Politecnico di MilanoScuola di Ingegneria Industriale e dell’Informazione

Corso di Laurea Magistrale in Mathematical Engineering

Topology optimization of self-assembling

anisotropic materials

Relatore:Prof. Marco Verani

Correlatori:Prof. Stefano BerroneProf. Nicola Parolini

Tesi di Laurea Magistrale di:Francesco Regazzoni

Matr. 837534

Anno Accademico 2015-2016

Form follows function.

Louis Sullivan, 1856–1924

Abstract

It is well established that in nature elastic bodies that have to resist tomechanical loading exhibit the spontaneous development of micro-structures(think for instance of the sponge structure of bones). In this thesis the possi-bility of realizing micro-structures by use of diblock copolymers, linear chainmolecules made of two subchains joined covalentely to each other, is con-sidered. When such binary mixtures undergo a critical temperature, theyspontaneously arrange in regular and periodic structures of some nanometrescharacteristic length. The process is described by a nonlocal version of theCahn-Hilliard equation, the Cahn-Hilliard-Oono equation, which can be re-garded as the gradient flow in a suitable Sobolev space of the Ohta-Kawasakifunctional. The numerical solution of this equation allows to predict andcontrol pattern formation.

A two-scales topology optimization problem is formulated: at the mi-croscale Cahn-Hilliard-Oono equation rules the pattern formation, while atthe macroscale the equilibrium configuration of the elastic body is deter-mined by the linear elasticity equation. Homogenization theory is employedto link the two scales. By controlling the parameters that rule the diblockcopolymers phase separation, the typology and orientation of these patternsis locally optimized, in order to maximize the elastic response of a macro-scopic body. At the same time the shape of the body is optimized, with theconstraint of the total mass.

To solve numerically the PDEs that govern the two scales, as well asthe PDEs associated with the homogenization of the stiffness tensor, thefinite elements method (FEM) is employed. To tackle discontinuities in thestate plane of diblock patterns, a multi-material problem is formulated, andconsequently a new algorithm that generalizes the Optimality Conditionsmethod for topology optimization to the multi-material case is developed.

I

II

Sommario

E ben noto che in natura, quando un corpo elastico deve sopportare uncarico meccanico, si assiste alla formazione spontanea di microstrutture (sipensi ad esempio alla struttura spugnosa delle ossa). In questa tesi vieneconsiderata la possibilita di realizzare delle microstrutture mediante l’utiliz-zo di diblock copolymer, molecole formate da due sotto catene legate fra diloro in modo covalente. Quando una miscela di diblock copolymer discen-de al di sotto di una temperatura critica, vengono a formarsi in manieraspontanea delle strutture ordinate e periodiche, con una lunghezza carat-teristica di qualche decina di nanometri. Questo processo e descritto dauna variante non locale dell’equazione di Cahn-Hilliard (equazione di Cahn-Hilliard-Oono), che puo essere letta come il flusso gradiente in un opportunospazio di Sobolev del funzionale di Ohta-Kawasaki. La simulazione nume-rica di quest’equazione permette di prevedere la formazione di pattern, maanche di controllarla mediante la scelta di alcuni parametri.

In questa tesi viene considerato un problema di ottimizzazione topologicaa due scale: alla microscala l’equazione di Cahn-Hilliard-Oono guida la for-mazione del pattern, mentre alla macroscala la configurazione di equilibriodel corpo elastico e determinata dall’equazione di elasticita lineare. Median-te la teoria dell’omogeneizzazione viene modellizzata la reciproca interazionefra le due scale. Controllando i parametri che regolano la formazione dellemicrostrutture, si vogliono ottimizzare localmente la topologia e l’orienta-mento dei pattern, in modo da massimizzare la rigidezza della struttura.Nello stesso tempo viene ottimizzata la forma del corpo, assegnata la massatotale.

Per la risoluzione numerica delle PDE che governano le due scale e del-le PDE legate all’omogeneizzazione del tensore di elasticita viene usato ilmetodo degli elementi finiti (FEM). Per gestire le discontinuita presentinel quadro di stato dei diblock copolymer viene formulato un problemamulti-materiale, per la cui risoluzione numerica viene sviluppato un algo-ritmo che generalizza il metodo delle condizioni di ottimalita (OC) al casomulti-materiale.

III

IV

Ringraziamenti

Mentre il marinaio sta entrando in porto dopo un viaggio durato cinqueanni, e naturale che si guardi indietro e ripensi a chi il viaggio lo ha resopossibile, e a chi ha fatto sı che fosse quello che e stato.

Desidero innanzitutto esprimere la mia piu profonda gratitudine a MarcoVerani e Nicola Parolini, che nell’ultimo anno e mezzo di navigazione sonostati validi insegnanti e preziose guide. Grazie per l’infinita disponibilita,per la vostra umanita e per avermi accompagnato nelle scelte che determin-eranno il mio immediato futuro. Un ringraziamento al Professor Berroneper la sua disponibilita.

Vorrei poi esprimere la mia riconoscenza a tutti i docenti che ho incon-trato sui ponti della Nave: al Prof. Verri, per avere cercato di spaventarci inquel lontano 19 settembre 2011; al Prof. Fuhrman, per il metodo Fuhrman;alla Prof. Paganoni, per la sua grinta; al Prof. Salsa, per il suo impareggia-bile carisma; al Prof. Grasselli, perche ARF e come Scilla e Cariddi: dopoaverla passata il viaggio e ancora lungo, ma tutto in confronto sembra unapasseggiata; al Prof. Tomarelli, perche it’s trivial ; e a tutti gli altri che sisono avvicendati al timone.

Grazie ai compagni di viaggio, per averlo reso divertente, stimolante,indimenticabile. Grazie Marcolino, perche senza di te sarebbe stata un’altracosa. Grazie Gab(b)ri, per la tua contagiosa allegria a ogni ora del giorno eper tutte le volte che mi hai ospitato in Via Frapolli. Grazie Cesare, complicedi mille trollate. Grazie Filippo Claudio Ugo Marcello Davide Sacco (ne hoazzeccati almeno un paio?) e Luca, fedeli compagni di magistrale. GrazieBea, per i momenti di nostalgia dei tempi in cui µ e ν si pronunciavano “mi”e “ni”. Grazie Simo e Gigi, insieme a me fidi vassalli del The King (unicomonarca insignito del doppio e bilingue articolo). Grazie poi alla compagniadei primi anni, purtroppo ormai decimata: EnricoLuca, Cace, Filice, Cris,Dome, Manuel e gli altri, per il Diritto alla Pausa, per i cartelloni, per itornei di scopone. Grazie ai compagni di ASP, e a tutte le altre persone chehanno condiviso con me momenti di questa navigazione.

Il pensiero va poi a chi c’e stato durante gli scali che il marinaio hacompiuto durante il viaggio, quando egli dalla Nave tornava sulla terraferma.Un grazie a tutti i miei amici per essere quello che sono. Grazie Pietro (aliasBanni), immancabile compagno d’avventura, per quelle che ci sono state

V

e per quelle che ci saranno (riusciremo a organizzare il viaggio di laurea).Grazie Lele, per esserci sempre. Grazie alle veduggesi Annale, Robbe e Calde(la prima e la terza qui eccezionalmente divise), a Spo, Simo e Teddy per levacanze e i momenti di amicizia condivisi. Grazie a Gege (altrimenti notacome il Magnifico), Giri, Eli, Cate e Titti per le sciate, i weekend elvetici ele imbucate. Grazie a Sbalocco per Nike+ Run Club e per la sua ospitalita,a Lucrella perche ha reso il pendolarismo piu divertente, ad Andrea perle lunghe chiacchierate. Grazie ai ragazzi della Scherma Lecco, per ognistoccata data e soprattutto per ogni ricevuta. Grazie ai Robies e ai ragazzidel coro, e a chi ho dimenticato in queste righe, perche cum quo aliquisiungitur talis erit.

Ma il grazie piu grande va ai miei genitori: se ora sto scrivendo questapagina lo devo soprattutto a voi. Impossibile riassumere in queste pocherighe tutti i motivi di gratitudine. Ne scrivero soltanto uno: grazie peravermi insegnato a mettere tutta la mia passione in quello che faccio. Graziea Luigi, per essere oltre che un fratello il mio migliore amico. Grazie allaNonna Elia, per essere la mia piu grande fan.

E grazie a te, per tutto quello che per me rappresenti.

Il marinaio e ormai entrato in porto. Sorride.

Ma no! Ne puo la nera nave al fischiodel vento dar la tonda ombra di pino.E pur non vuole il rosichıo del tarlo,

ma l’ondata, ma il vento e l’uragano.Anch’io la nube voglio, e non il fumo,

il vento, e non il sibilo del fuso,non l’ozıoso fuoco che sonnacchia,

ma il cielo e il mare che risplende e canta.

