Single shooting arecompared. While single shooting requires fewer function evalu-ations, direct transcription arrives at a better solution.

Method Objective Func-tion value

F-Count

Single Shooting 29.0786 8178

Direct Transcrip-tion

19.1916 78064

5 CONCLUSIONSMaterials used in engineering design are most often hard

materials. Challenges associated with conceptual understandingand mathematical modeling of soft rheologically complex ma-terials, such as viscoelastic materials, may be important factorsin the limited use of soft materials in engineering design. Thisarticle aims to improve this situation by connecting system-leveldesign with generalized kernel targets that can then be used toguide material-level design. A more generalized treatment ofK(t) supports more complete and flexible design space explo-ration, and we postulate that it may help reduce design fixationand enhance design innovation.

Taking the example of linear viscoelastic systems, a sys-tematic modeling procedure was laid out and the constitutiveintegro-differential equations were discussed in detail. The chal-lenges in designing a generalized viscoelastic material with no apriori assumptions on its mathematical model were elaborated.The design approach proposed here is qualitatively similar to the


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X(t)X(t) with Maxwell parameters

(b)FIGURE 8: Results for optimizing the problem using (a) Single Shooting and (b) Direct Transcription. Direct Transcription achievesa better optimum than single shooting, but requires more function evaluations. The optimal amplitude trajectories are compared to theones obtained using a Maxwell parameterization [44]. While it is an early stage design which may not result in a physically feasibleK(t), the results show promise of bettering the results achieved by a Maxwell viscoelastic model.

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(b)FIGURE 9: Optimal K(t) trajectories are shown for (a) Single Shooting and (b) Direct Transcription. While these trajectories maynot be directly physically realizable, they provide initial insights into modeling material classes that exhibit optimal target functions.Subsequent problem reformulations can help guide results toward physically realizable material behavior targets.

top-level problem in an analytical target cascading (ATC) designframework. Finally, the motivation to use direct optimal con-trol in generalized viscoelastic design was presented. The appli-cation of direct optimal control was demonstrated by means ofa vibration isolator problem. The findings show that adding ageneralized viscoelastic element helps attenuate vibration, whileallowing for design freedom. Of the direct optimal control meth-ods used, direct transcription achieves a better performance atthe cost of additional function evaluations. Improvements in DT

for IDEs could help reduce overall expense. While the resultingK(t) target obtained does not fit into existing parameterizationsreadily, it provides insights into the possibilities of using variousclasses to materials to achieve an optimal target.

As a subsequent step, we identify using a state-space ap-proximation of the convolution integral as another importantstrategy to investigate as a simultaneous approach to viscoelasticmaterial design. Cumulatively, a benchmarking of the the numer-ical methods with respect to efficiency and convergence would be


required. The other crucial part in our future exploration is to un-derstand the physical realizations of the optimal target K(t). Byunderstanding the relationship between the target and physicallyfeasible materials, we can potentially go on to create a bettermathematical model of a generalized viscoelastic system.

6 ACKNOWLEDGEMENTSThe authors would like to thank Prof. Randy Ewoldt and Re-

becca Corman of the Mechanical Science and Engineering de-partment at the University of Illinois at Urbana-Champaign fortheir extensive support and collaboration. The authors wouldalso like to thank Anand Deshmukh and Daniel Herber, graduatestudents at University of Illinois, for their help and suggestionspertaining to optimal control formulation and solution.

REFERENCES[1] Shadwick, R., 1999. “Mechanical Design in Arteries”. Journal of

Experimental Biology, 202(23), pp. 3305–3313.[2] Ewoldt, R. H., Clasen, C., Hosoi, A. E., and McKinley, G. H.,

2007. “Rheological Fingerprinting of Gastropod Pedal Mucus andSynthetic Complex Fluids for Biomimicking Adhesive Locomo-tion”. Soft Matter, 3, pp. 634–643.

[3] Denny, M. “The Role of Gastropod Pedal Mucus in Locomotion”.Nature, 285, pp. 160–161.

[4] Ewoldt, R. H., Hosoi, A. E., and McKinley, G. H., 2009. “Non-linear Viscoelastic Biomaterials: Meaningful Characterization andEngineering Inspiration”. Integrative and comparative biology,49(1), pp. 40–50.

[5] Ewoldt, R. H., 2014. “Extremely Soft: Design with RheologicallyComplex Fluids”. Soft Robotics, 1(1), pp. 12–20.

