FREIE MATERIALOPTIMIERUNG FUR CHALEN UND LATTEN

FREE MATERIAL OPTIMIZATION

FOR SHELLS AND PLATES

FREIE MATERIALOPTIMIERUNG FUR SCHALEN UND PLATTEN

Der Naturwissenschaftlichen Fakultat

der Friedrich-Alexander-Universitat Erlangen-Nurnberg

zur

Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von

Stefanie Gaile

aus Munchen

Als Dissertation genehmigt

von der Naturwissenschaftlichen Fakultatder Friedrich-Alexander-Universitat Erlangen-Nurnberg

Tag der mundlichen Prufung: 27. Mai 2011

Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink

Erstberichterstatter: Prof. Dr. Gunter Leugering

Zweitberichterstatter: Prof. Dr. Michael Stingl

Abstract

Within this thesis we develop mathematical models and numerical methods for the Free Ma-terial Optimization problem for shells and plates. In Chapter 1 we provide a motivation arisingfrom structural engineering to address this problem and classify the Free Material Optimizationapproach within other common methods in structural design optimization.

In Chapter 2 we introduce the foundations of differential geometry and continuum mechanicsnecessary for a reliable prediction of the shell’s elastic behavior. The chosen description is basedon the theory of Cosserat continua, a direct approach to the shell as a two-dimensional midsur-face in physical space endowed with director vectors to provide additional degrees of freedom inorder to model bending and shear deformations. Thus the displacements consist of a translationof the points on the midsurface and a rotation of the associated director vectors, in which drilling– the rotation around their own axis – is neglected as the director vectors are considered to beinfinitely thin. We restrict ourselves to displacements that fulfill the Reissner-Mindlin kinemat-ical assumption: the material lines, that are represented by the director vectors, remain straightand unstretched during deformation. This requirement leads to a first-order approximation ofthe three-dimensional elasticity theory including shear effects, which is known as Naghdi’s shellmodel. The material tensorsC and D of the shell, whose entries can be depicted as springconstants connecting all possible directions, are directly derived from the three-dimensionalelasticity tensor by including monoclinic material symmetry as well as the basic assumption ofvanishing normal stresses commonly used in shell theory. Moreover, in the special case of aplanar midsurface the formulas for the membrane, bending and shear strains of this model canbe considerably simplified leading to the Reissner-Mindlinplate model.

Chapter 3 is dedicated to the development of a Free Material Optimization formalism forNaghdi shells. Free Material Optimization is a subbranch ofstructural optimization and accord-ingly deals with the problem of finding the stiffest structure for a given design domain and apredefined set of loads constructed from a limited amount of material. To this end we considerthe entire elasticity tensorsC andD in their most general form as optimization variables andinclude only the basic requirements for linear elastic material in the constraints. This freedom inthe design space leads to the ultimately best design, although the optimal material typically doesnot preexist in nature and approximations of the optimal structure have to be intricately manu-factured e.g. by using tapelayering techniques or constructing composites. Moreover, since wehave no information about the density of an arbitrary material, we require another measure forthe amount of used material and employ a summed trace of the elasticity tensors for this purpose.Based on the minimum potential energy principle specifyingthe equilibrium of Naghdi shellswe are thus able to state the minimum compliance problem. Itssaddlepoint structure allows usto show existence of at least one optimal solution via a Minimax theorem. Furthermore we provethat this problem is equivalent to the dual of a nonlinear convex semidefinite program, which werefer to as the primal problem formulation. To this end we employ Lagrange duality and show alocal maximum principle for matrices that guarantees the existence as well as necessary charac-teristics of suitable Lagrange multipliers. Within this proof we also obtain information about the

This research was funded by the European Commission within the Sixth Framework Programme STREP 30717PLATO-N.

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structure of the optimal material tensor at the upper and lower material bounds. In contrast to theoriginal saddlepoint formulation the primal problem is a convex problem, moreover the materialmatrices, that tremendously increase the dimension of the discretized problem, are hidden in theprimal problem as Lagrange multipliers. Accordingly the primal problem is considerably bettersuited for a numerical approach than the saddlepoint problem and the equivalence proof ensuresthat the obtained solutions are also optimal with respect tothe original problem formulation.Moreover we introduce the minimum weight problem formulation which has recently gainedmuch attention due to the development of numerical solvers that render this problem’s structurecomputationally tractable.

In Chapter 4 we focus on the numerical solution of the preceding problem formulations. Tothis end we apply a finite element method to obtain discretized versions of the previously intro-duced optimization problems. After a sensitivity analysiswe are able to compute solutions byemploying the nonlinear semidefinite programming code PENSCP, an efficient solver for prob-lems arising from Free Material Optimization. We test our software on a collection of numericaltest examples frequently used in the structural optimization of shells and show the validity ofour results by comparing them with solutions originating from other prominent material opti-mization approaches. Our software does not only provide reliable results including extensiveinformation about the optimal material properties, but is also designed for realistic applicationsas it allows the simultaneous optimization of structures composed of solids as well as shells.Moreover it is possible to include condensed data structures in order to decrease the problemdimension by a static condensation of segments that are constructed from fixed material or tosimulate the characteristics of element types which are notcomprised in our software package.

In Chapter 5 we discuss the extension of the Free Material Optimization problem for shellsby multidisciplinary optimization constraints. We consider linear displacement constraints in adiscrete context, which can be utilized to manipulate the shape of the deformed structure. Inthe case of stress constraints we distinguish between in-plane stresses and out-of-plane stressesand apply them in order to avoid material damage or even failure due to high stresses. Forthe formulation of eigenfrequency constraints we introduce a dynamic model describing thefree vibrations of Naghdi shells. Therefrom we deduce a semidefinite matrix constraint thatcan be employed to raise the natural frequency of the structure to prohibit resonance excitationand ultimately a resonance disaster. The final type of constraints regarded by us are bucklingconstraints, which address the susceptibility of shells togeometrical imperfections and loadperturbations. The resulting sudden deformations are captured by a nonlinear strain model,which is used for the derivation of nonlinear semidefinite matrix constraints in order to avoidbuckling behavior.

The thesis is concluded by a summary emphasizing continuative questions for future researchin Chapter 6.

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Zusammenfassung

Im Rahmen dieser Dissertation entwickeln wir mathematische Modelle und numerische Metho-den, um das freie Materialoptimierungsproblem fur Schalen und Platten zu losen. In Kapitel 1stellen wir die aus der Bautechnik stammende Motivation fur dieses Problem vor und ordnenden Ansatz der freien Materialoptimierung in Bezug zu anderen gebrauchlichen Methoden inder Strukturoptimierung ein.

Im 2. Kapitel stellen wir die Grundlagen der Differentialgeometrie und der Kontinuumsme-chanik vor, die fur eine zuverlassige Vorhersage des elastischen Verhaltens der Schale notigsind. Die gewahlte Beschreibung basiert auf der Theorie der Cosserat-Medien, einem direk-ten Zugang zur Schale in Form einer zweidimensionalen Mittelflache im physischen Raum, diemit Direktorvektoren ausgestattet ist. Die dadurch zur Verfugung gestellten zusatzlichen Frei-heitsgrade dienen dazu, Biege- und Scherdeformationen zu modellieren. Demzufolge bestehendie Verschiebungen aus der Translation der Punkte auf der Mittelflache und einer Rotation derzugehorigen Direktorvektoren, wobei das Drillen – die Rotation der Vektoren um ihre eige-ne Achse – vernachlassigt wird, da man annimmt, dass die Direktorvektoren unendlich dunnsind. Wir beschranken uns auf Verschiebungen, die die kinematischen Annahmen der Reissner-Mindlinschen Theorie erfullen: die durch die Direktorvektoren reprasentierten Materiallinienbleiben wahrend der Deformation gerade und verandern dabei auch nicht ihre Lange. DieseBedingung fuhrt zu einer Naherung erster Ordnung der dreidimensionalen Elastizitatstheorie,bei der Schereffekte berucksichtigt werden und die als Naghdis Schalenmodell bekannt ist. DieMaterialtensoren der SchaleC und D, deren Eintrage als Federkonstanten interpretiert werdenkonnen, die alle moglichen Richtungen miteinander verknupfen, werden direkt vom dreidimen-sionalen Elastizitatstensor hergeleitet. Dazu wird zum einen die monokline Materialsymmetrieals auch die in der Schalentheorie ublicherweise verwendete Annahme der verschwindendenNormalspannungen herangezogen. Im Spezialfall einer ebenen Mittelflache konnen die For-meln fur die Membran-, Biege- und Scherverzerrungen deutlich vereinfacht werden, was zumReissner-Mindlinschen Plattenmodell fuhrt.

Das 3. Kapitel ist der Entwicklung eines Formalismus fur die freie Materialoptimierung vonNaghdi-Schalen gewidmet. Freie Materialoptimierung ist ein Teilgebiet der Strukturoptimierungund beschaftigt sich dementsprechend mit dem Problem, fur einen gegebenen Designbereich undfestgelegte Lastfalle die steifste Struktur zu finden, dieaus einer beschrankten Menge an Ma-terial gebaut ist. Zu diesem Zweck betrachten wir die kompletten ElastitzitatstensorenC undDin ihrer allgemeinsten Form als Optimierungsvariablen undberucksichtigen nur die grundlegen-den Bedingungen fur lineares elastisches Material in den Nebenbedingungen. Diese Freiheit imEntwurfsraum erlaubt das Erreichen der ultimativ besten Struktur, jedoch existiert das optimaleMaterial typischerweise nicht in der Natur. Daher mussen Naherungen der optimalen Strukturaufwendig gefertigt werden, beispielsweise durch die Verwendung von Tapelayering-Technikenoder durch die Konstruktion von Verbundwerkstoffen. Daruber hinaus besitzen wir keine In-formation uber die Dichte eines beliebigen Materials und benotigen daher ein alternatives Maßfur die Menge des verwendeten Materials. Zu diesem Zweck verwenden wir eine kombinierteSpur der Elastizitatstensoren. Ausgehend vom Prinzip dervirtuellen Arbeit, das das statischeGleichgewicht der Naghdi-Schale festlegt, sind wir somit in der Lage, das Problem der mini-

v

malen Nachgiebigkeit aufzustellen. Dessen Sattelpunktstruktur erlaubt es uns, die Existenz vonmindestens einer optimalen Losung mithilfe eines Minimax-Satzes zu zeigen.Uberdies bewei-sen wir, dass dieses Problem zum dualen eines nichtlinearenkonvexen semidefiniten Programmsaquivalent ist, das wir als primales Problem bezeichnen. Dazu verwenden wir die Lagrangedua-litat und zeigen ein lokales Maximumprinzip fur Matrizen, das sowohl die Existenz als auch dienotwendigen Eigenschaften geeigneter Lagrangemultiplikatoren garantiert. Im Rahmen diesesBeweises erhalten wir außerdem Informationen uber den Aufbau des optimalen Materialtensorsan den oberen und unteren Materialschranken. Im Gegensatz zu der originalen Sattelpunktfor-mulierung ist das primale Problem konvex, daruber hinaus sind die Materialmatrizen, die dieDimension des diskreten Problems immens vergroßern, in der primalen Problemformulierungals Lagrangemultiplikatoren versteckt. Demzufolge ist das primale Problem deutlich besser fureinen numerischen Zugang geeignet als das Sattelpunktsproblem, wobei derAquivalenzbeweissicherstellt, dass die erhaltenen Losungen auch fur die ursprungliche Problemformulierung op-timal sind. Zudem fuhren wir das Problem des minimalen Gewichts ein, das in letzter Zeit anRelevanz gewonnen hat, da numerische Loser entwickelt wurden, die mit dieser Problemformu-lierung umgehen konnen.

In Kapitel 4 beschaftigen wir uns mit einer numerischen Losung der vorangegangenen Pro-blemformulierungen. Dazu verwenden wir eine Finite-Elemente-Methode, um diskrete Versio-nen der zuvor eingefuhrten Optimierungsprobleme zu erhalten. Nach einer Sensitivitatsanalysesind wir in der Lage, mithilfe des nichtlinearen semidefiniten Losers PENSCP, der effizienteMethoden fur die aus der freien Materialoptimierung stammenden Probleme besitzt, Losungenzu berechnen. Wir uberprufen unsere Software mithilfe einer Ansammlung von haufig verwen-deten Testbeispielen aus der Strukturoptimierung von Schalen und zeigen die Stichhaltigkeit un-serer Resultate, indem wir sie mit Losungen vergleichen, die aus anderen etablierten Methodender Materialoptimierung stammen. Unsere Software liefertnicht nur zuverlassige Ergebnisse,die ausfuhrliche Informationen uber die optimalen Materialeigenschaften enthalten, sondern istdaruber hinaus auch fur praktische Anwendungen geeignet, da sie die simultane Optimierungvon Strukturen erlaubt, die sowohl aus solidem Material alsauch aus Schalen aufgebaut sind.Desweiteren ist es moglich, kondensierte Datenstrukturen einzubinden, um die Problemdimen-sion durch eine statische Kondensation von Teilstrukturenmit festgelegtem Material zu verklei-nern oder um die Eigenschaften von Elementtypen zu simulieren, die in unserem Softwarepaketnicht enthalten sind.

Im 5. Kapitel diskutieren wir die Erweiterung des freien Materialoptimierungsproblems furSchalen durch zusatzliche Nebenbedingungen. Wir betrachten lineare Nebenbedingungen furdie Verschiebungen, die dazu verwendet werden konnen, dieForm der deformierten Strukturzu beeinflussen. Im Fall der Nebenbedingungen fur Spannungen unterscheiden wir zwischenden Spannungen in und außerhalb der Ebene, und benutzen sie,um Materialschaden oder sogarMaterialversagen aufgrund zu hoher Spannungen zu vermeiden. Zur Formulierung der Neben-bedingungen fur die Eigenfrequenzen der Struktur stellenwir ein dynamisches Modell vor, dasdie freien Schwingungen von Naghdi-Schalen beschreibt. Daraus leiten wir eine semidefiniteMatrixnebenbedingung her, die dazu verwendet werden kann,die naturlichen Eigenfrequenzender Struktur anzuheben, um so die Anregung durch Resonanz und schlußendlich eine Reso-nanzkatastrophe zu verhindern. Die letzte Art von Nebenbedingungen, die wir betrachten, sind

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Knicknebenbedingungungen, die sich mit der Anfalligkeitvon Schalen fur geometrische Un-regelmaßigkeiten und Laststorungen befassen. Die daraus resultierenden plotzlich auftretendenVerformungen werden durch ein nichtlineares Verzerrungsmodell beschrieben, das wir zur Her-leitung von nichtlinearen Matrixnebenbedingungen verwenden, um so das Knickverhalten zuvermeiden.

Im 6. Kapitel wird diese Doktorarbeit durch eine Zusammenfassung abgeschlossen, die aufweiterfuhrende Fragestellungen fur zukunftige Forschungsprojekte eingeht.

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Contents

1. Introduction 1

2. Elastic Shell Models 52.1. Geometrical Description of the Midsurface . . . . . . . . . .. . . . . . . . . 52.2. Cosserat shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92.3. Kinematic measures for shells . . . . . . . . . . . . . . . . . . . . .. . . . . 112.4. Linear elastic shells including shear effects . . . . . . .. . . . . . . . . . . . 14

2.4.1. Naghdi shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.2. Reissner-Mindlin plates . . . . . . . . . . . . . . . . . . . . . . .. . 15

2.5. Linear elastic shells neglecting shear effects . . . . . .. . . . . . . . . . . . . 162.5.1. Kirchhoff-Love shells . . . . . . . . . . . . . . . . . . . . . . . . .. 172.5.2. Kirchhoff plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6. Material tensors for shells . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 182.6.1. Continuum Mechanics for Solids . . . . . . . . . . . . . . . . . .. . 182.6.2. From Solids to Shells . . . . . . . . . . . . . . . . . . . . . . . . . . .192.6.3. Isotropic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20

3. Free Material Optimization for Shells 233.1. The Elasticity Tensor as a Design Variable . . . . . . . . . . .. . . . . . . . . 243.2. Minimum Compliance Problem Formulation . . . . . . . . . . . .. . . . . . . 27

3.2.1. Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . .. 293.2.2. Free Material Optimization for Reissner-Mindlin Plates . . . . . . . . . 31

3.3. Primal Minimum Compliance Problem Formulation . . . . . .. . . . . . . . . 313.4. Minimum Weight Problem Formulation . . . . . . . . . . . . . . . .. . . . . 443.5. Free Material Optimization for Multiple Load Cases . . .. . . . . . . . . . . . 453.6. Free Material Optimization for Isotropic Naghdi Shells . . . . . . . . . . . . . 493.7. Free Material Optimization for Kirchhoff-Love Shells. . . . . . . . . . . . . . 50

4. Numerical Approach 534.1. Finite Element Discretization . . . . . . . . . . . . . . . . . . . .. . . . . . . 54

4.1.1. Polygonal Approximation of the Midsurface . . . . . . . .. . . . . . 54

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4.1.2. Discrete Continuum Mechanics . . . . . . . . . . . . . . . . . . .. . 584.1.3. Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . .. 624.1.4. Assembly of the Global Stiffness Matrix . . . . . . . . . . .. . . . . . 63

4.2. Discretized Free Material Optimization Problems for Plates and Shells . . . . . 644.3. PENSCP – A Solver for Nonlinear SDPs . . . . . . . . . . . . . . . . .. . . . 674.4. Numerical Testcases and Results . . . . . . . . . . . . . . . . . . .. . . . . . 69

4.4.1. Structures Combined of Solids and Shells . . . . . . . . . .. . . . . . 744.4.2. Inclusion of Condensed Data Structures . . . . . . . . . . .. . . . . . 75

5. Multidisciplinary Optimization Constraints 835.1. Displacement Constraints . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 845.2. Stress and Strain Constraints . . . . . . . . . . . . . . . . . . . . .. . . . . . 885.3. Constraints on the Fundamental Eigenfrequency . . . . . .. . . . . . . . . . . 955.4. Global Stability Constraints . . . . . . . . . . . . . . . . . . . . .. . . . . . . 100

6. Summary and Outlook 105

A. Appendix 109A.1. Technical Remarks on Theorem 3.3.1 . . . . . . . . . . . . . . . . .. . . . . 109

Bibliography 115

Acknowledgments 123

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Contents

List of Frequently Used Symbols

Symbol Name or description Place of definitionor first occurence

a determinant of first fundamental formaαβ (2.20)ai covariant basis vectors of the surface (2.1)aαβ first fundamental form of a surface (2.2)Aγ

m discretized dyadic strain matrix for the strainγ inthe elementm

(4.40)

Aχm discretized dyadic strain matrix for the strainχ in

the elementm(4.41)

Aζm discretized dyadic strain matrix for the strainζ in

the elementm(4.42)

bαβ second fundamental form of a surface (2.13)Bγ

i discretized strain matrix for the strainγ at nodei (4.28)Bχ

i discretized strain matrix for the strainχ at nodei (4.29)

Bζi discretized strain matrix for the strainζ at nodei (4.30)

cαβ third fundamental form of a surface (2.19)Cαβλ µ elasticity tensor for membrane and bending (2.45)C matrix notation of the elasticity tensorCαβλ µ (2.49)CDC coefficient matrix for displacement constraints (5.1)C set of admissible elasticity matrices Sect. 3.1C set of sym. positive semidefinite matrices (3.3)dDC right hand side for displacement constraints (5.1)Dαλ elasticity tensor for shear (2.46)D matrix notation of the elasticity tensorDαλ (2.49)f external force resultant density (2.47)gu external traction resultant density (2.47)gθ external moment resultant density (2.47)H mean curvature of the midsurface (2.17)K Gaussian curvature of the midsurface (2.18)k shear correction factor (2.47)L set of admissible Lagrange multipliers (3.28)M set of sym. positive semidefinite matrices with

bounded trace(3.39)

P increase in total potential energy (5.44)sipe upper bound for in-plane strains Sect. 5.2 (PMW-StC)

soope upper bound for out-of-plane strains Sect. 5.2 (PMW-StC)

sipσ upper bound for in-plane stresses Sect. 5.2 (PMW-SC)

soopσ upper bound for out-of-plane stresses Sect. 5.2 (PMW-SC)

S midsurface of the shell Sect. 2.1t thickness of the shell Sect. 2.2u translational displacement (2.28)U set of admissible displacements (2.29)

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Contents

Symbol Name or description Place of definitionor first occurence

V upper global bound on the trace ofC andD (3.14)α Lagrange multiplier for volume constraint Sect. 3.3 (PP)βl Lagrange multiplier for lower box constraint Sect. 3.3 (PP)βu Lagrange multiplier for upper box constraint Sect. 3.3 (PP)γαβ membrane strain (tensor notation) (2.38)γ membrane strain (vector notation) (2.48)Γλ

αµ Christoffel symbol of a surface (2.10)δxJ(x,y)(δx) Gateaux derivative ofJ(x,y) w.r.t. x Sect. 3.3ζαβ shear strain (tensor notation) (2.40)ζ shear strain (vector notation) (2.48)θ rotational displacement (2.28)ϑi(r,s) bilinear 2D Lagrange shape function Sect. 4.1.2λ eigenvalue of the vibration problem (5.35)ξ space variableξ = (ξ1,ξ2,ξ3)

⊤ (2.28)Π(u,θ) potential energy of a shell (2.47)ρ+ upper local bound on the trace ofC andD (3.15)ρ− lower local bound on the trace ofC andD (3.15)τ time variable (5.27)χαβ bending strain (tensor notation) (2.39)χ bending strain (vector notation) (2.48)χω characteristic function of the setω (3.47)ω reference domain for the midsurfaceS Sect. 2.1ωm quadrangular element of the midsurface mesh Sect. 4.1.1∂ω Lipschitz boundary of midsurfaceω Sect. 2.2∂ω0 clamped part of the boundary∂ω Sect. 2.2∂ω1 free part of the boundary∂ω Sect. 2.2(),µ partial differentiation with respect to surface coor-

dinatesSect. 2.1

()|µ covariant differentiation with respect to first fun-damental form of a surface

(2.12)

〈 · , · 〉 inner product for matrices Sect. 3.3

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CHAPTER 1

Introduction

In recent decades the design of structures has experienced arevolution like never before. It hasbecome possible to create entirely new materials by the use of laminates and microstructures,thus providing properties that are not available from natural materials such as a negative Pois-son’s ratio or negative thermal expansion. Moreover we are now able to fabricate extremelyefficient materials as for example carbon fiber, which features tremendous stiffness in one di-rection, but almost none in the orthogonal directions. Clearly the use of artificial materials likecomposites is the future technology in the construction of efficient structures. Although a lot ofthese advanced materials are already available at present,many manufacturers are overextendedby the multitude of varieties and hold on to traditional fabrication standards.

As a result the modern manufacturing techniques have given rise to the discipline of mate-rial optimization. This subbranch of structural optimization is able to provide guidelines, whichkinds of new materials are useful and where they can be incorporated favorably, in order to facil-itate their utilization in industrial applications. The fundamental problem considered in materialoptimization consists in finding the stiffest structure subjected to a given set of loads and bound-ary conditions, when only a limited amount of material resources is available. There exist severalapproaches to the solution of this problem that differ in theset of admissible materials. One ofthe most established techniques is the SIMP (Solid Isotropic Material with Penalization) method[Ben89, Mle92, RZB92, Sig01]. It distributes a specific isotropic material in the design domainand strives for 0-1-designs with the aid of a penalization strategy. The prediction of the optimalmaterial distribution is also the central task for the inclusion of stiffeners and fillets as performedby [KM76, SD93]. The first step to include also the local material properties in the optimizationprocess was accomplished by the homogenization method [BK88, DB92]. It made the control ofmaterial properties on a macroscopic scale possible by modifying the dimensions of the underly-ing microstructure layout. At the same time the construction of orthotropic laminates motivatedthe optimization of layer thicknesses and fiber orientations [Ped91]. Nowadays this approachhas been generalized to the optimization of material, direction and thickness of the individual

1

1. Introduction

layers resulting in a wide range of available material parameters [GHH99, KLR09]. The set ofadmissible materials was further expanded to a finite numberof different material tensors bythe Discrete Material Optimization method [SL05]. It allows multiple material optimization forvarious applications based on a suitable extension of the SIMP weighting function.

All these material optimization methods include some kind of limitation with respect to theadmissible materials. In contrast to this [Rin93] proposedto abandon any restrictions and todemand only the basic requirements for an elastic material tensor. This idea laid the founda-tion for the Free Material Optimization method as introduced by [BGH+94, BD93]. The designvariable used in Free Material Optimization is the full material tensor at each point of the designdomain. Therefore it yields not only the optimal material distribution, but also the local materialproperties at each point and consequently the ultimately best design. A numerical solution ofthe Free Material Optimization problem by mathematical programming methods was enabled bythe problem formulation and analysis performed in [ZKB97, KZ02]. Based on the saddlepointformulation [Mac04] showed existence of a solution to the Free Material Optimization problemas well as convergence of the discretized problem formulation using finite elements. This wasalso accomplished by [Wer01], who employed Lagrange duality theory to transform the saddle-point formulation into a linear quadratically constrainedoptimization problem in the single loadcase. Numerical results for the obtained discretized problem formulation were computed withthe nonlinear SDP code PENNON [KS03]. An extension of the original Free Material Optimiza-tion problem to cover multiple load cases was achieved by [BKNZ99]. While the considerationof all possible anisotropic elasticity tensors allowed to exploit the entire freedom in the designspace, the resulting optimal materials do typically not preexist in nature. Accordingly a substan-tial effort must be put into a realization, which is close to the optimal structure using artificialmaterials, e.g. by tapelayering methods [HKW01]. However,this is worthwhile in fields likeaerodynamics with a huge demand of extremely light, yet stable structures.

In the past years multidisciplinary optimization constraints were introduced to the Free Mate-rial Optimization framework. These arise from various additional engineering requirements torealistic structures. The use of linear displacement constraints allows to prescribe the shape ofthe deformed structure [Koc97, KSZ08] and even to construct mechanical mechanisms [PBS01].Stress constraints on the other hand intend to avoid structural failure due to stresses beyond theyield strength of the material [DB98, KS06] and are thus frequently used in industrial applica-tions. Another important type of constraints are eigenfrequency constraints in order to suppressthe excitation of resonance vibrations by external forces.Taking free vibrations of the struc-ture into account during the optimization process leads to an optimal design, whose resonancefrequency lies above the range affected by external stimuli[Ped05, SKL09a]. Moreover it is pos-sible to reduce the susceptibility of the structure to geometrical imperfections of its initial shapeor disturbances of the applied loads which might cause sudden deformations known as buckling.This is achieved by introducing global stability constraints based on a nonlinear extension of thedeformations to the optimization problem [Olh89, CZS05, KS04].

All these problems have been solved for two- and three-dimensional solids in the context ofFree Material Optimization. However, many applications inthe design of light-weight structuresconsist mainly of thin-walled components like shells and plates because they are able to carryenormous loads compared to their relatively low self-weight. This is the major motivation for

2

Figure 1.1.: Natural shellsTwo different sea snail species: on the leftthe sinistral (left-handed) shell of Neptuneaangulata, on the right the dextral (right-handed) shell of Neptunea despecta [Wikb]

Figure 1.2.: Man-made shellsDome and cupola of the Cappella del Prese-pio (Santa Maria Maggiore in Rome) [Wika]

this thesis, that is focused on the development of mathematical models and numerical methodsfor a Free Material Optimization formulation based on an elastic shell model. Shells are om-nipresent in our surroundings and evolved naturally millions of years ago as for example in theform of sea shells and and snail shells. They provided the foundation for architectural master-pieces such as domes of churches and palaces and are nowadaysemployed for the constructionof extremely challenging applications as for example the fuselage of aircrafts or Formula 1 rac-ing cars. The theoretical background for a thorough description of shells has been laid in [Koi67]and [Nag72], whose proposed linear elastic shell models have established themselves as state ofthe art. Their analysis has been extended by [Nio85, Ber96, Cia98] to cover not only existenceand uniqueness of solutions, but also the description of thedynamic behavior and large deforma-tions of shells. The results of this thesis are based on Naghdi’s shell model [Nag72] and consistof the following chapters:

In Chapter 2we employ differential geometry for a description of the shell’s midsurface andaccordingly obtain kinematical measures and equilibrium conditions for Naghdi’s shell model.Furthermore we investigate the relation between the three-dimensional solid and the shell mate-rial tensors.

In Chapter 3we develop a minimum compliance problem formulation for theFree MaterialOptimization problem for shells. We prove existence of an optimal solution and equivalence tothe primal problem formulation, a convex nonlinear semidefinite program. Moreover we proposea minimum weight problem formulation as well as an approach to multiple load scenarios.

In Chapter 4we introduce a discretization of the Free Material Optimization problem forshells based on the finite element method. We provide a summary of the utilized optimizationalgorithm together with various numerical testcases from structural optimization. In addition

3

1. Introduction

we demonstrate how to include structures combined from solids and shells and condensed datastructures in the optimization process.

In Chapter 5we extend the Free Material Optimization problem formulation by several mul-tidisciplinary optimization constraints. We consider displacement and stress constraints in thediscrete context, deduce a dynamic description for the inclusion of vibration constraints as wellas a nonlinear expansion of the strains suited to the description of global stability constraints.

Chapter 6contains a summary of the results obtained in this thesis andan outlook concerningfuture research.

4

CHAPTER 2

Elastic Shell Models

In this chapter we present mathematical models to describe the special geometrical and elasticproperties of thin-walled structures such as shells and plates. These models are the foundationfor the formulation of the Free Material Optimization problem for shells and plates includingvarious multidisciplinary constraints and its numerical treatment, that we will introduce in thesubsequent chapters. While in the case of solid bodies the choice of the coordinate systemis usually up to the observer, shells establish their own “natural” coordinate system, which isdetermined by their shape. Hence, we start with an introduction to differential geometry char-acteristics necessary for a systematic description of a shell’s shape. Thereafter we examinekinematic measures for shells which exceed kinematic measures for solids in the sense that theyare not only able to distinguish between translational and shear strains, but also between trans-lational deformations parallel and normal to the midsurface of the shell referred to as membraneand bending behavior. These aspects are illustrated withinthe context of Cosserat media whichprovide an elegant approach to capture the geometrical and physical features of shell bodies.Finally we discuss the most common used linear shell models and their equilibrium equations,which allow to predict the behavior of shell bodies subjected to external loads.

2.1. Geometrical Description of the Midsurface

A shell is a three-dimensional body whose boundary surface has special geometric features[Nag72]. It is confined by two outer surfaces and the direction between these surfaces, whichis calledthickness, is small. The shell’s shape is distinguished by itsmidsurface– the lateralsurface lying in the middle of the two outer surfaces. A shellis said to becompleteif it hasno other boundary than the two outer surfaces, like e.g. a spherical hull. Otherwise the shell isbounded by a curve on the midsurface and a normal section along this curve. If the thicknessis much smaller than a characteristic length of the midsurface the shell is considered to bethin.Moreover, shells with plane midsurfaces are calledplates.

5

2. Elastic Shell Models

ξ

ξ

1

2 a

aa

1

23

thickness

midsurface

Figure 2.1.: Midsurface of a shell

A shell’s midsurfaceS is defined by an injective mapping from a reference domainω into thephysical spaceP containing the shell. Let the reference domainω ⊂ R

2 be open and boundedand denote byφ : R2 → R

3 a sufficiently smooth mapping fromω into the physical spaceP,e.g. φ ∈ W2,∞(ω). Then the midsurface is parametrized byS = φ(ω) [CB03]. In general,the midsurface may consist of a connected set of multiple surfaces defined by smooth injectivemappings from domains ofR2 into P. When performing a static analysis of this structure thesesurfaces are treated separately, whereas their individualenergies add up to the total energy of theshell. Hence, we concentrate without loss of generality on shells defined by a singular mapping.