Lecco, settembre 2016

VI

Contents

Abstract I

Sommario III

Contents VII

List of Figures X

List of Algorithms XIII

1 Introduction 1

1.1 Topology Optimization . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Micro-structures in topology optimization . . . . . . . . . . . 2

1.3 Self-assembling micro-structures . . . . . . . . . . . . . . . . . 3

1.4 Existing works . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6.1 Linear elasticity . . . . . . . . . . . . . . . . . . . . . 8

1.6.2 Isotropic constitutive law . . . . . . . . . . . . . . . . 9

I Problem formulation 11

2 Diblock copolymers 13

2.1 Diblock copolymers . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The Ohta-Kawasaki functional . . . . . . . . . . . . . . . . . 15

2.3 Cahn-Hilliard-Oono equation . . . . . . . . . . . . . . . . . . 17

2.3.1 Gradient flows . . . . . . . . . . . . . . . . . . . . . . 17

2.3.2 The Sobolev space H−1(Ω) . . . . . . . . . . . . . . . 18

2.3.3 CHO equation . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Finite Element discretization of the equation of Cahn-Hilliard-Oono . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Convex splitting of gradient flows . . . . . . . . . . . . 20

2.4.2 Finite element approximation . . . . . . . . . . . . . . 22

VII

2.5 Numerical simulations and phase plane in 2D . . . . . . . . . 23

2.5.1 Phase plane in 2D . . . . . . . . . . . . . . . . . . . . 25

2.5.2 Domain and boundary conditions . . . . . . . . . . . . 25

2.5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . 27

3 Homogenization: from the micro to the macro scale 31

3.1 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Two-scale asymptotic expansion . . . . . . . . . . . . 32

3.1.2 H-convergence . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Homogenized stiffness tensor of diblock copolymers patterns . 36

3.2.1 Interpolation between pure materials . . . . . . . . . . 36

3.2.2 Two-scales model . . . . . . . . . . . . . . . . . . . . . 38

3.2.3 Homogenization . . . . . . . . . . . . . . . . . . . . . 40

3.2.4 A simpler formulation . . . . . . . . . . . . . . . . . . 42

3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.1 Tensor rotation . . . . . . . . . . . . . . . . . . . . . . 46

4 Topology optimization problem formulation 49

4.1 Optimizing the shape of the body . . . . . . . . . . . . . . . . 49

4.1.1 Material distribution method . . . . . . . . . . . . . . 50

4.1.2 Minimum compliance problem . . . . . . . . . . . . . 51

4.2 Differentiability issues . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Existence issues and length-scale control . . . . . . . . . . . . 54

4.3.1 Existence of solutions . . . . . . . . . . . . . . . . . . 55

4.3.2 A revision of the two-scales model . . . . . . . . . . . 56

4.3.3 Length-scale control . . . . . . . . . . . . . . . . . . . 56

4.3.4 Existence result . . . . . . . . . . . . . . . . . . . . . . 59

II Solution method 63

5 Optimization algorithm 65

5.1 The Optimality Conditions method . . . . . . . . . . . . . . . 65

5.1.1 Generalization to the multi-material case . . . . . . . 68

5.2 Optimization algorithm for diblock copolymers bodies . . . . 76

5.2.1 Optimality conditions . . . . . . . . . . . . . . . . . . 76

5.2.2 Optimization of zh and mh . . . . . . . . . . . . . . . 78

5.2.3 Optimization of the micro-structure orientation . . . . 79

5.3 Implementation of length-scale control . . . . . . . . . . . . . 82

5.3.1 Filtering of ϑh . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Evaluation of the homogenized tensor and its derivatives . . . 88

5.4.1 Exact derivative of the homogenized tensor . . . . . . 89

5.4.2 Evaluation by interpolation . . . . . . . . . . . . . . . 90

VIII

6 Numerical implementation 936.1 Finite Elements discretization . . . . . . . . . . . . . . . . . . 93

6.1.1 Discrete problem formulation . . . . . . . . . . . . . . 936.1.2 Discrete optimization algorithm . . . . . . . . . . . . . 956.1.3 Discrete length-scale control . . . . . . . . . . . . . . . 97

6.2 Implementation details . . . . . . . . . . . . . . . . . . . . . . 996.2.1 Dealing with local minima . . . . . . . . . . . . . . . . 100

7 Numerical results 1037.1 Case study traction: convergence history . . . . . . . . . . . . 1047.2 Case study cantilever : mesh independence . . . . . . . . . . . 1067.3 Case study bridge: algorithm complexity . . . . . . . . . . . . 1087.4 Case study square: effect of the ratio EA/EB . . . . . . . . . 1107.5 Case study L-shape: a popular test case . . . . . . . . . . . . 110

8 Conclusions 113

A Useful results 117

Bibliography 119

IX

X

List of Figures

1.1 Internal structure of biological tissues. . . . . . . . . . . . . . 2

1.2 SBS block copolymer in TEM. Source: wikimedia.org . . . . 4

2.1 Top: a diblock copolymer macromolecule. Bottom: microphaseseparation of diblock copolymers into lamellae (left) and spheres(right). Source: [14] . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Stable configurations in three dimensions. Source: [14] . . . . 15

2.3 Phase plane in three dimensions. Source: [45] . . . . . . . . . 16

2.4 Stable configurations in two dimensions. In both cases γ =10. Source: [13] . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Phase plane in two dimensions. . . . . . . . . . . . . . . . . . 24

2.6 Periodicity cell for diblock copolymers tilings in 2D. . . . . . 26

2.7 Lamellae configurations. . . . . . . . . . . . . . . . . . . . . . 27

2.8 Computational mesh for the simulation of spots structures. . 27

2.9 Spots configurations over a rectangular elementary cell. . . . 29

2.10 Spots configurations over an hexagonal elementary cell. . . . 29

3.1 Homogenized stiffness tensor for the ratio EA/EB = 10. . . . 44

3.2 Homogenized stiffness tensor for the ratio EA/EB = 103. . . 45

3.3 Rotated homogenized tensor of lamellae patterns for m = 0,for different stiffness ratios between pure phases. . . . . . . . 47

3.4 Rotated homogenized tensor of spots patterns for EA/EB = 10. 48

5.1 Graphical representation of the strong Wolfe conditions. . . . 83

5.2 Entries of the homogenized stiffness tensor for different ratiosEA/EB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

7.1 Geometry and boundary conditions of the test traction. . . . 105

7.2 Result of the test traction. . . . . . . . . . . . . . . . . . . . . 105

7.3 Convergence history of the objective function in natural scale(top) and log-log scale (bottom). . . . . . . . . . . . . . . . . 106

7.4 Geometry and boundary conditions of the test cantilever. . . 107

7.5 Results of the test cantilever with increasing mesh resolutions 107

7.6 Result of the test cantilever with fine filter radius. . . . . . . 108

7.7 Execution times for the test bridge with different mesh sizes. 109

XI

7.8 Geometry and boundary conditions of the case study bridge(a) and square (b) . . . . . . . . . . . . . . . . . . . . . . . . 109

7.9 Results of test bridge. . . . . . . . . . . . . . . . . . . . . . . 1107.10 Results of test square. . . . . . . . . . . . . . . . . . . . . . . 1117.11 Geometry and boundary conditions of the test L-shape. . . . 1127.12 Results of test L-shape. . . . . . . . . . . . . . . . . . . . . . 112

XII

List of Algorithms

1 Multi-material Topology Optimization . . . . . . . . . . . . . 752 Update of (zh)h and (mh)h . . . . . . . . . . . . . . . . . . . 803 Line search algorithm . . . . . . . . . . . . . . . . . . . . . . 844 zoom function . . . . . . . . . . . . . . . . . . . . . . . . . . 84

XIII

Chapter 1

Introduction

The geometry and the topology of structures have a great impact ontheir performances. Moreover, the efficient use of material is crucial in manyfields of application, from automotive industry, to bioengineering or MEMSindustry. In these settings the optimization of the stiffness or other featuresof a structure, given the total mass, is of great interest. For this reason in thelast decades many algorithms aiming at this goal came to life. We distinguishbetween shape optimization problems, when the topology of a body is fixedand just its geometry is subject to optimization, and topology optimizationproblems when both the shape and the topology of the structure are to bedetermined. Of course the latter are of more interest, since they embrace amuch broader family of admissible designs.

1.1 Topology Optimization

We suppose to be given a reference domain Ω ⊂ Rd (where d = 2 or3), which the structure must be contained in, and a given bulk load band boundary traction t, acting on the Neumann portion of the boundaryΓN . We suppose that the body is anchored on the remaining part of theboundary ΓD = ∂Ω\ΓN , where homogeneous Dirichlet boundary conditionsare prescribed. Denoting by E(x) the stiffness tensor of the structure inthe framework of linear elasticity, the displacement field u(x) satisfies theequilibrium equation

−div σ = b in Ω

σ(x) = E(x)∇su(x) in Ω

σν = t on ΓN

u = 0 on ΓD ,

(1.1)

where σ(x) is the Cauchy tensor. Here and in the following∇s(·) = ∇(·)+∇(·)T2

denotes the symmetric part of the gradient operator.

1

(a) Human bones (b) Wood

Figure 1.1: Internal structure of biological tissues.

Generally speaking, the stiffness tensor is the design variable, since itsvalue in any single point of the domain depends on the layout of the struc-ture. For instance, if we consider a body occupying a region A ⊂ Ω andmade of a single material characterized by a tensor E0, the stiffness tensorin each point of the domain can be parametrized as E(x) = 1A(x)E0, where1A(x) denotes the indicator function of the set A. Otherwise, if we want toconsider multi-material layouts, the stiffness tensor can be parametrized asE(x) =

∑Nh=1 1Ah(x)Eh, where the N different materials of stiffness (Eh)h

occupy the regions (Ah)h.We are left to chose the objective functional to be minimized. In this

work the most popular choice in the context of topology optimization istaken into account, namely the minimization of the so-called compliance,defined as the total work of external forces (which coincides with the workof internal ones):

minimize

ˆΩ

b(x) · u(x) dx +

ˆΓN

t(x) · u(x) dx .