[6] Jansson, D. G., and Smith, S. M., 1991. “Design Fixation”. DesignStudies, 12(1), pp. 3 – 11.

[7] Linsey, J. S., Tseng, I., Fu, K., Cagan, J., Wood, K. L., and Schunn,C. D., 2010. “A Study of Design Fixation, Its Mitigation and Per-ception in Engineering Design Faculty”. Journal of MechanicalDesign, 132(4), April, p. 041003.

[8] Bharadwaj, N. A., Allison, J. T., and Ewoldt, R. H., 2013. “Early-stage Design of Rheologically Complex Materials via MaterialFunction Design Targets”. Urbana, 51, p. 61801.

[9] Ulrich, K. T., 2003. Product Design and Development. TataMcGraw-Hill Education.

[10] Kim, H. M., Michelena, N. F., Papalambros, P. Y., and Jiang, T.,2003. “Target Cascading in Optimal System Design”. Journal ofmechanical design, 125(3), pp. 474–480.

[11] Allison, J., Kokkolaras, M., Zawislak, M., and Papalambros, P. Y.,2005. “On the Use of Analytical Target Cascading and Collabo-rative Optimization for Complex System Design”. Proceedings of6th World Congress on Structural and Multidisciplinary Optimiza-tion.

[12] Biegler, L. T., 2010. Nonlinear Programming: Concepts, Algo-rithms, and Applications to Chemical Processes. SIAM.

[13] Allison, J. T., Guo, T., and Han, Z., 2014. “Co-Design of an ActiveSuspension Using Simultaneous Dynamic Optimization”. Journalof Mechanical Design, 136(8), p. 081003.

[14] Allison, J., and Herber, D., 2013. “Multidisciplinary Design Op-timization of Dynamic Engineering Systems”. AIAA Journal, ToAppear.

[15] Johnson, C. D., 1995. “Design of Passive Damping Systems”.Journal of Mechanical Design, 117(B), p. 171.

[16] Hilton, H. H., 2005. “Optimum Linear and Nonlinear ViscoelasticDesigner Functionally Graded Materials - Characterizations andAnalysis”. Composites Part A: Applied Science and Manufactur-ing, 36(10), pp. 1329–1334.

[17] Hilton, H. H., Lee, D. H., and El Fouly, A. R. A., 2008. “General-ized Viscoelastic Designer Functionally Graded Auxetic MaterialsEngineered/tailored for Specific Task Performances”. Mechanicsof Time-Dependent Materials, 12(2), pp. 151–178.

[18] Rahman, M., 2007. Integral Equations and Their Applications,1st ed. WIT Press.

[19] Wazwaz, A.-M., 2011. Linear and Nonlinear Integral Equations:Methods and Applications, 1st ed. Springer Publishing Company,Incorporated.

[20] Baker, C., 1977. The Numerical Treatment of Integral Equations.Monographs on Numerical Analysis Series. Oxford : ClarendonPress.

[21] Wang, Q., Wang, K., and Chen, S., 2014. “Least Squares Ap-proximation Method for the Solution of VolterraŁfredholm Inte-gral Equations”. Journal of Computational and Applied Mathe-matics, 272(0), pp. 141 – 147.

[22] Brunner, H., 1982. “A Survey of Recent Advances in the Numer-ical Treatment of Volterra Integral and Integro-differential Equa-tions”. Journal of Computational and Applied Mathematics, 8(3),pp. 213 – 229.

[23] Brunner, H., 1987. “Implicit Runge-Kutta-Nystram methodsfor general second-order Volterra integro-differential equations”.Computers & Mathematics with Applications, 14(7), pp. 549 –559.

[24] Baker, C. T., 2000. “A Perspective on the Numerical Treatmentof Volterra Equations”. Journal of Computational and AppliedMathematics, 125(12), pp. 217 – 249. Numerical Analysis 2000.Vol. VI: Ordinary Differential Equations and Integral Equations.

[25] Dixon, J. A., 1986. “Generalized Reducible Quadrature Methodsfor Volterra Integral and Integro-differential Equations”. Journalof Computational and Applied Mathematics, 16(1), pp. 27 – 42.

[26] Adomian, G., 1991. “A Review of the Decomposition Methodand Some Recent Results for Nonlinear Equations”. Computers &Mathematics with Applications, 21(5), pp. 101 – 127.

[27] Adomian, G., 1994. “Integral Equations”. In Solving FrontierProblems of Physics: The Decomposition Method, Vol. 60 of Fun-damental Theories of Physics. Springer Netherlands, pp. 224–227.