The coordinates used within the curvilinear coordinate system are labeled byξ i, i ∈ 1,2,3.Note that in accordance with standard notation in shell theory Latin indices run over 1, 2 and3, while Greek indices take the values 1 and 2. The mappingξ 1 7→ φ(ξ 1,ξ 2

fixed), whereξ 2 isrestrained to the constant valueξ 2

fixed, while ξ 1 is varied, describes aξ 1-coordinate curveonthe midsurface. Theξ 2-coordinate curve is analogously defined byξ 2 7→ φ(ξ 1

fixed,ξ 2). Thesecoordinate curves can be used to construct a local coordinate system at each pointP∈ S of themidsurface. A very straightforward approach consists in composing the coordinate system oftangent vectors to the coordinate curves through pointP, which is referred to ascovariant basis.Thus the covariant basis vectors on the midsurface are givenby

aα =∂φ

∂ξ α , a3 =a1×a2

‖a1×a2‖. (2.1)

As required by the definition of a basis these vectors have to be linearly independent. How-ever, they are usually not orthogonal. The basis vectorsa1 anda2 are used to build up thefirstfundamental form

aαβ = aα ·aβ (2.2)

of the surface. The first fundamental form is a symmetric second-order tensor, which resemblesthe metric tensor restricted to the midsurface. Moreover, it allows the conversion between thecovariant basis vectorsaα and thecontravariant basis vectors aα defined by

aα ·aβ = δ βα , (2.3)

where

δ βα :=

1 if α = β0 else

(2.4)

6

2.1. Geometrical Description of the Midsurface

represents the so-calledKronecker delta. According to (2.3) the contravariant basis vectors canbe pictured as normal vectors to the coordinate curves through point P. The transformationbetween the covariant and the contravariant basis vectors is described by

aα = aαβ aβ . (2.5)

This formula introduces theEinstein summation convention: a summation is carried out overidentical indices, which appear once as a subscript and onceas a superscript in the same expres-sion. Thus (2.5) has to be understood as

aα = aαβ aβ =2

∑β=1

aαβ aβ . (2.6)

The Einstein summation convention is used throughout the entire thesis.A shell’s metric has also to be taken into account when calculating derivatives of variables on

the midsurface. Letv be a given vector field on the midsurfaceS , which takes the form

v= vβ aβ. (2.7)

The partial derivative of the componentvβ with respect toξ α is denoted byvβ ,α . This equalsa derivation along a coordinate line in the reference domainω , which ignores the shape of theshell. For the calculation of the strains in section 2.3 we require a derivation along a curve onthe midsurface, that contains information about the dependence of the basis vectorsaβ and thusthe local coordinate system on the coordinatesξ α from the reference domainω . This derivativeis obtained via application of the product rule

∂v∂ξ α = vβ ,α aβ +vβ aβ

,α . (2.8)

By using the completeness of the coordinate system providedby a1,a2,a3 this expression isexpanded to

∂v∂ξ α = vβ ,α aβ +

(vµaµ

,α ·aβ)

aβ +(vµaµ

,α ·a3)

a3 , (2.9)

which can be simplified by inserting theChristoffel symbolΓµβα on the midsurface defined as

Γµβα = aβ ,α ·aµ

. (2.10)

Due to the equalityaβ ·aµ = δ µβ it holds thatΓµ

βα = −aβ ·aµ,α and the derivative along a curve

on the midsurface takes the form

∂v∂ξ α =

(

vβ ,α −Γµβαvµ

)

aβ +(vµaµ

,α ·a3)

a3 , (2.11)

Its tangential component is known assurface covariant derivativeand given by

vβ |α = vβ ,α −Γµβαvµ . (2.12)

7


Due to the flexural shape of the shell, which demands the use ofcurvilinear coordinates, itis reasonable to collect relevant information about the curvature in a curvature tensor, whichequals thesecond fundamental form bαβ of the shell. The covariant-covariant and the covariant-contravariant version of the second fundamental form are given by

bαβ = −a3,β ·aα , (2.13)

bαβ = −a3,β ·aα

. (2.14)

In fact it holds thata3 ·aα = 0, hence equation (2.14) can be reformulated asbαβ = a3 ·aα

,β andis employed to simplify (2.11) further to its final form

∂v∂ξ α = vβ |α aβ +bµ

αvµa3 . (2.15)

Moreover, (2.14) leads to the following expressions for derivatives ofa3

a3,α = a3,α =−bµ

αaµ =−bµαaµ. (2.16)

The eigenvalues of the second fundamental form are the principal curvatures, in other words theminimal and maximal curvatures of all coordinate curves through the pointP. Their arithmeticmean is referred to asmean curvatureH

H=12

tr(

bαβ

)

=12

(b1

1+b22

), (2.17)

while their product is namedGaussian curvature(or total curvature)K of the midsurface

K= det(

bαβ

)

= b11b2

2−b12b2

1 . (2.18)

Finally thethird fundamental formof the midsurface is defined as

cαβ = bλαbλβ . (2.19)

Note that all three fundamental forms are symmetric second-order tensors.Another matter of importance concerns the correlation between the differentialsdξ 1 anddξ 2

and the corresponding infinitesimal areadS on the surface. These quantities are linked by theJacobian of the associated coordinate transformation, which is in this case equal to the determi-nant of the first fundamental formaαβ . By introducing the notation

a= det(aαβ)= a11a22−a12

2 (2.20)

the infinitesimal surface areadS can be written as

dS = ‖a1×a2‖dξ 1dξ 2 =√

det(aαβ)

dξ 1dξ 2 =√

adξ 1dξ 2. (2.21)

A similar relationship exists for three-dimensional volume integrals. Letgi j be the metric tensorof the three-dimensional space and denote byg its determinant and bydR the infinitesimalvolume based on the differentialsdξ 1, dξ 2 anddξ 3. Then it holds that

dR =√

det(gi j )dξ 1dξ 2dξ 3 =√

gdξ 1dξ 2dξ 3. (2.22)

8

2.2. Cosserat shells

Figure 2.2.: Shell continuum Figure 2.3.: Shell built up from material lines

Figure 2.4.: Directors represent material lines Figure 2.5.: Cosserat shell

When comparing (2.21) and (2.22) the question arises on how the surface integrals and thethree-dimensional volume integrals are connected. As the first fundamental formaαβ can beinterpreted as the restriction ofgi j to the tangent plane, there exists an explicit expression for therelation between their determinants as follows

g= a(

1−2Hξ 3+K(ξ 3)2

)2. (2.23)

A derivation of this formula is given in [CB03, Section 2.2.3, Proof of (2.155)]. Furthermore itis possible to identify the relations between the three-dimensional basis vectorsgi and the basisvectors on the midsurfaceai as

gα =(

δ λα −ξ 3bλ

α

)

aλ , (2.24)

g3 = a3 . (2.25)

2.2. Cosserat shells

It was shown in the previous section that a shell’s position in the physical spaceP is veryefficiently parametrized by a surface, yet this descriptionis not sufficient to properly modelits physical behavior. After all a shell is a three-dimensional body, which extends also in thedirection orthogonal to the midsurface. The brothers Eugene and Francois Cosserat have facedthe problem on how to endow the shell’s midsurface with additional information to describeits thickness and beyond that also its characteristic behavior [CC09]. The basic idea of thetheory of Cosserat continua consists in attaching a director vector d to each pointx ∈ S ofthe midsurface. The additional degrees of freedom providedby the director vectors allow for athorough modeling of bending and shear effects, which can not be captured by a sole surface[CC09, Rub00]. These director vectors can be interpreted asmaterial fibres along the shell’sthickness as shown in Figures 2.2 – 2.5.

Note that the thicknesst need not be constant, a varying thickness profilet(ξ 1,ξ 2) can beused instead for all matters presented in this thesis as longas this thickness profile remains

9


fixed during optimization. However, for the sake of simplicity all results will be derived using aconstant thicknesst.

Suppose that the material lines represented by the directorvectors are orthogonal to the mid-surface in the shell’s undeformed configuration and denote by Ω the three-dimensional referencedomain defined as

Ω :=

(ξ 1,ξ 2

,ξ 3) ∈ R3

∣∣∣∣(ξ 1

,ξ 2) ∈ ω , ξ 3 ∈]

− t(ξ 1,ξ 2)

2,t(ξ 1,ξ 2)

2

[

. (2.26)

By introducing the mapping

Φ(ξ 1,ξ 2

,ξ 3) = φ(ξ 1,ξ 2)+ξ 3a3(ξ 1

,ξ 2) (2.27)

it is then possible to specify the subset of the Euclidean space occupied by the shell bodyBasB = Φ(Ω) [CB03]. In summary, Cosserat shells offer an excellent compromise between atwo-dimensional geometrical description capturing a shell’s unique shape and a modeling thatincludes all significant aspects to predict the behavior of the shell.

An essential task is the investigation of principles that determine the deformation of Cosseratshells under external loading. We assume that a basic rule ofshell theory holds, which is knownas theReissner-Mindlin kinematical assumption: material lines remain straight and unstretchedduring deformation. Hence, the displacements consist of a translation of all points on the mid-surface and a rotation of the associated director vectors. While the translational displacementscan be modeled straightforward by a three-dimensional vector u∈ [H1(ω)]3, it has to be kept inmind that the material fibres are regarded as infinitely thin.Thus rotations of the director vectorsaround their own axis – usually referred to as drilling – can be neglected. As the rotation of aninfinitely-thin straight material line is uniquely defined by a rotation vector normal to that line,the rotational displacementθ ∈ [H1(ω)]2 is introduced to represent the rotation via the surfacetensorθλ aλ [CB03]. On that account the displacements for shells take the form

U(ξ 1,ξ 2

,ξ 3) = u(ξ 1,ξ 2)+ξ 3θλ (ξ 1

,ξ 2)aλ (ξ 1,ξ 2) . (2.28)

Additionally the shell’s reference domain is considered tohave a Lipschitz boundary∂ω .This boundary∂ω is partitioned into two sets:∂ω0, where the shell is clamped and consequentlyDirichlet boundary conditions are applied, and the remainder ∂ω1, where the shell moves freelyand is subjected to forces and moments. Both sets∂ω0 and∂ω1 are open in∂ω and it holdsthat ∂ω = ∂ω0 ∪ ∂ω1 as well as∂ω0∩ ∂ω1 = /0. Furthermore there exists an index setI0 ⊆1,2,3,1,2 identifying the entries ofu andθ , which are affected by the Dirichlet boundaryconditions on∂ω0. With these definitions the set of admissible displacementscan be written as

U :=

(u,θ) ∈[H1(ω)

]5∣∣∣ ui = 0 andθα = 0 on ∂ω0 ∀i,α ∈ I0

. (2.29)

Obviously it holds that[H1

0(ω)]5 ⊂ U ⊂

[H1(ω)

]5.

10

2.3. Kinematic measures for shells


Next we continue with the study of kinematic measures for shells. These are required as dis-placement fields are not automatically accompanied by a deformation of the elastic body, theymight also originate from rigid body motions. In contrast tomotions the deformations of a bodyare manifested by the extension or compression of lines between points of the solid medium andby the change of angles between those lines [WT02]. The geometrical quantities necessary forthe description of the deformation at a point are collected in the so-calledstrain tensor. In factthere exist several versions of strain tensors adjusted to the needs of specific application fields,in this thesis we focus on theGreen-Lagrange strain tensorεi j . The strain tensorεi j is definedas the linearized change of the metric tensor, in other wordsthe linearization of the differencebetween the metric tensorgi j

∗ of the deformed configuration of the shell and the metric tensorgi j of the undeformed configuration

εi j =12

(gi j

∗−gi j)

lin =12

(giU, j +g jU,i

). (2.30)

To obtain formulas describing the shell strains the generaldisplacementU in equation (2.30) isreplaced by the displacements (2.28) required due to the Reissner-Mindlin kinematical assump-tion. The necessary derivatives can be calculated as

∂U∂ξ α =

∂u(ξ 1,ξ 2)

∂ξ α +ξ 3∂(θλ (ξ 1,ξ 2)aλ (ξ 1,ξ 2)

)

∂ξ α , (2.31)

where

∂u(ξ 1,ξ 2)

∂ξ α =∂

∂ξ α

(

uλ aλ +u3a3)

= uλ ,α aλ +uλ aλ,α +u3,αa3+u3a3

,α(2.15) and (2.16)

= uλ |α aλ +bλαuλ a3+u3,αa3−u3bλα aλ (2.32)

∂(θλ (ξ 1,ξ 2)aλ (ξ 1,ξ 2)

)

∂ξ α = θλ ,α aλ +θλ aλ,α

(2.15)= θλ |α aλ +bλ

αθλ a3 (2.33)

Moreover, it holds that

∂U∂ξ 3 = θλ (ξ 1

,ξ 2)aλ (ξ 1,ξ 2) . (2.34)

11


Figure 2.6.: Membrane strain Figure 2.7.: Bending strain Figure 2.8.: Shear strain

Replacing the three-dimensional basis vectorsgi by (2.24) and (2.25) and using the orthogonalityof the basis vectors (2.3) leads to the following shell strains

εαβ =12

(uα |β +uβ |α

)−bαβ u3

+ξ 3[

12

(

θα |β +θβ |α −bλβ uλ |α −bλ

αuλ |β)

+cαβ u3

]

+(ξ 3)2

[12

(

bλβ θλ |α +bλ

α θλ |β)]

, (2.35)

εα3 =12

(

θα +u3,α +bλαuλ

)

, (2.36)

ε33 = 0. (2.37)

These strains are now separated into several distinct deformations present within the scope ofa model based on the Reissner-Mindlin kinematical assumption. The term ofεαβ independentfrom the normal directionξ 3 is referred to as membrane strainγαβ , while the part linear inξ 3

represents the bending strainχαβ . The summand depending quadratically onξ 3 is neglecteddue to the assumption of a thin shell, while the shear strainζα for shells is equal to the originalshear strainεα3. In summary, this leads to the following shell strains

γαβ (u) =12

(uα |β +uβ |α

)−bαβ u3 , (2.38)

χαβ (u,θ) =12

(

θα |β +θβ |α −bλβ uλ |α −bλ

αuλ |β)

+cαβ u3 , (2.39)

ζα(u,θ) =12

(

θα +u3,α +bλαuλ

)

. (2.40)

The deformations associated with these strains are illustrated in a classical continuum mechani-cal perception in Fig. 2.6 – 2.8.

The introduced membrane, bending and shear strain depend linearly on the displacementsuandθ and are thus based on an approximation using small displacements. This assumption isvalid for almost all physical models used within this thesiswith one exception: the descriptionof buckling behavior in section 5.4. While shells are able tocarry enormous loads in a mem-brane state, they are very sensitive to geometrical imperfections or disturbances of the appliedloads which might cause sudden deformations of the shell [WT02]. As plates and shells arevery susceptible to this phenomenon known as snap-through buckling, the incorporation of thateffect is vital for thin-walled structures. However, buckling is not contained in a model using

12


only the linear terms of the strain-displacement relationship, hence its description requires theconsideration of higher-order terms in expression (2.30).

For the derivation of the non-linear strains it is assumed that the shell is in amembrane state,in other words the impact of the transverse shear forces and resulting changes of curvature areneglected [Nio85]. This idealization has proven to be effective in many practical applications andrequires only non-linear membrane strains, hence the expressions for the bending strains (2.39)and shear strains (2.40) remain valid. For the deduction of the membrane strains the propositionused for the derivation of the linear strains, namely that the deformation of the coordinate systemis insignificant and the equilibrium equations can be formulated in the undeformed configuration,is dropped. Thus it is necessary to evaluate the expressionsin formula (2.30) in the undeformedas well as the deformed state of the shell.

An entire deduction of the three-dimensional non-linear strains εi j is not necessary for thederivation of the non-linear membrane strainsγαβ : only entries in the index rangei, j ∈ 1,2,which are furthermore independent fromξ 3, contribute to the membrane strains. Hence (2.30)can be rewritten as

γαβ = εαβ∣∣ξ 3=0 =

12

(gαβ

∗−gαβ)∣∣ξ 3=0

=12

(gα

∗ ·gβ∗−gα ·gβ

)∣∣ξ 3=0

(2.24)=

12

(aα

∗ ·aβ∗−aα ·aβ

). (2.41)

As introduced in section 2.1 the undeformed configuration ofthe shell is parametrized byφ ,therefore the covariant basis vectorsaα are simply given by (2.1). Since the deformation of theshell is described by the displacementU satisfying condition (2.28) the deformed configurationof the shell can be written asφ∗ = φ +U [Cia98]. The covariant basis vectors in the deformedconfigurationaα

∗ are thus defined as

aα∗ =

∂φ∗

∂ξ α =∂ (φ +U)

∂ξ α =∂

∂ξ α

(

φ +uλ aλ +u3a3+ξ 3θλ aλ)

(2.32)(2.33)=

∂φ∂ξ α +uλ |αaλ +bλ

αuλ a3+u3,αa3−u3bλα aλ +ξ 3θλ |α aλ +ξ 3bλα θλ a3 .

Insertingaα andaα∗ into formula (2.41) while neglecting all terms depending onξ 3 provides

the non-linear membrane strains

γαβ =12

[(

aα +uλ |αaλ +bλαuλ a3+u3,αa3−u3bλα aλ

)

·(

aβ +uµ |β aµ +bµβ uµa3+u3,β a3−u3bµβ aµ

)

−aα ·aβ

]

=12

[

uα |β +uβ |α −2bαβ u3+aλ µ (uλ |α uµ |β −uλ |α bµβ u3−uµ |β bλα u3+bλαbµβ u32)

+ bλαuλ bµ

β uµ +bλαuλ u3,β +bµ

β uµu3,α +u3,αu3,β

]

= γαβ +12

aλ µ (uλ |α −bλαu3)(

uµ |β −bµβ u3)+

12

(

bλαuλ +u3,α

)(

bµβ uµ +u3,β

)

= γαβ (u)+12

aλ µ (γλα (u)−ψλα(u))(γµβ (u)−ψµβ (u)

)+

12

ϕα(u)ϕβ (u) , (2.42)

13


where

ψλα (u) :=12

(uα |λ −uλ |α

), (2.43)

ϕα(u) := u3,α +bλαuλ . (2.44)

This is the general form of the non-linear membrane strainsγαβ , which is used for the derivationof the buckling constraints in section 5.4. Note that there exist also simplifications to this generalform adapted to special cases such as small finite deflectionsand shallow shells [Ber96].

2.4. Linear elastic shells including shear effects

After the introduction of the various strains in the previous section we are now able to proceedto the equilibrium conditions predicting the behavior of shells under external loading. In themajor part of this thesis linear elastic shells are considered, in other words the strains dependlinearly on the displacements and the stresses are connected to the strains via a linear Hooke’slaw (leading to a bilinear form for the potential energy). The only exception to this are thenon-linear membrane strains allowing for the inclusion of buckling effects in section 5.4. Whilean elastic first-order approximation modeling is clear without ambiguity for three-dimensionalsolids, we have to distinguish in the case of shells whether shear phenomena are incorporated ornot. In this section we introduce the most prominent linear elastic shell model including shear,the so-called Naghdi’s shell model [Nag72, CB03, Ber96, Cia00], and the plate model derivedtherefrom known as Reissner-Mindlin plates.

2.4.1. Naghdi shells

To set up the equilibrium conditions for Naghdi shells we need to obtain a formula for thepotential energy of the shell. To this end we introduce the shell’s elasticity tensorsCαβλ µ andDαλ , who fulfill the following symmetry relations

Cαβλ µ =Cβαλ µ, Cαβλ µ =Cαβ µλ , Cαβλ µ =Cλ µαβ (2.45)

Dαλ = Dλα (2.46)

Furthermorek resembles the shear correction factor. In general it holds that k ∈ [0,1] ⊂ R+,however the exact value ofk is only known for special cases like isotropic material (k = 5

6) ororthotropic material (k= 2

3). Finally the force resultant density, the traction resultant density andthe moment resultant density are given byf ∈ [L2(ω)]3, gu ∈ [L2(∂ω1)]

3 andgθ ∈ [L2(∂ω1)]2,

respectively. Then according to [Nag72] the potential energy Π(u,θ) of the shell reads as

Π(u,θ) =12

∫

ωCαβλ µ

[

tγαβ γλ µ +t3

12χαβ χλ µ

]

+ tkDαλ ζαζλ dS

−∫

ωt f ·udS −

∫

∂ω1

gu ·u+gθ ·θ dl . (2.47)

14

2.4. Linear elastic shells including shear effects

A prominent way to rewrite this expression is the so-called Mandel notation as for example de-scribed by Pedersen [Ped98]. It includes the reformulationof strains as vectors and the elasticitytensors as matrices

γ =

γ11

γ22√2γ12

, χ =

χ11

χ22√2χ12

, ζ =

(ζ1

ζ2

)

, (2.48)

C=

C1111 C1122√

2C1112

C1122 C2222√

2C2212√

2C1112√

2C2212 2C1212

, D =

(D11 D12

D12 D22

)

. (2.49)

This notation leads to the following formula for the potential energyΠ(u,θ) of a Naghdi shell

Π(u,θ) =12

∫

ωtγ⊤Cγ +

t3

12χ⊤Cχ + tkζ⊤Dζ dS −

∫

ωt f⊤udS −

∫

∂ω1

gu⊤u+gθ

⊤θ dl .

(2.50)A shell exposed to external loads will strive for a balance ofthe internal and external forces, i.e.,it will assume a state where every change of the current deformation requires additional energy.This leads to the following condition for the static equilibrium of a Naghdi shell also known asminimum potential energy principle

min(u,θ )∈U

Π(u,θ) . (2.51)

Note that the vector-matrix representation conserves not only the potential energy and Hooke’slaw of the shell, but also the norm of the elasticity tensors.Indeed the latter is not true forwidespread engineering notations as for example the Voigt notation, which use different factorsfor single entries of the strain and elasticity tensors. This is an important issue, as the trace of theelasticity tensor serves as a norm in Chapter 3 in the construction of the optimization problem.

Moreover, the equilibrium condition can be formulated alternatively to the minimum potentialenergy principle as a variational problem. The relevance ofthis weak formulation lies in the factthat it serves as a basis for the finite element discretization of the static equilibrium in Section4.1. The variational formulation of the equilibrium condition for Naghdi shells reads as follows:Find (u,θ) ∈ U such that

∫

ωtγ⊤(u)Cγ(v)+

t3

12χ⊤(u,θ)Cχ(v,η)+ tkζ⊤(u,θ)Dζ (v,η)dS (2.52)

=∫

ωt f⊤vdS+

∫

∂ω1

gu⊤v+gθ

⊤η dl ∀(v,η) ∈ U .

2.4.2. Reissner-Mindlin plates

For the frequently needed special case of a planar midsurface the so-called Reissner-Mindlinplate model is deducted from Naghdi’s shell model. The absent curvature of a planar midsurfaceS leads to vanishing second and third fundamental forms

bαβ = 0 , cαβ = 0. (2.53)

15


Hence, the membrane, bending and shear strains of the Reissner-Mindlin plate problem assumea considerable reduced shape

γαβ (u1,u2) =12

(uα |β +uβ |α

),

χαβ (θ) =12

(θα |β +θβ |α

), (2.54)

ζα(u3,θ) =12(θα +u3,α) .

With these strains the potential energyΠ(u,θ) of the Reissner-Mindlin plate is calculated in thesame manner as for Naghdi shells

Π(u,θ) =12

∫

ωtγ⊤(u1,u2)Cγ(u1,u2)+

t3

12χ⊤(θ)Cχ(θ) (2.55)

+ tkζ⊤(u3,θ)Dζ (u3,θ)dS −∫

ωt f⊤udS −

∫

∂ω1

gu⊤u+gθ

⊤θ dl

and the equilibrium state of the plate is again found by solving the minimum potential energyproblem (2.51). Note that due to the simplified strains the potential energy is separated into partsonly depending onu1 andu2 and into other summands only depending onu3, θ1 andθ2. For thetask of static analysis of the plate it is thus possible to split problem (2.51) into the membraneproblem

min(u1,u2)∈Um

12

∫

ωtγ⊤(u1,u2)Cγ(u1,u2)dS −

∫

ωt ( f1u1+ f2u2) dS −

∫

∂ω1

(gu1u1+gu2u2) dl

and the so-called Reissner-Mindlin problem

min(u3,θ )∈URM

∫

ω

t3

24χ⊤(θ)Cχ(θ)+

t2

kζ⊤(u3,θ)Dζ (u3,θ)dS

−∫

ωt f3u3 dS −

∫

∂ω1

gu3u3+gθ⊤θ dl .

However, when performing an optimization of the material ofthe plate determined by the elastic-ity tensorsC andD these two problems are correlated and can not be separated anymore. Hencewe will only consider the entire minimization problem (2.55) for the optimization of plates inthe subsequent chapters.

2.5. Linear elastic shells neglecting shear effects

Besides Naghdi’s shell model there exists another very prominent linear elastic shell model incontinuum mechanics. It differs from the shell model introduced in the previous section as it ne-glects shearing behavior. This is achieved via a restriction on the Reissner-Mindlin kinematicalassumption: material lines remain straight, unstretched and orthogonal to the midsurface duringdeformation. This tighter condition is known as theKirchhoff-Love kinematical assumptionand

16

2.5. Linear elastic shells neglecting shear effects

is assumed to be valid only for thin shells. A shell is considered to be thin when the ratio betweena typical length scale of the shell such as for example the radius of curvature and its thicknessis greater than 20. Below we introduce the Kirchhoff-Love shell model and its associated platemodel, the Kirchhoff plate.

2.5.1. Kirchhoff-Love shells

The Kirchhoff-Love kinematical assumption is enforced viaa constraint connecting translationaland rotational displacements

θα =−u3,α −bλαuλ . (2.56)

This results not only in vanishing shear strainsζα = 0, furthermore it allows to replace therotational displacements in the kinematical equations andthe equilibrium condition obtaininga problem formulation depending only onu. Hence the membrane strainγαβ and the bendingstrainραβ appearing in Kirchhoff-Love shells are given by

γαβ (u) =12

(uα |β +uβ |α

)−bαβ u3 (2.57)

ραβ (u) = −χαβ (u) = u3|αβ +bλα |β uλ +bλ

αuλ |β +bλβ uλ |α −cαβ u3 (2.58)

It can be easily seen that the absence of the rotational displacement variableθ is traded forhigher order derivations of the remaining displacementu. A consideration of this fact leads tothe following modification of the admissible set for Kirchhoff-Love shells

UKL :=

u= (uα ,u3) ∈[H1(ω)

]2×H2(ω)∣∣∣ ui = 0 on∂ω0∀i ∈ I0

. (2.59)

This space is endowed with the norm

‖u‖H1(ω)×H2(ω) =

2

∑α=1

‖uα‖21,ω +‖u3‖2

2,ω

12

. (2.60)

The expression for the potential energy of a Kirchhoff-Loveshell is then obtained by eliminatingthe shear energy term and the term specifying the external moments from (2.47) resulting in

ΠKL(u) =12

∫

ωCαβλ µ

[

tγαβ γλ µ +t3

12ραβ ρλ µ

]

dS −∫

ωt f ·udS −

∫

∂ω1

gu ·udl . (2.61)

Again the elasticity tensorCαβλ µ is rewritten as a matrix according to (2.49) and the strainsγαβandραβ are remodeled as vectors

γ =

γ11

γ22√2γ12

, ρ =

ρ11

ρ22√2ρ12

(2.62)

bringing forth the following formula for the potential energy of a Kirchhoff-Love shell

ΠKL(u) =12

∫

ωtγ⊤Cγ +

t3

12ρ⊤Cρ dS −

∫

ωt f⊤udS −

∫

∂ω1

gu⊤udl . (2.63)

17


The equilibrium state of the Kirchhoff-Love shell is then determined by solving the minimumpotential energy problem

minu∈UKL

ΠKL(u) . (2.64)

2.5.2. Kirchhoff plates

To derive the Kirchhoff plate model from the Kirchhoff-Loveshells again the conditions for aplanar midsurface (2.53) are employed. The obtained strains are

γαβ (u1,u2) =12

(uα |β +uβ |α

)(2.65)

ραβ (u3) = u3|αβ (2.66)

These strains are inserted into the formula for the potential energy

ΠKL(u) =12

∫

ωtγ⊤(u1,u2)Cγ(u1,u2)+

t3

12ρ⊤(u3)Cρ(u3)dS −

∫

ωt f⊤udS −

∫

∂ω1

gu⊤udl ,

(2.67)which is then used to solve the minimum potential energy problem (2.64) to find the equilibriumstate of the Kirchhoff plate. This is again a problem formulation that can be separated into themembrane part depending onu1 andu2 and the remainder which depends only onu3. However,as both problems are coupled by the elasticity tensorC, this will not be relevant for the materialoptimization problems dealt with in this thesis.

2.6. Material tensors for shells

Within the derivation of elastic shell models in the previous sections the connection betweenthe three-dimensional strains and the membrane, bending and shear strains has been extensivelydiscussed. It is evident that there must also exist a relation between the solid elasticity tensorEi jkl and the shell elasticity tensorsCαβλ µ andDαλ , when the solid and the shell are constructedfrom the same material. This matter is addressed in this section beginning with a short summaryof relevant continuum mechanical formulas for solids.

2.6.1. Continuum Mechanics for Solids

The three-dimensional strain tensorεi j introduced in (2.30) is linked to the stress tensorσi j viaHooke’s law

σ i j (x) = Ei jkl (x)εkl(u(x)) , (2.68)

whereEi jkl is the three-dimensional elasticity tensor. Due to the symmetry properties of theintroduced tensors given byεi j = ε ji , σi j = σ ji and

Ei jkl = E jikl = Ei jlk = Ekli j (2.69)

18


these tensors can be rewritten as vectors and matrices according to [Ped98] as follows

ε = (ε11,ε22,ε33,√

2ε12,√

2ε13,√

2ε23)⊤, (2.70)

σ = (σ11,σ22

,σ33,√

2σ12,√

2σ13,√

2σ23)⊤ .

Furthermore the 21 independent entries ofEi jkl are written into a 6×6 matrix

E =

E1111 E1122 E1133√

2E1112√

2E1113√

2E1123

E1122 E2222 E2233√

2E2212√

2E2213√

2E2223

E1133 E2233 E3333√

2E3312√

2E3313√

2E3323√

2E1112√

2E2212√

2E3312 2E1212 2E1213 2E1223√

2E1113√

2E2213√

2E3313 2E1213 2E1313 2E1323√

2E1123√

2E2223√

2E3323 2E1223 2E1323 2E2323

, (2.71)

which is required to be symmetric and positive semidefinite.This reformulation provides notonly a vector equation formulation for Hooke’s law

σ = E · ε , (2.72)

but also a vector-matrix expression for the potential energy of the solid beneath the originaltensor terminology

Π(u) =∫

Ω

12

εi j (u(x))Ei jkl (x)εkl(u(x))dx−

∫

Γ f

f ·udx

=∫

Ω

12

ε⊤(u(x))E(x)ε(u(x))dx−∫

Γ f

f ·udx, (2.73)

where f ∈[L2(Γ f )

]3are the loads applied at the boundaryΓ f ⊂ ∂Ω.

2.6.2. From Solids to Shells

Shells are by definition elastic bodies, which are thin in onedirection. Hence a fundamentalassumption of shell theory states that the shell’s materialproperties do not change along thedirection normal to the midsurface. For this reason monoclinic material symmetry conditionshave to be taken into account when calculating the elastic properties of a shell constructed froma material characterized byEi jkl [CB03].

Eαβλ3 = Eα333= 0 ∀α ,β ,λ = 1,2. (2.74)

This considerably simplifies the constitutive equations for the solid medium, which take the form

σ αβ = Eαβλ µελ µ +Eαβ33ε33, (2.75)

σ α3 = 2Eα3λ3ελ3 , (2.76)

σ33 = E33λ µελ µ +E3333ε33. (2.77)

19


Moreover the disappearance of the normal stressesσ33= 0 – another basic principle of shell the-ory – is employed to obtain an expression forε33 from (2.77), which is substituted into (2.75).The resulting blocks of the elasticity matrix are identifiedwith the shell elasticity matrices bring-ing forth the constitutive equations for shells

σ αβ = Cαβλ µελ µ , (2.78)

σ α3 =12

Dαλ ελ3 (2.79)

and the sought-after transformation formulas for the material matrices

Cαβλ µ = Eαβλ µ − Eαβ33Eλ µ33

E3333 , (2.80)

Dαλ = 4Eα3λ3. (2.81)

Note that the factor 4 in relation (2.81) determiningDαλ is necessary to ensure that the shearenergy provided byζ⊤Dζ is equal to the shear energy terms ofε⊤Eε , which contain severalcoefficients according to (2.70) and (2.71).

2.6.3. Isotropic Material

A recurrent case of material symmetry is isotropic material, which is distinguished by its rota-tional invariant material tensor [Red97]. Hence the response of isotropic material is independentof the direction of the applied forces. The universal isotropic material tensor is given by

Ei jkliso = λgi j gkl +µ

(

gikg jl +gil g jk)

, (2.82)

whereλ andµ are the so-called Lame parameters specifying the entire isotropic material ten-sor. Therefore the general isotropic material tensor takesthe following form in the previouslypresented matrix notation

Eiso =

λ +2µ λ λ 0 0 0λ λ +2µ λ 0 0 0λ λ λ +2µ 0 0 00 0 0 2µ 0 00 0 0 0 2µ 00 0 0 0 0 2µ

. (2.83)

Remark2.6.1. Note that there are different possibilities in describing isotropic material witha pair of constants. While the Lame parametersλ and µ are typically used in mathematicsdue to the linear dependence of the material tensor onλ andµ , engineers hardly use the Lameparameters and instead focus on measurable quantities. Themost prominent pair of constantsare Young’s modulusE (a measure of stiffness for an isotropic material) and Poisson’s ratioν(measuring the tendency of a material, that is compressed inone direction, to expand in the othertwo directions). Transformation between these different notations can be performed by using theformulas

λ =Eν

(1+ν)(1−2ν), µ =

E2(1+ν)

. (2.84)

20


By means of the isotropic material tensor for solids (2.82) and the transformation formulas(2.80) and (2.81) the material tensorsCαβλ µ

iso and Dαλiso for a shell constructed from isotropic

material can be calculated. We exemplify this for the matrixentryC1111iso resulting in

C1111iso

(2.80)= E1111− E1133E1133

E3333

(2.82)= λ +2µ − λ ·λ

λ +2µ=

4µ(λ +µ)λ +2µ

(2.84)=

4 E2(1+ν)

(Eν

(1+ν)(1−2ν) +E

2(1+ν)

)

Eν(1+ν)(1−2ν) +2 E

2(1+ν)

=2E(2Eν +E(1−2ν))

(1+ν)(2Eν +2E(1−2ν))=

E(1+ν)(1−ν)

=E

1−ν2 . (2.85)

Executing these computations for all matrix entries leads to the isotropic material matrices forshells given by

Ciso =

E1−ν2

Eν1−ν2 0

Eν1−ν2

E1−ν2 0

0 0 E1+ν

, Diso =

( 2E1+ν 00 2E

1+ν

)

. (2.86)

These material matrices can be written as tensors, hence theisotropic material tensors for shellsbecome

Cαβλ µiso =

E2(1+ν)

(

aαλ aβ µ +aαµaβλ +2ν

1−νaαβ aλ µ

)

, (2.87)

Dαλiso =

2E(1+ν)

aαλ. (2.88)

Moreover, by introducing the two-dimensional Lame parameters

λ =Eν

(1+ν)(1−ν), µ =

E2(1+ν)

(2.89)

the isotropic elasticity matrices for shells can be reformed into a structure of high resemblanceto equation (2.83) as shown below

Ciso =

λ +2µ λ 0λ λ +2µ 00 0 2µ

, Diso =

(4µ 00 4µ

)

. (2.90)

21

CHAPTER 3

Free Material Optimization for Shells

In the previous chapter we presented a mathematical formalism suited for a thorough descriptionof the physical behavior of thin shells. Based on these models we will proceed with the core topicof this thesis: the material optimization of shells. To thisend we consider the entire elasticitytensor as an open variable and try to find not only the ultimately best material distribution, butalso the best local material properties. This approach known asFree Material Optimizationhas been introduced 1994 by Bendsøe et al. in [BGH+94] and is regarded as the most generalmethod possible for the optimization of linear elastic media.

In this chapter we develop a formalism for Free Material Optimization for shells based onthe originalminimum compliance formulationfor solids already presented in [BGH+94]. Inaddition we adapt several other problem formulations, which are very valuable as the obtainedproblem structure can be exploited by specific algorithms available to solve the semidefiniteprograms (SDP) arising from Free Material Optimization after a suitable discretization. On theone hand this concerns theprimal formulationderived from the saddle-point problem as givenin [Wer01], that results in an optimization problem with quadratic objective and bilinear matrixinequality constraints solvable e.g. by PENBMI [KS03]. On the other hand theminimum weightformulation leads to a nonlinear convex semidefinite programming problem treatable by thenonlinear SDP code PENSCP [SKL09b]. Moreover, we derive an extended problem formulationfor multiple load cases as introduced by [BKNZ99] as well as asimplified problem formulationfor Free Material Optimization of isotropic materials.