1.2 Micro-structures in topology optimization

It is well known that bodies exhibiting micro-structures are very efficientfrom the structural perspective. As a matter of fact there are plenty of exam-ples in nature of elastic bodies that, having to resist to mechanical loadings,present a fine-scale structure: bones, for instance, exhibit a sponge structure(see Figure 1.1a), and wood reveals an quasi-periodic micro-structure (seeFigure 1.1b). The underlying reason is that by introducing holes, withoutchanging the total volume, the efficiency of the structure is generally en-hanced. This observation has deep consequences both from a mathematicaland a numerical point of view.

2

From the analytical perspective the consequence is the ill-posedness ofthe topology optimization problem, which in general lacks of existence ofsolutions (see [1] for counterexamples). What typically happens is that it canbe found a minimizing sequence of designs (for instance a sequence os sets(Aj)j=1,..., and the corresponding solutions (uj)j=1,...) corresponding to aprogressive refinement of an initial design in finer and finer micro-structures.The sequence of uj is bounded H1

ΓD(Ω;Rd), and thus, by Theorem 4 in

Appendix A, there exist a limit solution u (weak limit in H1ΓD

(Ω;Rd) and

strong in L2(Ω) for a suitable subsequence), but in general there is not anyset A such that u is the solution associated to that design. Moreover, thesequence of indicator functions 1Aj is bounded in L∞(Ω), and thus we canonly deduce weak∗ convergence in L∞(Ω) (see Corollary 1 in Appendix A),a too weak notion of convergence to find an admissible design as limit ofthe sequence. Indeed, the limit is typically a function with intermediatevalues between 0 and 1, which can be interpreted as the “density” of themicro-structure, but does not correspond to any real design.

This translates into numerical instabilities when one tries to solve com-putationally the problem. When the problem is discretized, the resolutionof micro-structures is bounded by that of the computational mesh. Thus,when the mesh is refined, the solution may exhibit a refinement of the topol-ogy, hindering mesh independence. Moreover, numerical solution obtainedwith linear finite elements often exhibits checkerboard patterns, because ofa bad numerical modelling that overestimates the stiffness of this kind ofconfiguration [21], which resembles to micro-structures, but which have nophysical relevance and are mesh-dependent.

This kind of problems is typically tackled by restricting the set of ad-missible design ruling out the possibility for fine structures to formate. Thiscan be obtained by including into the objective functional the total varia-tion (TV ) of the density (the so-called perimeter penalization, see for in-stance [9]), or by filtering the density by means of a convolution kernel (seefor instance [41]). Although this family of solutions is satisfactory from apractical viewpoint, since it allows to retrieve existence of solution, mesh-independence and prevents numerical instabilities, on the other hand it rulesout micro-structures, which we know to be the most efficient way of mate-rial usage. We gained a well-posed problem, but we lost the most efficientdesigns.

1.3 Self-assembling micro-structures

One reason why the above mentioned solutions (not to allow arbitrar-ily fine scales in the design of the topology) have been well accepted inpractical applications, is that there are technical constraints in realizing themicro-structures predicted by the theory. Indeed, traditional manufactur-

3

Figure 1.2: SBS block copolymer in TEM. Source: wikimedia.org

ing methods do not allow the realization of this kind of patterns at a lowcost. However, at least two ways can be considered to fill this gap. The firstone is additive manufacturing : the additive nature of this printing processallows to realize objects with almost any kind of topology, at a surprisinglylevel of detail. The other way, which can bring to even finer resolutions,is the exploitation of self-assembling materials. Self-assembly is an excitingphenomenon by which a disordered arrangement of components re-arrangein an organized structure, as a consequence of local interactions (typicallyof chemical nature). The possibility of predict and even control the kindof pattern opens to a huge number of applications, which go far beyondthe structural field, thanks to the possibility of controlling the mechanical,magnetic, electronic and optical properties of the material.

Motivated by this ideas, in this work we study the possibility of formu-lating and numerically solving topology optimization problems for materi-als reporting micro-structures. We look for a formulation which optimizessimultaneously the local properties of the finer scale and the topology ofthe macroscopic scale. We focus on diblock copolymers, a family of self-assembling materials widely studied in the past, thanks to their good proper-ties of predictability of the resulting micro-structure [30]. When a disorderedsolution of diblock copolymers undergoes a critical temperature, phase sep-aration takes place, resulting in an ordered structure (see Figure 1.2). Thisprocess is modelled by a gradient flow in H−1 of the Ohta-Kawasaki func-tional [35], a Cahn-Hilliard-like free energy supplemented with a nonlocalterm.

The problem we are approaching presents some intrinsic difficulties that

4

are briefly listed here.

• Multi-scale nature of the problem.Typical length scales of diblock copolymers range from some units tosome tens of nanometres, while the problems of practical interest mayhave a length scale of hundreds of micrometres (MEMS devices) upto centimetres or even metres. For this reason a direct simulation ofthe whole body is infeasible, since it would require an incredibly highnumber of degrees of freedom. However, the periodicity of patternsformed by diblock copolymers can be exploited to get a two scaleformulation of the problem.

• Stiffness of Cahn-Hilliard like equations.The equation which models the formation of diblock copolymers pat-ters is a nonlocal version of the Cahn-Hilliard equation, a fourth-ordernonlinear PDE, rather stiff to be solved numerically. Moreover, the en-ergy functional associated with it is neither concave nor convex, whichmakes the choice of numerical scheme difficult. As we will see in Chap-ter 2, Eyre discovered a way to find unconditionally gradient stableschemes for this family of equations [22, 23]. However, in a two-scalesformulation the micro-scale equation has to be solved in each elementof the macroscopic mesh, once for each iteration of the optimizationalgorithm. To reduce the computational burden of the algorithm it isthen necessary to model in some way the map from the parametersruling pattern formation to the features of the resulting medium, inorder to build a database of all possible material properties.

• Discontinuities in phase plane of diblock copolymers stableconfigurations.As it will be shown in Chapter 2, diblock copolymers exhibit differ-ent classes of stable configurations. In two dimensions for instancethe micro-structure can exhibit lamellae or spots of one monomerdistributed regularly inside the other monomer. There are thresh-old values of the ratio of concentration between the two monomers atwhich the configuration minimizing the energy switches from one toanother. In other words, there are discontinuities in the map froma control parameters (the ratio between monomers) and the solution.Thus in proximity of those values the map lacks of differentiability,and moreover the optimization procedure is likely to be trapped intolocal minima.

In the implementation we will restrict ourselves to the two dimensionalcase, although the whole work can be generalized to three dimensions.

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1.4 Existing works

The connection between topology optimization and homogenization the-ory has a long history. As a matter of fact, the first algorithm devised fortopology optimization problems came under the name of homogenizationmethod (see the pioneering work of Bendsøe and Kikuchi [7]). In this set-ting homogenization theory is employed to enlarge the space of admissibledesigns by introducing all possible micro-structures (relaxation approach).When this first optimization phase is over, the micro-structures with ma-terial densities which are far from 0 or 1 are penalized, to recover a de-sign featuring only void regions and full-material regions. Therefore micro-structures described by the homogenization theory are employed merely inan intermediate step, and are not considered in the final configuration.

More recently the problem of optimizing the elastic properties of bodyexhibiting micro-structures has been addressed by some authors. Rodriguez,Guedes and Bendsøe himself worked on a hierarchical topology optimizationproblem [38, 16]. Here the authors optimize by the SIMP (Solid IsotropicMaterial with Interpolation) method the topology of the micro-structure,which is allowed to change from point to point. Similar are the works ofBarbaroise and Toader [46, 5, 4], with the difference that the latter employsthe notion of shape derivative and topological derivative to perform the op-timization at the microscale. In both cases, however, the periodic pattern ischosen among all possible micro-structures, while in the present case we haveto restrict ourselves to the much smaller class of micro-structures exhibitedby diblock copolymers.

The works of Conti et al. [18, 17] are in some sense closer to the purposesof the present work. They optimize the local features of a micro-structureof a given type (cells with single ellipsoidal holes, cell structures consist-ing of axes-aligned trusses, and so on). However the micro-structure isdescribed analytically, and thus the analytical computation of shape deriva-tives is possible, unlike the present case where the micro-structure is linkedto the solution of a PDE. In addition, in the present work different types ofmicro-structures, competing each other, should be considered, since diblockcopolymers patterns exhibit different topologies depending on the propor-tion between monomers.

Moreover some other differences between the present case and the ap-proaches available in literature can be pointed out. In this work we aimat a simultaneous optimization of the features of the micro-structure andof the topology of the macroscopic body, while in the past works only themicro-structure optimization has been addressed. At a closer look, the afore-mentioned works envision cells with very little material as admissible micro-structures, so that in the final design nearly void regions are present. In thissense we can say that the topology of the macroscopic body is optimizedas well; however this approach cannot be applied to the present case, since

6

the diblock copolymers micro-structures do not feature void regions, thenan alternative strategy should be addressed.

A further novelty with respect to the past works is that in this work thedesign variables do not determine directly the micro-structure, but there isan intermediate step consisting in a PDE. The design variables are in factcontrol variables for the Cahn-Hilliard-Oono equation.

1.5 Outline

The present work is structured as follows:

• In Chapter 2 an overview of the mathematical description of the phaseseparation phenomenon of diblock copolymers melts is provided. Thenthe finite elements discretization of the Cahn-Hilliard-Oono equation isaddressed, with a particular focus on time marching schemes. Finallythe results of some two dimensional simulations are reported.