[28] Hosseini, M., 2006. “Adomian Decomposition Method withChebyshev Polynomials”. Applied Mathematics and Computation,175(2), pp. 1685 – 1693.

[29] Hosseini, M., and Nasabzadeh, H., 2006. “On the Convergenceof Adomian Decomposition Method”. Applied Mathematics andComputation, 182(1), pp. 536 – 543.

[30] Batiha, B., Noorani, M., and Hashim, I., 2008. “Numerical Solu-tions of the Nonlinear Integro-differential Equations”. Int. J. OpenProblems Compt. Math, 1(1), pp. 34–42.

[31] VINCENT, J., 2012. Structural Biomaterials, stu - student edi-


tion ed. Princeton University Press.[32] Hu, H., Yang, Z., and Igarashi, H., 2007. “Aerodynamic Hysteresis

of a Low-reynolds-number Airfoil”. Journal of Aircraft, 44(6),pp. 2083–2086.

[33] Tompsett, G. A., Krogh, L., Griffin, D. W., and Conner, W. C.,2005. “Hysteresis and Scanning Behavior of Mesoporous Molecu-lar Sieves”. Langmuir, 21(18), pp. 8214–8225. PMID: 16114924.

[34] Lubich, C., 1988. “Convolution Quadrature and Discretized Oper-ational Calculus”. Numerische Mathematik, 52(4), pp. 413–425.

[35] Cuesta, E., Lubich, C., and Palencia, C., 2006. Convolu-tion Quadrature Time Discretization of Fractional Diffusion-waveEquations. Tech. rep., Math. Comp.

[36] Zhang, C., and Vandewalle, S., 2005. “General Linear Methodsfor Volterra Integro-differential Equations with Memory”. SIAMJ. Sci. Comput., 27(6), Dec., pp. 2010–2031.

[37] Peszynska, M., 1995. “Analysis of an Integro-differential EquationArising from Modelling of Flows with Fading Memory ThroughFissured Media”. Partial Diff. Eqs, 8, pp. 159–173.

[38] Peszynska, M., 1996. “Finite Element Approximation of DiffusionEquations with Convolution Terms”. Mathematics of Computationof the American Mathematical Society, 65(215), pp. 1019–1037.

[39] Bacelli, G., and Ringwood, J., 2014. “Numerical Optimal Controlof Wave Energy Converters”. Sustainable Energy, IEEE Transac-tions on, PP(99), pp. 1–9.

[40] Herber, D. R., and Allison, J. T., 2013. “Wave Energy Extrac-tion Maximization in Irregular Ocean Waves using Pseudospec-tral Methods”. In International Design Engineering TechnicalConferences, Portland, OR, USA, No. DETC2013-12600, ASME,p. V03AT03A018.

[41] Yu, Z., and Falnes, J., 1995. “State-space Modelling of a VerticalCylinder in Heave”. Applied Ocean Research, 17(5), pp. 265 –275.

[42] Breuer, S., and Onat, E. T., 1962. “On Uniqueness in Linear Elas-ticity”. Quarterly of Applied Mathematics, 19(4), pp. 355–359.

[43] Rabotnov, I. N., 1977. Elements of Hereditary Solid Mechanics.Mir, Moscow.

[44] Corman, R., Gururaja Rao, L., Allison, J., and Ewoldt, R. “De-sign with Rheologically Complex Materials via Material FunctionDesign Targets”. Journal of Mechanical Design, In review.

[45] Allison, J., Walsh, D., Kokkolaras, M., Papalambros, P. Y., andCartmell, M., 2006. “Analytical Target Cascading in Aircraft De-sign”. 44th AIAA aerospace sciences meeting and exhibit, pp. 9–12.

[46] Sussmann, H., and Willems, J., 1997. “300 Years of Optimal Con-trol: from the Brachystochrone to the Maximum Principle”. Con-trol Systems, IEEE, 17(3), Jun, pp. 32–44.

[47] Herber, D. R., 2014. “Dynamic System Design Optimization ofWave Energy Converters Utilizing Direct Transcription”. PhD the-sis, MS Thesis, Univ. of Illinois at Urbana-Champaign.

[48] Betts, J., 2010. Practical Methods for Optimal Control and Es-timation Using Nonlinear Programming, second ed. Society forIndustrial and Applied Mathematics.

[49] Ascher, U., and Petzold, L., 1998. Computer Methods for Ordi-nary Differential Equations and Differential-Algebraic Equations.Society for Industrial and Applied Mathematics.


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