First we have to decide which elastic shell model from the ones introduced in Chapter 2 isutilized for the deduction of these problem formulations. Since Naghdi’s shell model providesnot only more information about the optimal material (because it contains the transverse shearpart D of the elasticity tensor), but also covers a wider range of shells due to its validity for agreater thickness, it is the model of choice for the main partof this thesis. However we close thechapter with a problem formulation of Free Material Optimization for Kirchhoff-Love shells forcompleteness.

23

3. Free Material Optimization for Shells

3.1. The Elasticity Tensor as a Design Variable

We consider one of the fundamental problems of structural optimization: For a given set ofboundary conditions and a predefined set of loads, find the stiffest structure constructed from alimited amount of material that is able to carry the loads. Tothis end we search for the optimalmaterial to build up the contemplated structure. This problem has been addressed many timesand in various ways since there is no unique definition of “optimal material” – the solution ofthe problem depends entirely on the chosen set of admissiblematerials.

Well-known and successfully applied methods include the SIMP approach, which is basedon the density variation of a fixed isotropic material. This technique aims for 0-1-designs byusing a penalization strategy [Ben89, Mle92, RZB92] and hasbecome extremely favored due toits straightforward realization [Sig01]. Another procedure, that can be considered as one of thefoundations of structural optimization, is homogenization, where a modification of the elasticproperties of a structure is achieved by an optimization of the microstructure layout within thematerial [BK88, DB92]. A different approach, that also enjoys great popularity, is the design andoptimization of laminates, which make a wide range of material properties available by mod-ifying material, direction and thickness of the individuallayers [GHH99, KLR09]. Moreover,in recent years another very promising method called Discrete Material Optimization has beenintroduced, in which the admissible set consists of a finite number of different material tensors.The optimal material at each spot is then chosen via an extension of the SIMP weighting functionto a parametrization suited to multiple material optimization [SL05].

In this thesis we focus on the Free Material Optimization approach, which takes only the basicrequirements of a linear elastic continuum into account [BGH+94]. These comprise the sym-metry of the elasticity matrices described via the matricesC andD as introduced in Subsection2.4.1, which can be deduced from the symmetry properties of the strain and stress tensors

C=C⊤, D = D⊤

, (3.1)

and the positivity of the material’s elastic modulus that holds irrespective from the consideredstresses and strains

C 0 , D 0. (3.2)

No other constraints are imposed on the material or its symmetry class. From the point ofadmissible materials Free Material Optimization can be regarded as the most general materialoptimization method, as it simultaneously provides the ultimately best material distribution andlocal material properties.

In order to develop a consistent Free Material Optimizationformalism for shells we need tostart with the definition of the set of admissible materialsC . To admit holes and material-no-material situations the elasticity matrices are chosen from C∈ [L∞(ω)]3×3 andD ∈ [L∞(ω)]2×2.Therefore in its most general form the admissible set is given by

C :=

(C,D) ∈ [L∞(ω)]3×3× [L∞(ω)]2×2∣∣∣C=C⊤ 0, D = D⊤ 0

. (3.3)

Moreover, we require measures for the stiffness and weight of the shell structure. An establishedmeasure for stiffness is thecompliance, which describes how much the structure will deform

24

3.1. The Elasticity Tensor as a Design Variable

under a given set of loadsf , gu andgθ . It is defined as twice the negative potential energy instatic equilibrium [BS02] and hence is given by

comp(C,D) =

∫

ωt f⊤udS +

∫

∂ω1

(

g⊤u u+g⊤θ θ)

dl (3.4)

= max(u,θ )∈U

−∫

ω

(

tγ⊤Cγ +t3

12χ⊤Cχ + tζ⊤Dζ

)

dS

+2∫

ωt f⊤udS +2

∫

∂ω1

(

g⊤u u+g⊤θ θ)

dl (3.5)

= max(u,θ )∈U

−2ΠC,D(u,θ) =− min(u,θ )∈U

2ΠC,D(u,θ) .

Since the compliance quantifies how much a structure will yield to the applied forces, a mini-mization of the compliance with respect to the design variables will lead to the stiffest structurepossible.

In addition, we need to identify a measure for the amount of distributed material. Usuallythe weight of the structure is chosen for this task, which canbe obtained by integration of thematerial density over the entire structure. However, a specific material density is typically de-termined in an experimental measurement and can not be calculated from values given by thematerial tensor. As the optimal elasticity tensors provided by Free Material Optimization arenot limited to materials already existing in nature, there is no density information available forthem. Experimental data from laminates and composites suggest that the analytical relation be-tween elasticity tensor and density contains discontinuities and might not even fulfill the basicprerequisites for a well-defined mathematical function (inthe sense that two different compos-ites might have the same elasticity tensors, but differing densities). Hence, we require anothermeasure for the amount of used material and employ an invariant of the elasticity tensor for thispurpose. While in the original publication [BGH+94] this tensor invariant is kept general andearly publications about the topic contain different possibilities for the choice of this invariant[ZKB97], nowadays the trace of the tensor has proven to be an efficient option due to is rela-tively simple functional dependence which eases the structure of the arising numerical problems.The fact that the optimal material as proposed by Free Material Optimization does not alreadyexist in nature and that the measure provided by the trace might differ greatly from its densityis sometimes considered as a drawback of this method. Nevertheless it is a fundamental ideaof Free Material Optimization to overcome the boundaries oftraditional materials as well aslaminate manufacturing standards and to motivate the construction of entirely new materials bypointing out the high potential provided by advanced materials. All Free Material Optimizationresults should be understood from this point of view.

Our next step is the alignment of a trace formula that combines the elasticity tensorsC andDfor shells. We have to accomplish this consistently to the trace of the three-dimensional elasticitytensorE. This is extremely important because we want to solve problems containing solid aswell as shell elements in Section 4.4.1 and the results can only be meaningful if we use similarcosts for the spent material resources. To this end let us recapitulate the trace of the three-dimensional elasticity tensor for solids, which is identical to the trace of the elasticity matrixE

25


due to the chosen Mandel matrix notation

tr(

Ei jkl)

=3

∑i, j=1

Ei ji j = E1111+E2222+E3333+2E1212+2E1313+2E2323

=6

∑i=1

Eii = tr(E) . (3.6)

The traces of the shell elasticity tensors coincide also with the traces of the corresponding elas-ticity matricesC andD, which we convert to entries of the elasticity matrixE for comparativepurposes by using the results obtained in Subsection 2.6.2,

tr(

Cαβλ µ)

= tr(C) = C1111+C2222+2C1212 (3.7)

2.80= E1111+E2222+2E1212− E11332+E22332+2E12332

E3333 (3.8)

tr(

Dαλ)

= tr(D) = D11+D22 2.81= 4E1313+4E2323

. (3.9)

When we compare the equations (3.6) and (3.9) it is obvious that we have to scale tr(D) by afactor of 1

2, if we like to have a measure combining the traces ofC andD, that is conform withthe trace ofE. Moreover, note that the combined shell trace is a surface measure, while the traceof the three-dimensional elasticity matrixE is a volume measure. Thus the combined shell tracehas to be multiplied by a factort reflecting a preceding integration over the shell’s thicknessdirection, for which the material distribution is assumed to be constant in this direction. Insummary we measure the amount of material used at a certain spot x∈ ω by

t tr(C(x))+12

t tr(D(x)) (3.10)

and the amount of material used for the entire structure by

vol(C,D) :=∫

ωt

(

tr(C(x))+12

tr(D(x))

)

dS. (3.11)

In addition, we can reconstruct the trace of the three-dimensional elasticity matrixE for solidsfrom the combined shell trace by

tr(E) = tr(C)+12

tr(D)+E11332+E22332+E33332+2E12332

E3333 (3.12)

and vice versa

t

(

tr(C)+12

tr(D)

)

= t

(

tr(E)− E11332+E22332+E33332+2E12332

E3333

)

. (3.13)

Note that during the conversion of the shell elasticity matricesC andD to the solid elasticity ma-trix E according to formula (3.12) four matrix entries can be arbitrarily chosen. It is tempting to

26

3.2. Minimum Compliance Problem Formulation

set these entries to zero leading to equivalence of the three-dimensional trace and the combinedshell trace, however one needs to be careful as this would result in materials, that do not opposenormal forces at all.

For some of the presented optimization problems it is beneficial to add the constraints on thetraces of the elasticity tensors directly to the admissiblesetC . On the one hand this concernsthe volume constraint

∫

ωt

(

tr(C)+12

tr(D)

)

dS≤V , (3.14)

because only limited material resources are available for the structure. On the other hand nu-merical experience has shown that the inclusion of box constraints

0≤ ρ− ≤ t tr(C(x))+12

t tr(D(x))≤ ρ+ a.e. inω , (3.15)

to avoid arbitrarily high material concentrations at single points brings forth much more realisticand useful designs. Finally we combine the original admissible set (3.3) with the constraints(3.14) and (3.15) to obtain another version of the set of admissible elasticity tensors

C :=

(C,D) ∈ [L∞(ω)]3×3× [L∞(ω)]2×2

∣∣∣∣∣∣∣∣

C =C⊤ 0D = D⊤ 0∫

ω t(tr(C(x))+ 1

2tr(D(x)))

dS≤V0≤ ρ− ≤ t tr(C(x))+ 1

2 t tr(D(x))≤ ρ+

which is obviously a subset ofC . For simplicity of notation we will assumeρ− = 0. But notethat all statements presented in this thesis are also true for positiveρ−. Moreover, we will oftenencounter situations that require a sensible choice of the constantsV, ρ+ andρ−. In general weassume that

ρ−|ω | ≤V ≤ ρ+|ω | , (3.16)

but additionally the constants have to be chosen such that the set of admissible material tensors isnon-empty and conditions like for example (3.22) and (3.75)are fulfilled. As the box constraintshave no real physical motivation and are often used to adjustthe optimal design to the specificneeds of an application, they can always be chosen to meet theabove mentioned criteria.


The first problem formulation considered by us is the minimumcompliance problem formu-lation, which has already been established for solids in thefirst paper introducing Free Mate-rial Optimization [BGH+94]. It has been the prevalent problem formulation for many years[ZKB97, BKNZ99, Wer01, KZ02, Mac04, GLS09], thus it will be the first problem formulationthat we extend to the shell case. As the name implies the compliance (3.4) is taken as objectivefunction and minimized to obtain the stiffest design for thestructure. Besides the equilibriumcondition (2.52) an additional volume constraint (3.14) limiting the available material resourcesis added to the problem formulation. Without this constraint the stiffest solution would be ob-tained by filling the entire design domain with material hence leading to unusable and costly

27


designs. The problem formulation is completed by box constraints (3.15) in order to avoid arbi-trarily high material concentrations at single points

min(C,D)∈C(u,θ )∈U

∫

ωt f⊤udS+

∫

∂ω1

gu⊤u+gθ

⊤θ dl (PMC)

subject to∫

ωtγ⊤(u)Cγ(v)+

t3

12χ⊤(u,θ)Cχ(v,η)+ tkζ⊤(u,θ)Dζ (v,η)dS

=∫

ωt f⊤vdS+

∫

∂ω1

gu⊤v+gθ

⊤η dl ∀(v,η) ∈ U

∫

ωt

(

tr(C)+12

tr(D)

)

dS≤V

ρ− ≤ t tr(C(x))+12

t tr(D(x))≤ ρ+ a.e. inω

This problem formulation (PMC) can be simplified by hiding the constraints on the combinedshell tracet tr(C)+ 1

2 t tr(D) in the admissible setC . Moreover, the equilibrium condition can beincluded in the objective function by using the alternativeexpression (3.5) for the compliance.However, the price for the absence of the equilibrium condition is a more sophisticated problemformulation known as thesaddle point problem

min(C,D)∈C

comp(C,D)(3.5)=

= min(C,D)∈C

max(u,θ )∈U

−∫

ωtγ⊤(u)Cγ(u)+

t3

12χ⊤(u,θ)Cχ(u,θ)+ tkζ⊤(u,θ)Dζ (u,θ)dS

+2∫

ωt f⊤udS+2

∫

∂ω1

gu⊤u+gθ

⊤θ dl . (3.17)

By introducing the saddle point function1

JD((C,D),(u,θ)) :=− 12

∫

ωtγ⊤(u)Cγ(u)+

t3

12χ⊤(u,θ)Cχ(u,θ)+ tkζ⊤(u,θ)Dζ (u,θ)dS

+∫

ωt f⊤udS+

∫

∂ω1

gu⊤u+gθ

⊤θ dl (3.18)

optimization problem (3.17) can be equivalently formulated as

min(C,D)∈C

max(u,θ )∈U

JD((C,D),(u,θ)) . (PD)

Strictly speaking problem (PD) is not the minimum compliance problem anymore, as the saddlepoint functionJD((C,D),(u,θ)) is identical to the compliance divided by two. However, bothproblems are equivalent and yield the same optimal points, therefore we will ignore this factor of

1The subscript D of the saddle point functionJD and the problem formulation (PD) refers todual, as problem(PD) will be identified as the dual of a non-linear semidefinite program (PP) in the following section. Accordingly thesubscript P introduced for problem (PP) indicates theprimal problem formulation.

28


two in many problem formulations to obtain saddle point functions that are equal to the negativepotential energy of the shell. While problem (PD) is not suited for a numerical approach, itstill enjoys great popularity within publications concerning Free Material Optimization. Thereason for this is its great suitability for theoretical analysis such as the duality results presentedin Section 3.3, but also the existence of solutions of the Free Material Optimization problemaddressed in the next subsection.

3.2.1. Existence of Solutions

We now want to show existence of at least one optimal solutionfor the single load prob-lem for shells (PD). In this case the optimal solution equals a saddle point of the functionalJD((C,D),(u,θ)). Theory and existence of saddle points are covered by the Minimax-Theorem,a well-known result from convex analysis [ET76]. However, the standard formulation of theMinimax-Theorem requires both optimization variables to stem from reflexive Banach spaces.This condition is obviously violated by the setC , asL∞ is not a reflexive space. The remedy forthis problem is provided by [Mac04], where a modified Minimax-Theorem is introduced, thatwill enable us to prove the existence of an optimal point for problem (PD).

Theorem 3.2.1. Problem (PD) has an optimal solution((C∗,D∗),(u∗,θ∗)) ∈ C ×U .

Proof. C ⊆ L∞(ω ;S3)× L∞(ω ;S2) is a convex, non-empty set for appropriately chosen con-stantsV, ρ+ andρ−. As it lies in a norm ball ofL∞(ω ;S3)×L∞(ω ;S2), it is weak*–compactand thus closed and bounded in the weak*–topology.U on the other hand is convex, closedand non-empty in the

[H1(ω)

]5–topology. According to the Minimax-Theorem of [Mac04] it is

now sufficient to show

(i) for all (C,D) ∈ C : (u,θ) 7→ JD((C,D),(u,θ)) is concave and continuous,

(ii) for all (u,θ) ∈ U : (C,D) 7→ JD((C,D),(u,θ)) is convex and continuous,

(iii) there exist(C0,D0) ∈ C such that

JD((C0,D0),(u,θ)) −→−∞ ∀(u,θ) ∈ U with ‖(u,θ)‖H1(ω) −→ ∞

in order to prove the existence of at least one saddle point((C∗,D∗),(u∗,θ∗)) ∈ C ×U of JD.Note that the original form of the theorem as formulated in [Mac04] can be obtained by

multiplying JD((C,D),(u,θ)) with −1. Using the Schwarz inequality we get

∣∣JD((C,D),(u,θ))

∣∣ =

∣∣∣∣−1

2

∫

ωtγ⊤Cγ +

t3

12χ⊤Cχ + tkζ⊤Dζ dS

+

∫

ωt f⊤udS+

∫

∂ω1

gu⊤u+gθ

⊤θ dl

∣∣∣∣

≤∣∣∣∣

12

∫

ωtγ⊤Cγ dS

∣∣∣∣+

∣∣∣∣

12

∫

ω

t3

12χ⊤Cχ dS

∣∣∣∣+

∣∣∣∣

12

∫

ωtkζ⊤Dζ dS

∣∣∣∣

+

∣∣∣∣

∫

ωt f⊤udS

∣∣∣∣+

∣∣∣∣

∫

∂ω1

gu⊤udl

∣∣∣∣+

∣∣∣∣

∫

∂ω1

gθ⊤θ dl

∣∣∣∣

29


∣∣JD((C,D),(u,θ))

∣∣ ≤ t

2‖C‖L∞(ω)‖γ‖2

L2(ω)+t3

24‖C‖L∞(ω)‖χ‖2

L2(ω)

+t2

k‖D‖L∞(ω)‖ζ‖2L2(ω)+ t‖ f‖L2(ω)‖u‖H1(ω)

+‖gu‖L2(∂ω1)‖u‖H1(∂ω1)+‖gθ‖L2(∂ω1)‖θ‖H1(∂ω1) .

As φ ∈W2,∞(ω) we haveaαβ ∈W1,∞(ω) and thusΓαµλ ∈ L∞(ω) as well asbαβ ∈ L∞(ω). This

implies

‖γ‖2L2(ω) = ‖γ11‖2

L2(ω)+‖γ22‖2L2(ω)+2‖γ12‖2

L2(ω)

= ‖u1|1−b11u3‖2L2(ω)+‖u2|2−b22u3‖2

L2(ω)+2

∥∥∥∥

12

u1|2+12

u2|1−b12u3

∥∥∥∥

2

L2(ω)

= ‖u1,1−Γ111u1−Γ2

11u2−b11u3‖2L2(ω)+‖u2,2−Γ1

22u1−Γ222u2−b22u3‖2

L2(ω)

+12

∥∥u1,2−Γ1

12u1−Γ212u2+u2,1−Γ1

21u1−Γ221u2−2b12u3

∥∥

2L2(ω)

≤ ‖u1,1‖L2(ω)+‖Γ111‖L∞(ω)‖u1‖L2(ω)+‖Γ2

11‖L∞(ω)‖u2‖L2(ω)

+‖b11‖L∞(ω)‖u3‖L2(ω)+ . . .

≤ cγ‖u‖H1(ω) .

with cγ > 0. Likewise we can conclude‖χ‖2L2(ω)

≤ cχ‖u‖H1(ω) and‖ζ‖2L2(ω)

≤ cζ‖u‖H1(ω).

Together with the continuity of the trace operator inH1(ω)

‖(u,θ)‖H1(∂ω1) ≤ c1‖(u,θ)‖H1(ω)

this leads to

∣∣JD((C,D),(u,θ))

∣∣ ≤ t

2‖C‖L∞(ω)cγ‖u‖H1(ω)+

t3

24‖C‖L∞(ω)cχ‖(u,θ)‖H1(ω)

+t2

k‖D‖L∞(ω)cζ‖(u,θ)‖H1(ω)+ t‖ f‖L2(ω)‖u‖H1(ω)

+‖gu‖L2(∂ω1)‖u‖H1(ω)+‖gθ‖L2(∂ω1)‖θ‖H1(ω) .

This yields the continuity required in (i) and (ii). As the linearity of the mapping(C,D) 7→JD((C,D),(u,θ)) implies convexity, (ii) is valid. The concavity required in(i) follows fromthe linear dependence of the strains on the displacements, the quadratic dependence ofJD onthe strains and from the positive semidefiniteness ofC and D. Finally let us show (iii): theoriginal publications on the ellipticity of Naghdi’s shellmodel for isotropic material [BCM94]and for nonhomogeneous anisotropic material [FL98] are based on the assumptionφ ∈C3(ω).In [BL01] it was proven that this assumption can be weakened to φ ∈ W2,∞(ω) in the case ofisotropic materials. Consequently it suffices to choose isotropic matrices

C0 =

2µ + 2λ µλ+2µ

2λ µλ+2µ 0

2λ µλ+2µ 2µ + 2λ µ

λ+2µ 00 0 2µ

and D0 =

(4µ 00 4µ

)

(3.19)

30

3.3. Primal Minimum Compliance Problem Formulation

fulfilling the constraints on the traces given in (3.14) and (3.15) in order to show that (iii) isvalid. The summed trace is given by

t tr(C)+12

t tr(D) = t

(

10µ +4λ µ

λ +2µ

)

= t

(10λ µ +20µ2+4λ µ

λ +2µ

)

λ∈R+µ∈R++

< t

(14λ µ +28µ2

λ +2µ

)

= 14tµ (3.20)

and a lower bound on the trace is

t tr(C)+12

t tr(D) = t

(

10µ +4λ µ

λ +2µ

)

= t

(

14µ − 8µ2

λ +2µ

)

λ∈R++

> t

(

14µ − 8µ2

2µ

)

= t (14µ −4µ) = 10tµ . (3.21)

Hence a possible choice is

λ ∈R++ ,ρ−

10t≤ µ ≤ min

V

14t|ω | ,ρ+

14t

. (3.22)

This proves the existence of a saddle point of the functionalJD((C,D),(u,θ)).

3.2.2. Free Material Optimization for Reissner-Mindlin Pl ates

The minimum potential energy problem (2.55) represents thestarting point for the deductionof the Free Material Optimization problem for Reissner-Mindlin plates. A partition into themembrane and the Reissner-Mindlin problem is not possible,as both problems are coupled bythe elasticity matricC, which is used as the design variable in Free Material Optimization. Dueto the similarity of the equilibrium conditions for Reissner-Mindlin plates (2.55) and for Naghdishells (2.51) the compliance and the combined trace for Reissner-Mindlin plates are identicalto the introduced quantities for shells presented in (3.4) and (3.10). Hence, all optimizationproblems established in this thesis remain also valid for Reissner-Mindlin plates, though theyare based on the simplified formulas for the plate strains according to (2.54). Note that theminimum compliance problem formulation for Free Material Optimization for Reissner-Mindlinplates was already given in [BD93], where a solution is obtained by analytic derivation of theoptimal material properties.


Although the introduced minimum compliance problem formulation (PD) is easily deductedfrom physical principles and is furthermore well suited to the application of saddle point theo-rems, it is a rather ineligible problem formulation with respect to a numerical solution. Whenfaced with such difficult problem structures it is a common approach to check whether the struc-ture of the dual problem is more convenient for computational purposes. However, as the op-timization variablesC andD lie in a subspace ofL∞, which is a non-reflexive Banach space,

31


the dual variables stem from a measure space [DS58]. Workingwith these measures is a so-phisticated task, that has been discussed with respect to applications in optimal control theory[Tro05]. We will instead employ an idea, that has already been successfully used for the the-oretical analysis of the Free Material Optimization problem for solids: we assume that (PD) isthe dual problem formulation and seek for the associated primal problem. We benefit from thisapproach, asL∞ is regarded as the space of the dual variables – the associated primal variablesare thus elements ofL1, which is much more convenient to work with compared to a measurespace. Based on this concept a duality statement for the FreeMaterial Optimization problem forsolid material has been proven in [Wer01], i.e., the saddle point problem is equivalent to a linearquadratically constrained optimization problem.

Within this section we show that a similar technique can be applied to the Free Material Opti-mization problem for shells resulting in a convex nonlinearsemidefinite program instead of thesaddle-point problem (PD), which is numerically tractable by non-linear SDP codes asfor exam-ple PENBMI [KS03]. Analogous to [Wer01] the strategy for theproof is an initial well-chosenguess of the primal optimization problem and its Lagrange multipliers followed by a verificationof the required properties. However, the proof given in [Wer01] profits from the equivalenceof the Free Material Optimization problem for solids and theVariable Thickness Sheet problem[BGH+94], hence only a scalar variable has to be considered. In thepresent case this is not pos-sible due to the different strains appearing in shells, thusthe following proof is entirely basedon situations with multiple linearly independent strains,where a reduction to scalar variablesis not possible. As a result the proof of Theorem 3.3.1 is alsoapplicable to the multiple loadcase for solids as well as for shells, where the duality properties of the primal Free MaterialOptimization problem are only proven for the finite-dimensional optimization problem obtainedafter a Finite Element discretization [BKNZ99]. Accordingly the following proof closes the gapto the infinite-dimensional Free Material Optimization problem with multiple load cases besidesthe verification of the primal problem formulation for shells.

Theorem 3.3.1. Problem (PD) is equivalent to the Lagrange dual problem of

max(u,θ )∈U

α∈R+

βu,l∈L1(ω)βu,l≥0

∫

ωt f⊤udS+

∫

∂ω1

(gu⊤u+gθ

⊤θ)dl −αV −ρ+∫

ωβudS

subject tot2

γ(u)γ(u)⊤+t3

24χ(u,θ)χ(u,θ)⊤ − t(α +βu−βl)13 0 a. e. inω (PP)

t2

kζ (u,θ)ζ (u,θ)⊤ − t2(α +βu−βl)12 0 a. e. inω

where1n denotes the unit matrix inRn.

Before we continue with the proof of Theorem 3.3.1, let us point out some properties andadvantages of the primal problem formulation (PP). The initial guess of problem (PP) is derivedby calculating the dual from (PD), while ignoring all difficulties due to non-reflexive spaces.As the displacement(u,θ) belongs to a subset ofH1(ω), it follows that γ(u,θ) ∈ [L2(ω)]3

and thus we obtainγγ⊤ ∈ L1(ω ,S). The same statement holds for the other dyadic matrices

32


composed of the strain vectorsχ andζ . Hence, for anα , βu andβl chosen fromL1(ω), the entiresemidefinite matrix constraints possess the same regularity. Moreover,βu andβl are regardedas the Lagrange multipliers for the upper and the lower box constraint on the combined shelltrace, respectively, whileα is perceived as the multiplier for the volume constraint. Asall theseconstraints are inequality constraints, this motivates the positivity of the introduced variablesleading toβu ∈ L1(ω), βu ≥ 0 andβl ∈ L1(ω), βl ≥ 0. As the volume constraint is a scalarvalued constraint,α itself is a positive scalarα ∈ R+ giving rise to the problem formulationpresented in Theorem 3.3.1.

The advantages of problem (PP) compared to the original problem formulation (PD) are es-pecially relevant for the numerical solution of the Free Material Optimization problem. On theone hand the obtained problem formulation (PP) is preferred over the saddle-point problem (PD)as the material matricesC andD – whose discrete counterparts highly increase the number ofoptimization variables in the optimization problem – are hidden in the problem formulation asLagrange multipliers of the matrix inequality constraints. Hence the number of variables re-quired for the discrete version of (PP) is significantly lower than the number of variables neededfor the solution of the discretized saddle point problem (PD). More important, problem (PP) is aconvex optimization problem, more precisely a convex nonlinear semidefinite program (SDP).Hence any solution of problem (PP) fulfilling the local optimality conditions is also a globaloptimal solution to that problem. In addition, there exist numerical codes specialized on theefficient solution of SDP problems, which can be employed to obtain discrete solutions, e.g.the non-linear SDP code PENNON [KS03]. After this summary onthe relevance of problemformulation (PP) we continue with the proof of Theorem 3.3.1.

Proof of Theorem 3.3.1.We start with constructing the Lagrange function of problem(PP)

L((u,θ),(C,D),α ,β ) =∫

ωt f⊤udS+

∫

∂ω1

(gu⊤u+gθ

⊤θ)dl

−αV −ρ+∫

ωβudS (3.23)

−∫

ω〈C, t

2γγ⊤+

t3

24χχ⊤− tα13− tβu13+ tβl13〉dS

−∫

ω〈D,

t2

kζζ⊤− t2

α12−t2

βu12+t2

βl12〉dS.

The Lagrange multipliers for the symmetric semidefinite matrix constraints have to be symmetricpositive semidefinite matrices. As discussed above the matrix constraints exhibitL1-regularity,therefore the Lagrange multipliers are chosen from the dualspace ofL1, which leads toC ∈L∞(ω ,S3) andD ∈ L∞(ω ,S2) and allows us to identifyC andD with the material matrices ofthe original problem formulation (PD). The inner matrix product used for the construction of theLagrange function〈 · , · 〉 is defined in (A.3) and has to be understood as a scalar productof thetype 〈 · , · 〉L∞,L1. Finally βu andβl are united in the vectorβ = (βu,βl )

⊤ ∈ [L1(ω)]2 to allow amore concise and clear notation. We now sort the terms in the Lagrangian and use a property ofthe inner matrix product (A.1) to obtain

33


L((u,θ),(C,D),α ,β ) =−∫

ω〈C, t

2γγ⊤〉dS−

∫

ω〈C, t3

24χχ⊤〉dS

−∫

ω〈D,

t2

kζζ⊤〉dS+∫

ωt f⊤udS+

∫

∂ω1

(gu⊤u+gθ

⊤θ)dl

+α∫

ωt tr(C)+

t2

tr(D)dS−αV (3.24)

+∫

ωβu

(

t tr(C)+t2

tr(D)−ρ+)

dS

−∫

ωβl

(

t tr(C)+t2

tr(D))

dS.

Next we define the saddle point functionJL((C,D),(u,θ ,α ,β )), which is equal to the Lagrangefunction L((u,θ),(C,D),α ,β ) of the observed problem after utilization of another property ofthe inner matrix product (A.2)

JL((C,D),(u,θ ,α ,β )) := −12

∫

ωtγ⊤Cγ +

t3

12χ⊤Cχ + tkζ⊤Dζ dS+

∫

ωt f⊤udS

+

∫

∂ω1

gu⊤u+gθ

⊤θ dl +α∫

ωt trC+

t2

trDdS−αV (3.25)

+∫

ωβu

(

t trC+t2

trD−ρ+)

dS−∫

ωβl

(

t trC+t2

trD)

dS

= L((u,θ),(C,D),α ,β ) .

This leads to a saddle point problem based on the Lagrange function

sup(u,θ )∈U

α∈R+


inf(C,D)∈L∞

C=C⊤0D=D⊤0

JL((C,D),(u,θ ,α ,β )) . (PL)

It now remains to show that problem (PL) is equivalent to problem (PD) in the sense thateach saddle-point((C∗,D∗),(u∗,θ∗)) of (PD) can be extended byα ∈ R+ andβu,l ∈ L1(ω) toa saddle-point((C∗,D∗),(u∗,θ∗,α∗,βl ,u

∗)) of (PL) and that on the other hand each saddle-point((C∗,D∗),(u∗,θ∗,α∗,βl ,u

∗)) of (PL) defines a saddle-point((C∗,D∗),(u∗,θ∗)) of (PD). When wehave verified the validity of the saddle-point properties for a point((C∗,D∗),(u∗,θ∗,α∗,βl ,u

∗)),we are not only able to replace the sup inf in problem (PL) by maxmin , but we are also allowedto change the order to minmax [ET76]. The existence of a saddle-point implies also a vanish-ing duality gap, which guarantees that the solutions of the dual problem (PL) and the primalproblem (PP) are equivalent. The following proof extends the ideas of [Wer01, Theorem 3.3.1],from scalar valued constraints to matrix valued constraints. We begin with a characterization ofsaddle-points by using the following Lemma, whose proof is given in appendix A.1:

Lemma 3.3.2. We consider the saddle-point problem

minp∈A

maxu∈B

J(p,u) (3.26)

34


whereA = C , B = U and J= JD. Then(p∗,u∗) ∈ A ×B is a saddle-point of J if and only if

δpJ(p∗,u∗)(p− p∗) ≥ 0 ∀ p∈ A ,

δuJ(p∗,u∗)(u−u∗) ≤ 0 ∀u∈ B . (3.27)

An analogous result is true forA = C , B = U ×L and J= JL.

Here we have used the following definition

L := (α ,β ) ∈ R+× [L1(ω)]2 | β ≥ 0 . (3.28)

Moreover recall the following sets

C :=

(C,D) ∈ L∞(ω ;S3)×L∞(ω ;S2)

∣∣∣∣∣∣∣∣

C 0D 0∫

ω t(tr(C(x))+ 1

2tr(D(x)))

dS≤V0≤ ρ− ≤ t tr(C(x))+ 1

2 t tr(D(x)) ≤ ρ+

C :=(C,D) ∈ L∞(ω ;S3)×L∞(ω ;S2) |C 0 , D 0 . (3.29)

For the given saddle-point functionalsJD((C,D),(u,θ)) of problem (PD) as defined in (3.18)and andJL((C,D),(u,θ ,α ,β )) of problem (PL) as given in (3.25) the Gateaux derivatives canbe computed as

δC,DJD((C,D),(u,θ))(δC,δD) = −12

∫

ωtγ⊤δCγ +

t3

12χ⊤δCχ

+ tKζ⊤δDζ dS, (3.30)

δu,θ JD((C,D),(u,θ))(δu,δθ) = −∫

ωtγ⊤(u)Cγ(δu)+

t3

12χ⊤(u,θ)Cχ(δu,δθ)

+ tKζ⊤(u,θ)Dζ (δu,δθ)dS+∫

ωt f⊤δudS

+

∫

∂ω1

gu⊤δu+gθ

⊤δθ dl , (3.31)

δC,DJL((C,D),(u,θ ,α ,β ))(δC,δD) = −12

∫

ωtγ⊤δCγ +

t3

12χ⊤δCχ + tKζ⊤δDζ dS

+α∫

ωt trδC+

t2

trδDdS

+

∫

ωβu

(

t trδC+t2

trδD)

dS

−∫

ωβl

(

t trδC+t2

trδD)

dS (3.32)

35


and

δu,θ ,α ,β JL((C,D),(u,θ ,α ,β ))(δu,δθ ,δα ,δβ ) =

−∫

ωtγ⊤(u)Cγ(δu)+

t3

12χ⊤(u,θ)Cχ(δu,δθ)+ tKζ⊤(u,θ)Dζ (δu,δθ)dS

+∫

ωt f⊤δudS+

∫

∂ω1

gu⊤δu+gθ

⊤δθ dl +δα(∫

ωt trC+

t2

trDdS−V

)

+∫

ωδβu

(

t trC+t2

trD−ρ+)

dS−∫

ωδβl

(

t trC+t2

trD)

dS, (3.33)

respectively.Thus the saddle-point conditions for problem (PD) can be written as

δC,DJD((C∗,D∗),(u∗,θ∗))(C−C∗

,D−D∗) = (D1)

= −12

∫

ωtγ⊤(C−C∗)γ +

t3

12χ⊤(C−C∗)χ + tKζ⊤(D−D∗)ζ dS≥ 0 ∀(C,D) ∈ C ,

δu,θ JD((C∗,D∗),(u∗,θ∗))(u−u∗,θ −θ∗) = (D2)

= −∫

ωtγ⊤(u∗)C∗γ(u−u∗)+

t3

12χ⊤(u∗,θ∗)C∗χ(u−u∗,θ −θ∗)

+ tKζ⊤(u∗,θ∗)D∗ζ (u−u∗,θ −θ∗)dS

+

∫

ωt f⊤(u−u∗)dS+

∫

∂ω1

gu⊤(u−u∗)+gθ

⊤(θ −θ∗)dl ≤ 0 ∀(u,θ) ∈ U

and the saddle-point conditions for problem (PL) can be formulated as

δC,DJL((C∗,D∗),(u∗,θ∗

,α∗,β ∗))(C−C∗

,D−D∗) = (L1)

= −12

∫

ωtγ⊤(C−C∗)γ +

t3

12χ⊤(C−C∗)χ + tKζ⊤(D−D∗)ζ dS

+α∗∫

ωt tr(C−C∗)+

t2

tr(D−D∗)dS

+

∫

ωβu

∗(

t tr(C−C∗)+t2

tr(D−D∗))

dS

−∫

ωβl

∗(

t tr(C−C∗)+t2

tr(D−D∗))

dS≥ 0 ∀(C,D) ∈ C

δu,θ ,α ,β JL((C∗,D∗),(u∗,θ∗

,α∗,β ∗))(u−u∗,θ −θ∗

,α −α∗,β −β ∗) = (L2)

−∫

ωtγ⊤(u∗)C∗γ(u−u∗)+

t3

12χ⊤(u∗,θ∗)C∗χ(u−u∗,θ −θ∗)

+ tKζ⊤(u∗,θ∗)D∗ζ (u−u∗,θ −θ∗)dS+∫

ωt f⊤(u−u∗)dS

+

∫

∂ω1

gu⊤(u−u∗)+gθ

⊤(θ −θ∗)dl +(α −α∗)

(∫

ωt trC∗+

t2

trD∗dS−V

)

+

∫

ω(βu−βu

∗)(

t trC∗+t2

trD∗−ρ+)

dS−∫

ω(βl −βl

∗)(

t trC∗+t2

trD∗)

dS

≤ 0 ∀(u,θ ,α ,β ) ∈ U ×L .