• Chapter 3 deals with the two-scales nature of the problem consideredin this work. First of all, a brief theoretical background on the homog-enization theory is provided, and then its application to the presentcase is addressed. The outcome is a two-scales equation for the equi-librium configuration of a body obtained by self-assembling diblockcopolymers. This formulation allows to treat the present problem withreasonable computational effort.

• In Chapter 4 a minimum compliance problem for diblock copolymersbodies is formulated. Some modifications are then proposed to makethe problem more suitable for an optimization algorithm, and finallythe existence of solutions is proved.

• In Chapter 5 an algorithm to solve the topology optimization problemintroduced in Chapter 4 is proposed.

• Chapter 6 deals with the numerical implementation of the algorithmintroduced in the previous chapter. First of all, a finite elements dis-cretization is proposed, and some details about the implementationare given.

• In Chapter 7 some numerical results showing the capabilities of theproposed method are presented.

1.6 Preliminaries

In this section we recall some basic definitions, results and notationswhich will be useful in this work.

7

1.6.1 Linear elasticity

In the following we will denote by Ω a connected open and boundedsubset of Rd, where d = 2, 3 is the dimension of the space. We consider apartition of the boundary ∂Ω into a Dirichlet and a Neumann boundary,respectively denoted by ΓD and ΓN . We require the Dirichlet portion to benon-empty. Let M2 be the space of symmetric second order tensors in Rd,and M4 the space of fourth order symmetric tensors acting on symmetricsecond order tensors in Rd:

M2 =A = (aij)1≤i,j≤d s.t. aij = aji

M4 =

A = (aijkl)1≤i,j,k,l≤d s.t. aijkl = ajikl = aklij

.

We denote by I2 the identity tensor inM2 and by I4 the identity tensorinM4. Moreover, we introduce the subspace ofM4 of coercive tensors andof coercive tensors with coercive inverse:

M4α =

A ∈M4 s.t. Aξ : ξ ≥ α|ξ|2 ∀ξ ∈M2

M4

α,β =A ∈M4 s.t.

Aξ : ξ ≥ α|ξ|2A−1ξ : ξ ≥ β|ξ|2 ∀ξ ∈M2

,

where α and β are positive constants. An admissible Hook’s law is a tensorfield E belonging to L∞(Ω;M4

α,β).

We observe that the set M4α,β is bounded. Indeed, given a tensor E ∈

M4α,β and a generic test tensor ξ ∈ M2, and denoting η = Aξ, we have to

the following chain of inequalities:

β|Aξ|2 = β|η|2 ≤ A−1η : η = ξ : (Aξ) ≤ |ξ||Aξ| ,

which yields to |Aξ| ≤ β−1|ξ|.We denote by H1

ΓD(Ω;Rd) the Sobolev space of Rd-valued L2 functions

with distributional gradient in L2 and vanishing on ΓD. We consider a bulkload b belonging to the dual space of H1

ΓD(Ω;Rd), which will be denoted by

H−1(Ω;Rd), and a boundary traction t ∈ H−1/2(ΓN ).We can now state the weak formulation of the problem (1.1):

find u ∈ H1ΓD

(Ω;Rd) s.t.´Ω E∇su : ∇sv = 〈b,v〉H−1,H1

ΓD

+ 〈t,v〉H−1/2,H1/2 ∀v ∈ H1ΓD

(Ω;Rd) .(1.2)

When the forcing terms b and t are smooth (L2 on the respective do-mains), the right hand side can be rewritten by the equalities:

〈b,v〉H−1,H1ΓD

=

ˆΩ

b · v

〈t,v〉H−1/2,H1/2 =

ˆΓN

t · v .

8

By Lax-Milgram lemma and Korn’s inequality (see Appendix A) thesolution of problem (1.2) exists unique and satisfies the following stabilityestimate for some constant C:

‖u‖H1(Ω;Rd) ≤ C(‖b‖H−1(Ω;Rd) + ‖t‖H−1/2(ΓN )

). (1.3)

1.6.2 Isotropic constitutive law

A special class of Hook’s laws is given by isotropic media. The isotropyconstraint reduces the number of degrees of freedom in the constitutive lawto a couple. The most common choices are the Lame moduli (λ, µ) or thecouple Young’s modulus E together with Poisson’s coefficient ν (dimension-ally speaking, the latter is non-dimensional, while the others are pressures).The relationships between the couples of parameters are the following:

E = µ(2µ+3λ)µ+λ

ν = λ2(µ+λ)

µ = E

2(1+ν)

λ = Eν(1+ν)(1−2ν) .

(1.4)

The admissible values are summarized below:E > 0

0 ≤ ν < 12 (non-auxetic materials)

−1 < ν < 0 (auxetic materials)

(1.5)

µ > 0

λ > 0 (non-auxetic materials)

2µ+ 3λ > 0 , λ < 0 (auxetic materials).

(1.6)

Denoting by ε = ∇su the stain tensor, and by σ = Eε the stress tensor,the direct and inverse isotropic constitutive laws read as follows:

σ = 2µε+ λ tr(ε) I2 = 2µ∇su + λ div (u) I2

ε =1 + ν

Eσ − ν

Etr(σ) I2 ,

(1.7)

or, by components:

σij = 2µεij + λ εkk δij

εij =1 + ν

Eσij −

ν

Eσkk δij .

(1.8)

From the above definitions, it follows that the stiffness matrix E and itsinverse, the compliance matrix C = E−1, can be written as follows:

E = 2µ I4 + λ I2 ⊗ I2

C =1 + ν

EI4 −

ν

EI2 ⊗ I2 .

(1.9)

9

10

Part I

Problem formulation

11

Chapter 2

Diblock copolymers

In this chapter we provide a brief theoretical background on diblockcopolymers, with the goal of characterizing the micro-structure that origi-nates after phase separation. Since the main focus of this work is on theoptimization of the micro-structure, we do not give full details on some ofthe topics of this chapter, but we refer to the mentioned literature for fullproofs and details.

In Section 2.1 the phenomenon of phase separation of diblock copolymersis introduced. From the mathematical viewpoint, this process tends to min-imize the so-called Ohta-Kawasaki functional (2.4), which is the subject ofSection 2.2, and it is modelled by the Cahn-Hilliard-Oono (CHO) equation(2.17), introduced in Section 2.3. In Section 2.4 the finite elements dis-cretization of the CHO equation is addressed, and in Section 2.5 the resultsof numerical simulations are reported.

2.1 Diblock copolymers

A diblock copolymer is a linear chain molecule made of two subchains,joined covalentely to each other, consisting respectively on NA monomers oftype A and of NB monomers of type B. When the temperature is above acritical value Tc, the disordered state is stable, but when the temperatureundergoes this threshold, the repulsion between monomers of type A and Bmakes the subchains segregate into A-rich and B-rich domains. However,because of chemical bonds between subchains, a macroscopic segregationcannot take place, and the phase separation is bounded at a mesoscopicscale, i.e. at a scale λ such that l λ L, where l is the molecular scaleand L is the macroscopic scale. (see Figure 2.1). When this process is over,the resulting mesoscopic domains exhibit highly regular periodic patterns.In three dimensions the observed patterns are of four types: lamellae, doublegyroids, hexagonally packed cylinders, and spheres (see Figure 2.2). Thebook by Hamley [25] provides a complete background on this matter, and

13

Figure 2.1: Top: a diblock copolymer macromolecule. Bottom: microphaseseparation of diblock copolymers into lamellae (left) and spheres (right).Source: [14]

many works can be found in the literature (see for instance [35, 15, 14]).Microphase separation involve three dimensionless parameters (χ, N ,

fA) and a dimensional one (l):

1. χ, the Flory-Huggins interaction parameter, a measure of the incom-patibility between monomers of type A and B;

2. N = NA + NB, the index of polymerization, i.e. the total number ofmonomers in each macromolecule;

3. fA = NAN , the molecular fraction of monomers of type A in each macro-

molecule (or equivalently fB = 1− fA = NBN );

4. l, the Kuhn statistical length, corresponding to the average monomerslength.

The first mathematical attempt to predict the equilibrium configurationrelies on the self-consistent mean-field theory (SCMFT) [28]. In this frame-work, the interactions between monomers of the two types are simulated bymeans of self-consistent external fields. In this way, one does not have tointegrate the contribution of the interaction among all monomers, but havejust to consider the contribution of one polymer in a self-consistent field.By assuming a symmetry ansatz, the energy is minimized among all theconfigurations satisfying the symmetry assumptions. Finally, by comparingthe energy of configurations corresponding to different ansatze, it is possibleto fill the phase plane.

If we neglect thermal fluctuations, according to SCMFT the parame-ters dependence reduces from the four parameters previously listed to theproduct χN and fA. In Figure 2.3 the phase plane obtained by means of

14

(a) lamellae (b) gyroids

(c) cylinders (d) spheres

Figure 2.2: Stable configurations in three dimensions. Source: [14]

this theory is reported. The agreement between theoretical predictions andexperimental observations is satisfactory [26].

2.2 The Ohta-Kawasaki functional

By linearising the dependence of the monomer density on the exter-nal fields, one gets a density functional theory, first proposed by Ohta andKawasaki [35] (see [15] for further details). Let Ω be a domain in Rd. Weconsider an order parameter ϕ(x), defined as the difference between themass fractions of the two monomers (denoted respectively be ϕA and ϕB):

ϕ(x) = ϕA(x)− ϕB(x) . (2.1)

We have ϕ(x) ∈ [−1, 1], ϕ(x) = −1 in pure B zones and ϕ(x) = 1 inpure A zones. We define the following quantity:

m = fA − fB = 2fA − 1 . (2.2)

From this definition it follows:

Ωϕ(x) dx =

1

|Ω|

ˆΩϕ(x) dx = m. (2.3)

15

Figure 2.3: Phase plane in three dimensions. Source: [45]

Given the previous definitions, the Ohta-Kawasaki functional, definedover the set ϕ ∈ H1(Ω) s.t.

fflΩ ϕ = m reads as follows:

H(ϕ) =

ˆΩ

(δ2

2|∇ϕ|2 + F (ϕ)

)dx

+τ

2

ˆΩ

ˆΩG(x,y)(ϕ(x)−m)(ϕ(y)−m) dx dy .

(2.4)

The function F is a double-well potential, with minima correspondingto pure states of monomers A or B. The most common choice is:

F (s) =(1− s2)2

4. (2.5)

By G we denote the Green’s function of −∆ on Ω with periodic orno-flux boundary conditions (in the first case we look for periodic phaseconfigurations). It has been proved in [15] that the parameters δ and τ (ofphysical dimensions [δ] = L, [τ ] = L−2) scale as follows:

δ2 ∼ l2

fAfBχ(2.6)

τ ∼ 1

f2Af

2BχN

2l2. (2.7)

16

We now show that when the determination of the phase plane is ad-dressed, the number of parameters involved in the equations can be reduced,as in the case of SCMFT.

In [12] it has been proved that the minimum energy associated with theOhta-Kawasaki functional scales as δ2/3τ1/3 = (δ

√τ)2/3.

Moreover, in [32] it has been proved that the characteristic length ofphase patters is δ1/3τ−1/3, while the characteristic length of interfacial thick-ness is of order δ. The degree of phase separation, measured by the ratiobetween the two length is then of order (δ

√τ)−2/3. Also in this case the

dependence of the solution on the combination δ√τ is evident.

Finally, introducing the dimensionless spatial coordinate

x =√τx, Ω =

√τΩ ,

we get the dimensionless version of (2.4) (where G denotes the Green func-tion associated to the domain Ω):

H(ϕ) =

ˆΩ

((δ√τ)2

2|∇ϕ|2 + F (ϕ)

)dx

+1

2

ˆΩ

ˆΩG(x, y)(ϕ(x)−m)(ϕ(y)−m) dx dy .

(2.8)

Once again it is evident the dependence on the combination δ√τ . We

are then lead to introduce the parameter γ = (δ√τ)−1. By the relations

(2.6)-(2.7), we get γ ∼ χN . It is now clear that, in the determination of thephase plane, the couple (fA, χN) is equivalent to (m, γ).

2.3 Cahn-Hilliard-Oono equation

As we stated before, the Ohta-Kawasaki functional (2.4) allows to findequilibrium configurations in the phase separation of diblock copolymers asminima of the functional itself. Additionally, it provides a way to simulatethe time evolution of the segregation process, writing a gradient flow for thefunctional in a proper space.

2.3.1 Gradient flows

Generally speaking, given a functional H and an Hilbert space V , wecan define an evolution equation as follows:

∂tϕ = −∇VH(ϕ) , (2.9)

where ∇H denotes the gradient in the space H, i.e. the Riesz representativeof the Gateaux derivative of H with respect to its argument ϕ. In otherwords, equation (2.9) should be interpreted as follows:

(∂tϕ,ψ)V = −∂H∂ϕ

[ψ] , (2.10)

17

where (·, ·)V denotes the scalar product in V , and the right-hand side denotesthe Gateaux derivative applied to the increment ψ.

It can be easily seen the time evolution results into a decrease of thefunctional H:

dHdt

=∂H∂ϕ

[∂tϕ]

= −∥∥∂tϕ∥∥2

V≤ 0 . (2.11)

2.3.2 The Sobolev space H−1(Ω)

We introduce the subspace of H1 of null-average functions:

H1(Ω) := u ∈ H1(Ω) s.t. (u, 1)L2(Ω) = 0 .

Thanks to Poincare inequality, H1(Ω) is endowed by the following scalarproduct:

(u, v)H1(Ω) =

ˆΩ∇u · ∇v dx .

We denote by H−1(Ω) the dual space of H1(Ω). Given ϕ ∈ H−1(Ω), itsRiesz representative is the solution u of the following problem:

(u, v)H1 = 〈ϕ, v〉H−1,H1 ∀v ∈ H1(Ω) , (2.12)

which reads, in strong form:−∆u = ϕ in Ω∂νu = 0 on ∂Ωffl

Ω u = 0 .(2.13)

For this reason, we denote by (−∆)−1 : H−1(Ω) → H1(Ω) the Rieszoperator.

The following chain of equalities holds true:

(ϕ1, ϕ2)H−1 =((−∆)−1ϕ1, (−∆)−1ϕ2

)H1 = 〈ϕ1, (−∆)−1ϕ2〉H−1,H1 .

2.3.3 CHO equation

We can now derive the gradient flow in the space H−1(Ω) of the Ohta-Kawasaki functional. The first term of the energy, which coincides withthe Cahn-Hilliard functional translates into the traditional Cahn-Hilliardequation [27]. As to the nonlocal term, first of all we observe that it can berewritten by introducing an auxiliary function vϕ, solution of the problem:

−∆vϕ = ϕ−m in Ω∂νvϕ = 0 on ∂Ωffl

Ω vϕ = 0 .(2.14)

18

Then it holds:ˆ

Ω

ˆΩG(x,y)(ϕ(x)−m)(ϕ(y)−m) dx dy =

ˆΩvϕ(ϕ−m) dx

=

ˆΩ‖∇vϕ‖2 dx

=∥∥ϕ−m∥∥2

H−1 .

From the previous relation it follows that the Riesz representative inH−1(Ω) of the derivative of the nonlocal term is the difference ϕ−m itself,and thus this contribution translates into a reaction term. To sum up,the gradient-flow in H−1 of the Ohta-Kawasaki functional is given by thefollowing equation, known as Cahn-Hilliard-Oono equation (CHO) [3]:

∂tϕ = ∆µ− τ(ϕ−m)µ = −δ2∆ϕ+ F ′(ϕ) ,

(2.15)

where µ is called chemical potential. The equation has to be equipped byan initial condition and by boundary conditions. In this framework, naturalboundary conditions are of no-flux type:

∂νϕ = ∂νµ = 0 on ∂Ω .

Another common choice are boundary conditions of periodic type, in-spired by the observation that diblock copolymers spontaneously form pe-riodic structures. As a matter of fact it is possible to repeat the abovediscussion replacing Neumann homogeneous boundary conditions with peri-odic boundary conditions. For the well-posedness and the existence of globaland exponential attractors we refer to [31] and [10].

For both the choices of boundary conditions mentioned before, CHOequation satisfies a mass conservation property, which justifies the choice ofthe spaceH−1 (which is, thanks to this fact, often referred as mass-preservingspace). Integrating (2.15) over Ω we get:

d

dt

Ωϕdx =

Ω∂tϕdx =

Ω

∆µ dx− τ

Ω(ϕ−m) dx =

= −τ(

Ωϕdx−m

).

(2.16)

Then, if the initial average value of ϕ equals m (as in the case of diblockcopolymers, see (2.3)), this property is maintained as time goes by. This isconsistent with the physical interpretation of the equation, since this entailsthat we assist to a mere rearrangement of monomers, and not to their cre-ation or destruction. In case the condition (2.3) is not satisfied at the initialtime,

fflΩ ϕdx converges exponentially to m.

19

Summarizing, the full CHO system is here reformulated for future refer-ences:

∂tϕ = ∆µ− τ(ϕ−m) in ΩT := Ω× [0, T ]

µ = −δ2∆ϕ+ F ′(ϕ) in ΩT

∂νϕ = ∂νµ = 0 in ∂Ω× [0, T ] or ϕ, µ Ω-periodic

ϕ = ϕ0 in Ω× t = 0 .

(2.17)

2.4 Finite Element discretization of the equationof Cahn-Hilliard-Oono

Cahn-Hilliard-Oono equation (2.17) allows to find minima of the Ohta-Kawasaki functional (2.4) by solving a parabolic PDE. However, it is wellknown that equations of the Cahn-Hilliard family are rather stiff. Indeed, ifwe employ Euler’s method with a scaling parameter of order δ ∼ 10−2, toget a stable scheme we are forced to choose a time step of order 10−7, whilethe fastest time scales are just of order 10−4, so that thousands of time stepsare needed to see even the fastest dynamics. On the other hand, if implicitEuler’s method is applied, the stability threshold raises to 10−5, but to haveunique solvability of the nonlinear system the time steps should not exceed10−6 [23].

To overcome this kind of difficulties Eyre proposed a general approach,based on a convex splitting of the energy, to derive unconditionally sta-ble (in a sense that we will recall later) methods for equations of gradientflow kind [22], which he later successfully applied to Cahn-Hilliard equation[23]. In 2013 Aristotelous applied Eyre’s approach to derive a DiscontinuousGalerkin scheme for Cahn-Hilliard-Oono equation [2], but his analysis canbe applied to a standard Finite Element approximation too.