36


Now we show, that (L1) and (L2) imply (D1) and (D2), thus a saddle-point ((C∗,D∗),(u∗,θ∗))of (PD) is defined by each saddle-point((C∗,D∗),(u∗,θ∗,α∗,β ∗)) of problem (PL):

Settingα = α∗ and additionallyβ = (βu,βl )⊤ = (βu

∗,βl

∗)⊤ = β ∗ ∈ [L1(ω)]2 in (L2) yields(D2). Furthermore the positivity ofα and β together with the existence of the saddle-point((C∗,D∗),(u∗,θ∗,α∗,β ∗)) (and thus existence of a finite solution) implies

∫

ωt trC∗+

t2

trD∗dS ≤ V ,

0≤ t trC∗+t2

trD∗ ≤ ρ+ a.e. inω .

Thus(C∗,D∗) ∈ C . Two cases can be distinguished:α∗ = 0 or α∗ > 0. In the latter case take alook at (L2) foru= u∗, θ = θ∗ andβ = β ∗:

(α −α∗)

(∫

ωt trC∗+

t2

trD∗ dS−V

)

︸︷︷︸

≤0

≤ 0 ∀α ∈ R+ .

While this is always true forα∗ = 0, in the caseα∗ > 0 we can chooseα = α∗2 ∈R++ and get

−α∗

2︸︷︷︸

<0

(∫

ωt trC∗+

t2

trD∗ dS−V

)

︸︷︷︸

≤0

≤ 0.

Obviously the only way to fulfill this inequality in the caseα∗ > 0 for all α ∈ R+ is∫

ωt trC∗+

t2

trD∗dS=V . (3.34)

The same idea works forβ : Set the variablesu, θ , α andβl in (L2) equal to their optimal valuesu∗, θ∗, α∗ andβl

∗ exceptβu. This yields∫

ω(βu−βu

∗)(

t trC∗+t2

trD∗−ρ+)

dS≤ 0 ∀βu ∈ L1(ω),βu ≥ 0.

In the caseβu∗(x) = 0 a.e. inω this condition is satisfied due to the positivity ofβu andt trC∗+

t2 trD∗−ρ+ ≤ 0. Otherwise there exists a subsetω ⊂ ω , |ω |> 0 with βu

∗(x) > 0 a.e. inω andβu

∗(x) = 0 a.e. inω \ ω . Hence set

βu =

βu∗ in ω \ ω

βu∗

2 in ω

⇒∫

ω(βu−βu

∗)(

t trC∗+t2

trD∗−ρ+)

dS =

∫

ω−βu

∗

2︸︷︷︸

<0

(

t trC∗+t2

trD∗−ρ+)

︸︷︷︸

≤0

dS ≤ 0.

37


As it holds thatβu∗(x)> 0 a.e. inω this can only be true for

t trC∗+t2

trD∗ = ρ+ a.e. inω . (3.35)

Analogously we derive for a similar subsetω ⊂ ω , |ω | > 0 with βl∗(x) > 0 a.e. inω the

following condition has to hold

t trC∗+t2

trD∗ = 0 a.e. inω . (3.36)

It remains to show (D1). Comparison with (L1) leads to the conclusion that it suffices to show

α∗∫

ωt tr(C−C∗)+

t2

tr(D−D∗)dS+∫

ωβu

∗(

t tr(C−C∗)+t2

tr(D−D∗))

dS

−∫

ωβl

∗(

t tr(C−C∗)+t2

tr(D−D∗))

dS≤ 0 ∀(C,D) ∈ C . (3.37)

The last two terms are non-positive due to (3.35) and (3.36),while the first term is non-positivedue to (3.34). This yields (D1) and thus this direction of theproof is completed.The last part of the proof consists of showing that each saddle-point((C∗,D∗),(u∗,θ∗)) of prob-lem (PD) can be extended to a saddle-point((C∗,D∗),(u∗,θ∗,α∗,β ∗)) of (PL) or, in other words,(D1) and (D2) induce (L1) and (L2). To this end we first show a local maximum principle formatrices.

Lemma 3.3.3 (Local maximum principle for matrices). If the maximum of∫

ω s1⊤Ms1+s2

⊤Ms2+. . .+sp

⊤MspdS is attained at M= M∗ (for M ∈ [L∞(ω)]n×n, M = M⊤ 0, ρ− ≤ trM ≤ ρ+ a.e.in ω ,

∫

ω trM dS≤ V and si ∈[L2(ω)

]n, i = 1, . . . , p), then there exists a Lagrange multiplier

α∗ ∈ R such that the following statements hold:

(i)∫

ω〈M−M∗

,−p

∑i=1

sisi⊤+α∗1n〉dS≥ 0 for all M ∈ M , (3.38)

whereM := M ∈ [L∞(ω)]n×n |M = M⊤ 0, ρ− ≤ tr(M)≤ ρ+ . (3.39)

(ii) Define by∑nj=1λ jejej

⊤ the eigenvalue decomposition of∑pi=1sisi

⊤ with ej⊤ek = δ jk and

‖ej‖L2 = 1∀ j = 1, . . . ,n. Furthermore set

J := j ∈ 1, . . .n|ej 6∈ ker(M∗) . (3.40)

Then for each subsetω ⊂ ω , |ω | > 0 with trM∗ > 0 a.e. there exists at least one ej 6∈ker(M∗) whose associated eigenvalueλ j > 0 a.e. onω .

(iii) For each subsetω ⊂ ω , |ω |> 0 with

maxj∈J

λ j < α∗ ⇒ trM∗(x) = ρ− a.e. inω . (3.41)

38


(iv) For each subsetω ⊂ ω , |ω |> 0 with

minj∈J

λ j > α∗ ⇒ trM∗(x) = ρ+ a.e. inω . (3.42)

The proof is given in appendix A. For a given saddle-point((C∗,D∗),(u∗,θ∗)) of (PD) we nowdefine

s1 =

1√2γ(u∗)00

, s2 =

t√24

χ(u∗,θ∗)

00

, s3 =

000√

Kζ (u∗,θ∗)

(3.43)

and the matrix

M =

(tC 00 1

2t D

)

. (3.44)

Thus in the present case we havep= 3 andn= 5, as a result the matrix∑pi=1sisi

⊤ does not havefull rank and there exist eigenvaluesλi = 0. Then according to Lemma 2 there exists anα∗ ∈R

such that for eachω ⊂ ω , |ω |> 0 with

maxj∈J

λ j < α∗ ⇒ trM∗ = ρ− a.e. inω ,

minj∈J

λ j > α∗ ⇒ trM∗ = ρ+ a.e. inω .

Define

βu∗ =

max

j=1,...,nλ j −α∗ if min

j∈Jλ j > α∗

0 else(3.45)

βl∗ =

α∗− max

j=1,...,nλ j if max

j∈Jλ j < α∗ or 6 ∃ej 6∈ ker(M∗)

0 else(3.46)

Consider now the caseα∗ > 0. Suppose∫

ω trM∗dS<V and there existsω ⊂ ω , |ω | > 0 withtrM∗ < ρ+ a.e. onω (otherwise the design would be entirely filled with materialimplying anunfavorable choice ofV andρ+). Assume further that there existsω ⊂ ω, |ω |> 0 with trM∗ > 0a.e. onω . Due to Lemma 2 we know there exists aλk > 0 with the associated eigenfunctionek 6∈ ker(M∗). Define the matrix

M = M∗+ χωekek⊤ 1|ω | min

(V −

∫

ωtrM∗dS

),(ρ+− trM∗)

,

whereχω is the characteristic function of the setω :

χω(x) =

1 if x∈ ω0 else

. (3.47)

39


Obviously we haveM = M⊤ 0,∫

ω trM dS≤V and 0≤ ρ− ≤ trM ≤ ρ+ a.e. onω . ThusM iscontained in the admissible set. Now take a look at the objective

−∫

ω〈M,

n

∑j=1

λ jejej⊤〉dS=

= −∫

ω〈M∗

,

p

∑i=1

sisi⊤〉dS−

∫

ω

1|ω |︸︷︷︸

>0

min

(V −

∫

ωtrM∗dS

),(ρ+− trM∗)

︸︷︷︸

>0

λk︸︷︷︸

>0

dS

< −∫

ω〈M∗

,

p

∑i=1

sisi⊤〉dS.

This contradicts the optimality ofM∗. If the additional assumption of a subsetω with trM∗ > 0a.e. onω does not hold then we have trM∗ = 0 a.e. onω . Here two cases must be distinguished:eitherλ j = 0∀ j ∈ 1, . . . ,n a.e. onω , which implies that for all subsets withλ j > 0 for somej ∈ 1, . . . ,n we have a.e. trM∗ = ρ+ – in other words a entirely filled design except for regionswithout strains who do not contribute to the compliance. Therefore this would again indicatean inappropriate choice ofV andρ+. If we have on the other handλ j > 0 a.e. onω for somej ∈ 1, . . . ,n the proof given above still holds forλk = λ j . Thus forα∗ > 0 we must have∫

ω trM∗dS=V. The caseα∗ < 0 is not taken into consideration as minj=1,...,n λ j ≥ 0, thereforeα∗ < 0 would result in trM∗ = ρ+ a.e. onω . With the above definition ofβu

∗ andβl∗ it follows

that

(α −α∗)

(∫

ωtrM∗dS−V

)

+∫

ω(βu−βu

∗)(trM∗−ρ+

)dS

−∫

ω(βl −βl

∗)trM∗dS≤ 0 ∀α ∈ R+ , β ∈[L1(ω)

]2, β ≥ 0. (3.48)

Together with (D2) this yields (L2). To show (L1) we again refer to Lemma 2. According to(3.38) we have

∫

ω〈M−M∗

,−p

∑i=1

sisi⊤+α∗1n〉dS≥ 0 ∀M ∈ M .

We want to show (L1)∫

ω〈M−M∗

,−p

∑i=1

sisi⊤+(α∗+βu

∗−βl∗)1n〉dS≥ 0

for all M ∈ [L∞(ω)]n×n, M = M⊤ 0. It is sufficient to prove this statement for all subsetsω ⊂ ω , |ω |> 0. Here we distinguish three different cases:

1. trM∗ = ρ+ a.e. onω .

According to Lemma 2 (ii) there exists at least one eigenfunction ek such thatek⊤M∗ek =

µk > 0. The value of the trace is independent from the chosen coordinate system thus wehave

n

∑j=1

ej⊤M∗ej = trM∗ = ρ+ = ∑

j∈J

ej⊤M∗ej . (3.49)

40


If there exists only one eigenfunctionek 6∈ ker(M∗) this yieldsµk = ρ+ and forM∗ it holds

thatM∗ = ρ+vv⊤ with ‖v‖ = 1. Due toρ+ = ek⊤M∗ek = ek

⊤ρ+vv⊤ek = ρ+(v⊤ek

)2we

getv⊤ek =±1 and as∑nj=1

(v⊤ej

)2= 1 it follows thatv⊤ej = 0∀ j 6= k and thusv=±ek

resulting inM∗ = ρ+ekek⊤.

If there exist more than one eigenfunction not in the kernel of M∗, for example

0< ek⊤M∗ek = µk < ρ+

, 0< el⊤M∗el = µl ≤ ρ+−µk ,

we can assume without loss of generality thatλk ≥ λl holds. Consider the matrix

M = M∗−µl el el⊤+µlekek

⊤ ∈ M .

Inserting this into (3.38) yields

∫

ω〈M−M∗

,

n

∑j=1

(−λ j +α∗)ejej⊤〉dS

=

∫

ω〈−µl el el

⊤+µl ekek⊤,

n

∑j=1

(−λ j +α∗)ejej⊤〉dS

=

∫

ω−µl (−λl +α∗)+µl (−λk+α∗)dS

=

∫

ωµl︸︷︷︸

>0

(λl −λk)︸︷︷︸

≤0

dS≥ 0.

This can only be true forλk = λl =: λ . As such a matrixM can be constructed for any pairof eigenfunctionsej ,ek 6∈ ker(M∗), this results inλ j = λ ∀ j ∈ J . Suppose thisλ wouldnot be the maximal eigenvalue, in other words:

∃λm = maxj=1,...,n

λ j > λ .

Then setMm = ρ+emem⊤ ∈ M and inserting this into (3.38)

∫

ω〈Mm−M∗

,

n

∑j=1

(−λ j +α∗)ejej⊤〉dS=

=

∫

ω(−λm+α∗)em

⊤Mmem︸︷︷︸

=ρ+

−(−λ +α∗) ∑j∈J

ej⊤M∗ej

︸︷︷︸

=ρ+

dS

=

∫

ωρ+(λ −λm)dS< 0

contradicts the optimality ofM∗. Thus we get

λ = maxj=1,...,n

λ j = maxj∈J

λ j .

41


We now distinguish two cases. Either we have minj∈J λ j > α∗ which results inβu∗> 0

andβl∗ = 0 a.e. onω . Then we get

∫

ω〈M−M∗

,

n

∑j=1

(−λ j +α∗+ βu∗

︸︷︷︸

=λ−α∗

)ejej⊤〉dS=

=∫

ω

n

∑j=1

(λ −λ j)︸︷︷︸

≥0

ej⊤Mej

︸︷︷︸

≥0

− ∑j∈J

(λ − λ)︸︷︷︸

=0

ej⊤M∗ej dS

≥ 0 ∀M ∈ C.

The other possibility is minj∈J λ j ≤ α∗ ≤ maxj∈J λ j and thusβu∗ = βl

∗ = 0 a.e. onω .But as minj∈J λ j = maxj∈J λ j = λ this yieldsλ = α∗ and results in

∫

ω〈M−M∗

,

n

∑j=1


=∫

ω

n

∑j=1

(−λ j +α∗)︸︷︷︸

=−λ j+λ≥0

ej⊤Mej

︸︷︷︸

≥0

− ∑j∈J

(−λ +α∗)︸︷︷︸

=0

ej⊤M∗ej dS

≥ 0 ∀M ∈ C.

This part of the proof has a physical interpretation: when the upper box constraint is active,the material tensor is built up as a dyadic product of the strain vector associated with thelargest eigenvalueλ . If there exist several strains with the same eigenvalue, the materialtensor is built up as sum of their associated dyadic matrices. A distribution of material tostrains with lower eigenvalues is considered ineffective at spots, where the upper materiallimit is reached. The next case can be interpreted analogously.

2. trM∗ = ρ− a.e. onω .

First assumeρ− > 0. In this case we either haveλ j = 0 ∀ j = 1, . . . ,n, then unlessα∗ = 0 (which results in

∫

ω〈M − M∗,∑nj=1(−λ j + α∗)ejej

⊤〉dS= 0) we haveβl∗ =

α∗−maxj=1,...,n λ j = α∗−0= α∗ and thus

∫

ω〈M−M∗

,

n

∑j=1

(−λ j +α∗−βl∗)

︸︷︷︸

=0

ejej⊤〉dS= 0 ∀M ∈ C. (3.50)

On the other hand there might existλ j > 0, then it follows as in Lemma 2 (ii) that at leastone eigenfunctionej 6∈ ker(M∗). If only one eigenfunctionek 6∈ ker(M∗) exists we getagain (compare to 1.) thatM∗ = ρ−ekek

⊤. Analogously we also derive that for more thanone eigenfunctionej 6∈ ker(M∗) the associated eigenvalues have to be equal and are themaximal eigenvalue of∑p

i=1sisi⊤:

∀ j ∈ J : λ j = maxj=1,...,n

λ j = λ .

42


Again two cases are distinguished. When maxj∈J λ j < α∗ it follows thatβu∗ = 0, βl

∗ =

α∗− λ and this results in∫

ω〈M−M∗

,

n

∑j=1

(−λ j +α∗−βl∗)ejej

⊤〉dS=

=

∫

ω

n

∑j=1

(−λ j + λ )︸︷︷︸

≥0

ej⊤Mej

︸︷︷︸

≥0

− ∑j∈J

(−λ + λ)︸︷︷︸

=0

ej⊤M∗ej dS

≥ 0 ∀M ∈ C .

The other case is minj∈J λ j ≤ α∗ ≤ maxj∈J λ j resulting inβu∗ = βl

∗ = 0. But due tomin j∈J λ j = maxj∈J λ j = λ it holds thatλ = α∗ and thus

∫

ω〈M−M∗

,

n

∑j=1


=

∫

ω

n

∑j=1

(−λ j + λ )︸︷︷︸

≥0

ej⊤Mej

︸︷︷︸

≥0

− ∑j∈J

(−λ + λ)︸︷︷︸

=0

ej⊤M∗ej dS

≥ 0 ∀M ∈ C .

In the special caseρ− = 0 it holds thatM∗ = 0. Thus there is noej 6∈ ker(M∗). This resultsin βu

∗ = 0 andβl∗ = α∗−maxj=1,...,n λ j = α∗− λ . This yields

∫

ω〈 M︸︷︷︸

0

−0,n

∑j=1

(−λ j +α∗−α∗+ λ)︸︷︷︸

≥0

ejej⊤〉dS≥ 0 ∀M ∈ C .

3. trM∗ ∈ (ρ−,ρ+)

This results inβu∗ = βl

∗ = 0 and due to (3.38) we already know∫

ω〈M−M∗

,

n

∑j=1

(−λ j +α∗)ejej⊤〉dS≥ 0 ∀M ∈ M .

Assume there exists aM ∈ C \M with trM > ρ+ such that∫

ω〈M−M∗

,

n

∑j=1

(−λ j +α∗)ejej⊤〉dS< 0.

Consider the matrix

M = M∗+(M−M∗)ρ+− trM∗

trM− trM∗ .

As ρ+−trM∗

trM−trM∗ < 1 it follows thatM ∈ M . Inserting into (3.38)∫

ω〈M−M∗

,

n

∑j=1


=∫

ω

ρ+− trM∗

trM− trM∗ 〈M−M∗,

n

∑j=1

(−λ j +α∗)ejej⊤〉dS< 0

43


yields a contradiction to the optimality ofM∗. The same is true when assuming that thereexists aM ∈ C \M with trM < ρ− such that

∫

ω〈M−M∗

,

n

∑j=1


Then defining the matrix

M = M∗+(M−M∗)trM∗−ρ+

trM∗− trM∈ M

yields again a contradiction to the optimality ofM∗:

∫

ω〈M−M∗

,

n

∑j=1


=

∫

ω

trM∗−ρ+

trM∗− trM〈M−M∗

,

n

∑j=1


Thus we get the desired result∫

ω〈M−M∗

,

n

∑j=1

(−λ j +α∗)ejej⊤〉dS≥ 0 ∀M ∈ C .

This completes the proof.

3.4. Minimum Weight Problem Formulation

In contrast to the previously considered minimum compliance problem formulations it is alsopossible to use the material measure vol(C,D) introduced in (3.11) as objective function andensure the stiffness of the structure via a compliance constraint of the form

∫

ωt f⊤udS+

∫

∂ω1

gu⊤u+gθ

⊤θ dl ≤ c. (3.51)

This leads to the following minimum weight problem formulation

min(u,θ )∈U(C,D)∈C

∫

ωt · trC+

t2· trDdS

subject to ρ− ≤ t · trC+t2· trD ≤ ρ+ (3.52)

∫

ωt f⊤udS+

∫

∂ω1

gu⊤u+gθ

⊤θ dl ≤ c

∫

ωtγ⊤(u)Cγ(v)+

t3

12χ⊤(u,θ)Cχ(v,η)+ tkζ⊤(u,θ)Dζ (v,η)dS=

=

∫

ωt f⊤vdS+

∫

∂ω1

gu⊤v+gθ

⊤η dl ∀(v,η) . ∈ U

44

3.5. Free Material Optimization for Multiple Load Cases

From a physical point of view there is no big difference between problem (3.52) and the mini-mum compliance problem formulation (PD), as in both cases the goal of a stiff and light-weightstructure is achieved. However, the minimum weight problem(3.52) is preferred by manufac-turers, as the applied load cases and consequentially an adequate stiffness sufficient to carrythese loads is already constituted in the problem definition. Hence the compliance constraintcan be used to ensure the minimal stiffness prescribed by norms and regulations, while the en-tire amount of material incorporated into the structure, which is closely related to the monetarycosts of the structure, is minimized.

However, while codes for the efficient solution of problem (PD) have been developed over adecade ago, a satisfying numerical approach to problem (3.52) has been developed only recentlyin the scope of the EU-project PLATO-N. An outline of this novel optimization algorithm PEN-SCP as introduced by [SKL09b] is given in Section 4.3. It is important to note that PENSCPfactorizes the global stiffness matrix and therefore tolerates no eigenvalues, that are equal tozero. Accordingly we have to replace the lower bound on the summed trace of the shell materialtensorsC andD defined in (3.15) by a lower bound on the eigenvalues ofC andD to guaranteethe positive definiteness of the global stiffness matrix. This finally leads to the following mini-mum weight formulation for the Free Material Optimization problem for shells (PMW), which isa non-convex non-linear semidefinite program

min(u,θ )∈U(C,D)∈C

∫

ωt · trC+

t2· trDdS

subject to

(tC 00 t

2D

)

− ε15 0 (PMW)

t · trC+t2· trD ≤ ρ+

∫

ωt f⊤udS+

∫

∂ω1

gu⊤u+gθ

⊤θ dl ≤ c

∫

ωtγ⊤(u)Cγ(v)+

t3

12χ⊤(u,θ)Cχ(v,η)+ tkζ⊤(u,θ)Dζ (v,η)dS=

=

∫

ωt f⊤vdS+

∫

∂ω1

gu⊤v+gθ

⊤η dl ∀(v,η) ∈ U .

Obviously a positive parameterε ensures, that all material matrices are positive definite, thoughthis also implies that no holes can be created in the structure in contrast to previous problemformulations.


So far we have only considered problem formulations with a single load case. While this issufficient for many academic testcases, this is of course nottrue for industrial applications. Astructure optimized with respect to one single load case might be very unstable when a differentload is applied, even tiny loads might result in a collapse ofthe structure. In order to avoidthis scenario and to create robust structures the consideration of large numbers of load cases

45


is common practice for industrial problem settings. The wings of an aircraft for example haveto withstand entirely different load cases during take off,flight and landing; additionally theyshould not succumb to unpredictable incidents such as the influences of wind or air pockets. Afrequently used approach to deal with multiple load cases isan optimization in the worst-casesense. It has been introduced with regard to Free Material Optimization in [BKNZ99], where themultiple load problem is even combined with a multiple obstacle problem, and further discussedin [KZ02].

We begin the adaption of this procedure to Naghdi shells withthe definition of the multipleload cases( f 1,g1

u,g1θ ), . . . ,( f L,gL

u,gLθ ) from L2(ω)×L2(∂ω1)×L2(∂ω1), whose number is de-

noted byl = 1, . . . ,L. Moreover, the force term of the potential energy is given bythe linearmapping

F l (ul,θ l ) :=

∫

ωt f l ·udS+

∫

∂ω1

glu ·ul +gl

θ ·θ l dl . (3.53)

The displacements associated with thel -th load case are chosen from(ul ,θ l )∈U . Furthermorewe introduce the vectorsu := (u1, . . . ,uL)⊤ andθθθ := (θ1, . . . ,θL)⊤ to allow for a more compactand concise notation. Note that it is possible to use varyingadmissible setsU l depending onthe load casel , which allows different boundary conditions for each load case. This is primarilynecessary when working with contact boundary conditions asin [BKNZ99], though this couldalso be employed for Dirichlet boundary conditions if required. However in any case it has tohold thatU l 6= /0 ∀l ∈ 1, . . . ,L.

As the compliance of the load casel is a measure on how much a structure will deform underthe loads( f l ,gl

u,glθ ), the worst-case compliance is the one which leads to the largest deforma-

tion of the structure. Thus, the optimization of the worst-case compliance is achieved via aminimization of the compliance associated with the load case yielding the maximal compliance,i.e.

min(C,D)∈C

maxl=1,...,L

max(ul ,θ l )∈U

− 12

∫

ω

[

tγ(ul )⊤Cγ(ul )+t3

12χ(ul

,θ l )⊤Cχ(ul,θ l ) (3.54)

+ tkζ (ul,θ l )⊤Dζ (ul

,θ l )]

dS+F l (ul,θ l ) .

The discrete maximization over the load cases is one of the issues that renders problem (3.54) adifficult optimization problem. Fortunately it can be resolved by replacing the discrete variablel by a weight vector(λ1, . . . ,λL)

⊤ = λλλ ∈ Λ running over the unit simplex

Λ :=

λλλ ∈ RL

∣∣∣∣∣

L

∑l=1

λl = 1 , λl ≥ 0 ∀l = 1, . . . ,L

. (3.55)

This leads to the following continuous optimization problem

min(C,D)∈C

max(u,θθθ)∈U

λλλ∈Λ

L

∑l=1

− λl

2

∫

ω

[

tγ(ul )⊤Cγ(ul )+t3

12χ(ul

,θ l )⊤Cχ(ul,θ l ) (3.56)

+ tkζ (ul,θ l )⊤Dζ (ul

,θ l )]

dS+λl Fl (ul

,θ l )

46


which is a reformulation of problem (3.54) due to the fundamental theorem of Linear Program-ming [BJS05]. In the next step we address the dependence of the objective function with respectto the introduced variableλλλ . If the objective function would be concave in(uuu,θθθ ,λλλ ), the entireoptimization problem would exhibit a convex-concave structure and thus the application of sad-dle point theory is possible. This desirable concave dependence is not yet present, but we areable to achieve this by performing a variable transformation. For this purpose some prearrange-ments are necessary. On the one hand we exchange the min-max in (3.56) by an inf-sup andreplace the unit simplexΛ by the half open setΛ0, which is defined as

Λ0 := λλλ ∈ Λ | λl > 0 ∀l = 1, . . . ,L . (3.57)

Note that the optimal value of problem (3.56) remains the same, when choosingλλλ ∈ Λ0, as(3.56) is now an inf-sup problem. On the other hand we substitute the displacement variablesusing the following schemevl := λl ul andη l := λl θ l . Hence the variables(u,θθθ ,λλλ ) associatedwith the supremum become(v,ηηη ,λλλ ) and thereby problem (3.56) is converted into

inf(C,D)∈C

sup(v,ηηη ,λλλ )∈V

JML ((C,D),(v,ηηη ,λλλ )) . (PML-MC )

The saddle point function of the multiple load problem is defined as

JML ((C,D),(v,ηηη ,λλλ )) :=L

∑l=1

− 12λl

∫

ω

[

tγ(vl )⊤Cγ(vl )+t3

12χ(vl

,η l)⊤Cχ(vl,η l ) (3.58)

+ tkζ (vl,η l )⊤Dζ (vl

,η l )]

dS+F l (vl,η l )

and the convex setV is

V :=

(v,ηηη ,λλλ )∣∣vl = λl u

l, η l = λl θ l

, (u,θθθ ) ∈ U , λλλ ∈ Λ0

. (3.59)

The saddle point functionJML ((C,D),(v,ηηη ,λλλ )) is linear and thus convex with respect to thematerial matricesC andD. Furthermore it is concave with respect to the variable(v,ηηη ,λλλ ) – incontrast to the objective functions of the previous problemformulations (3.54) and (3.56). Thelatter follows from the concavity of the function− x2

y on the setR×R++, which can be verifieddirectly. Due to this functional dependence the existence of a solution to problem (PML-MC ) canbe deducted from a saddle point theorem.

Theorem 3.5.1. There exists a pair(C,D) ∈ C such that

sup((v,ηηη ,λλλ ))∈V

JML ((C∗,D∗),(v,ηηη ,λλλ )) = min

(C,D)∈Csup

((v,ηηη ,λλλ ))∈V

JML ((C,D),(v,ηηη ,λλλ )) . (3.60)

Moreover, it holds that

inf(C,D)∈C

sup((v,ηηη ,λλλ ))∈V

JML ((C,D),(v,ηηη ,λλλ )) = sup((v,ηηη ,λλλ ))∈V

inf(C,D)∈C

JML ((C,D),(v,ηηη ,λλλ )) . (3.61)

47


Proof. According to [Mor64] the statement holds if we can show that

(i) V is a convex set,

(ii) JML ((C,D),(v,ηηη ,λλλ )) is concave for fixed(C,D) ∈ C ,

(iii) C is convex and weak∗-compact and

(iv) JML ((C,D),(v,ηηη ,λλλ )) is convex and lower semicontinuous onC (equipped with the weak∗-topology ofL∞(ω)) for fixed (v,ηηη ,λλλ ) ∈ V .

Property (i) follows from the convexity ofU and the unit simplexΛ0 together with the linearityof the mapping transforming(u,θθθ ) into (v,ηηη). The concavity ofJML with respect to(v,ηηη ,λλλ )as required in (ii) has already been discussed prior to Theorem 3.5.1. The convexity and weak∗-compactness ofC (and consequently (iii)) as well as the convexity ofJML with respect to (C,D)have already been shown in the proof of Theorem 3.2.1. The continuity of the saddle pointfunction given in Theorem 3.2.1 with respect to(C,D) is also valid for the saddle point functionJML , as it is a linear combination of the original saddle point function JD. Since the definitionof the weak∗-topology yields the continuity in this topology as demanded in statement (iv), theproof is complete.

As discussed in [BKNZ99] problems (3.54), (3.56) and (PML-MC ) have the same objectivevalue, however (PML-MC ) only works on the restricted setΛ0. It is possible to augmentΛ0 in(PML-MC ) to Λ, which requires an extended-valued version ofJML . As we are primarily interestedin the optimal design variables(C∗,D∗), these technicalities are of secondary importance andwill not be investigated any further here.

Instead we conclude this section with a duality result for the case of multiple loads. Analo-gously to Theorem 3.3.1 we introduce a primal problem formulation (PML-P) and show that thedual of this problem is equivalent to the previously obtained problem (PML-MC ).

Theorem 3.5.2. Problem (PML-MC) is equivalent to the Lagrange dual problem of

sup(v,ηηη ,λλλ )∈V

α∈R+


L

∑l=1

F l (vl,η l)−αV −ρ+

∫

ωβudS+ρ−

∫

ωβl dS (PML-P)

subject toL

∑l=1

t2λl

(

γ(vl )γ(vl )⊤+t2

12χ(vl

,η l )χ(vl,η l )⊤

)

− t(α +βu−βl)13 0 a. e. inω

L

∑l=1

tk2λl

ζ (vl,η l )ζ (vl

,η l )⊤− t2(α +βu−βl)12 0 a. e. inω

where1n denotes the unit matrix inRn.

Proof. The proof follows closely the proof of Theorem 3.3.1 with marginal changes. Due toproblem formulation (PML-MC ) the maximum is relaxed to a supremum, for which the same

48

3.6. Free Material Optimization for Isotropic Naghdi Shells

saddle point theorems hold. In the entire proof of Theorem 3.3.1 we never used the fact that thesum of the dyadic matrices composed from the strain vectors has a smaller rank than the materialmatrices. Instead Lemma 3.3.3 has been formulated already in a rather general fashion to coveralso the case of multiple loads, where we work with three strain vectors per load case. Thusthe entire proof is not only valid for a single triple of strains, but also forL times the numberof strain triples. The prefactors1λl

have no effect on the statements of the proof, as they fulfillthe positivity conditionλl > 0 and can simply be included in the vectorssi in the application ofLemma 3.3.3. Hence the proof of Theorem 3.3.1 can be adopted for the multiple load case.

3.6. Free Material Optimization for Isotropic Naghdi Shell s

The fundamental motivation for Free Material Optimizationis the search for the ultimately bestmaterial tensor, regardless of symmetry, fibre direction etc.. We have already pointed out thatthe obtained optimal structures are hard to manufacture, yet Free Material Optimization shouldalso encourage manufacturers to reconsider and improve their production process. Nonethelessit might be useful for some applications to enforce specific symmetry properties on the materialparameters.

An often requested symmetry class is orthotropic material,as many sophisticated structuresare fabricated from laminates consisting of several orthotropic layers. However open orthotropicfibre directions introduce trigonometric functions into the problem such that the advantageousmathematical structure of the Free Material Optimization problem is immediately lost and isreplaced by a non-convex optimization problem with a multitude of local optima. Moreover, asthe problem still features semidefinite matrix constraintsthere are no numerical solvers availableto solve the resulting problem. In this case a discrete approach as chosen in Discrete MaterialOptimization [SL05] is considerably more promising, in which different material directions areincluded by adding a finite number of rotations of the same material tensor to the admissible set.