2.4.1 Convex splitting of gradient flows

We consider a functional H, and the corresponding gradient flow on aHilbert space V (see Section 2.3.1):

∂tϕ = −∇VH(ϕ) . (2.18)

We have already observed that this equation dissipates the energy H.From this observation it follows the definition:

Definition 1 (Gradient Stability). A one-step time marching scheme forthe equation (2.18) is said to be gradient stable if for all initial data:

H(ϕn+1) ≤ H(ϕn) ∀n ≥ 1 .

20

We observe that gradient stability is the discrete counterpart of the en-ergy dissipation property (2.11). It is interesting to note that when H is con-vex, there exists a unique minimum, and the flow is said to be contractive;on the contrary, if H is concave, the flow can expand. Simple calculationsshow that in the former case implicit Euler’s method results necessarily intoa decrease of the energy, while in the latter case this property is satisfiedby explicit Euler. This observation suggests to split the energy into a con-tractive part, to be treated implicitly, and an expansive part, to be treatedexplicitly. We suppose that the functional can be written as:

H = Hc −He ,

where both Hc and He are convex on V . We therefore consider the followingtime marching scheme:

ϕn+1 − ϕn

∆t= −∇VHc(ϕn+1) +∇VHe(ϕn) . (2.19)

By convexity assumption it follows:

Hc(ϕn+1) ≤ Hc(ϕn) +(∇VHc(ϕn+1), ϕn+1 − ϕn

)V

He(ϕn+1) ≥ He(ϕn) +(∇VHe(ϕn), ϕn+1 − ϕn

)V.

The above inequalities entail that energy dissipates as time goes by:

H(ϕn+1)−H(ϕn) =[Hc(ϕn+1)−Hc(ϕn)

]−[He(ϕn+1)−He(ϕn)

]≤(∇VHc(ϕn+1)−∇VHe(ϕn), ϕn+1 − ϕn

)V

= −‖ϕn+1 − ϕn‖2V

∆t≤ 0 .

Example 1. The heat equation with homogeneous Dirichlet boundary con-ditions can be interpreted as a gradient flow of the functional

E =1

2

ˆΩ‖∇u‖2 dx

with respect to the L2 scalar product:

(∂tu, v)L2 = −∂E∂u

[v] = −ˆ

Ω∇u · ∇v dx =

ˆΩ

∆uv dx ∀v ∈ H10 (Ω) .

Being the energy E convex, we recover the well known result of stabilityof implicit Euler’s method for the heat equation.

21

Coming back to the Ohta-Kawasaki functional, when the double-wellpotential F has the form (2.5) it is simple to find a convex splitting of theenergy:

Hc =δ2

2

∥∥∇ϕ∥∥2

L2 +1

4

∥∥ϕ∥∥4

L4 +1

4|Ω|+ τ

2

∥∥ϕ−m∥∥2

H−1

He =1

2

∥∥ϕ∥∥2

L2 .

(2.20)

This choice leads to a nonlinear scheme, since in the contractive term,to be treated implicitly, we have a term of fourth degree, whose differentialis of third degree. To overcome this inconvenient, Eyre himself suggests thefollowing splitting:

Hc =δ2

2

∥∥∇ϕ∥∥2

L2 +1

4|Ω|+ τ

2

∥∥ϕ−m∥∥2

H−1 +a

2

∥∥ϕ∥∥2

L2

He =1 + a

2

∥∥ϕ∥∥2

L2 −1

4

∥∥ϕ∥∥4

L4 .

(2.21)

The contractive part is convex, while the second is convex just on thesubspace of functions such that ‖ϕ‖L∞ ≤ 1+a

3 . Since according to our phys-ical interpretation ϕ(x) ∈ [−1, 1], choosing a ≥ 2 the inequality is satisfied.It is worth noticing that the contractive terms contains only terms of degreeless than two, and thus it translates into a linear scheme as it will be shownin the following section.

2.4.2 Finite element approximation

To complete the discretization of problem (2.17) we are left to addressspace discretization. In his original paper Eyre resorts to finite differences,but in this work we will exploit a Finite Element discretization following[2]. The basic tool to derive the Finite Element approximation is the weakformulation of the problem (2.17):

〈∂tϕ,ψ〉+ (∇µ,∇ψ) + τ(ϕ,ψ) = τ(m,ψ) ∀ψ ∈ V(µ, ξ) = δ2(∇ϕ,∇ξ) + (F ′(ϕ), ξ) ∀ ξ ∈ Vϕ(·, 0) = ϕ0(·) ,

(2.22)

where we denote by (·, ·) the L2 scalar product, and by 〈·, ·〉 the dualitypairing in H1. The space V coincides with H1(Ω) in the case of homogeneousNeumann boundary conditions, or with its subspace of periodic functions inthe case of periodic boundary conditions.

We then consider a triangulation Th of the spatial domain Ω, and a finiteelements subspace of V , that we denote by Vh. We denote by tnn=0,... thesequence of discrete times, and by ∆tn = tn+1 − tn the time steps. We look

22

for finite elements approximations of ϕ and µ such that:

ϕnh(·) ' ϕ(·, tn), ϕnh ∈ Vh ∀n = 1, ...

µnh(·) ' µ(·, tn), µnh ∈ Vh ∀n = 1, ... .(2.23)

Non-linear splitting scheme

We now consider the splitting (2.20). This choice leads to the followingscheme:

(ϕn+1h −ϕnh

∆tn , ψh)

+(∇µn+1

h ,∇ψh)

+ τ(ϕn+1h , ψh

)= τ

(m,ψh

)∀ψh ∈ Vh(

µn+1h , ξh

)= δ2

(∇ϕn+1

h ,∇ξh)

+((ϕn+1

h )3, ξh)−(ϕnh, ξh

)∀ ξh ∈ Vh

ϕ0h = ϕ0,h ,

(2.24)where ϕ0,h ∈ Vh is a suitable approximation of the initial datum.

We observe that, due to the presence of the term((ϕn+1h

)3, ξh), this is a

nonlinear scheme. For proofs of unconditional solvability and stability, andfor error estimates we refer to [2].

Linear splitting scheme

Employing the choice (2.21) to split the energy, we get the followinglinear scheme:

(ϕn+1h −ϕnh

∆tn , ψh)

+(∇µn+1

h ,∇ψh)

+ τ(ϕn+1h , ψh

)= τ

(m,ψh

)∀ψh ∈ Vh(

µn+1h , ξh

)= δ2

(∇ϕn+1

h ,∇ξh)

+((ϕnh)3, ξh

)− (1 + a)

(ϕnh, ξh

)+ a(ϕn+1h , ξh

)∀ ξh ∈ Vh

ϕ0h = ϕ0,h .

(2.25)

2.5 Numerical simulations and phase plane in 2D

We now give some details about the FreeFem++ implementation intwo dimensions of the schemes described in the previous sections. We con-sider P2 elements defined over a triangle mesh for the discretization of boththe unknowns ϕ and µ.

For the solution of the nonlinear problem (2.24) we use Newton method,employing the difference between successive iterations as stopping criterion,with tolerance 10−6. Up to our experience, the Newton solver convergesin average in 2 to 3 iterations (this figure is in accordance with the 2.3iterations reported in average by Eyre for the finite difference solution ofCahn-Hilliard equation [23]).

23

(a) lamellae (m = 0) (b) spots (m = 0.3)

Figure 2.4: Stable configurations in two dimensions. In both cases γ = 10.Source: [13]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.52

2.5

3

3.5

4

4.5

5

m

γ

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

5

10

15

20

25

m

γ

Fig. 3.1. (These figures need to be viewed in colour) Numerically computed phase diagram.(Bottom) Complete diagram. (Top) Detail for γ close to 2. Blue crosses: Lamellae. Red circles:Hex packed spots. Black diamonds: disorder. The red dashed-dotted lines mark the linear stabilityboundary of spots, the blue dashed-dotted line marks the linear stability boundary of lamellae, theblack dashed-dotted line marks the linear stability boundary of the disordered sate, and the solidblack lines mark the global stability regions of lamellae and spots respectively.

7

Figure 2.5: Phase plane in two dimensions. Blue crosses: Lamellae. Red cir-cles: hexagonally packed spots. Black diamonds: disorder. The red dashed-dotted lines mark the linear stability boundary of spots, the blue dashed-dotted line marks the linear stability boundary of lamellae, the black dashed-dotted line marks the linear stability boundary of the disordered sate, andthe solid black lines mark the global stability regions of lamellae and spotsrespectively. Source: [13]

24

2.5.1 Phase plane in 2D

It is well known that the minima of the Ohta-Kawasaki functional ex-hibits periodic structures [14, 13]. On the ground of this a priori knowledgeof the solution, we consider a domain with periodic boundary conditions,aiming at obtaining the fundamental cell of the pattern. In [13] a exhaus-tive investigation of the two dimensional phase plane of the Ohta-Kawasakifunctional is performed. According to the authors, three kinds of minimizerscan occur:

• disordered state, i.e. the uniform state u ≡ m;

• lamellae, i.e. the one-dimensional configuration of Figure 2.4a, report-ing parallel stripes of pure A-phases and pure B-phases;

• hexagonally packed spots, i.e. periodic array of circles arranged in anhexagonal lattice (see Figure 2.4b).

By asymptotic analysis corroborated by numerical simulations, the bound-ary between the stability regions of the possible states are computed. Theresults can be observed in Figure 2.5, where just one half of the phase planeis reported. The other half can be obtained by replacing one monomer withthe other.

To our purposes, i.e. predicting the pattern according to the controlvariable m, the fundamental observation is that for a given value of γ (whichis fixed once the kind of monomers are chosen, and thus cannot be regardedas a control parameter) we have four regions:

1. m ∈ [−1,−m2) ∪ (m2, 1] disorder;

2. m ∈ (−m2,−m1) spots of material A inside material B (denoted inthe following by A-spots);

3. m ∈ (−m1,m1) lamellae;

4. m ∈ (m1,m2) spots of material B inside material A (B-spots).

Since we are interested in a regular structure, we will restrict our interestto spots and lamellae configurations. In the following we will consider thecase when γ = 20, and the corresponding thresholds m1 = 0.2, m2 = 0.6.

2.5.2 Domain and boundary conditions

As we mentioned before, we exploit the a priori knowledge about theperiodicity of the solution in order to obtain numerically the elementary cellof the pattern.

In the region m ∈ (−m1,m1), the pattern is mono dimensional, and anyrectangular domain is suitable to reproduce it (see Figure 2.6a). We have

25

(a) (b) (c)

Figure 2.6: Periodicity cell for diblock copolymers tilings in 2D. In case ofspots micro-structures two different kinds of elementary cell can be chosen.

only to pay attention to the domain width in the direction orthogonal tothe stripes, since it will force the period of the pattern. In [13] an a prioriestimate of the period is given, and up to our experience it is accurateenough to obtain just one period in the domain. Of course, to obtain thetrue minimizer one should rescale the domain to get the global minimizer;however, as we will see in Chapter 3, this step can be avoided since for ourpurposes the knowledge of the exact period is irrelevant.

In the regions m ∈ (−m2,−m1) and m ∈ (m1,m2) the patterns thatminimize the energy are the hexagonally packed array of spots shown inFigure 2.4b. To reproduce this kind of tiling two kinds of domain can beexploited, namely an hexagonal-shaped domain (Figure 2.6b) or a rectan-gular one, with height-to-width ratio equals to

√3 (Figure 2.6c). As to the

choice of the domain size, the same remarks of the previous case apply.

Remark 1. Besides the stiffness of Cahn-Hilliard-Oono equation (2.17), afurther complication lies in the fact that the energy landscape of the Ohta-Kawasaki functional (2.4) is highly non-convex, being characterized by localminimizers as well as metastable configuration, at which gradient-flow dy-namics are very slow. To help the time evolution to exit these states, weadd a noise term to the equation, which is periodically switched off to letthe state reach a stable configuration.

Remark 2. When an equilibrium solution for a given value of m is available,to compute the solution for a similar value of m it is possible to follow acontinuation approach, employing as initial condition the available solution.Indeed, thanks to (2.16), the system will converge to a solution with thedesired average monomers composition. This operation saves much compu-tational time. Moreover the continuation approach is useful to obtain thedesired configuration when the value of m is close to the stability boundariesof phase plane.

26

(a) m=-0.2 (b) m=0 (c) m=0.2

Figure 2.7: Lamellae configurations.

Figure 2.8: Computational mesh for the simulation of spots micro-structures.

2.5.3 Numerical results

In all the simulations reported in this section the parameters are set toδ = 0.04 and τ = 25/16, so that γ = 20. The time step is set initially to10−2, and it is progressively raised as the evolution becomes slower. Simu-lations are run until the normalized difference between successive iterations(‖ϕn+1

h − ϕnh‖/‖ϕnh‖) undergoes the threshold of 10−5.

Numerical results show that, as one could expect, schemes (2.24) and(2.25) leads to the same final configurations. As a matter of fact, stableconfigurations for both schemes satisfy the same equation in discrete form.Because the latter is more efficient (up to three times faster), it was employedto obtain the results presented in this section.

27

Lamellae

To obtain numerically the equilibrium configuration shown in Figure2.4a a square mesh of unitary length is considered. The mesh is made of7200 triangles. The obtained configuration are shown in Figure 2.7. Blueregions represent pure A-phase, while the red ones are regions reporting onlymonomers of type B. Intermediate colours represent hybrid compositions.

Spots on rectangular cell

The mesh considered in this section is a rectangle with base 0.7√

3, andwith height 0.7 · 3. The mesh is made of 3060 elements. Figure 2.9 showsthe results of the simulations.

Spots on hexagonal cell

A regular hexagon of side-length equal to 0.7 is considered. The mesh,made of 5138 triangles, is shown in Figure 2.8. The result of the simulationsare shown in Figure 2.10.

28

(a) m=-0.6 (b) m=-0.4 (c) m=-0.2

(d) m=0.2 (e) m=0.4 (f) m=0.6

Figure 2.9: Spots configurations over a rectangular elementary cell.

(a) m=-0.6 (b) m=-0.4 (c) m=-0.2

(d) m=0.2 (e) m=0.4 (f) m=0.6

Figure 2.10: Spots configurations over an hexagonal elementary cell.

29

30

Chapter 3

Homogenization: from themicro to the macro scale

The length-scale of diblock copolymers patterns is typically comprisedbetween 1 and 50 nanometres, many orders of magnitude smaller than thescale of practical applications. A direct simulation of the whole domain isthus infeasible, since it would exceed the computation power of any calcula-tor. For this reason an up-scaling strategy is required. As a matter of fact,homogenization can be regarded as a mathematical tool that allows to de-scribe the macroscopic properties of a body given its microscopic structure.

In Section 3.1 the basic results of homogenization theory which are rel-evant to our purposes are reported. Then, in Section 3.2, a two-scales for-mulation of the linear elasticity problem for diblock copolymers body isintroduced. Proposition 1 allows to apply the homogenization theory tothe present case, yielding to a formula for the homogenized stiffness ten-sor of diblock copolymers media (see equation (3.25)). Finally, in Section3.3, the numerical results of the previous chapter are employed to computenumerically the homogenized tensor for some values of the parameters.

3.1 Homogenization

Homogenization theory allows to derive macroscopic effective propertiesof microscopically heterogeneous media. In the framework of elasticity, givena microscopic structure of known elastic properties, it aims at determiningthe so-called homogenized tensor, i.e. a stiffness tensor which models themacroscopic behaviour of the micro-structure. As such, homogenizationtheory provides a theoretical framework for the mathematical modelling ofcomposites materials, which are obtained by mixing at a very fine scaledifferent materials.

This operation is performed by embedding the original two-scales prob-lem in a family of problems with increasing separation between the two

31

scales. Each problem is parametrized by the microscopic length-scale ε, andthe limit for ε → 0 is taken into account. The goal of homogenization isto characterize the limit problem. The questions are: is the limit of thesolutions solution of a problem? If so, of which problem?

The case of periodic micro-structures can be treated by means of a two-scales asymptotic expansion, which provides an explicit formula for the ho-mogenized tensor. This analysis, however, is not rigorous from a mathemat-ical point of view, since its derivation is just heuristic. Nevertheless, it ispossible to justify rigorously its results in the framework of the more generaltheory of H-convergence.

The most general theory in homogenization is based on the notion ofH -convergence, introduced by Spagnolo under the name of G-convergence[43, 42], and later generalized by Tartar and Murat [24, 33]. This definesan adequate topology for the notion of convergence of problems as ε goes tozero.

3.1.1 Two-scale asymptotic expansion

This section is devoted to the derivation of an explicit formula for thehomogenized tensor of an elastic body exhibiting a (locally) periodic micro-structure. We consider a domain Ω ⊂ Rd occupied by an elastic body. Therescaled periodicity cell is denoted by Y ⊂ Rd. We consider a family ofproblems, parametrized by the length scale ε, supposing that the stiffnesstensor of each problem can be written as

Eε(x) = E(x,

x

ε

), (3.1)

where E is Y -periodic in the variable y = xε . Formula (3.1) models a locally

periodic medium: at a mesoscopic level the body is periodic of period ε, butthe periodicity pattern changes slowly in x.

We suppose E ∈ L∞(Ω × Y ;M4α,β). Denoting by b an external bulk

load, we consider the linear elasticity equation for the Hook’s law Eε:

− div[E(x,

x

ε

)∇suε(x)

]= b(x) . (3.2)

Thanks to the periodicity of the medium, it is reasonable that the dis-placement field exhibits a locally periodic structure. Thus we consider thefollowing ansatz : we suppose that the family of solutions uεε can be writ-ten in the same form as the stiffness tensor (3.1), and that it can be expandedin series with respect to ε:

uε(x) =

+∞∑n=0

εnun(x,

x

ε

), (3.3)

where each un is Y -periodic in its second argument. In the limit ε→ 0 theleading term is u0. We hope that this terms does not depend on y: if so, it

32

is representative of the behaviour of the solution when the micro scale andthe macro scale are fully distinct, i.e. in the limit ε→ 0.

This series expansion is plugged into equation (3.2), and by the chainrule

d

dxψ(x,

x

ε

)=

∂

∂xψ(x,

x

ε

)+ ε−1 ∂

∂yψ(x,

x

ε

),

we can write the problem in series in ε:

− ε−2[div y

(E∇s

yu0

)](x,

x

ε

)− ε−1

[div y

(E∇s

xu0 + E∇syu1

)+ div x

(E∇s

yu0

)](x,

x

ε

)−[

div x

(E∇s

xu0 + E∇syu1

)+ div y

(E∇s

xu1 + E∇syu2

)](x,

x

ε

)− b(x)

−

+∞∑n=1

εn[...]

= 0 .