In contrast to orthotropic material the isotropic materialsymmetry does not cause such a dras-tic effect on the mathematical structure of the Free Material Optimization problem for shells.When replacing the generic anisotropic material matrices by rotationally invariant isotropic ma-terial matrices for shells as introduced in (2.90) the problem retains its convex-concave saddlepoint structure. To verify this fact we set up the Free Material Optimization problem for isotropicshells by inserting the isotropic material matrices (2.90)in the minimum compliance problemformulation (PD). The trace derived from the isotropic material matrices (2.90) is

triso(C,D) = triso(λ , µ) := 2λ +10µ (3.62)

and subsequently the admissible set for the elasticity matrices can be written as

Ciso :=

(λ , µ) ∈ [L∞(ω)]2

∣∣∣∣∣∣

λ ≥ 0 , µ ≥ 0∫

ω 2tλ +10tµ dS≤V0≤ ρ− ≤ 2tλ +10tµ ≤ ρ+

. (3.63)

49


A separation of the isotropic material matrixCiso according to

Ciso =

λ +2µ λ 0λ λ +2µ 00 0 2µ

= 2µ

1 0 00 1 00 0 1

+ λ

1 1 01 1 00 0 0

(3.64)

inserted into the saddle point functionJD as defined in (3.18) leads to the following saddle pointfunction for the isotropic case

Jiso((λ , µ),(u,θ)) :=−∫

ωµ[

t‖γ(u)‖22+

t3

12‖χ(u,θ)‖2

2 +2tk‖ζ (u,θ)‖22

]

dS

− 12

∫

ωλ[

t(γ11(u)+ γ22(u))2+

t3

12(χ11(u,θ)+ χ22(u,θ))2

]

dS

+∫

ωt f⊤udS+

∫

∂ω1

gu⊤u+gθ

⊤θ dl (3.65)

and therefore to the Free Material Optimization problem forisotropic shells

min(λ ,µ)∈Ciso

max(u,θ )∈U

Jiso((λ , µ),(u,θ)). (PMC-iso)

Although this problem lacks the matrix variables used in (PD) it is still a convex-concave opti-mization problem and we can show existence of at least one optimal solution via a saddle pointtheorem.

Theorem 3.6.1. Problem (PMC-iso) has an optimal solution((λ ∗, µ∗),(u∗,θ∗)) ∈ Ciso×U .

Proof. Existence of an optimal solution for problem (PMC-iso) follows from the Minimax-Theo-rem given in [Mac04]. As problem (PMC-iso) is closely related to the original minimum com-pliance problem (PD) we can adopt several statements from the proof of Theorem 3.2.1. Firstof all the definition of the setU and thus all its properties required for the proof are identical.As Ciso is a closed subset of the compact spaceC , it is also compact. Furthermore it is convexand non-empty for a reasonable choice ofV, ρ+ andρ−. The conditions (i) and (iii), which arerequired for the Minimax-Theorem as stated in the proof of Theorem 3.2.1, follow directly fromthe proof of Theorem 3.2.1 asCiso is a subset ofC and moreover(C0,D0) ∈ Ciso. It remainsto show property (ii), i.e. that for all(u,θ) ∈ U the mapping(λ , µ) 7→ Jiso((λ , µ),(u,θ)) isconvex and continuous. This holds, asJiso((λ , µ),(u,θ)) depends linearly onλ andµ , thus theprerequisites of the Minimax-Theorem in [Mac04] are fulfilled and the existence of an optimalsolution for problem (PMC-iso) is shown.

3.7. Free Material Optimization for Kirchhoff-Love Shells

So far we have concentrated on the Free Material Optimization problem for Naghdi shells, asthis model is not only valid for thin shells and takes the transverse shear part of the materialtensor into account. It is therefore the optimal compromisebetween a linear elastic model pre-serving the convex-concave structure and a maximal information about the optimal material

50

3.7. Free Material Optimization for Kirchhoff-Love Shells

tensor. However it is certainly possible to formulate a FreeMaterial Optimization problem foranother shell model such as the Kirchhoff-Love shell as introduced in (2.5.1), which is relevantfor applications with thin shells that fulfill the Kirchhoff-Love kinematical assumption. To thisend we define a saddle point function equal to the negative potential energy of a Kirchhoff-Loveshell

JKL (C,u) :=−12

∫

ωtγ⊤Cγ +

t3

12ρ⊤Cρ dS+

∫

ωt f⊤udS+

∫

∂ω1

gu⊤udl . (3.66)

The compliance of a Kirchhoff-Love shell is equal to twice the maximum ofJKL (C,u) withrespect tou ∈ UKL , though we will ignore the factor of two in the setup of the saddle pointproblem following the same approach we used for Naghdi shells. The design variable is thematerial matrixC chosen fromL∞(ω ,S3) and the amount of material spent at a single spotx∈ ωis simply measured by tr(C(x)). This implies the following form of the volume constraint

∫

ωt tr(C(x))dS≤V (3.67)

and the box constraints0≤ ρ− ≤ t tr(C(x)) ≤ ρ+ a.e. onω . (3.68)

Combining these restrictions yields the set of admissible material matrices in the Kirchhoff-Lovecase

CKL :=

C∈ L∞(ω ,S3)

∣∣∣∣

∫

ω t tr(C(x)) dS≤V0≤ ρ− ≤ t tr(C(x)) ≤ ρ+ a.e. onω

. (3.69)

Thus the Free Material Optimization problem for Kirchhoff-Love shells is given by

minC∈CKL

maxu∈UKL

JKL (C,u) . (PMC-KL )

This is again a convex-concave saddle point problem and we are able to prove existence of atleast one optimal solution via the Minimax-Theorem provided by [Mac04].

Theorem 3.7.1. For a midsurface parametrized byφ ∈C3(ω) problem (PMC-KL) has an optimalsolution(C∗,u∗) ∈ CKL ×UKL.

Proof. The existence of a saddle point and thus an optimal solution of problem (PMC-KL ) followsfrom the Minimax-Theorem given in [Mac04].UKL is non-empty, convex and a closed subspaceof [H1(ω)]2×H2(ω). CKL lies in a norm ball ofL∞(ω ,S3). As a result it is weak∗-compact andthus closed and bounded in the weak∗-topology. Additionally the setCKL is convex and non-empty for appropriately chosen constantsV, ρ− andρ+, hence the involved setsUKL andCKL

fulfill all necessary prerequisites for the application of the Minimax-Theorem. As the strainsγ andρ depend linearly on the displacementsu and since the strain energy term ofJKL (C,u)depends quadratically on these strains (which are then multiplied by a negative factor) this partof the saddle point function is concave and continuous with respect to the displacementsu.The force term is linearly dependent onu and hence concave and continuous with respect to thedisplacements, which holds accordingly for the entire saddle point functionJKL (C,u). MoreoverJKL (C,u) depends linearly on the material matrixC and is thus convex and continuous with

51


respect toC. The ellipticity ofJKL (C,u) has already been proven in [CM92, BCM94] for a shellmidsurface with inherentC3(ω) regularity and an isotropic material tensor of the form

Cαβην0 =

2λ µλ +2µ

aαβ aην +µ(

aαηaβν +aανaβη)

. (3.70)

TheC3(ω) regularity has already been postulated in Theorem 3.7.1. The ellipticity has to holdonly for a singleC0 ∈ CKL . We rewrite (3.70) in the Mandel matrix notation

C0 =

2λ µλ+2µ +2µ 2λ µ

λ+2µ 02λ µ

λ+2µ2λ µ

λ+2µ +2µ 00 0 2µ

(3.71)

hence the trace ofC0 is given by

tr(C0) =4λ µ

λ +2µ+6µ =

4λ µ +6λ µ +12µ2

λ +2µ

λ∈R+µ∈R++

<10µλ +12µ2+8µ2

λ +2µ

=10µ(λ +2µ)

λ +2µ= 10µ . (3.72)

We are also able to estimate a lower bound for the trace ofC0

tr(C0) =10λ µ +12µ2

λ +2µ=

10λ µ +20µ2

λ +2µ− 8µ2

λ +2µλ∈R++

>10µ(λ +2µ)

λ +2µ− 8µ2

2µ(3.73)

= 10µ −4µ = 6µ . (3.74)

Consequentially it holds for

λ ∈ R++ andρ−

6t≤ µ ≤ min

V

10t|ω | ,ρ+

10t

(3.75)

thatC0 ∈ CKL . Thus we have found aC0 for which the required ellipticity holds, which yieldsthe existence of an optimal point(C∗,u∗) of the optimization problem (PMC-KL ).

52

CHAPTER 4

Numerical Approach

In the preceding chapter we have deducted an analytical formulation of the Free Material Opti-mization problem for shells. As we aim for a numerical solution of this optimization problem weaddress ourselves to the task of developing a discrete approximation of the previously presentedinfinite-dimensional optimization problems in this chapter. Our approach is based on the finiteelement method, an established technique in structural engineering for the numerical solutionof partial differential equations. To this end the structure’s geometry, which is in the case ofshells defined by the midsurface, is partitioned into polygons – we will focus on quadranglesand triangles in this thesis. The design variables (the material matricesC andD) are approxi-mated by a piecewise constant function, while the state variables (the displacementsu andθ ) aredescribed by a piecewise linear function. Hence they are represented either by discrete valuesin the polyangular elements or by values at the nodes. The integrals appearing in the optimiza-tion problem are decomposed into element-wise integrals, that are converted into integrals overa reference element by a variable transformation. The final integration is then substituted by asummation over Gauss points leading to a finite-dimensionaldiscrete problem formulation.

The obtained discrete Free Material Optimization problem is a nonlinear nonconvex semidef-inite program (SDP), that can be transformed into a nonlinear convex SDP by eliminating thedisplacements from the problem formulation via an exploitation of the static equilibrium condi-tion. For the numerical solution of the problem we have developed codes within the softwarepackage FMOKernel, that is part of the PLATO-N software platform. We have implemented allnecessary finite element routines and function and gradientevaluations for the minimum compli-ance as well as the minimum weight problem formulation. Moreover the nonlinear SDP solverPENSCP developed by M. Stingl and M. Kocvara [SKL09b] is part of the software platform. Wehave employed this code for the solution of several testexamples from the field of material opti-mization of thin-walled structures and provide a short summary of the functioning of PENSCPas well as a comparison of our results with various other established optimization approaches instructural design.

53

4. Numerical Approach

Moreover we discuss several improvements of this software package to make it more suitableto engineering needs. On the one hand this concerns the simultaneous optimization of structurescombined from solids and shells. In most practical applications shells do not appear as stand-alone structures, instead they are attached to a framework constructed from solids or trusses. Weachieve this combined optimization on a discrete level, where we profit much from the carefulchoice of consistent material measures for shells and solids in Subsection 2.6.2. On the otherhand we introduce condensed data structures, that arise from the separation of the structure intoseveral parts in the context of static analysis. Thereby it is possible to condense parts of thestructure, that contain only fixed material, in a Free Material Optimization problem. This is notonly a possibility to solve high-dimensional problems originating from large structures, but alsoto include parts of the mesh built up from element types that are not native in the FMOKernelsuch as springs or junction elements.

4.1. Finite Element Discretization

In this chapter we dedicate ourselves to the numerical solution of the Free Material Optimiza-tion problem for shells. The numerical software package, that has been developed to this end,consists of two main components: a code, that simulates the physical behavior of the shell forthe current material distribution, and the optimizer, thatcalculates a new material distributionbased on the data from the simulator in order to improve the objective function. This sectionfocuses on the simulation software. For this purpose we employ a finite element method, whichis nowadays state-of-the-art for the approximate solutionof partial differential equations. It re-places the geometry as well as the appearing physical quantities by discretized counterparts andthereby transforms the weak formulation of the equilibriumcondition (2.52) into a linear systemof equations. A thorough description of finite element methods for shells can for example befound in [Ber96, Bat02, CB03]. The finite element routines described in this section have beenrealized within theFELib of the PLATO-N software platform.

4.1.1. Polygonal Approximation of the Midsurface

The first step consists in a geometrical discretization of the shell body. As the shape of theshell is determined by its midsurface, this is realized by a partition of the midsurface intopolygonal elements. This polygonal approximation of the midsurfaceS is referred to asmeshand its shape is defined by a set of nodes and elements. Here theelements provide an ap-proximation of the original domain, however they have to be disjoint and element nodes arenot allowed to lie on edges of adjacent elements. We denote each node by an indexn, thatruns from n = 1, . . . ,N, whereN is the total number of nodes contained in the mesh. Inthe FELib nodes are stored as objects of the classAFMOKNode, which contains theirx-,y- and z-coordinate together with their indexn. Accordingly the classAFeDataSet con-tains an array consisting of all nodesADynamicArray<AFMOKNode> m dANodes, thatallows access to a single node via the functionconst AFMOKNode & Node(int nIdx)and provides the total number of nodes byint NumNodes(). The elements on the otherhand are labeled by an indexm running overm= 1, . . . ,M, thus the subsetωm of the refer-

54


Figure 4.1.: MidsurfaceS of the shell Figure 4.2.: Mesh (polygonal approximation)

ence domainω is associated with the elementm. Elements are stored in objects of the classAFMOKShellElement, that is inherited fromAFMOKFiniteElement. Its members in-clude an arrayIm containing the indices of the nodes confining the element, the index ofthe element itself, the element’s thicknesstm, its material type (isotropic or anisotropic) to-gether with the associated material coefficients and a boolean indicating whether the elementis a design element or if its material properties are fixed. These elements are arranged inthe list ADynamicArray<AFMOKFiniteElement*> m dAElements within the classAFeDataSet and accessed via the functionsAFMOKFiniteElement & Element(intnIdx) or AFMOKShellElement & ShellElement(int nIdx). The number of to-tal elements as well as the number of elements of a specific type is extracted by callingintNumElements(),int NumSolidElements()orint NumShellElements(). Thisdata, that specifies the geometry of the shell, is part of the problem definition and provided by aninput file in the BDF file format. BDF is a format that is widely-used by industry in conjunctionwith the finite element software NASTRAN and is especially popular for aerospace applications.The NASTRAN element types considered within this thesis, that are also recorded by a membervariable of the classAFMOKShellElement, are primarily the quadrangular CQUAD4 ele-ments and sporadically the triangular CTRIA3 elements in order to ease the partition in the caseof very complicated geometries.

In Chapter 2 we have defined the local coordinate system of themidsurface (2.1). For thediscretization we calculate a local coordinate system for each elementm by means of the func-tion AFeDataSet::PrepareBasis(). In the case of a CQUAD4 element this functioncalculates the first two basis vectors according to

am1 =

14(rm

3 + rm4 − rm

1 − rm2 ) , (4.1)

am2 =

14(rm

2 + rm3 − rm

1 − rm4 ) , (4.2)

wherermn is the position vector of the node, that is then-th list entry of the index setIm asso-

ciated with elementm as depicted in 4.3. Note that the four nodes of a CQUAD4 element donot have to lie in the same plane. Thus the orientation of the plane spanned by the basis vectors

55


a

a

a

r

rr

r1

2

3

2

1

3

4

Figure 4.3.: Basis for CQUAD4 elements Figure 4.4.: Basis for CTRIA3 elements

am1 andam

2 is an intermediate orientation compared with the orientations of all triangles confinedby nodes of the considered CQUAD4 element. For a CTRIA3 element these basis vectors arecalculated in the following way

am1 = rm

3 − rm1 , (4.3)

am2 = rm

2 − rm1 . (4.4)

Afterwards the basis vectoram3 is computed as normalized cross product ofam

1 andam2

am3 =

am1 ×am

2

‖am1 ×am

2 ‖. (4.5)

The basis vectorsam1 andam

2 are not normalized, as the information about their length isrelevantfor the transformation between the original element and thereference element in Subsection4.1.3. The discrete local basis is then stored as a member of eachAFMOKShellElement. Thebasis is accessed via the functionADynamicArray<double> GetBasis()and its inversevia void GetInverseBasis(ADynamicArray<double> & dAInvBasis). Whilethe former functions are required for the transformation tothe reference element, the func-tion ADynamicArray<double> GetNormalisedBasis() allows simple rotations ofthe element.

Figure 4.5.: Normal vectorat nodes

The basis vectoram3 provides for each elementm a normal di-

rection. However, for the calculation of the fundamental forms werequire information about the change of the normal vectors overthe entire surface. To this end a normal map for the nodes is cal-culated by averaging the normals of all adjacent elements inthevoid functionAFeDataSet::PrepareNormals(). Essen-tially this function generates a list with a normal vector for eachnodean

3, that is initially set to zero. Then it performs a loop overall elements and adds the element’s normal vectoram

3 to the nor-mal of the nodesan

3 belonging to the element. Thereby an av-erage normal vector is calculated as depicted in Fig. 4.5, asallelement normal vectorsam

3 are normalized. Afterwards the newnormal vectors associated with nodes are normalized and foreachelement the normals of the incorporated nodes are stored as a

56


1 2 3

4 5 6

Element 1:

1 4 5 2

Element 2:

3 6 5 2

a

a

1

2

3

3

a3

n

Figure 4.6.: Incorrect normal calculation

1 2 3

4 5 6

Element 1:

1 4 5 2

Element 2:

2 5 6 3

a a1 23 3

a3

n

Figure 4.7.: Proper normal calculation

member ofAFMOKShellElement. They are accessible for later use through the functionADynamicArray<double> GetNormal().

However, the sorting of the nodes within the element definition strongly affects the calcu-lation of the normals. The reason for this is the derivation of the basis vectors as shown inFig. 4.3 and Fig. 4.4, that depends entirely on the arrangement of the nodes. As a result wemight obtain errors for the normal calculation in the case ofa inconsistent orientation of thenodes. An example is depicted in Fig. 4.6, where the oppositeorientation of two neighboringparallel elements leads to vanishing normals. The correct result for the normals at the nodesis given in Fig. 4.7. As a remedy for this problem we have incorporated a sorting routine toavoid a faulty normal calculation. Based on the orientationof the first element thevoid func-tion AFeDataSet::SortFlatElements() rearranges the sequence of the nodes in theneighboring elements in order to ensure a consistent alignment. This procedure is conductedover the entire mesh. Note that industrial meshing tools such as e.g. HyperMesh or ANSYScreate meshes with a consistent element orientation for single surfaces. Hence the previouslymentioned problem occurs only for structures composed of several parts, where the differentsegments exhibit different element orientations.

Next we calculate the derivative of the normal vector with aids of a finite difference scheme.To this end we compute the center point of the elementm, that containsNm nodes inIm, as

rmc =

1Nm

∑n∈Im

rn , (4.6)

which is associated with the element normal vectoram3 , and moreover we define normalized

basis vectors

amα =

amα

‖amα‖

. (4.7)

The normal vector at the noden is denoted byan3. Then the discretized derivative of the normal

vector is calculated as

am3,α =

1Nm

[

∑n∈Im

(an3−am

3 )⊤am

α(rn− rm

c )⊤am

α

]

amα . (4.8)

The derivatives of the normal vector are stored as members ofthe classAFMOKShellElementand accessible via the functionADynamicArray<double> GetNormalDerivative().

57


Finally we calculate the necessary fundamental forms of thediscretized midsurface. As theseare based on the local basis of a specific element, the fundamental forms are considered tobe constant throughout each single element. For the computation of the second fundamentalform bα

β of the elementm we require the first fundamental formaαβ , that is provided by thevoid function GetFirstConConFundamentalForm(ADynamicArray<double> &dAFirstFund). Due to (2.3) we employ an orthogonalization procedure within this functionin order to construct the contravariant basis vectors as

a1m = am

1 − am1⊤am

2

am2⊤am

2

am2 , (4.9)

a2m = am

2 − am1⊤am

2

am1⊤am

1

am1 . (4.10)

After a normalization according to

a1m =

a1m

‖a1m‖

, a2m =

a2m

‖a2m‖

, (4.11)

the first fundamental formaαβ of the elementm is obtained by

aαβ =

(

a1m⊤

a1m a1

m⊤

a2m

a2m⊤

a1m a2

m⊤

a2m

)

. (4.12)

Following the definitions (2.13) and (2.14) the second fundamental forms are computed as

bαβ =

(

−am3,1

⊤am1 −am

3,2⊤am

1

−am3,1

⊤am2 −am

3,2⊤am

2

)

, (4.13)

bαβ =

(

−am3,1

⊤a1λ amλ −am

3,2⊤a1λ am

λ−am

3,1⊤a2λ am

λ −am3,2

⊤a2λ amλ

)

, (4.14)

and accessible for the computations of the strains in the next subsection via thevoid functionsGetSecondCoCoFundamentalForm(ADynamicArray<double> & dAarray)andGetSecondCoConFundamentalForm(ADynamicArray<double> & dAarray).

cαβ =

(b1

1b11+b21b21 b1

1b12+b21b22

b12b11+b2

2b21 b12b12+b2

2b22

)

(4.15)

finally is the recipe to calculate the third fundamental formas introduced in (2.19) and is realizedin thevoid functionGetThirdCoCoFundamentalForm(ADynamicArray<double>& dAThirdFund).

4.1.2. Discrete Continuum Mechanics

The next step after the discretization of the midsurface is the discretization of all physical quan-tities appearing in the equilibrium condition. We start with the displacements(u,θ) ∈U , whose

58


first derivative is required for the computation of the strains. Therefore the approximation of thedisplacements should be at least linear and fulfill the Reissner-Mindlin kinematical assumptionat all nodes of the mesh. To this end we introduce the bilinearLagrange shape functionsϑi(ξ ,η)from the setQ1 of piecewise-linear functions for quadrilateral elements

ϑ1(ξ ,η) =14(1−ξ )(1−η) , (4.16)

ϑ2(ξ ,η) =14(1−ξ )(1+η) , (4.17)

ϑ3(ξ ,η) =14(1+ξ )(1+η) , (4.18)

ϑ4(ξ ,η) =14(1+ξ )(1−η) (4.19)

and from the setP1 of piecewise-linear functions for triangular elements

ϑ1(ξ ,η) = 1−η −ξ , (4.20)

ϑ2(ξ ,η) = η , (4.21)

ϑ3(ξ ,η) = ξ . (4.22)

As a result we are able to formulate the following expressionfor the displacements as the discretecounterpart to (2.28)

Un =Nm

∑i=1

ϑi(ξ ,η)(

uni +ztm2

θni

)

, (4.23)

that leaves all material fibres straight and unstretched during deformation and hence guaranteesconsistency with the Reissner-Mindlin kinematical assumption. The discrete version of(u,θ)is then given by the vector(u1, . . . ,uN,θ1, . . . ,θN) ∈ R

6N, whereun and θn correspond to thedisplacements associated with the noden (accordingly entries 3(n−1)+1, 3(n−1)+2 and 3nof the displacement vector arrays). Note that we consider three rotational degrees of freedomand hence a total of six degrees of freedom for each node. The reason for this is that the dis-placements are given with respect to the global coordinate system and the transformation fromthe local to the global coordinates blows up the rotational degrees of freedom to a vector ofdimension three. However, in order to avoid the inclusion ofdrilling into the discrete model wetake the additional condition

an3⊤θn = 0 ∀n∈ 1, . . . ,N (4.24)

for each node during the assembly of the global stiffness matrix in Subsection 4.1.4 into account.This formulation will also allow to incorporate a differenttype of elements with six degrees offreedom via condensed data structures to the problem as described in Section 4.4.2.

Together with the displacements also the homogeneous Dirichlet boundary conditions aretranslated into the discrete context. Fixed displacementson the Lipschitz boundary∂ω0 areconverted into fixed degrees of freedom at single nodes lyingon the discrete version of theboundary∂ω0. As the values of these displacements are fixed, we do not haveto regard them asfree parameters during the solution of the static system. Instead these displacements are entirely

59


removed from the problem and set to their prescribed values at the end of the optimizationroutine. This is accomplished by a multiplication of the displacement vector with the matrixTactive∈R

DOFNr×6N, that simply removes all entries describing inactive degrees of freedom fromthe displacement vector leading to a vector of length DOFNr,the number of all active degreesof freedom

(un,θn) := Tactive· (u1, . . . ,uN,θ1, . . . ,θN) (4.25)

Hence(un,θn) ∈ RDOFNr has to be understood as the discrete displacement vector containing

only active degrees of freedom and(u1, . . . ,uN,θ1, . . . ,θN) ∈ R6N as the discrete displacement

vector containing all degrees of freedom. Accordingly we define

Un :=

(un,θn) ∈ RDOFNr

∣∣∣an

3⊤θn = 0 ∀n= 1, . . . ,N

. (4.26)

The opposite mapping transforming(un,θn) into the original vector(u1, . . . ,uN,θ1, . . . ,θN) isachieved via a multiplication with the matrixTactive

⊤, that sets all entries associated with fixeddegrees of freedom to the value zero. The classAFeDataSet contains several arrays thatprovide information about the correlation between nodes and their associated degrees of free-dom and the indices in the global list of active degrees of freedom. On the one hand thisis the arraym dAnDOFList, that is accessible via theADynamicArray<int> functionDOFList() and contains for each noden the index of the first degree of freedom belong-ing to that node. Hence the degrees of freedom associated with the noden lie in the interval[m dAnDOFList[n], m dAnDOFList[n+1]-1]. The arraym dAnDOFMap on the otherhand possesses six entries for each noden, which are identified with the three translationaldegrees of freedom and the three rotational degrees of freedom. If the considered degree offreedom is active, then the corresponding entry is set to theassociated index of the global list ofdegrees of freedom. If the degree of freedom is fixed due to Dirichlet boundary conditions, theentry is set to -1 and if the degree of freedom does not exist (as for example rotational degreesof freedom for nodes only connected to solid elements), it isset to -2. This array is provided bytheADynamicArray<int> functionDOFMap().

This reduction of the considered degrees of freedom effectsalso the discrete load vector. Dur-ing the transition to the discrete problem the loadsf , gu and gθ are converted into a vector( f1, . . . , fN) ∈ R

6N containing the discrete load values for each separate node and its associateddegrees of freedom. However, loads applied at fixed degrees of freedom are not able to cause adeformation and thus they can simply be neglected. This is again accomplished by multiplica-tion with the matrixTactive, hence the discrete load vectorfn ∈ R

DOFNr containing only entriesassociated with active degrees of freedom is given by

fn := Tactive· ( f1, . . . , fN) . (4.27)

The loads are stored in the arraym dALoads within the classAFeDataSet. The number ofload cases is obtained from theint function NumLoadCases(). The weight and the loadvector of a specific load case are then extracted via the functions GetLoadWeight (intl) andGetLoadVector (int l).

Next we proceed with the derivation of the discrete strains.In the continuous context thestrains consist of derivatives of the displacements(u,θ). In the discrete problem formulation

60


this translates to derivatives of the shape functions, as all spatial dependence has been transferredfrom (u,θ) to ϑi(ξ ,η) in the displacement definition (4.23). The following definition of the

strain-displacement matricesBγi , Bχ

i andBζi associated with the membrane strainsγ , the bending

strainsχ and the shear strainsζ , respectively,

Bγi =

ϑi|1 0 −b11ϑi 0 0 00 ϑi|2 −b22ϑi 0 0 0

1√2ϑi|2

1√2ϑi|1 −

√2b12ϑi 0 0 0

, (4.28)

Bχi =

−b11ϑi|1 −b2

1ϑi|1−b1

2ϑi|2 −b22ϑi|2 . . .

− 1√2(b1

2ϑi|1+b11ϑi|2) − 1√

2(b2

2ϑi|1+b21ϑi|2)

c11ϑi ϑi|1 0 0. . . c22ϑi 0 ϑi|2 0√

2c12ϑi1√2ϑi|2

1√2ϑi|1 0

, (4.29)

Bζi =

( 12b1

1ϑi12b2

1ϑi12ϑi,1

12ϑi 0 0

12b1

2ϑi12b2

2ϑi12ϑi,2 0 1

2ϑi 0

)

, (4.30)

allows to deduct the strains corresponding to the shape function ϑi by a multiplication of thestrain-displacement matrix with the displacement vector(un,θn). Note that this definition of thestrain-displacement matrices refers to the reference element that is the domain of definition forthe shape functions and specified in the following subsection. As a result we have to insert abasis transformationSm derived from the local element basisam

1 ,am2 ,a

m3 in between in order to

guarantee that the strain-displacement matrices and the displacement vector are observed in thesame coordinate system. The membrane strain vectorγm for the elementm is thus given by

γm(x) =Nm

∑i=1

Bγi (x)Sm(uni ,θni ) (4.31)

and expressions for the bending and shear strains can be defined accordingly. Note further, thatthe factors of

√2 necessary due to the Mandel notation are consistently included in the formulas

for Bγi andBχ

i .We proceed with the discretization of the material matricesC and D. These are assumed

to be element-wise constant, thus they are approximated by vectors, whosem-th entry is thematerial matrix for elementm. Hence the discretized material matrix vector(Cm,Dm) has to beunderstood as(C1, . . . ,CM,D1, . . . ,DM), whereCm ∈ R

3×3 andDm ∈ R2×2 for all m= 1, . . . ,M

and we define the admissible set for the discrete material matrices as

Cm =(Cm,Dm) ∈R

9M∣∣Cm ∈ S

3, Dm ∈ S

2 ∀m= 1, . . . ,M, (4.32)

whereSd is the space of symmetricd× d-matrices. The thickness is also considered to beelement-wise constant, thus we have a real positive valuetm associated with each elementm.Finally we have to decide on the value of the shear correctionfactor k. This factor lies inthe interval [0,1], however it can only be exactly determined for some special cases such as

61


isotropic material (k = 56) or orthotropic material (k = 2

3). We can not predict the materialsymmetry obtained by Free Material Optimization beforehand, however it is known that theoptimal material approaches isotropic material symmetry with an increasing number of loadcases. These are typical for the industrial testcases we consider, hence we set the shear correctionfactor to the valuek= 5

6.

4.1.3. Numerical Integration

After the discretization of the physical quantities appearing in the equilibrium condition we willproceed with a numerical method to approximate the integrals. First we decompose the integralover the entire reference domainω into separate integrals over the single elementsωm

∫

ωf (x,y)dxdy=

M

∑m=1

∫

ωm

f (x)dx. (4.33)

By using a variable transformation the integrals overωm are then converted into integrals overthe reference elementωref. The reference element for CQUAD4 elements consists of the square[−1,1]× [−1,1], while the reference element for CTRIA3 elements consists of the triangle withthe corner points (0,0), (1,0) and (0,1). We denote byJref =

∂ (x,y)∂ (ξ ,η) the Jacobi determinant for

the variable transformation between the global coordinatesystem and the reference elementcoordinate system, hence the integral associated with the elementmcan be written as

∫

ωm

f (x,y)dxdy=∫

ωref

f (x(ξ ,η),y(ξ ,η))Jref dξ dη =

∫

ωref

f (ξ ,η)Jref dξ dη . (4.34)

For the integral over the reference element we now employ a numerical integration procedureknown as Gauss quadrature. It replaces the integral by a weighted sum over function evaluationsat single points known as Gauss pointsξg. Thus we obtain

∫

ωref

f (ξ ,η)Jref dξ dη ≈GPNr

∑g=1

wg f (ξg)Jref , (4.35)

wherewg is the weight of the Gauss pointg and GPNr is the total number of Gauss points.There exist various possibilities on the number and position of used Gauss points, where theapproximation accuracy increases with the number of used Gauss points. Within this thesis wewill employ the following integration rules:

CQUAD4 elementGPNr= 4

g wg ξg

1 1.0 (− 1√3,− 1√

3)

2 1.0 (− 1√3,+ 1√

3)

3 1.0 (+ 1√3,+ 1√

3)

4 1.0 (+ 1√3,− 1√

3)

CTRIA3 elementGPNr= 1

g wg ξg

1 1.0 (+13,+

13)

62


4.1.4. Assembly of the Global Stiffness Matrix

We are now able to perform the assembly of the global stiffness matrix. To this end we deductthe element stiffness matrices of the elementωm as

Kγ(Cm) =GPNr

∑g=1

Nm

∑i, j=1

tmSm⊤Bγ

i⊤(ξg)CmBγ

j (ξg)SmJref ,

Kχ(Cm) =GPNr

∑g=1

Nm

∑i, j=1

tm3

12Sm

⊤Bχi⊤(ξg)CmBχ

j (ξg)SmJref , (4.36)

Kζ (Dm) =GPNr

∑g=1

Nm

∑i, j=1

tmkSm⊤Bζ

i

⊤(ξg)DmBζ

j (ξg)SmJref .

We also define for further use

Kshell(Cm,Dm) :=EltNr

∑m=1

Kγ(Cm)+Kχ(Cm)+Kζ (Dm) . (4.37)

During the assembly of the global stiffness matrixKshell(C,D) we have to include condition(4.24), where the vector product can be written elaboratelyas

(an3)1(θn)1+(an

3)2(θn)2+(an3)3(θn)3 = 0. (4.38)

This condition is included via a penalty function to the minimum potential energy formulationof the equilibrium problem

min(un,θn)∈Un

Πp(un,θn) =12(un,θn)

⊤Kshell(Cm,Dm)(un,θn)− fn⊤(un,θn)

+12

p((an3)1(θn)1+(an

3)2(θn)2+(an3)3(θn)3)

2 (4.39)

wherep is a penalty parameter. The equilibrium condition∇uΠp(un,θn) = 0 then translates into

Kshell(Cm,Dm)(un,θn)+ pN

∑n=1

(an3)1 · (an

3)1 (an3)1 · (an

3)2 (an3)1 · (an

3)3

(an3)2 · (an

3)1 (an3)2 · (an

3)2 (an3)2 · (an

3)3

(an3)3 · (an

3)1 (an3)3 · (an

3)2 (an3)3 · (an

3)3

(θn)1

(θn)2

(θn)3

= 0.

Moreover we define the discrete version of the dyadic products γγ⊤, χχ⊤ andζζ⊤ as

Aγm(un) =

GPNr

∑g=1

Nm

∑i, j=1

wgBγj (ξg)Sm(un,θn)(un,θn)

⊤Sm⊤(Bγ

i )⊤(ξg) , (4.40)

Aχm(un,θn) =

GPNr

∑g=1

Nm

∑i, j=1

wgBχj (ξg)Sm(un,θn)(un,θn)

⊤Sm⊤(Bχ

i )⊤(ξg) , (4.41)

Aζm(un,θn) =

GPNr

∑g=1

Nm

∑i, j=1

wgBζj (ξg)Sm(un,θn)(un,θn)

⊤Sm⊤(Bζ

i )⊤(ξg) . (4.42)

63


4.2. Discretized Free Material Optimization Problems forPlates and Shells

The discretization introduced in the previous section allows to convert the optimization problemspresented in Chapter 3 to a discrete framework. This is achieved by replacing all quantities bytheir discrete counterparts. The obtained discrete problem formulations are then suited for anumerical solution by using nonlinear semidefinite programming solvers.

We begin with the primal Free Material Optimization problemfor shells (PP). After a transi-tion to the discrete context we obtain its discrete version (Pm

P )

max(un,θn)∈U\

α∈R+

βu,βl∈RM+

f⊤n (un,θn)−Vα −M

∑m=1

(ρ+β m

u −ρ−β ml

)(Pm

P )

subject tot2

Aγm(un)+

t3

24Aχ

m(un,θn)− t(α +β mu −β m

l )13 0 , m= 1, . . . ,M ,

t2

kAζm(un,θn)−

t2(α +β m

u −β ml )12 0 , m= 1, . . . ,M .

Problem (PmP ) is a finite-dimensional nonlinear semidefinite program. Due to its convexity each

KKT-point of problem (PmP ) is also a global minimizer. The material matrices(Cm,Dm) are hid-

den in this problem formulation as Lagrange multipliers of the semidefinite matrix constraints.This leads to a considerable reduction of the problem dimension, because instead of the mate-rial matrices, that are resembled by 9 variables per elementand thus by a total of 9M variables,this problem formulation contains the skalar variableα and the variablesβu,βl ∈ R

M+ , that are

described by 2M+1 variables.Next we transfer the minimum weight problem formulation (PMW) to the discrete context and

obtain problem (PmMW) as

min(un,θn)∈Un(Cm,Dm)∈Cm

M

∑m=1

tm · trCm+tm2· trDm (Pm

MW)

subject to Cm 0; Dm 0 ∀m= 1, . . . ,M(

tmCm 00 tm

2 Dm

)

− ε15 0 ∀m= 1, . . . ,M

tm · trCm+tm2· trDm ≤ ρ+ ∀m= 1, . . . ,M

Kshell(Cm,Dm)(un,θn) = fnfn⊤(un,θn) ≤ c.

This is a nonlinear nonconvex semidefinite program. As we ensure by the lower bound on theeigenvalues that the material matricesCm andDm are positive definite for allm= 1, . . . ,M, thisresults in a positive definite stiffness matrixKshell(Cm,Dm). This guarantees the existence of aninverse matrixKshell

−1(Cm,Dm). Accordingly we are able to use the equilibrium condition in

64

4.2. Discretized Free Material Optimization Problems for Plates and Shells

order to substitute the displacements(un,θn) by

Kshell(Cm,Dm)(un,θn) = fn ,

⇒ (un,θn) = Kshell−1(Cm,Dm) fn . (4.43)

By inserting formula (4.43) into problem (PmMW) we remove the state variable(un,θn) from the

optimization problem and obtain a reduced minimal weight problem formulation (PredMW)

min(Cm,Dm)∈Cm

M

∑m=1

tm · trCm+tm2· trDm (Pred

MW)

subject to Cm 0; Dm 0 ∀m= 1, . . . ,M(

tmCm 00 tm

2 Dm

)

− ε15 0 ∀m= 1, . . . ,M


fn⊤Kshell

−1(Cm,Dm) fn ≤ c.

This problem is convex, hence we favor it over the previous problem formulation (PmMW). The

Free Material Optimization problem for shells in the reduced minimum weight formulation isthus a convex nonlinear semidefinite program.

Finally we consider the minimum compliance formulation (PMC) and obtain its discrete ver-sion (Pm

MC) by inserting the discretized counterparts of the quantities in the problem formulation


fn⊤(un,θn) (Pm

MC)

subject to Cm 0; Dm 0 ∀m= 1, . . . ,MM

∑m=1

tm · trCm+tm2· trDm ≤ V

(tmCm 0

0 tm2 Dm

)

− ε15 0 ∀m= 1, . . . ,M


Kshell(Cm,Dm)(un,θn) = fn .

Again we are able to employ the positivity of the global stiffness matrixKshell(Cm,Dm) in orderto transform the nonconvex nonlinear semidefinite program (Pm

MC) into its reduced form

min(Cm,Dm)∈Cm

fn⊤Kshell

−1(Cm,Dm) fn (PredMC)

subject to Cm 0; Dm 0 ∀m= 1, . . . ,MM

∑m=1


(tmCm 0

0 tm2 Dm

)

− ε15 0 ∀m= 1, . . . ,M


65


and thus obtain a convex nonlinear semidefinite program.

The numerical implementation of these problem definitions is realized in theFMOlib. Forthe minimum weight formulation the volume objective is calculated in thedouble functionAFMOVolumeObjective::Eval according to

fvol(Cm,Dm) :=M

∑m=1

tm · trCm+tm2· trDm. (4.44)

In the minimum compliance formulation the volume is employed as constraint. Its evaluation isperformed in thedouble functionAFMOVolumeConstraint::Eval

gvol(Cm,Dm) :=M

∑m=1

tm · trCm+tm2· trDm−V . (4.45)

The gradient for both functions is identical. We calculate the derivative with respect to the entryin the p-th row andq-th column of the material matrix associated with the element m as

∂ fvol(Cm,Dm)

∂ (Cm)p,q=

∂gvol(Cm,Dm)

∂ (Cm)p,q= δpqtm , (4.46)

∂ fvol(Cm,Dm)

∂ (Dm)p,q=

∂gvol(Cm,Dm)

∂ (Dm)p,q= δpq

tm2. (4.47)

The implementations of these derivatives are accessible via theAFMOKSparseGradient*functionsAFMOVolumeObjective::GradandAFMOVolumeConstraint::Grad. Onthe other hand the compliance objective for the minimum compliance formulation is calculatedin thedouble functionAFMOComplianceObjective::Eval as

fcomp(Cm,Dm) := fn⊤(un,θn) . (4.48)

In the case of the minimum weight formulation the complianceconstraint is evaluated accordingto

gcomp(Cm,Dm) := fn⊤(un,θn)−c. (4.49)

in thedouble function AFMOComplianceConstraint::Eval. Again the gradients ofthe objective and the constraint are identical, as their difference is only the constant value of thecompliance bound. For the calculation of the derivatives intheAFMOKSparseGradient*functionsAFMOVolumeObjective::Grad andAFMOVolumeConstraint::Grad wehave to keep in mind, that we calculate derivatives with respect to the optimization variables(Cm,Dm). Thus we require a derivation of the state variable(un,θn) with respect to the materialmatrices, which we obtain be employing formula (4.43). Hence the derivative of the compliancewith respect to the entry in thep-th row andq-th column of the material matrix associated with

66

4.3. PENSCP – A Solver for Nonlinear SDPs

the element ˆm is given by

∂ fcomp(Cm,Dm)

∂ (Cm)p,q=

∂gcomp(Cm,Dm)

∂ (Cm)p,q= fn

⊤ ∂ (un,θn)

∂ (Cm)p,q

=− fn⊤Kshell

−1(Cm,Dm)∂(Kγ

m(Cm)+Kχm(Cm)

)

∂ (Cm)p,q(un,θn) , (4.50)

∂ fcomp(Cm,Dm)

∂ (Dm)p,q=

∂gcomp(Cm,Dm)

∂ (Dm)p,q= fn

⊤ ∂ (un,θn)

∂ (Dm)p,q

=− fn⊤Kshell

−1(Cm,Dm)∂Kζ

m(Dm)

∂ (Dm)p,q(un,θn) . (4.51)

4.3. PENSCP – A Solver for Nonlinear SDPs

For the solution of the discretized optimization problems presented in the previous section thenonlinear semidefinite solvers PENBMI [KS03] and PENSCP [SKL09b] are available, whichare well suited to the specific problem structure arising from the Free Material Optimizationproblem formulation. Before the recent development of PENSCP the most successful methodto solve Free Material Optimization problems was provided by PENBMI. However this solverrequires a particular problem structure: a semidefinite program with a quadratic objective andbilinear matrix inequality constraints. Therefore it is only viable for the solution of problem (Pm

P )and not able to solve Free Material Optimization problems with multidisciplinary optimizationconstraints as for example the optimization problems presented in Chapter 5. This is not theonly drawback of PENBMI in comparison with PENSCP: the computational complexity of thenumerical method employed by PENBMI depends cubically on the number of load cases. Thusonly a limited number of load cases can be included, which is not sufficient for industrial appli-cations, where the problem definition typically contains a large number of load cases. As thisrestriction does not apply to the method implemented withinPENSCP, it is considerably bettersuited for the solution of realistic problems. As a result PENSCP has become the state-of-the-artsolver for the solution of Free Material Optimization problems and we employ it for the calcu-lation of the numerical testcases in the following section.PENSCP has been developed by M.Stingl and M. Kocvara. In this section we give an overview ofits functioning as described in[SKL09b].

PENSCP has been developed to solve the following general type of optimization problems

minY∈S

f (Y) (P)

subject to gk(Y)≤ 0, k= 1,2, . . . ,K ,

Yi Sdi Yi Sdi Yi , i = 1,2, . . . ,m,

67


whereSd denotes the space of symmetricd×d-matrices. Accordingly the definitions

S :=Sd1 ×S

d2 × . . .×Sdm ,

Y :=(Y1,Y2, . . . ,Ym) ,

(d1,d2, . . . ,dm) ∈Nm

I :=1,2, . . . ,m

are used and the relationA Sd B has to be understood as 0Sd B−A, in other words,B−Abelongs toSd

+: the cone of all positive semidefinite matrices inSd. PENSCP has been inspired by

the codes CONLIN by Fleury [Fle89], the method of moving asymptotes by Svanberg [Sva87]and SCP by Zillober [Zil01]. All of these methods have been effectively applied to topologyoptimization problems with scalar design variables such asthe density or the thickness, andemploy separable convex first order approximations of nonlinear functions for the numericalsolution of the optimization problem. The fundamental ideaof PENSCP is the generalizationof this concept to functions, which are defined on matrix spaces, in order to make this methodapplicable for the matrix variables used in Free Material Optimization problems.

To this end a block-separable convex approximation scheme is proposed. For the functionf : S→ R , which is continuously differentiable on a subsetB ⊂ S, the differential operatorsare entry-wise defined as

(∇i f)

p,q :=

(∂ f∂Yi

)

p,q, 1≤ p,q≤ di . (4.52)

The projections of∇i f (Y) ontoSdi+ andSdi

− are indicated by∇i+ f (Y) and∇i

− f (Y), respectively.Moreover denote by〈·, ·〉

Sdi the standard inner product onSdi , that is given by〈A,B〉Sdi = tr(AB)

for anyA,B∈ Sdi . For a set of non-negative real parametersτ := τ1,τ2, . . . ,τm and asymptotes

L = (L1,L2, . . . ,Lm)⊤ andU = (U1,U2, . . . ,Um)

⊤, that fulfill

Li ≺S

di+

Yi ≺S

di+

Ui ∀i ∈ I ,

the hyperbolic approximationf L,U,τY

of f atY is defined as

f L,U,τY

(Y) := f (Y)+m

∑i=1

〈∇i+ f (Y),(Ui −Yi)(Ui −Yi)

−1(Ui −Yi)− (Ui −Yi)〉Sdi

−m

∑i=1

〈∇i− f (Y),(Yi −Li)(Yi −Li)

−1(Yi −Li)− (Yi −Li)〉Sdi

+m

∑i=1

τi〈(Yi −Yi)2,(Ui −Yi)

−1+(Yi −Li)−1〉

Sdi . (4.53)

This hyperbolic approximation is not only convex, furthermore it holds that

f (Y) = f L,U,τY

(Y) ,∂

∂Yif (Y) =

∂∂Yi

f L,U,τY

(Y) ∀i ∈ I .

68

4.4. Numerical Testcases and Results

By employing the functionf L,U,τY

(Y) a local approximation of problem (P) is given by

minY∈S

f L j,U j

,τ j

Y j (Y) (P j )

subject to gk(Y)≤ 0, k= 1,2, . . . ,K ,

Yij

Sdi Yi Sdi Yij, i = 1,2, . . . ,m.

This gives rise to the following sequential convex programming algorithm for matrix-valuedfunctions:

Algorithm 4.3.1. Let a starting point Y1 ∈ S and initial multipliers(v1,V1)∈Rk+×S+ be given.

(S1) Set j= 1.

(S2) Choose the asymptotes Lj ∈ L and Uj ∈ U together withτ ≥ τ j1,τ

j2, . . . ,τ

jm ≥ τ > 0.

(S3) Solve problem (Pj ). Denote the solution by Y+ ∈ S and the associated Lagrange multipli-ers by(v+,V+) ∈ R

k+×S+.

(S4) Chooseα j = min1, α, whereα = argminα∈R+f (Y j +α(Y+−Y j)).

(S5) (Y j+1,v j+1,V j+1) = (Y j ,v j ,V j)+α j((Y+,v+,V+)− (Y j ,v j ,V j)).

(S6) If Yj+1 is stationary for problem (P): STOP. Otherwise put j= j +1 and go to (S2).

The subproblem (P j ) can be solved by a generalized augmented Lagrangian methodfor thesolution of nonlinear semidefinite programs as for instancethe solver PENNON [KS03]. Notethat it is possible to show global convergence of Algorithm 4.3.1 with some additional assump-tions. The original paper [SKL09b] does not only provide a more general description of theseconcepts, that is also valid for multiple load cases, but contains also explicit information aboutthe choice of the asymptotesL andU as well as their admissible setsL andU .


We have implemented the numerical methods presented in the previous sections within theFMOKernel of the PLATO-N software platform. In this section we discussthe numericalresults obtained by this software for several testexamplesthat are frequently employed in thestructural optimization of shells and plates.

Testcase 1: Simply supported square plate under a central load.The first testcase is a square plate with a transverse load at the center of the square. The plate

is supported at all edges, thus homogeneous Dirichlet boundary conditions are applied to thetranslational degrees of freedomu as depicted in Fig. 4.8. This testexample has been employedby [SD93] in order to compare the results of a design optimization for a thin-walled structuredescribed by the Reissner-Mindlin plate model, the Kirchhoff plate model or a two-dimensionalmodel. The obtained density for the Reissner-Mindlin platemodel is similar to the optimal

69


cross−section view

top view

t

L

L

y

x

L

f

Figure 4.8.: Problem definition Figure 4.9.: Optimal stiffness distribution

Figure 4.10.:In-plane strain

Figure 4.11.:Out-of-plane strain

Figure 4.12.:In-plane stress

Figure 4.13.:Out-of-plane stress


top view

t

L

L

L

y

x

f 1

f 3

f

f 3

1

f

f 4

2

f

f

4

2






70


top view

t

f

L

L

y

x






stiffness distribution, that we obtained according to Fig.4.9. The structure forms a stiff centralpart to withstand the applied force and choses the shortest way to the edges in order to reach thefixed boundary. As a result the structure is connected to the central points of the edges, howeverthere is almost no material distributed in the corner regions. Fig. 4.10 – 4.13, which present thestrain and stress distribution, show that for the given lateral lengthL = 1000mmand thicknesst = 10mm the material is mostly used to increase the stiffness with respect to membrane andbending behavior.

Testcase 2: Simply supported square plate under multiple load cases.We consider again a square plate with fixed translational displacementsu= 0 at the edges of

the plate. Four symmetrical distributed loads are applied to the plate, which act independently.A solution for the resulting multiple load case problem was obtained in [DLS95] by employingcomposites up to rank three. The solution provided by Free Material Optimization as givenin Fig. 4.15 displays the same density distribution. In a multiple load scenario there exists aseparate displacement field and accordingly also strains and stresses for each load case. Thestrains and stresses shown in Fig. 4.16 – 4.17 belong to the load in the lower left corner. Due tothe symmetry of the problem formulation the strains and stresses of the other load cases have anidentical shape. It can be seen that the strains and stressesstrive to the nearest boundary as thisis the easiest way to avoid large displacements.

Testcase 3: Corner supported square plate under a central load.This testexample is another variation of the square plate, that has e.g. been addressed in

[OC02]. A transverse load is applied in the center of the plate, however it is only fixed at thecorners. As a result the optimal structure (Fig. 4.21) consists in a stiff segment at the center,which objects the force and is connected to the corners in order to adhere the structure to thefixed parts of the boundary. This is also reflected in the plot of the in-plane stresses in Fig. 4.24.that are naturally strongly affected by the material distribution.

71



top view

t

yL

L

L

1

2

2

f

x






top view

f

x

y

t

L

L

H






72


L

L

f

x

yz






Testcase 4: Simply supported rectangular plate under a central load.In this example we investigate the influence of a loss of the symmetric geometry of the square

by replacing it with a rectangle. As predicted by [Lip94, SD93] the material distribution willstrive to the closest part of the boundary, that is fixed by Dirichlet boundary conditions. Accord-ingly the optimal stiffness distribution in Fig. 4.27 possesses a strongly developed connectionto the center of the long edges. A close look at these links reveals the formation of several ribsbetween the center of the structure and the edge. [Pet99] deals with the theoretical backgroundfor this behavior, which is known as the checkerboard problem. It is a frequently appearingnumerical effect and can be avoided by using higher-order elements.

Testcase 5: Corner supported parabolic shell under a central load.We consider a parabolic shell, that is pinned at the corners due to homogeneous Dirichlet

conditions and subjected to a downward force at the center. This testexample has been optimizedusing Discrete Material Optimization by [SL05], who obtained a similar stiffness distribution tothe Free Material Optimization design depicted in Fig. 4.33. Similar to Testcase 3 the structureconnects the central section to the four corners. However, due to the initial curved shape of theshell it will react to the applied force not only with a downward movement of the central part,but also with an increase of the curvature in segments between the center and the edges.

Testcase 6: Corner supported half cylinder under a central load.This testexample has been inspired by the Scordelis-Lo roof, a cylinder segment loaded by

its self-weight, that is an often used testcase for shells [CB03] and their optimization [Mau98].However, the definition of self-weight is unclear in the caseof Free Material Optimization,because we have no information about the weight of the optimal structure and estimate it by

73


using the trace of the material tensor. For this reason the testexample was modified by replacingthe self-weight by a central downward force. It is noteworthy, that the optimal structure doesnot only develop connections to the corners, that are fixed byhomogeneous Dirichlet boundaryconditions. In contrast to the previously considered testexample the optimal structure creates acircular strip to avoid a sidewards bending due to the applied force, therefore this testexamplerepresents not only material optimization, but also topology optimization, as the optimal designcontaines holes in the shell structure.

4.4.1. Structures Combined of Solids and Shells

The previous testexamples demonstrate the capabilities ofour approach. Beyond the stiffnessdistributions depicted in the presented figures the solution consists of the optimal material ten-sors for each element, hence we obtain detailed informationabout symmetries and directions ofthe optimal material. However, in realistic problems shells usually do not occur as standalonestructures. They are attached to a framework consisting of solid material or trusses. This moti-vates the optimization of structures, that are built up as a combination of solids and shells, whichis referred to as the research field of elastic junctions. Thetheoretical background for this prob-lem [Cia90, Naz97, BC98] is dominated by the discontinuities arising from the combination ofthe different structures. However, we will approach this problem on a discrete level and employa simple, yet functional technique: we merge the translational degrees of freedom of the shelland the solid in shared nodes. The rotational degrees of freedom of the shell are allowed tomove freely, however it is possible to assess a certain valueto the rotational degrees of freedomat shared nodes by using Dirichlet boundary conditions. Theimplementation of this approach isachieved by a reasonable administration of the degrees of freedom. An illustration is providedby the following testexample.

74


Figure 4.44.: Optimal solid stiffness Figure 4.45.: Optimal shell stiffness

Testcase 7: Torsionbox.

The considered box is fixed via homogeneous Dirichlet boundary conditions at the loweredges and subjected to loads at the upper edges, that induce atorsion of the box. A comparisonof the optimal stiffness distribution of the solid interiorand the cover, that is modelled by shells,reveals that the solid material is primarily used to fortifythe vertical edges of the box. Dueto the kink of the surface the shells are not able to provide stiffness in these regions. On theother hand they are concentrated in the upper regions of the sides of the box in order to maintainthe distance between the corners. As this is mostly achievedby membrane effects shells arethe superior structures for this task. Note that reasonableresults are only possible due to ourconsistent choice of the combined shell trace, which we havecarefully deducted in Section2.6.2.

4.4.2. Inclusion of Condensed Data Structures

Another useful extension of our software is the inclusion ofcondensed data structures. Theseare typically employed in static analysis in order to reducethe dimension of the equilibriumproblem. Although this procedure is not applicable to elements that belong to the design space, itallows to incorporate parts of the structure constructed from a fixed material without increasingthe problem dimension. An even more appealing usage of condensed data structures is theconsideration of element types, that are not included in oursoftware. The following explanationof static condensation is illustrated by means of a two-dimensional example.

75


Figure 4.46.: Simple beam example

Original Testexample

We consider a simple beam modelled by 10× 2 CQUAD4 elements. The beam is clamped onthe left-hand side (all nodes on the left side of the mesh are fixed in x- andy-direction) and adownward force is applied on the lower right corner of the beam.

This testexample is a two-dimensional example with no degrees of freedom outside of thex-y-plane. The elements are rectangular elements with four bilinear Lagrange shape functionsgiven by

ϑ1(ε ,η) =14(1− ε)(1−η) ,

ϑ2(ε ,η) =14(1+ ε)(1−η) ,

ϑ3(ε ,η) =14(1+ ε)(1+η) ,

ϑ4(ε ,η) =14(1− ε)(1+η)

associated with the four nodes of the element. Numerical integration is conducted by using thefollowing integration points and weights (with respect to the standard reference element):

Integration point Integration weight(

− 1√3,− 1√

3

)

1.0(

1√3,− 1√

3

)

1.0(

1√3,

1√3

)

1.0(

− 1√3,

1√3

)

1.0

The elements colored in red have fixed material properties defined by the material tensor

Efixed =

60000.0 18000.0 0.018000.0 60000.0 0.0

0.0 0.0 10000.0

Thus only the material tensors of the white elements (number6 – 9 and 16 – 19) are optimizedusing Free Material Optimization.

76


Condensation of the Testexample

The next step is the static condensation of this testexample. Only the elements with fixed mate-rial (colored in red) are condensed, so the elements with free material remain in the mesh. Notethat in this case also the boundary conditions and the loads are condensed – the final (condensed)testcase will have elements with free material, a condensedstiffness matrix and a condensed loadvector, but no Dirichlet boundary conditions or external loads.

The data needed for the calculation of the condensated stiffness matrix and the conden-sated load vector is prepared by the FMOKernel and transferred to the external matlab routinemembranetestcondensation.m. This routine assembles a stiffness matrix containingonly contributions from elements with fixed material. Aftera permutation this matrix has ablock structure containing a blockKaa associated with DOFs that are connected to elementswith free material and a blockKcc with DOFs that will be condensed. The load vectorf is alsosplit into two partsfa and fc containing either entries associated with DOFs also remaining inthe condensed example or entries associated with the condensed part of the mesh. With thesevectors and matrices the original equilibrium equation takes the form

(Kaa Kac

Kca Kcc

)(ua

uc

)

=

(fafc

)

(4.54)

Condensation is achieved by using the second line of equation (4.54) and transforming it into aterm foruc:

uc = Kcc−1( fc−Kcaua) (4.55)

Inserting expression (4.55) into the first line of (4.54) yields the condensed equilibrium equation

(Kaa−KacKcc−1Kca)

︸︷︷︸

condensed stiffness matrix

ua = fa−KacKcc−1 fc

︸︷︷︸

condensed load vector

(4.56)

The previously mentioned matlab routinemembranetestcondensation.muses these for-mulas for the calculation of the condensed stiffness matrixand the condensed load vector anddumps them into the filemembranetestcondenseddata.txt in a bdf compatible for-mat. This allows to construct the bdf-file for the problem definition of the condensed examplemembranecondtest.bdf, which will be solved in the next section.

Condensated Testexample

With the data produced in the previous section we are now ableto set up the condensed test ex-ample. A look into the brd- and the frc-file shows that there are no Dirichlet boundary conditionsand external forces, this data is now hidden in the data contained in the csm-file. Thus the meshcontains only the elements with free material (number 6 – 9 and 16 – 19) as shown in Figure4.47.

It is now possible to run the test example containing condensed elements with the FMOKernel.To make sure that the problems are equivalent we can compare the output from the original testexample with output from the condensed testexample. This output is again produced within

77


Figure 4.47.: Non-condensed part of the beam

the functionAFeDataSet::AssembleGlobalStiffMat(double* rDesign) in theFMOKernel.

To check whether the data of the original testproblem is equivalent to the data of the condensedtestproblem we can use the matlab routinemembranetestcomparison.m. This file readsin the data of both testcases. First it compares the reduced format of the original stiffness matrix

(Kaa Kac

Kca Kcc

)

︸︷︷︸

original global stiffness matrixassembled over ALL elements

⇒ OriginalGlobalStiffness= Kaa−KacKcc−1Kca

︸︷︷︸

global stiffness matrixin reduced format

with the stiffness matrix of the condensed testexample obtained by summing up the global stiff-ness matrix for the elements with free material and the condensed stiffness matrix

CondensedGlobalStiffness= Kcond. exampleglobal + m

︸︷︷︸

mat. scaling

Kcond. examplecondensed .

Execution ofmembranetestcomparison.m shows that the difference of those two matri-ces lies 10 orders of magnitude below the entries of these matrices – from a numerical pointof view they are identical. Furthermore the matlab routine displays the condensed load vec-tor obtained by using the data from the original problem and the condensed load vector of thecondensed testexample. Obviously those two vectors are identical. Thus the original and thecondensed testexample are equivalent.

Results

Finally we can compare the optimization progression as wellas the optimization results for theoriginal and the condensed testexample. To this end both testexamples defined by the workspacefiles membranetest.fmw andmembranecondtest.fmw have been solved within FMSby the FMOKernel. The matlab routinemembranetestresults.m reads in the resulting h5-files and presents the optimal material tensors and the associated displacements for the originaland the condensed testcase. A comparison of the results for the optimal material shows that thesolutions are almost identical (except for tiny differences due to numerical imprecision).

78


Element number Original Testcase Condensed Testcase

6

72693 −1115 7572−1115 167 −116

7572 −116 941

72693 −1115 7572−1115 167 −116

7572 −116 941

7

56267 938 7629938 166 128

7629 128 1188

56267 938 7629938 166 128

7629 128 1188

8

39413 −822 7707−822 167 −1617707 −161 1663

39413 −822 7707−822 167 −1617707 −161 1663

9

22724 861 7036861 183 268

7036 268 2343

22724 861 7036861 183 268

7036 268 2343

16

72659 −1128 −7626−1128 168 119−7626 119 953

72659 −1128 −7626−1128 168 119−7626 119 953

17

56108 956 −7664956 167 −131

−7664 −131 1200

56108 955 −7664956 167 −131

−7664 −131 1200

18

39792 −668 −7335−668 162 124

−7335 124 1508

39792 −668 −7335−668 162 124

−7335 124 1508

19

22464 323 −7615323 155 −110

−7615 −110 2749

22465 323 −7615323 155 −110

−7615 −110 2749

79


Furthermore we can compare the displacements of the obtained optimal designs. To this endwe extract the original displacements from the original testcase results and the displacementsua from the results of the condensed testcase. As we receive only the displacements of DOFspresent in the condensed testexample, we have to calculate the displacementsuc of the DOFshidden in the condensed data matrices by using (4.55). The following table displays for everynode and associated DOF its original displacement and either ua or uc. Obviously the obtaineddisplacements of both testcases are identical.

Node DOF Orig. disp. ua uc

2 X -0.0009 -0.00092 Y -0.0012 -0.00123 X -0.0018 -0.00183 Y -0.0040 -0.00404 X -0.0025 -0.00254 Y -0.0085 -0.00855 X -0.0031 -0.00315 Y -0.0143 -0.01436 X -0.0037 -0.00376 Y -0.0213 -0.02137 X -0.0040 -0.00407 Y -0.0290 -0.02908 X -0.0044 -0.00448 Y -0.0375 -0.03759 X -0.0048 -0.00489 Y -0.0467 -0.0467

10 X -0.0051 -0.005110 Y -0.0566 -0.056611 X -0.0052 -0.005211 Y -0.0672 -0.067213 X 0.0000 0.000013 Y -0.0011 -0.001114 X -0.0000 -0.000014 Y -0.0040 -0.004015 X 0.0000 0.000015 Y -0.0084 -0.008416 X -0.0000 -0.000016 Y -0.0142 -0.014217 X -0.0000 -0.000017 Y -0.0212 -0.0212

Node DOF Orig. disp. ua uc

18 X 0.0000 0.000018 Y -0.0290 -0.029019 X 0.0000 0.000019 Y -0.0374 -0.037420 X -0.0000 -0.000020 Y -0.0467 -0.046721 X 0.0000 0.000021 Y -0.0566 -0.056622 X -0.0000 -0.000022 Y -0.0671 -0.067124 X 0.0009 0.000924 Y -0.0012 -0.001225 X 0.0018 0.001825 Y -0.0040 -0.004026 X 0.0025 0.002526 Y -0.0085 -0.008527 X 0.0031 0.003127 Y -0.0143 -0.014328 X 0.0037 0.003728 Y -0.0213 -0.021329 X 0.0040 0.004029 Y -0.0290 -0.029030 X 0.0044 0.004430 Y -0.0375 -0.037531 X 0.0048 0.004831 Y -0.0467 -0.046732 X 0.0051 0.005132 Y -0.0566 -0.056633 X 0.0052 0.005233 Y -0.0671 -0.0671

The fact that both testcases are consistent and produce the same results can also be seen in thevisualization of the original and the condensed design. Thefollowing pictures show the optimalmaterial distribution for both testcases - obviously the elements with free material have identicalmaterial distributions. Thus we conclude that the handlingof condensed data matrices withinFMK works correctly and produces reasonable results.

80


Original example Condensed example

81

CHAPTER 5

Multidisciplinary Optimization Constraints

In the previous chapters we have developed models and methods to find stiff and light-weightshell structures by using Free Material Optimization. However, the requirements for shells inreal-world applications exceed these two basic characteristics by far. The desired properties ofthe structure might include a specific shape, encouragementor prevention of a certain behavioror influence on particular physical quantities of the shell.While some of these attributes can notbe captured by a scientific expression as for example an aesthetical appearance, others can bedescribed via mathematical formalisms and even be appendedto the Free Material Optimizationproblem. Structural optimization problems that aim to fulfill several different criteria in the sameproblem formulation are referred to asmultidisciplinary optimization problems. In the followingsections we will consider four specific multidisciplinary constraints, which are directly relatedto typical industrial demands in structural engineering.

The first type of constraints discussed by us are lineardisplacement constraints, which havebeen introduced to the Free Material Optimization problem for solids by [KSZ08]. Displacementconstraints are introduced to a problem formulation to prescribe the shape of the deformed shell.On the one hand this might allow to avoid deformation into areas that are not part of the designdomain due to a dedication to other purposes, e.g. the wire harness, or external influences suchas rigid obstacles. On the other hand displacement constraints can be used to ensure movementin a desired direction, although this contradicts the original tendency of the structure. This highlycomplex task is known as the construction of mechanical mechanisms.

The next constraint type arestress constraints. These address a problem that has been ne-glected until now: the linear elastic models used in wide areas of structural engineering and alsoin the most parts of this thesis are only valid for small displacements and an according stressvalue. For significantly larger stresses the structure might concede to the applied forces and fi-nally get ripped apart. But even below this point the structure’s behavior can not be captured bylinear elasticity anymore – instead permanent deformations accompanied by irreversible flow-ing processes occur, which belong to the subject of plasticity. Stress constraints are employed to

83

5. Multidisciplinary Optimization Constraints

keep the structure below these phenomena in a save range of operation and hence avoid materialfailure due to high stresses. Our approach for dealing with stress constraints in Free MaterialOptimization follows the ideas from [KS06], where they wereintroduced for the solid case.However, in the problem formulation for shells we have to distinguish between in-plane andout-of-plane stresses to account for the special elastic features of thin-walled structures.

Another interesting type of constraints considered by us are eigenfrequency constraints. Tothis end we have to look beyond the static analysis of a structure and take its dynamic propertiesinto account. In a dynamic context the behavior of elastic structures is characterized by vibra-tions [Ber96]. Note that every elastic body has its own natural vibrations, which it performs inthe absence of external alternating forces. These free vibrations are determined by the eigenfre-quencies of the structure. When the elastic body is exposed to external loads, whose frequencyis close to its eigenfrequency, these vibrations are amplified and might lead to tremendous os-cillations and even the disintegration of the structure. This phenomenon is known as resonancedisaster. Eigenfrequency constraints are implemented to avoid this effect by shifting the eigen-frequencies to a different range that is far from the frequency range occupied by external influ-ences. [SKL09a] have introduced eigenfrequency constraints to the Free Material Optimizationproblem for solids. We will adapt the presented methods to Naghdi shells in order to obtainstructures less susceptible to vibration resonance.

The last type of constraints considered in this thesis areglobal stability constraints. Properlydeployed shells can carry enormous loads compared to their low self-weight. However, they arevery susceptible to geometrical imperfections of their initial shape or disturbances of the appliedloads and react to these influences with sudden deformationsknown as snap-through buckling.A mathematical description of this effect is obtained by a nonlinear extension of Naghdi’s shellmodel followed by an investigation of the stability of the resulting static equilibrium [Nio85].For load perturbations above thecritical buckling parameterthere exist multiple solutions tothe equilibrium problem, between which the structure can jump. Global stability constraintsare introduced to raise the critical buckling parameter such that the occurring loads lie belowthis critical limit. According to [Koc02] the constraint on the critical buckling parameter can betransformed into a nonlinear semidefinite matrix constraint, that is compatible with the semidef-inite problem structure of the Free Material Optimization problem for shells.

5.1. Displacement Constraints

We begin with the expansion of the Free Material Optimization problem by displacement con-straints. Displacement constraints are used to affect the deformed shape of the shell. The firstproblems including displacement constraints were motivated by design domains with rigid ob-stacles [BKNZ99]. In the deformed state the structure is allowed to touch the obstacle, but it cannever push it aside. Accordingly a force equilibrium is present at the contact surface betweenelastic body and obstacle. Meanwhile displacement constraints are employed for various othergoals [Koc97, BLR94]. They are utilized to guarantee minimal, maximal or fixed displacementsin subsets of the design domain or for certain nodes of the finite element mesh. With their aidit is for instance possible to avoid deformation of the structure into areas that are dedicated toother purposes, e.g. the wire harness, electronic equipment or the like. Moreover, the prescrip-

84


tion of a specific deformed shape allows to use parts of the structure as reflectors or antennasor to improve the aerodynamic features of the structure. Another more sophisticated applicationof displacement constraints is the assertion of movement ina certain direction for some parts ofthe structure. Thereby one can construct structures that deform in a fashion that at first glancecontradicts structural engineering intuition. Typical examples are the force inverter, where theexertion of a force on one side of the design domain will lead to a displacement in the oppositedirection on the other side of the design domain, or the gripper, that transforms a horizontal forceon one side of the design domain in opposing vertical displacements on the other side, whichallow to hold items in between these opposing parts of the structure. These types of structuresare described by the collective termmechanical mechanisms[PBS01].

After this overview on the applications of displacement constraints we turn to their practicalimplementation. We focus on linear displacement constraints, which are sufficient to handlethe examples discussed above. The first consideration of displacement constraints in connectionwith Free Material Optimization for solid material was performed by [BKNZ99], who con-centrated on unilateral boundary conditions arising from obstacles in the design domain. Thisapproach was generalized in [KSZ08] by universally stated discrete linear displacement con-straints allowing a greater bandwidth of applications. We will expand the ideas from [KSZ08] toplates and shells and accordingly treat displacement constraints exclusively in a discrete context.Hence we introduce the following linear displacement constraint for a given matrixCDC ∈R

r×n,the displacement vector(un,θn)

1 and a vectordDC ∈Rr

CDC(un,θn)≤ dDC . (5.1)

Thus we obtain the Free Material Optimization problem for shells with displacement constraintsby appending (5.1) to the original discrete problem formulation for shells (Pm

MC)


fn⊤(un,θn) (PMC-DC)

subject toEltNr

∑m=1


(tmCm 0

0 tm2 Dm

)

− ε15 0 ∀m= 1, . . . ,EltNr

tm · trCm+tm2· trDm ≤ ρ+ ∀m= 1, . . . ,EltNr

Kshell(Cm,Dm)(un,θn) = fnCDC(un,θn) ≤ dDC

Analogously to the original discrete problem formulation (PmMC) the Free Material Optimiza-

tion problem for shells including displacement constraints (PMC-DC) is a nonlinear nonconvexsemidefinite program. Yet there is a crucial difference to problem (Pm

MC) concerning existence ofsolutions. The chosen displacement constraints might be too restrictive, hence it is not possible

1Recall that(un,θn) is the discrete displacement vector as introduced in (4.25), which should not be confusedwith a functional dependence onun andθn.

85


to create a structure from the limited material resources, that fulfills this constraint. As a resultthe volume and the displacement constraint can not be valid simultaneously, one of them willalways be violated. In this case there exists no optimal solution to problem (PMC-DC).

Furthermore we would like to emphasize the difference between the displacement constraintsused here and contact conditions (as found for example in [BKNZ99]). As explained in [Koc97]we obtain unilateral contact conditions by adding linear displacement constraints to the minimum-potential-energy formulation (PD). The difference between the two problems becomes apparentwhen the structure shares a common interface with an obstacle inside or at the boundary of thedesign domain. While the normal stresses at the interfaces vanish in the case of displacementconstraints this is not necessary for unilateral contact conditions. Here the reaction forces atthe interface do not have to be equal to zero as long as the normal stresses fulfill the staticequilibrium equations. Thus displacement constraints area special case of unilateral contactconditions.

Additionally to the previous consideration of the minimum compliance problem with dis-placement constraints the constraints (5.1) can also be added to the discrete minimum weightformulation (Pm

MW). Again we are able to transform the obtained nonlinear nonconvex semidef-inite program into a nonlinear convex semidefinite program by eliminating the displacements(un,θn) through a utilization of the equilibrium equation. This leads to the reduced minimumweight problem with displacement constraints

min(Cm,Dm)∈Cm

EltNr

∑m=1

tm · trCm+tm2· trDm (PMW-red-DC)

subject to

(tmCm 0

0 tm2 Dm

)

− ε15 0 ∀m= 1, . . . ,EltNr


fn⊤Kshell

−1(Cm,Dm) fn ≤ c

CDCKshell−1(Cm,Dm) fn ≤ dDC

Furthermore both problem formulations can be extended to multiple load scenarios. In thiscase there exist separate displacement constraints for each load case, hence the displacementconstraint (5.1) is replaced by

CDC(unl,θn

l )≤ dDC ∀ l = 1, . . . ,L (5.2)

and in the reduced formulation the following displacement constraint is used

CDCKshell−1(Cm,Dm) fn

l ≤ dDC ∀ l = 1, . . . ,L . (5.3)

The displacement constraint and its derivative are implemented in a straight-forward fash-ion. Because the inequality has to be understood as a component-wise expression, the con-straint consists ofr separate inequalities defined by a vector(CDC)k∗ (resemling thek-th lineof CDC) and a value(dDC)k. The constraint evaluation is performed in thedouble function

86


AFMODisplacementConstraint::Eval. The value of thek-th line of the displacementconstraint is calculated as

gDC(Cm,Dm) := (CDC)k∗ · (un(Cm,Dm),θn(Cm,Dm))− (dDC)k . (5.4)

The required gradient evaluation is performed in theAFMOKSparseGradient* functionAFMODisplacementConstraint::Gradaccording to the following formulas for the deri-vation with respect to the entry in thep-th row andq-th column of the material matrix associatedwith the element ˆm

∂gDC(Cm,Dm)

∂ (Cm)p,q= (CDC)k∗ ·

∂ (un,θn)

∂ (Cm)p,q

=−(CDC)k∗ Kshell−1(Cm,Dm)

∂(Kγ

m(Cm)+Kχm(Cm)

)

∂ (Cm)p,q(un,θn) , (5.5)

∂gDC(Cm,Dm)

∂ (Dm)p,q= (CDC)k∗ ·

∂ (un,θn)

∂ (Dm)p,q

=−(CDC)k∗ Kshell−1(Cm,Dm)

∂Kζm(Dm)

∂ (Dm)p,q(un,θn) . (5.6)

Both functions are part ofFMOlib. For an efficient computation we use functions providedby theFELib. The vector(CDC)k∗ Kshell

−1(Cm,Dm) is equal to a solution of the static sys-tem, where the right hand side is replaced by(CDC)k∗, hence it is calculated in theint func-tionAFeDataSet::SolveSystemWithV. The result together with the displacement vector(un,θn) is then inserted into theint function AFeDataSet::DirDerDisplacements.The latter function provides the inner product of the two vectors with a matrix, that representsthe derivation of the global stiffness matrix with respect to a single entry of the material matrixof a specific element. We discuss the results of the implementation by means of the followingtest example.

Testcase 8: Rectangular plate on pins with displacement constraints.

Figure 5.1.: Unconstrained plateCompliance: 6.5692Max. displacement: 4291.53

Figure 5.2.: Displacement constrained plateCompliance: 6.5812Max. displacement: 3899.99

87


We consider a rectangular plate, that is fixed on two pins on either one of the long sides of therectangle. Downward forces are applied on the entire lengthof the short sides of the rectangle.Figure 5.1 displays the optimal material distribution and displacement for the minimum com-pliance problem formulation without displacement constraints. As expected the structure movesdownward at the ends of the plate, where the forces are applied, and bends upwards between thepins. The maximal upward displacement occurs at the rims, its maximal value is 4291.53.

Our intention is now to reduce this upward bending by introducing a displacement constraintto the problem. Therefore we add the constraintu3 ≤ 3900 for all boundary nodes between thepins to the optimization problem and solve the resulting Free Material Optimization problemwith displacement constraints. It is clearly visible that the structure reacts to the constraint bymoving material to the central area to stiffen it and avoid the upward bending. As a result themaximal displacement is smaller than in the original example. Its value is 3899.99, thus thedisplacement constraint is fulfilled. Of course this additional requirement results in a highercompliance of the structure: while the original problem hasa compliance of 6.5692, the dis-placement constrained problem has an increased compliancevalue of 6.5812.

5.2. Stress and Strain Constraints

The next type of constraints considered by us are stress constraints. These are introduced inorder to avoid the appearance of high stresses in the optimalstructure. In an extreme casehigh stresses might cause structural failure as the material is not able to withstand the appliedforces anymore and rips apart. Even below this range high stresses are critical because forstresses above the yield stress the structure will deform according to a plastic model resulting inirreversible deformations. Hence all predictions based onan elastic model are not sufficient todescribe the behavior of the structure. The major goal of stress constraints is to limit the stressesappearing in the optimal structure such that neither plasticity nor material failure occur.

The handling of stress constraints in material optimization problems is a complicated task.First of all there exists no general criterion to determine material failure due to high stresses.Instead various criteria are used depending on the specific structure, for example the maximumshear stress (Tresca) criterion or the distortional energy(von Mises) criterion are often employedas failure criteria for isotropic layers, but in the case of fiber-reinforced orthotropic layers theusage of the Tsai-Hill or Tsai-Wu criterion is much more common [GHH99]. This diversityis discussed in [DB98], where von Mises criteria for rank 2 microstructures and for power-law materials are established. The latter is employed in a topology optimization procedurefollowing the SIMP approach, moreover the so-called singularity problem is discussed, thatoften occurs when dealing with stress constraints. This phenomenon, that is also addressed in[SS01, AK08], arises due to local stresses of elements with vanishing material, that neverthelesstend to finite values. This results in discontinuities of thestress constraint at zero density andin this case the associated design domain is nonconvex and may contain degenerate parts withzero measure. Often the global optimum is located in these degenerated subsets, thus the Slaterconstraint qualification is violated and standard optimization algorithms based on the Kuhn-Tucker condition are unable to reach the optimal structure.A practical remedy for the singularityproblem consists of theε-relaxation approach, where the stress constraints are replaced by a

88


relaxed version that leads to a design space without any degenerate parts and whose solutionconverges to the original solution forε → 0 [CG97].

The expansion of the Free Material Optimization problem with stress constraints has beenfirst discussed in [KS06]. As the optimal material provided by Free Material Optimization cannot be assigned to a specific material class prior to the optimization, none of the specializedfailure criteria mentioned above is advisable. Instead a norm of the stress tensor integrated overthe finite element is used as a measure for the stress constraint and bounded from above in thediscretized problem formulation. Consequently an evaluation of stresses at points, that is knownto be rather inexact, is avoided and the problem is kept computationally tractable. Furthermorethe stress constraints are normalized by the measure of the element|ωm| and the upper bound onthe trace of the materialρ+.

We address stress constraints in a similar fashion, howeverwe have to consider three differentstresses in the case of shells: membrane, bending and shear stresses. This gives rise to thequestion on whether to combine the constraints for these stresses or to treat them in separateconstraints. To answer this a look at deformations causing the different stresses is helpful. InFig. 5.3 a compression in the midplane causing membrane stresses is shown, while in Fig. 5.4a typical bending deformation is depicted. Although the resulting stresses are different, theirdirections stay in the plane provided by the midsurface. Therefore we regard these stresses asin-plane stresses. In contrast to these stresses a look at the shear stresses in Fig. 5.5 reveals,that they are out-of-plane stresses. Thus we will add a combined constraint for the in-planestresses bounded bysip

σ and a single constraint for the out-of-plane stresses bounded bysoopσ to

the optimization problem. Analogously to [KS06, KSZ08] we focus on the minimum weightformulation (Pm

MW) in the context of stress constraints, therefore we obtain the following FreeMaterial Optimization problem for shells with stress constraints


EltNr

∑m=1

tm · trCm+tm2· trDm (PMW-SC)

subject to

(tmCm 0

0 tm2 Dm

)

− ε15 0 ∀m= 1, . . . ,EltNr


GpNr

∑g=1

‖CmBγm,g(un,θn)+ tCmBχ

m,g(un,θn)‖2Mises ≤ sip

σ (ρ+)2|ωm| ∀m= 1, . . . ,EltNr

GpNr

∑g=1

‖DmBζm,g(un,θn)‖2

Mises ≤ soopσ (ρ+)2|ωm| ∀m= 1, . . . ,EltNr

Kshell(Cm,Dm)(un,θn) = fnfn⊤(un,θn) ≤ c

The von Mises norm used in this problem formulation is deducted from the three-dimensionalvon Mises norm, that is defined for the three-dimensional stress vectorσ⊤

m

σ⊤m = (σ11

m ,σ22m ,σ33

m ,√

2σ12m ,

√2σ13

m ,√

2σ23m )

89


Figure 5.3.: Membrane stress Figure 5.4.: Bending stress Figure 5.5.: Shear stress

of them-th element in the following way

‖σm‖2Mises :=

GpNr

∑g=1

σ⊤m,gMσm,g , where M :=

2 −1 −1 0 0 0−1 2 −1 0 0 0−1 −1 2 0 0 0

0 0 0 6 0 00 0 0 0 6 00 0 0 0 0 6

. (5.7)

As the in-plane and out-of-plane stresses appearing in the shell case can be associated with partsof the three-dimensional stress vector, the von Mises norm for the shell stresses consists of theappropriate blocks of the matrixM. Precisely the in-plane stresses are constructed according to

Cm[Bγm(un,θn)+ tBχ

m(un,θn)](2.35)= Cmε ip

m(2.78)= σ ip

m =(

σ11m ,σ22

m ,√

2σ12m

)⊤, (5.8)

hence the in-plane part of the von Mises norm is given by

‖σ ipm‖2

Mises :=GpNr

∑g=1

σ ipm,g

⊤Mipσ ip

m,g , where Mip :=

2 −1 0−1 2 0

0 0 6

. (5.9)

On the other hand the out-of-plane stresses

DmBζm(un,θn)

(2.36)= Dmεoop

m(2.79)= 2σoop

m =√

2(√

2σ13m ,

√2σ23

m

)⊤(5.10)

lead to the following out-of-plane part of the von Mises norm

‖DmBζm(un,θn)‖2

Mises:=GpNr

∑g=1

(un,θn)⊤Bζ

m,g⊤

DmMoopDmBζm,g(un,θn) , whereMoop :=

(3 00 3

)

.

It is also possible to consider stress constraints for multiple load scenarios. This requiresseparate stress constraints for each load case as given by

GpNr

∑g=1

‖CmBγm,g(un

l,θn

l )+ tCmBχm,g(un

l,θn

l )‖2Mises ≤ sip

σ (ρ+)2|ωm| ∀m=1,...,EltNr∀ l=1,...,L (5.11)

GpNr

∑g=1

‖DmBζm,g(un

l,θn

l )‖2Mises ≤ soop

σ (ρ+)2|ωm| ∀m=1,...,EltNr∀ l=1,...,L (5.12)

90


Moreover the stress constraints can also be appended to the reduced problem formulation byusing the following modified expressions

GpNr

∑g=1

‖Cm(Bγm,g+ tBχ

m,g)K−1shell(Cm,Dm) fn

l‖2Mises ≤ sip

σ (ρ+)2|ωm| ∀m=1,...,EltNr∀ l=1,...,L (5.13)

GpNr

∑g=1

‖DmBζm,gK−1

shell(Cm,Dm) fnl‖2

Mises ≤ soopσ (ρ+)2|ωm| ∀m=1,...,EltNr

∀ l=1,...,L (5.14)

Function and gradient evaluation for the in-plane and out-of-plane stress constraints are imple-mented in theFMOlib. The value of the in-plane stress constraints is calculatedin thedoublefunctionAFMOStressConstraint::Eval according to the formula

gipSC(Cm,Dm) :=

GpNr

∑g=1

(un,θn)⊤(Bγ

m,g+ tBχm,g)

⊤CmMipCm(Bγm,g+ tBχ

m,g)(un,θn)

−sipσ (ρ+)2volume(ωm) . (5.15)

Consequently the gradient of the in-plane stress constraint of the elementm with respect to theentry in thep-th row andq-th column of the material matrix associated with the element m isevaluated in theAFMOKSparseGradient* functionAFMOStressConstraint::Gradas

∂gipSC(Cm,Dm)

∂ (Cm)p,q=−

(

2GpNr

∑g=1

(un,θn)⊤(Bγ

m,g+ tBχm,g)


m,g)

)

· (5.16)

·Kshell−1(Cm,Dm)

∂(Kγ

m(Cm)+Kχm(Cm)

)

∂ (Cm)p,q(un,θn)

+2δmm

GpNr

∑g=1

(un,θn)⊤(Bγ

m,g+ tBχm,g)

⊤CmMipIp,q(Bγm,g+ tBχ

m,g)(un,θn) ,

∂gipSC(Cm,Dm)

∂ (Dm)p,q=−

(

2GpNr

∑g=1

(un,θn)⊤(Bγ

m,g+ tBχm,g)


m,g)

)

· (5.17)

·Kshell−1(Cm,Dm)

∂Kζm(Dm)

∂ (Dm)p,q(un,θn) ,

whereδmm is the Kronecker delta, that is equal to 1 form= mand 0 otherwise andIp,q is a matrix,whose entry in thep-th row andq-th column as well as the entry in theq-th row and thep-th col-umn are equal to 1 and all other entries are set to 0. The out-of-plane stress constraints are calcu-lated in a similar fashion in thedouble functionAFMOShearStressConstraint::Eval

goopSC (Cm,Dm) :=

GpNr

∑g=1

(un,θn)⊤Bζ

m,g⊤

DmMoopDmBζm,g(un,θn)−soop

σ (ρ+)2volume(ωm) (5.18)

91


and the corresponding gradient evaluation is provided by theAFMOKSparseGradient* func-tion AFMOShearStressConstraint::Grad

∂goopSC (Cm,Dm)

∂ (Cm)p,q=−

(

2GpNr

∑g=1

(un,θn)⊤Bζ

m,g⊤

DmMoopDmBζm,g

)

· (5.19)

·Kshell−1(Cm,Dm)

∂(Kγ

m(Cm)+Kχm(Cm)

)

∂ (Cm)p,q(un,θn) ,

∂goopSC (Cm,Dm)

∂ (Dm)p,q=−

(

2GpNr

∑g=1

(un,θn)⊤Bζ

m,g⊤

DmMoopDmBζm,g

)

· (5.20)

·Kshell−1(Cm,Dm)

∂Kζm(Dm)

∂ (Dm)p,q(un,θn)

+2δmm

GpNr

∑g=1

(un,θn)⊤Bζ

m,g⊤

DmMoopIp,qBζm,g(un,θn) .

These evaluations are performed by various member functions of the classAFeDataSetwithinthe FELib. Thedouble function AFeDataSet::StressNorm allows access to the in-plane von Mises stress norm of a specific elementmand load casel as required for the evaluationof gip

SC(Cm,Dm). The out-of-plane counterpart used for the computation ofgoopSC (Cm,Dm) is pro-

vided by thedouble functionAFeDataSet::ShearStressNorm. For the computationof the gradients according to the above specified formulas the classAFeDataSet possessesthevoid functionAFeDataSet::StressDerVec, that calculates the derivatives of the in-plane strain norm via thevoid functionAFMOKBOperator::InPlaneStressDerVec.Analogously thevoid functionAFeDataSet::TransverseShearStressDerVec em-ploys thevoid function AFMOKBOperator::TransverseShearStressDerVec forthe computation of the derivatives of the out-of-plane stresses. The following testexample, thatis a modified version of the popular L-shaped domain testexample, demonstrates the results ofthis implementation.

Testcase 9: Variation of the L-shaped domain.The L-shaped domain is an often employed testcase for stressconstraints in the case of two-

dimensional elasticity. We modify it slightly by clamping the shell at the lower left edge, hencethe translational displacementsu as well as the rotational displacementsθ have to be zero there.The forces, that are applied at the lower corner of the right part of the structure, are composed

Figure 5.6.: Optimalstiffness distribution



92


Figure 5.9.: Optimalstiffness distribution



of an in-plane component, that is similar to the two-dimensional example, and an out-of-planecomponent in order to neglect a pure membrane example. Fig. 5.6 displays the optimal stiff-ness distribution without any stress constraints. It is a mixture of the arch known from thetwo-dimensional example and a stiff part around the inner corner of the structure that tries toavoid out-of-plane displacements. As shells are much more susceptible to out-of-plane loads,much more material is assigned to this task. The magnitude ofthe maximal in-plane stress isapproximately 49 as depicted in Fig. 5.7.

For the solution displayed in Fig. 5.9 we added an stress constraint to the problem formulationthat limits the in-plane stress to 40. Fig. 5.10 demonstrates that this goal is achieved, the in-planestress value is 37. Naturally this leads to an increase in theout-of-plane stress. A comparison ofFig. 5.8 and 5.11 shows that the out-of-plane stress was raised from a value of 51 to a value of56. Moreover, note that the optimal stiffness distributionin Fig. 5.9 forms a bow at the verticalinner edge in order to redirect the stresses and thus avoid high stress concentrations.

It is also possible to include constraints on the strains instead of the stresses. This might behelpful in the case of low material densities, which allow large deformations and strains with-out violating the stress constraints. Analogous to the stress constrained problem (PMW-SC) theintegral over the norms of the strains are bounded from abovein the discretized problem. As wediffer again between in-plane and out-of-plane behavior, this leads to the following Free MaterialOptimization problem for shells with strain constraints inthe minimum weight-formulation


EltNr

∑m=1

tm · trCm+tm2· trDm (PMW-StC)

subject to

(tmCm 0

0 tm2 Dm

)

− ε15 0 ∀m= 1, . . . ,EltNr


GpNr

∑g=1

‖Bγm,g(un,θn)+ tBχ

m,g(un,θn)‖2 ≤ sipe (ρ+)2|ωm| ∀m= 1, . . . ,EltNr

GpNr

∑g=1

‖Bζm,g(un,θn)‖2 ≤ soop

e (ρ+)2|ωm| ∀m= 1, . . . ,EltNr

Kshell(Cm,Dm)(un,θn) = fnfn⊤(un,θn) ≤ c

93


Of course it is possible to extend this formulation to multiple load cases by replacing the dis-placements(un,θn) and the forcefn with load case specific versions(un

l ,θnl ) and fnl and by

listing separate strain constraints for each load case. Moreover a reduced formulation can beaccomplished by replacing(un

l ,θnl ) with Kshell

−1(Cm,Dm) fnl and simultaneously removing theequilibrium condition from the problem definition.

We proceed with the function and gradient evaluations of thestrain constraints within theFMOlib. The evaluation of the in-plane strain constraints is performed in thedouble functionAFMOStrainConstraint::Eval according to the following formula

gipStC(Cm,Dm) :=

GpNr

∑g=1

(un,θn)⊤(Bγ

m,g+ tBχm,g)

⊤(Bγm,g+ tBχ

m,g)(un,θn)

−sipe (ρ+)2volume(ωm) . (5.21)

The actual computation of this function evaluation is carried through in thedouble functionAFeDataSet::StrainNorm, that is part of theFELib and calls thedouble functionAFMOKBOperator::InPlaneStrainNorm. The gradient of the in-plane strain constraintassociated with the elementm with respect to the entry in thep-th row andq-th column ofthe material matrix of the element ˆm is calculated in theAFMOKSparseGradient* functionAFMOStrainConstraint::Grad by using the subsequent expressions

∂gipStC(Cm,Dm)

∂ (Cm)p,q=−

(

2GpNr

∑g=1

(un,θn)⊤(Bγ

m,g+ tBχm,g)

⊤(Bγm,g+ tBχ

m,g)

)

· (5.22)

·Kshell−1(Cm,Dm)

∂(Kγ

m(Cm)+Kχm(Cm)

)


∂gipStC(Cm,Dm)

∂ (Dm)p,q=−

(

2GpNr

∑g=1

(un,θn)⊤(Bγ

m,g+ tBχm,g)

⊤(Bγm,g+ tBχ

m,g)

)

· (5.23)

·Kshell−1(Cm,Dm)

∂Kζm(Dm)

∂ (Dm)p,q(un,θn) .

For this gradient evaluation we employ thevoid functionAFeDataSet::StrainDerVecand itsvoid subroutineAFMOKBOperator::InPlaneStrainDerVec from theFELibin order to compute the in-plane strain norm derivatives efficiently. On the other hand thedouble functionAFMOShearStrainConstraint::Evalprovides the out-of-plane strainnorm according to

goopStC(Cm,Dm) :=

GpNr

∑g=1

(un,θn)⊤Bζ

m,g⊤

Bζm,g(un,θn)−soop

e (ρ+)2volume(ωm) . (5.24)

Here we use thedouble function AFeDataSet::ShearStrainNorm and thedoublefunction AFMOKBOperator::TransverseShearStrainNorm for the computation ofthe out-of-plane strain norm value. Finally the gradient ofthis strain norm is computed in the

94

5.3. Constraints on the Fundamental Eigenfrequency

AFMOKSparseGradient* functionAFMOShearStrainConstraint::Grad as

∂goopStC(Cm,Dm)

∂ (Cm)p,q=−

(

2GpNr

∑g=1

(un,θn)⊤Bζ

m,g⊤

Bζm,g

)

· (5.25)

·Kshell−1(Cm,Dm)

∂(Kγ

m(Cm)+Kχm(Cm)

)


∂goopStC(Cm,Dm)

∂ (Dm)p,q=−

(

2GpNr

∑g=1

(un,θn)⊤Bζ

m,g⊤

Bζm,g

)

·Kshell−1(Cm,Dm)

∂Kζm(Dm)

∂ (Dm)p,q(un,θn) . (5.26)

For the calculation the gradient of the out-of-plane strainnorm we made use of thevoid func-tion AFeDataSet::TransverseShearStrainDerVec and moreover thevoid func-tion AFMOKBOperator::TransverseShearStrainDerVec from theFELib. Hencewe are also able to solve problems with strain constraints instead of stress constraints.


In this section we focus on vibration constraints. A structure subjected to periodic forces, whosefrequency is nearby the structure’s natural frequency, omits resonance behavior. It gains energyfrom the external excitation and its amplitude will increase until the structure rips apart. One ofthe most prominent examples of this behavior known as resonance disaster is Angers bridge, asuspension bridge leading across the Maine river in France,who collapsed 1850 under soldiersmarching in cadence.

An effective strategy to avoid structural failure due to vibrational resonance, which we willemploy in this article, is raising a structure’s fundamental eigenfrequency. The eigenfrequenciesare determined by solving the dynamic problem in the absenceof external loading providing thefree vibrations of the structure. Thus we need the dynamic linear elasticity problem for Naghdishells, but up to now we only worked with the static elasticity problem. This problem formu-lation was obtained by considering the general three-dimensional static elasticity problem andintroducing the approximations of Naghdi’s shell model. Hence it is natural to start now fromthe general three-dimensional dynamic elasticity problemand use the same approximations. Thethree-dimensional dynamic elasticity problem is well-known and according to [Ber96] given by

Find a functionτ →U(ξ ,τ) of [0,T] 7→ V such that

∫

Rρ∗ ∂ 2U

∂τ2 V dR+

∫

RE∗ i jkl γ∗i j (U)γ∗kl(V)dR = 0 ∀V ∈ V (5.27)

U(ξ ,0) =U0(ξ ) ,∂U∂τ

(ξ ,0) =U1(ξ )

whereU denotes the displacement in three dimensions, depending ontime τ and the three-dimensional space variableξ . ∂R0 denotes the clamped part of the boundary∂R of the shell

body R, hence the displacementsU lie in V :=

U ∈[H1(R)

]3∣∣∣ U |∂R0

= 0

, where it is

95


assumed that|∂R0| > 0. Furthermoreγ∗i j is the covariant strain tensor andE∗ i jkl is the con-travariant elasticity tensor relative to the local three-dimensional basisg1,g2,g3 of the shellandρ∗ is its mass density.

It is now possible to insert the assumptions of Naghdi’s shell model into the general three-dimensional problem formulation (5.27) and integrate overthe shell’s thickness to obtain two-dimensional dynamic equations for the vibrations of Naghdishells. The bidimensional approxi-mation for the latter term was already introduced in equation (2.50) as

∫

RE∗ i jkl γ∗i j (U)γ∗kl(V)dR ≈ a((u,θ),(v,η)) (5.28)

where

a((u,θ),(v,η)) =∫

ωtγ⊤(u)Cγ(v)+

t3

12χ⊤(u,θ)Cχ(v,η)+ tkζ⊤(u,θ)Dζ (v,η)dS

=

∫

ωtCαβλ µ

[

γαβ (u)γλ µ (v)+t2

12χαβ (u,θ)χλ µ (v,η)

]

+ tkDαλ ζα(u,θ)ζλ (v,η)√

adξ 1dξ 2 (5.29)

Thus only a bidimensional approximation of the first term hasto be found. To this end the ap-proximation for the displacements of Naghdi shells (2.28) and the definition of the infinitesimalvolumedR as given in (2.22) are inserted

∫

Rρ∗ ∂ 2U

∂τ2 V dR =

∫

Rρ∗ ∂ 2

∂τ2

(u+ξ 3θαaα)

(

v+ξ 3ηβ aβ)√

gdξ 1dξ 2dξ 3

Now it is possible to employ the relation between the three-dimensional metricgi j and the firstfundamental formaαβ of the midsurface from equation (2.23):

∫

Rρ∗ ∂ 2U

∂τ2 V dR =

=

∫

Rρ∗ (u+ξ 3θαaα)

(

v+ξ 3ηβ aβ)(

1−2Hξ 3+K(ξ 3)2

)√adξ 1dξ 2dξ 3

=∫

Rρ∗(

uv+ξ 3uηβ aβ +ξ 3vθα aα +(ξ 3)2 θα ηβ aαβ

)(

1−2Hξ 3+K(ξ 3)2

)

dξ 3dS

=∫

ω

∫ t2

− t2

ρ∗(

uv+ξ 3(

uηβ aβ +vθαaα)

+(ξ 3)2 θα ηβ aαβ −2ξ 3H uv

−2(ξ 3)2

H

(


−2(ξ 3)3H θα ηβ aαβ +

(ξ 3)2

K uv

+(ξ 3)3

K

(


+(ξ 3)4

K θα ηβ aαβ)

dξ 3dS

The next step is integration over the thickness of the shell using the formulas

∫ t2

− t2

(ξ 3)2n

dξ 3 =2

(2n+1)22n+1 t2n+1,

∫ t2

− t2

(ξ 3)2n+1

dξ 3 = 0

96


under the assumption of a constant material densityρ in the normal direction resulting in∫

Rρ∗ ∂ 2U

∂τ2 V dR =∫

ωρ(

tuv+t3

12θαηβ aαβ −2

t3

12H

(


+t3

12K uv+

t5

80K θαηβ aαβ

)

dS

As the shell’s thickness is considered to be small the last term includingt5 can be neglected andleads together with the differential geometrical formulas

uv= aαβ uαvβ + u3v3 , uηβ αβ = aαβ uα ηβ , vθα aα = vβ aαβ θα

to the following approximation∫

Rρ∗∂ 2U

∂τ2 V dR ≈ b((u,θ),(v,η))

where

b((u,θ),(v,η)) =

∫

ωρt

(

1+Kt2

12

)(

aαβ uαvβ + u3v3

)

+ρt3

12aαβ (θα ηβ −2H

[uα ηβ + θαvβ

])dS (5.30)

Thus the bidimensional dynamic elasticity equations for Naghdi shells in the absence of externalloading take the form

Find a functionτ →(u(ξ 1,ξ 2,τ),θ(ξ 1,ξ 2,τ)

)of [0,T] in U such that

a((u,θ),(v,η))+b((u,θ),(v,η)) = 0 ∀(v,η) ∈ U (5.31)

whereU is defined as in (2.29). The solutions to this problem, which describes vibrations of theshell without any effects of external loads and under kinematic boundary conditions, which aretime-independent, are called free vibrations. According to [Ber96, RR04] they may be expressedas (

u(ξ 1,ξ 2

,τ),θ(ξ 1,ξ 2

,τ))

= (α cosΩt +β sinΩt)(u(ξ 1

,ξ 2), θ (ξ 1,ξ 2)

). (5.32)

HereΩ is the vibration frequency. Inserting (5.32) into (5.31) and substituting ˜u andθ by u andθ for simplicity yields the following problem

Find triples(Ω,u,θ) ∈ R++×U such that

a((u,θ),(v,η))−Ω2 b((u,θ),(v,η)) = 0 ∀(v,η) ∈ U (5.33)

where

b((u,θ),(v,η)) =

∫

ωρt

(

1+Kt2

12

)(

aαβ uαvβ +u3v3

)

+ρt3

12aαβ (θα ηβ −2H

[uα ηβ +θαvβ

])dS (5.34)

Definingλ = Ω2 the vibration problem for shells has been transformed into atime-independentgeneralized eigenvalue problem:

97


Find (λ ,u,θ) ∈ R++×U such that

a((u,θ),(v,η)) = λ b((u,θ),(v,η)) ∀(v,η) ∈ U (5.35)

As shown in [SKL09a] the solutions to this problem can be usedto increase the stability of astructure with respect to vibration phenomena. This is doneby raising the structure’s funda-mental eigenfrequency or equivalently the smallest well defined eigenvalue. Thus the frequencyrange for resonance behaviour is also shifted and removed from frequency ranges of externalexcitations. To this end let the smallest well defined eigenvalueλmin(C,D) of problem (5.35) formaterial matrices(C,D) from C be

λmin(C,D) := min

λ∣∣∃(u,θ) ∈ U : (λ ,u,θ) solves (5.35) and(u,θ) 6∈ ker(b)

(5.36)

whereker(b) =

(w,ν)

∣∣ b((w,ν),(v,η)) = 0 ∀(v,η) ∈ U

.

To raise the structure’s fundamental eigenfrequency the constraint

λmin(C,D)≥ λ (5.37)

is added to the Free Material Optimization problem for shells, whereλ is a prescribed positivelower bound. According to [Ber96] the following statement holds true:

Theorem 5.3.1. Assume thatΦ ∈[C2(ω)

]3and that all points on the midsurfaceS = Φ(ω)

are smooth. Then the eigenvalues of problem (5.35) form an increasing sequence0< λ1 ≤ λ2 ≤. . .≤ λn ≤ . . . tending to+∞, each of the eigenvalues having a finite multiplicity. Furthermore,there exists an orthonormal basis composed of the eigenvectors associated with the eigenvaluesλ j satisfying

a((u j ,θ j),(v,η)) = λ j b((u j ,θ j),(v,η)) ∀(v,η) ∈ U

andb((ui ,θi),(u j ,θ j)) = δi j .

Together with Corollary A.3 from [SKL09a] it is possible to equivalently reformulate the eigen-value constraint (5.37) as

inf(v,η)∈U‖(v,η)‖=1

a((v,η),(v,η))− λ b((v,η),(v,η)) ≥ 0 (5.38)

Thus the minimum compliance Free Material Optimization problem for shells (PD) with vibra-tion constraints reads as

min(C,D)∈C

max(u,θ )∈U

−12

∫

ωtγ⊤Cγ +

t3

12χ⊤Cχ + tKζ⊤Dζ dS

+

∫

ωt f⊤udS +

∫

∂ω1

gu⊤u+gθ

⊤θ dl

such that∫

ωt · trC+

12

t · trDdS ≤V (5.39)

ρ− ≤ t · trC+12

t · trD ≤ ρ+


a((v,η),(v,η))− λ b((v,η),(v,η)) ≥ 0

98


It is also possible to add the vibration constraints to the minimum weight formulation (PMW)resulting in

min(u,θ)∈U(C,D)∈C

∫

ωt · trC+

t2· trDdS

subject to C 0; D 0(

tC 00 t

2D

)

− ε15 0

t · trC+t2· trD ≤ ρ+ (5.40)

∫

ωt f⊤udS+

∫

∂ω1

gu⊤u+gθ

⊤θ dl ≤ c


a((v,η),(v,η))− λ b((v,η),(v,η)) ≥ 0

∫

ωtγ⊤(u)Cγ(v)+

t3

12χ⊤(u,θ )Cχ(v,η)+ tkζ⊤(u,θ )Dζ (v,η)dS =

=

∫

ωt f⊤vdS +

∫

∂ω1

gu⊤v+gθ

⊤η dl ∀(v,η) ∈ U

To solve problem (5.40) numerically a discrete version of the vibration constraint has to beadded to the discrete problem formulation (Pm

MW). Thus a discrete formulation for the termb((u,θ),(u,θ)) is required. To this end a consistent mass matrix as in [EKM97, LH06] isintroduced. Thus define a matrixM entirely filled with blocks of the type

c1a11 c1a12 0 c2a11 c2a12

c1a21 c1a22 0 c2a21 c2a22

0 0 c1 0 0c2a11 c2a12 0 c3a11 c3a12

c2a21 c2a22 0 c3a21 c3a22

where the abbreviationsc1 := 1+K t2

12, c2 :=−2H t2

12 andc3 := t2

12 are used. Together with vectorsVm,g ∈ R

N, m= 1,2, . . . ,EltNr, g= 1,2, . . . ,GpNr, withϑ j(xgm), j ∈ Dm at the j-th position and

zeros otherwise, the mass matrix for Naghdi shells can be defined as

M(C,D) =EltNr

∑m=1

Mm(C,D) ,

Mm(C,D) =(

tm · trCm+tm2· trDm

)

Mm, (5.41)

Mm =EltNr

∑g=1

Vm,gMVm,g⊤.

Thus together with the stiffness matricesKγ , Kχ andKζ the vibration constraint can be writtenas

inf(v,η)∈Un‖(v,η)‖=1

(v,η)⊤(

EltNr

∑m=1

Kγ(Cm)+Kχ(Cm)+Kζ (Dm)− λMm(C,D)

)

(v,η) ≥ 0 (5.42)

99


or in a more compact fashion as


(v,η)⊤(

Kshell(C,D)− λM(C,D))

(v,η)≥ 0 (5.43)

leading to the discrete Free Material Optimization problemfor shells in the primal minimalweight formulation:

min(u,θ )∈Un(C,D)∈Ch

EltNr

∑m=1

tm · trCm+tm2· trDm

subject to Cm 0; Dm 0 ∀m= 1, . . . ,EltNr(

tmCm 00 tm

2 Dm

)

− εm15 0 ∀m= 1, . . . ,EltNr


(EltNr

∑m=1

Kγ(Cm)+Kχ(Cm)+Kζ (Dm)

)

(u,θ) = fn

fn⊤(u,θ) ≤ c


(v,η)⊤(

Kshell(C,D)− λM(C,D))

(v,η) ≥ 0

As in [SKL09a] this is a mathematical programming problem with linear matrix inequality con-straints and standard nonlinear constraints, which can be turned into a semidefinite program andcan be solved with the methods described in [SKL09a].

5.4. Global Stability Constraints

The last type of constraints we investigate are global stability constraints. Constraints of thiskind are used to avoid equilibrium states that can not be regarded as stable. Consider a shellin an equilibrium configuration(SI ) implying a displacement field(u,θ) ∈ U . As described in[Ber96] there are three basic types of equilibrium states depending on the sign of the increasein total potential energyP((u,θ),(v,η)), when the shell is subjected to sufficiently small dis-placements(v,η) ∈ U shifting it into the neighboring state(SII ). The increase in total potentialenergy is defined as

P((u,θ),(v,η)) = Π(u+v,θ +η)−Π(u,θ) . (5.44)

100


Using a taylor expansion for the first term this can also be written as

P((u,θ),(v,η)) =

= Π(u,θ)+Π′(u,θ)(v,η)+12

Π′′(u,θ)(v,η)2 +‖(v,η)‖2ε(v,η)−Π(u,θ)

= Π′(u,θ)︸︷︷︸

=0 in equilibrium

(v,η)+12

Π′′(u,θ)(v,η)2 +‖(v,η)‖2ε(v,η)

=12

Π′′(u,θ)(v,η)2 +‖(v,η)‖2ε(v,η)

where lim(v,η)→0 ε(v,η) = 0. Thus the sign of the increase in total potential energy depends onthe termΠ′′(u,θ)(v,η)2, which can be used to identify the equilibrium type as

• stable equilibrium, when the displacement field(u,θ) is a strict local minimum of thepotential energyΠ(u,θ). Under the assumption, thatΠ(u,θ) is twice continuously differ-entiable with respect to(u,θ), this is equivalent to the existence of a constantc1 > 0 suchthat

Π′(u,θ) = 0 (equilibrium of(SI ))

Π′′(u,θ)(v,η)2 ≥ c1‖(v,η)‖2 ∀(v,η) ∈ U

• unstable equilibrium, if there exists at least one displacement field(v,η) ∈ U such that


Π′′(u,θ)(v,η)2< 0

• neutral equilibrium, which implies the following conditions


∃c2 > 0 such that Π′′(u,θ)(v,η)2 ≥ 0 ∀(v,η) ∈ U , ‖(v,η)‖ ≤ c2

∃(vi ,ηi) such thatΠ′′(u,θ)(vi ,ηi)2 = 0 i = 1,2, . . . (5.45)

where(vi ,ηi) ∈ U \0. These specific displacements are called buckling modes of thestructure.

It is our goal to investigate whether the structure will omitbuckling behaviour. As a linearmodel does not contain the effect of buckling we have to expand our shell model using nonlinearterms to be able to describe these properties [Ram87]. Buckling usually appears in a membranestate, hence nonlinear terms are only added to the membrane strains [Nio85]. This results in thenonlinear membrane strains

γαβ (u) = γαβ (u)+12

aκν (γκα(u)−ψκα(u))(γνβ (u)−ψνβ (u)

)+

12

ϕα(u)ϕβ (u) (5.46)

101


whereγαβ (u) are the linear membrane strains already introduced in (2.38) and the remainingfunctions are defined as

ψκα(u) :=12

(uα |κ −uκ |α

), (5.47)

ϕα(u) := u3,α +bκαuκ . (5.48)

For the bending strainsχαβ (u,θ) and the shear strainsζα(u,θ) the linear formulas as given in(2.39) and (2.40) are used. Using these kinematic equationswe follow the calculations presentedin [Ber96] and are hence able to calculate the increase of total potential energyP((u,θ),(v,η))as defined in (5.44)

P((u,θ),(v,η)) =12

∫

ωtCαβλ µ γαβ (u+s·v)γλ µ(u+s·v)− tCαβλ µ γαβ (u)γλ µ (u)

+t3

12Cαβλ µ χαβ (u+s·v,θ +s·η)χλ µ(u+s·v,θ +s·η)

+ tkDαλ ζα(u+s·v,θ +s·η)ζλ(u+s·v,θ +s·η)

− t3

12Cαβλ µ χαβ (u,θ)χλ µ (u,θ)− tkDαλ ζα(u,θ)ζλ (u,θ)dS

−∫

ωt f · (u+s·v)− t f ·udS

−∫

∂ω1

gu · (u+s·v)+gθ · (θ +s·η)−gu ·u−gθ ·θ dl

The potential energy has already been given in (2.50). Furthermore let(u,θ) ∈ U , (v,η) ∈ U

ands∈ R+. Employing the equilibrium conditionΠ′(u,θ)(v,η)!= 0 leads to

∫

ωtCαβλ µ

(

γαβ (u)γλ µ (v)+12

aκνaστ(γκα (u)−ψκα(u))(γνβ (u)−ψνβ (u)) ·

· (γσλ (u)−ψσλ (u))(γτµ (v)−ψτµ(v))+12

ϕα(u)ϕβ (u)ϕλ (u)ϕµ(v)+

+ γαβ (u)ϕλ (u)ϕµ(v)+12

γαβ (v)ϕλ (u)ϕµ (u)+ γαβ (u)aκν (γκλ (u)−ψκλ (u)) ·

· (γν µ(v)−ψν µ(v))+12

γαβ (u)aκν (γκλ (u)−ψκλ (u))(γν µ (u)−ψν µ(v))+

+ϕα(u)ϕβ (u)aκν (γκλ (u)−ψκλ (u))(γν µ(v)−ψν µ(v))+

+ϕα(u)ϕβ (v)aκν (γκλ (u)−ψκλ (u))(γν µ (u)−ψν µ(u))

)

+

+t3

12Cαβλ µ χαβ (u,θ)χλ µ (v,η)+ tkDαλ ζα(u,θ)ζλ (v,η)dS =

=

∫

ωt f ·vdS +

∫

∂ω1

gu ·v−gθ ·η dl ∀(v,η) ∈ U

102


granting access to the quantity of interest

Π′′(u,θ)(v,η)2 =12

∫

ωnαβ (u)[aκν (γκα(v)−ψκα(v))(γνβ (v)−ψνβ (v))+ϕα(v)ϕβ (v)]

+ tCαβλ µγαβ (v)γλ µ (v)+t3

12Cαβλ µ χαβ (v,η)χλ µ (v,η)

+ tkDαλ ζα(v,η)ζλ (v,η)dS , (5.49)

where we have used the definitionnαβ (u) = tCαβλ µ γαβ (u) for the initial stress term of the state(SI ). Throughout the buckling analysis it is assumed that the loads acting on the shell are givenby P+ λQ, whereP is a permanent load case applied to the initial configuration(SI ) andλQis a fluctuating perturbation of the originally stable equilibrium associated withP. The criticalload P+ λcQ is defined as the load with the smallest valueλc under which buckling occurs.The determination of the buckling modes is more accurate fora permanent loadP, which isclose to the critical load. In this case one may make a linear approximation of the stressesnαβ = pαβ +λqαβ , wherepαβ (uP) andqαβ (uQ) are the stresses associated with the loadsP andQ, respectively. This approximation, that allows for a linear calculation from the displacementsuP anduQ and also the resulting stressespαβ (uP) andqαβ (uQ) from the loadsP andQ, is knownas linear buckling.Defining now the bilinear forms

a((v,η),(w,ϑ)) :=∫

ωtCαβλ µ γαβ (v)γλ µ (w)+

t3

12Cαβλ µ χαβ (v,η)χλ µ (w,ϑ)

+ tkDαλ ζα(v,η)ζλ (w,ϑ)dS (5.50)

buQ((v,η),(w,ϑ)) :=∫

ωqαβ (uQ)[a

κν(γκα(v)−ψκα(v))(γνβ (w)−ψνβ(w))

+ϕα(v)ϕβ (w)]dS (5.51)

cuP((v,η),(w,ϑ)) :=∫

ωpαβ (uP)[a

κν(γκα (v)−ψκα(v))(γνβ (w)−ψνβ(w))

+ϕα(v)ϕβ (w)]dS (5.52)

it becomes apparent through comparison with the buckling condition (5.45) and the decisiveterm of the increase in total potential energy (5.49), that the buckling modes can be determinedas eigenvectors(vc,ηc) of the generalized eigenvalue problem

Find the eigenvalue of the smallest moduleλc and the associated eigenvector(vc,ηc)which are solutions to the equation

a((vc,ηc),(w,ϑ))+cuP((vc,ηc),(w,ϑ))+λcbuQ((vc,ηc),(w,ϑ)) = 0 ∀(w,ϑ) ∈ U . (5.53)

The corresponding loadP+λcQ is called the critical load of the shell.We now want to add a constraint to the discretized Free Material Optimization problem in theminimum weight-formulation (PMW) to avoid buckling behavior. As in [KS04] we setP = 0resulting inuP = 0 andpαβ = 0. We already know that the discretized form ofa((u,θ),(v,η))is given by(u,θ)⊤Kshell(C,D)(v,η). Introducing the geometry stiffness matrixGshell the term

103


buQ((u,θ),(v,η)) can be rewritten in its discretized version as(u,θ)⊤Gshell(C,D,u,θ)(v,η).Thus we are able to formulate a discretized global stabilityconstraint by demanding that noneof the eigenvalues of the generalized eigenvalue problem

Kshell(C,D)(vc,ηc)+λcGshell(C,D,u,θ)(vc,ηc) = 0 (5.54)

lie in the interval[0,1]. According to [Koc02] this condition can be reformulated as a matrixconstraint of the form

Kshell(C,D)+Gshell(C,D,u,θ) 0 (5.55)

Hence we obtain the Free Material Optimization problem for shells with a global stability con-straint

min(u,θ )∈Un(C,D)∈Ch

EltNr

∑m=1

tm · trCm+tm2· trDm (5.56)

subject to Cm 0; Dm 0 ∀m= 1, . . . ,EltNr(

tmCm 00 tm

2 Dm

)

− εm15 0 ∀m= 1, . . . ,EltNr


(EltNr

∑m=1

Kγ(Cm)+Kχ(Cm)+Kζ (Dm)

)

(u,θ) = fn

fn⊤(u,θ) ≤ c

Kshell(C,D)+Gshell(C,D,u,θ) 0

which is a nonlinear SDP with nonlinear matrix constraints.

104

CHAPTER 6

Summary and Outlook

In this thesis we have developed mathematical models and numerical methods in order to solvethe Free Material Optimization problem for shells. In Chapter 2 we have introduced Naghdi’sshell model, a first order approximation including shear deformations, which is based on a differ-ential geometrical description of the shell’s midsurface.In order to employ this model the shellis considered as a Cosserat continuum. Instead of approaching the shell as a three-dimensionalbody, it is depicted as a surface in the physical space and endowed with additional degrees offreedom at each point to capture physical aspects that exceed pure membrane behavior.

This linear elastic model represents the basis of the Free Material Optimization formalism forshells and plates, that we have developed in Chapter 3. It focuses on one of the fundamentalproblems of structural optimization: find the stiffest structure in a given design domain and fora prescribed set of loads constructed from a limited amount of material. To this end we considerthe entire anisotropic material tensors as optimization variables and use their trace as measurefor the amount of used material. Within this thesis we have regarded several different problemformulations for this task: we started with the minimum compliance problem, that is directlymotivated by physical aspects. We exploit the saddlepoint structure of this problem in order toprove existence of an optimal solution for the Free MaterialOptimization problem for shellsusing a Minimax-Theorem. However, this problem formulation is not suited for numerical pur-poses. For this reason we introduce the primal problem formulation, whose dual is equivalent tothe original saddlepoint formulation. This primal problemfeatures several advantages: foremostit is a convex problem, more precisely a nonlinear convex semidefinite program. For this kind ofproblem there exist numerical solvers and their further development is an active research area.Moreover the material matrices, that are responsible for the huge dimension of the discretizedversion of the saddlepoint problem, are hidden in the primalproblem formulation as Lagrangemultipliers of semidefinite matrix constraints. These facts clearly state the importance of theequivalence proof given in Subsection 3.3, whose components such as the local maximum prin-ciple for matrices are deliberately proven in a general context such that the proof covers also

105

6. Summary and Outlook

the equivalence statement in the case of multiple load scenarios for shells as well as for solids.The final problem formulation presented in this thesis is theminimum weight formulation, thathas become the dominant problem formulation due to the availability of the novel optimizationalgorithm PENSCP.

There exist several links within the obtained results for future research. An extension of con-siderable interest for the structural engineering community is the combination of several layersof shells referred to as laminates. The reason is that many realizations of thin-walled struc-tures are actually built up as laminates in practice, as these constitute an efficient and versatileway to construct lightweight material with manipulable material properties [GHH99]. For thedescription of laminates two alternatives are possible: they can either be accessed as a three-dimensional medium based on a theory derived from three-dimensional solid elasticity [Red97],or as a Cosserat medium, where the multiple layers are captured by multiple director vectors ateach point of the midsurface [KJ02]. By allowing asymmetriclaminates it is even possible toconstruct composites with entirely new properties such as membrane-bending coupling. How-ever, the primal benefit is the detailed insight into the layout of the ultimately optimal laminatedesign, that is of tremendous value for manufacturers and encourages the use of novel fabricationmethods.

From a theoretical point of view an extension to a nonlinear elastic model promises to be aninteresting research field [Ant05]. The obvious candidate for this is the von Karman plate model,that describes large deflections of thin flat plates [Kar10,Cia97]. It is one of the most establishednonlinear plate models with a firm theoretical background and there exist approved numericalmethods for its simulation [SK97]. However, the essential challenge consists in proving theexistence of an optimal solution and in the investigation ofthe duality properties of the obtainedoptimization problem. Moreover this model covers a range ofoperation where the currentlyemployed linear elastic models do not provide a reliable prediction of the plate’s behavior.

In Chapter 4 we have employed a finite element method togetherwith the nonlinear semidefi-nite solver PENSCP to obtain a computational solution for the Free Material Optimization prob-lem for shells. These numerical procedures have been realized within the PLATO-N softwareplatform. Numerical tests with examples widely-used in structural optimization of shells showthat our code provides reliable and reasonable results, which contain detailed information aboutthe material properties of the optimal design. Moreover we are able to perform Free MaterialOptimization for structures combined of solids and shells,which is required for almost everyrealistic application, and our software allows to include condensed data structures for parts ofthe structure containing only fixed material in order to reduce the problem dimension or to in-corporate element types that are not part of the PLATO-N software platform.

The numerical realization of Naghdi’s shell model, that we employ in this thesis, is knownto display shear locking behavior for bending dominated testcases. This undesirable effect hasbeen neglected as Free Material Optimization is primarily used in the predesign phase and shearlocking is included in later steps of the design circle. However a possibility to incorporate aremedy for this problem in the PLATO-N software platform is the use of MITC elements asintroduced by [BD86, Bat02, CB03], that utilize mixed interpolation of tensorial componentsin order to prohibit locking behavior and thus to provide reliable results for bending dominatedproblems.

106

Moreover we have discussed the optimization of structures combined of solids and shells inSection 4.4.1. This has been achieved on a discrete level by merging the translational degrees offreedom. However, there exists a more precise theory for thedescription of junctions of elasticbodies [Cia90, Naz97, BC98], which can be employed to perform the combination of shells andsolids on an analytical level and to obtain a more detailed description of the effects caused bysuch multi-structure junctions.

Finally we have investigated the inclusion of multidisciplinary optimization constraints to theFree Material Optimization problem for shells in Chapter 5.We have considered displacementconstraints in order to affect the shape of the deformed structure as well as stress constraintsto avoid material failure due to high stresses. For the investigation of vibration constraints wehave deducted a dynamic model for the free vibrations of Naghdi shells. The resulting eigenfre-quency constraints can be employed to raise the structure’snatural frequencies and thus preventa resonance disaster.

Buckling constraints constitute another type of multidisciplinary constraints, for which wehave provided a theoretical description within this thesis, but no numerical realization withinthe software platform as the optimizer is not yet able to provide reliable results for this sce-nario. However, buckling constraints are of extreme importance for thin-walled structures,as these are very susceptible to imperfections of the initial geometry or perturbations of theapplied loads and typically react with sudden deformationsknown as snap-through behavior[Bus85, Olh89, LL10]. Hence the development of accurate numerical procedures for bucklingconstraints represents another promising field for future research.

107

APPENDIX A

Appendix

A.1. Technical Remarks on Theorem 3.3.1

Lemma A.1.1 (Technical trace lemma). Consider a matrix C∈ L∞(ω ,Sn), a vectorγ ∈[L2(ω)

]n,

scalarsα ∈ R, t ∈ R++ and denote by1n the unit matrix inRn. Then the following statementshold

i)〈C, tα13〉= tr(C · tα13) = tα tr(C) , (A.1)

ii)∫

ω〈C, tγγ⊤〉dS=

∫

ωtr(C · tγγ⊤)dS=

∫

ωtγ⊤Cγ dS. (A.2)

Proof. According to [WSV00] the inner matrix product for two symmetric matricesA,B∈Rn×n

is defined as

〈A,B〉= tr(A ·B) =n

∑i, j=1

Ai j Bi j . (A.3)

Statement i) follows directly from this definition. As it holds, that

γ⊤Cγ =n

∑i, j=1

γiCi j γ j =n

∑i, j=1

γiγ jCi j =n

∑i, j=1

(γγ⊤

)

i jCi j

we can also deduct ii).

Lemma 3.3.2 We consider the saddle-point problem

minp∈A

maxu∈B

J(p,u) (3.26)

109

A. Appendix

whereA = C , B = U and J= JD. Then(p∗,u∗) ∈ A ×B is a saddle-point of J if and only if

δpJ(p∗,u∗)(p− p∗) ≥ 0 ∀ p∈ A ,

δuJ(p∗,u∗)(u−u∗) ≤ 0 ∀u∈ B . (3.27)

An analogous result is true forA = C , B = U ×L and J= JL.

Proof. We use the following result: if for two reflexive Banach spaces V andZ the followingassumptions hold:

(I) A ⊂ V andB ⊂ Z are convex, closed and non-empty,

(II) the functionJ : A ×B → R satisfies

∀ p∈ A u 7→ J(p,u) is concave and upper semi-continuous,

∀u∈ B p 7→ J(p,u) is convex and lower semi-continuous,

(III) and in addition

∀ p∈ A u 7→ J(p,u) is Gateaux-differentiable,

∀u∈ B p 7→ J(p,u) is Gateaux-differentiable,

then(p∗,u∗) ∈ A ×B is a saddle-point ofJ according to [ET76, Chapter IV, Proposition 1.6] ifand only if

δpJ(p∗,u∗)(p− p∗) ≥ 0 ∀ p∈ A ,

δuJ(p∗,u∗)(u−u∗) ≤ 0 ∀u∈ B .

The proof is given for reflexive spaces in [ET76, Chapter IV, Proposition 1.6], however itturns out that the proof can directly be generalized to non-reflexive Banach spaces such asV = L∞(ω ;S3)× L∞(ω ;S2). Therefore it suffices to show that conditions (I) to (III) are sat-isfied for problem (PD) and (PL). In the caseA = C , B = U andJ = JD the assumptions (I)and (II) are valid due to the proof of Theorem 1. ForA = C , B = U ×L andJ = JL as-sumption (I) is fulfilled asC – the cone of semidefinite matrices inV – is convex, closed andnon-empty. This is also true forL and thus forU ×L . As the functionalJL is composed asthe sum of linear functionals toJD the properties required for (II) are preserved. (III) followsobviously from (3.30) to (3.33).

Proof of Lemma 3.3.3. This proof is an extension to matrices of the original proof in [CM70].It makes again use of the notation introduced in (3.39) and (3.40):

M = M ∈ [L∞(ω)]n×n |M = M⊤ 0, ρ− ≤ tr(M)≤ ρ+ ,J = j ∈ 1, . . .n|ej 6∈ ker(M∗) .

110


(i) First define a linear continuous mappingφ(M) : [L∞(ω)]n×n → R2:

φ(M) =

( ∫

ω s1⊤Ms1+s2

⊤Ms2+ . . .+sp⊤MspdS

∫

ω trM dS

)

=

(ξ1

ξ2

)

. (A.4)

The range ofφ is denoted by

G := (ξ1,ξ2) ∈R2 |(ξ1,ξ2) = φ(M) , M ∈ [L∞(ω)]n×n

, (A.5)

M = M⊤ 0, ρ− ≤ tr(M)≤ ρ+ .The setG is closed, convex and bounded. The point(ξ1

∗,ξ2

∗) such thatξ1∗ is maximal for

ξ2∗ =V corresponds toM∗. Thus(ξ1

∗,ξ2

∗)∈ ∂G and there exists a hyperplane supportingG at (ξ1

∗,ξ2

∗) due to [Roc70, Theorem 11.6]. Defining this hyperplane by thevector(1,−α∗) results in

ξ1∗−α∗ξ2

∗ ≥ ξ1−α∗ξ2 ∀ξ1,ξ2 ∈ G. (A.6)

Inserting the definition ofξ1 andξ2 yields∫

ω

p

∑i=1

si⊤M∗si dS−α∗

∫

ωtrM∗dS≥

∫

ω

p

∑i=1

si⊤Msi dS−α∗

∫

ωtrM dS (A.7)

for all M ∈ M . Using the notation of the inner product this yields (3.38) and thus (i):∫

ω〈M∗

,

p

∑i=1

sisi⊤−α∗1n〉dS≥

∫

ω〈M,

p

∑i=1

sisi⊤−α∗1n〉dS ∀M ∈ M . (A.8)

(ii) Suppose there exists a subsetω ⊂ ω , |ω | > 0 where trM∗ > ρ− a.e.. We first show thatthere exists at least one eigenvalueλk > 0, k ∈ 1, . . . ,n. Supposeλ j = 0 ∀ j = 1, . . . ,n.Without loss of generality we can assume that there exists another subsetω ⊂ ω , |ω |> 0,ω ∩ ω = /0 with trM∗ < ρ+ a.e. inω and at least oneλk > 0, k∈ 1, . . . ,n (which is truefor non-vanishingsi and a reasonable choice ofρ+). Consider now the matrix

M = M∗− χωtrM∗−ρ−

trM∗ M∗+ χω min

1|ω |

∫

ωtrM∗−ρ−dS,ρ+− trM∗

ekek⊤,

whereχω(x) and χω(x) are the characteric functions of the setsω and ω , respectively.Thus it follows thatM ∈ [L∞(ω)]n×n, M = M⊤ 0,

∫

ω trM dS≤V, ρ− ≤ trM ≤ ρ+ andwe get

∫

ω〈M,

p

∑i=1

sisi⊤〉dS =

∫

ω〈M∗

,

p

∑i=1

sisi⊤〉dS

−∫

ω

trM∗−ρ−

trM∗ 〈M∗,

n

∑j=1

λ jejej⊤〉

︸︷︷︸

=0

dS

+

∫

ωmin

1|ω |

∫

ωtrM∗−ρ−dS,ρ+− trM∗

︸︷︷︸

>0

· λk︸︷︷︸

>0

dS

>

∫

ω〈M∗

,

p

∑i=1

sisi⊤〉dS.

111

A. Appendix

This obviously contradicts the optimality ofM∗, thus for eachω with trM∗ > ρ− thereexists at least one positive eigenvalue. Assume now that forall j ∈ 1, . . . ,n with λ j > 0ej ∈ ker(M∗) holds. Consider another matrix

M =

M∗ on ω \ ω

trM∗ekek⊤ on ω .

In this case we get

∫

ω〈M,

p

∑i=1

sisi⊤〉dS =

∫

ω\ω〈M∗

,

p

∑i=1

sisi⊤〉dS

+

∫

ω〈M∗

,

n

∑j=1

λ jejej⊤〉dS

︸︷︷︸

=0

+

∫

ωdS.

Again this contradicts the optimality ofM∗. Thus for everyω with trM∗ > ρ− a.e. thereexists at least oneej 6∈ ker(M∗) with a positive eigenvalueλ j > 0. This shows (ii). Notethat the optimality ofM∗ remains when neglecting all components belonging to the kernelof M∗:

∫

ω〈M∗

,

n

∑j∈J

−λ jejej⊤〉dS=

∫

ω〈M∗

,

n

∑j=1

−λ jejej⊤〉dS

=∫

ω〈M∗

,

p

∑i=1

−sisi⊤〉dS≤

∫

ω〈M,

p

∑i=1

−sisi⊤〉dS

= −∫

ω〈M,

n

∑j∈J

−λ jejej⊤〉dS−

∫

ω〈M,

n

∑j=1

ej∈ker(M∗)

−λ jejej⊤〉

︸︷︷︸

>0∀M∈C

dS

≤ −∫

ω〈M,

n

∑j∈J

−λ jejej⊤〉dS.

If all ej 6∈ ker(M∗) have the same eigenvalueλ > 0 (this includes the case when there existsonly oneej 6∈ ker(M∗) with positive eigenvalue), then the problem can be projected ontothe lower-dimensional spaceP spanned by these eigenvectors. LetdP be the dimensionof P, then it follows that

n

∑j∈J

λejej⊤ = 1dP

⇒ 〈M,

n

∑j∈J

−λejej⊤〉= λ trPM . (A.9)

Hence this case resembles the scalar case presented in [CM70]. Thus in the followingwe will assume that there exist at least two eigenfunctionsek,el 6∈ ker(M∗) with differenteigenvalues, one of them positive:λk > λl ≥ 0.

112


(iii) Assume there exists a subsetω ⊂ ω , |ω |> 0 with

maxj∈J

λ j < α∗ and trM∗> ρ− a.e. inω . (A.10)

According to (ii) there exists at least one eigenfunctionek 6∈ ker(M∗) with λk > 0, k ∈1, . . . ,n. Furthermore we assume, there exists at least one other eigenfunction el 6∈ker(M∗) andλk > λl ≥ 0, l ∈ 1, . . . ,n, otherwise this would represent the scalar casefrom [CM70]. Define the matrixM by

M = M∗+ χω(x)mintrM∗−ρ−,el

⊤M∗el(

ekek⊤−el el

⊤)

.

This yields not only trM = trM∗, but alsoM ∈ M . Consider now

∫

ω〈M,

p

∑i=1

sisi⊤−α∗1n〉dS

=∫

ω〈M∗

,

p

∑i=1

sisi⊤〉−α∗trM dS

+

∫

ωχω mintrM∗−ρ−

,el⊤M∗el〈ekek

⊤−el el⊤,

n

∑j=1

λ jejej⊤〉dS

=∫

ω〈M∗

,

p

∑i=1

sisi⊤〉−α∗trM∗dS+

∫

ωmintrM∗−ρ−

,el⊤M∗el

︸︷︷︸

>0

(λk−λl)︸︷︷︸

>0

dS

>

∫

ω〈M∗

,

p

∑i=1

sisi⊤−α∗1n〉dS.

As this is a contradiction to the optimality ofM∗, this yields (iii).

(iv) The proof for the case minj∈J λ j > α∗ works analogously with the matrix

M = M∗+ χω(x)minρ+− trM∗,el

⊤M∗el(

ekek⊤−el el

⊤)

.

Although the case trM∗ < ρ+ includes trM∗ = ρ−, no problems occur, as either there is acontradiction to minj∈J λ j > α∗ or there exists an eigenvalueλk > 0 with an associatedeigenfunctionek 6∈ ker(M∗) and the proof of (iv) again holds.

2

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121

Acknowledgments

During the last years I have not only learned a variety of things about mathematics and myself,I also experienced the support of several wonderful and knowledgeable people, which I willalways remember with immense thankfulness. I want to express my special gratitude to

• Prof. Dr. Gunter Leugeringfor his willingness to introduce a physicist to the field of opti-mization, his enthusiasm for the research projects I proposed, his comprehensive scientificstore of knowledge, that I could draw upon when I was at loss and his confidence in mywork,

• Prof. Dr. Michael Stingl, my mentor in the field of Free Material Optimization, whoprovided tremendous help for the theoretical as well as for the numerical aspects of thisthesis. On uncountable occasions I entered his office with questions and always receivedpatient and valuable advice – this thesis would not have beenpossible without his aid,

• Prof. Dr. Mathias Stolpe, who took over my supervision during my research visit inDenmark, for his exemplary enthusiasm for science, that constantly instilled me with mo-tivation, his assistance during the first implementation ofthe software, that prevented a lotof headaches and his surpassing dedication as a mentor for structural optimization as wellas the tactics of squash,

• Prof. Dr. Martin Bendsøefor his neverending encouragement regarding my research, forfacilitating my research visit at DTU, that had a huge positive influence on my thesis, andfor introducing me to many experts of the structural optimization community and theirgreat ideas, which has been and always will be a very inspiring network,

• my colleagues from the Institute of Applied Mathematics II at the University of Erlangen-Nurnberg, especiallyProf. Dr. Wolfgang AchtzigerandProf. Dr. Johannes Jahn, whoI consider as role models due to their expertise in science and teaching as well as theirdiligence, moreoverDr. Gabriele EichfelderandDr. Wigand Rathmannfor huge amountsof moral support and entirely enjoyable teaching teamwork,Dr. Friedrich Graef for beinga skillful and good-natured system admistrator at the same time and finally all my fellowresearchers and Ph.d. students, who are likewise responsible for the pleasant and kindatmosphere at our institute,

123

Acknowledgments

• all PLATO-N project partners, who contributed in various ways to a successful and enjoy-able workflow, in particularProf. Dr. Michal Kocvara for sharing his enormous knowl-edge on structural optimization as well as his humor,Herbert Hornlein for providingextremely valuable insights on Michell trusses and compliance potatoes,Gabor Bodnar,Richard Boyd, Julian Keller and Markus Wagnerfor their testing of the FMOKernel,which revealed several hidden bugs, andSlobodan Veljovicfor his practical implementa-tion of our test case library,

• my colleagues at the Institute for Mathematics during my time at DTU, especiallyGe-offrey Lefebvre, Sonja Lehmann, Dr. Eduardo MunozandMonika Rotthaus, who wereprimarily responsible for making it an entirely positive experience and created “hyggelig”surroundings thousand kilometers away from home,

• Prof. Dr. Kai-Uwe Bletzinger, Prof. Dr. Tomasz LewinskiandProf. Dr. Jan Sokołowskifor inviting me to several international conferences, which provided a huge amount of newideas and motivation for my own research,

• Dr. Tobias Moroder, who endured all my ups and downs during the development of thisthesis without complaining even once and instead provided me with a constant supply oflove, encouraging words and utter confidence,

• my family: my motherTheresia Gailefor her caring nature and her belief in my abilities,my fatherClaus Gailefor his structural engineering genes as well as his unconditionalsupport and my brotherThomas Gailefor being exactly the way he is,

• and finally my friends, who were happy for me in good times, cheered me up in bad timesand were simply there, when I needed them. I will be always indebted toDr. SusanneHeld and Tina Pooschke, who are much better best friends than I could ever hope for.Additionally I consider myself lucky to be a part of the livesof Dr. Oliver Erlenbach,Dr. Stephan Geier, Dr. Wolfgang MeyerandDr. Julian Zeitvogland will never forget themultitude of adventures that we experienced since our first day at university.

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