(3.4)

Since the whole series is equal to zero, each coefficient must vanish. Thisyields to a cascade of equations, and we will see that the first three ones areenough for our purposes. The equation associated with ε−2 reads

− div y

(E(x,y)∇s

yu0(x,y))

= 0 . (3.5)

Here we can consider x as a parameter, and it turns out to be an ho-mogeneous equation in y, equipped with periodic boundary conditions. ByFredholm’s alternative the only solutions of this equations are constants(since the variable x is a parameter, the value of the constant can dependon x). Then there exists a function u such that

u0(x,y) = u(x) .

The equation associated to ε−1, since ∇syu0 = 0, reads:

− div y

(E(x,y)∇s

yu1(x,y))

= div y

(E(x,y)∇s

xu(x)). (3.6)

Once again, it can be regarded as an equation for the unknown u1 inthe variable y, equipped with periodic boundary conditions. Note that theright hand side depends linearly on the strain tensor ∇s

xu(x).

Finally, the ε0 equation reads

− div x

(E(x,y)∇s

xu(x) + E(x,y)∇syu1(x,y)

)− div y

(E(x,y)∇s

xu1(x,y) + E(x,y)∇syu2(x,y)

)= b(x) .

(3.7)

33

Integrating all terms over Y , it turns out that, thanks to the divergencetheorem and to periodic boundary conditions, the terms under the operatordiv y vanish, and the equation reduces to

− div x

( 1

|Y |

ˆYE(x,y)

[∇s

xu(x) +∇syu1(x,y)

]dy)

= b(x) . (3.8)

Since, as we noticed before, u1 depends linearly on the strain tensor∇s

xu(x), equation (3.8) can be regarded as an equation for the unknown u.We are left to express explicitly this dependence, in order to simplify into(3.8).

For this purpose, we consider the canonical basis of Rd, which we denoteby ei1≤i≤d, and the following basis for the space of symmetric second ordertensors M2:

eij =1

2

(ei ⊗ ej + ej ⊗ ei

). (3.9)

For each element of the basis and for each x ∈ Ω we consider the followingcell problem for the unknown wij(x,y) (seen as a function of y):

−div y

(E(x,y)∇s

ywij(x,y))

= div y

(E(x,y)eij

)in Y

y 7→ wij(x,y) Y -periodic(3.10)

Now it is possible to express the unknown u1 as a linear combination ofthe cell functions wij :

u1(x,y) =d∑

i,j=1

1

2

(∂ui∂xj

(x) +∂uj∂xi

(x))wij(x,y) . (3.11)

Finally, plugging the previous relation into (3.8) it turns out that theunknown u solves an equation of the same kind as the original one:

− div(E∗(x)∇su(x)

)= b(x) , (3.12)

where the tensor E∗ is defined by its entries:

E∗ijkl(x) =1

|Y |

ˆY

[Eijkl(x,y) +

(E(x,y)∇s

ywij(x,y))kl

]dy

=1

|Y |

ˆYE(x,y)

(eij +∇s

ywij)

:(ekl +∇s

ywkl)

dy .

(3.13)

The tensor E∗ describes the effective (or homogenized) properties of theheterogeneous medium, since it does not depend on the choice of the domain,load or boundary conditions. It is thus called homogenized tensor.

34

3.1.2 H-convergence

To provide the derivation of the homogenized stiffness tensor (3.13) witha solid mathematical background it is necessary to rely on the notion ofH-convergence. Here we introduce the basic definition and results of H-convergence in the linear elasticity setting, following the works of [1, 24].

We consider a family of Hook’s laws Eε ∈ L∞(Ω;M4α,β), where ε is a

sequence of positive real numbers going to zero. We remark that ε canbe interpreted as a length scale, but it is not required to assume any pre-cise physical role. Moreover, to introduce the notion of H-convergence noassumptions of periodicity are necessary.

We consider the linear elasticity problem associated the Hook’s law Eε:find uε ∈ H1

ΓD(Ω;Rd) s.t.´

Ω Eε∇suε : ∇sv = 〈b,v〉H−1,H1ΓD

+ 〈t,v〉H−1/2,H1/2 ∀v ∈ H1ΓD

(Ω;Rd) .(3.14)

Being the family of Hook’s laws Eε uniformly coercive, thanks to thestability estimate (1.3) the family of solutions uε is bounded in H1

ΓD(Ω;Rd).

Then, by the relative compactness of bounded sets with respect to the weaktopology in H1

ΓD(Theorem 4 in Appendix A), there exists a subsequence

weakly converging in H1ΓD

to some limit u ∈ H1ΓD

(Ω;Rd).Arguing as before, the sequence of stress tensors σε = Eε∇suε is bounded

in L2(Ω;M2), being the family Eε uniformly bounded. Then, there existsa subsequence weakly converging to a limit stress tensor σ ∈ L2(Ω;M2). Anatural question which arises from this argument is whether or not there isa relationship between the limit strain and the limit stress tensor. We thenintroduce the following definition:

Definition 2. A sequence of Hook’s laws Eε ∈ L∞(Ω;M4α,β) is said to H-

converge (or to converge in the sense of homogenization) to a limit Hook’slaw E∗ ∈ L∞(Ω;M4

α,β) if, for any forcing term (b, t) ∈ H−1(Ω;Rd) ×H−1/2(ΓN ), the sequence uε of solutions of the problem (3.14) satisfies:

uε u in H1ΓD

(Ω;Rd)Eε∇suε E∗∇su in L2(Ω;M2) ,

where u is the unique solution to the homogenized problem:find u ∈ H1

ΓD(Ω;Rd) s.t.´

Ω E∗∇su : ∇sv = 〈b,v〉H−1,H1ΓD

+ 〈t,v〉H−1/2,H1/2 ∀v ∈ H1ΓD

(Ω;Rd) .(3.15)

In other words, we are considering a family of problems (3.14) and look-ing for a notion of “limit problem”. To do this we consider the sequence ofthe solutions, and define the limit problem as the problem whose solution isthe limit solution. We have the following compactness result [1]:

35

Theorem 1. For any sequence of Hook’s Laws Eε belonging to L∞(Ω;M4α,β)

there exists a subsequence H-converging to a homogenized Hook’s law E∗ ∈L∞(Ω;M4

α,β).

The notion of H-convergence gives a mathematical justification to theheuristic derivation of formula (3.13). Indeed, a crucial result states that inthe case of periodic homogenization (i.e. a sequence of Hook’s laws definedas in (3.1)), the sequence Eε H-converges to the tensor defined in (3.13).We refer to [1] for details.

3.2 Homogenized stiffness tensor of diblockcopolymers patterns

We now go back to diblock copolymers melts, and address the problemof finding an homogenized tensor capable of describing the medium at amacroscopic level. As mentioned in Chapter 2, when a diblock copolymerssolution undergoes a critical temperature, phase separation takes place, end-ing up with an ordered structure consisting of regions rich of material A ormaterial B. Equilibrium configurations can be recovered as minima of theOhta-Kawasaki functional (2.4), and are described by the order parameterϕ(x) ∈ [−1, 1], defined as the difference between the relative density of thetwo monomers. Thus we have ϕ = −1 in pure B zones and ϕ = 1 in pure Azones.

3.2.1 Interpolation between pure materials

We suppose that pure materials A and B can be modelled as linear elasticmedia, described respectively by the Hook’s laws EA and EB (see Section1.6.2):

EA = 2µA I4 + λA I2 ⊗ I2

EB = 2µA I4 + λB I2 ⊗ I2 .(3.16)

We remark that such laws belong to the space M4α,β for some α, β > 0,

as it is shown in the following Proposition. This result will allow us to applythe results of Section 3.1.2 to the present case.

Proposition 1. Consider the isotropic Hook’s law E = 2µ I4 + λ I2 ⊗ I2

in dimension d = 2 or 3, with elastic moduli taking the admissible valuessummarized in (1.5)-(1.6). Then E ∈ M4

α,β, where the coercivity constants

36

α, β > 0 are given by:

α =

2µ non-auxetic materials

2µ+ dλ auxetic materials

β =

1−(d−1)ν

E non-auxetic materials1+νE auxetic materials.

(3.17)

Proof. The symmetry constraints are trivially satisfied. We now show thecoercivity of E in the case of non-auxetic materials (µ > 0, λ ≥ 0). For anygiven ε ∈M2 it holds:

Eε : ε = 2µ|ε|2 + λ tr(ε)2 ≥ 2µ|ε|2 .

In the case of auxetic materials (µ > 0, −23µ < λ < 0) we have:

Eε : ε = 2µ|ε|2 + λ tr(ε)2

= 2µ

d∑i,j=1

ε2ij︸ ︷︷ ︸∑d

j=1 ε2jj+2

∑i<j ε

2ij

+λ( d∑j=1

εjj

)2

︸ ︷︷ ︸∑dj=1 ε

2jj+2

∑i<j εiiεjj

= 4µ∑i<j

ε2ij + (2µ+ λ)

d∑j=1

ε2jj + 2λ

∑i<j

εiiεjj︸ ︷︷ ︸≤ 1

2(ε2ii+ε

2jj)

≥ 4µ∑i<j

ε2ij + (2µ+ λ)

d∑j=1

ε2jj + λ

∑i<j

(ε2ii + ε2

jj)︸ ︷︷ ︸(d−1)

∑dj=1 ε

2jj

= 4µ∑i<j

ε2ij + (2µ+ dλ)

d∑j=1

ε2jj ≥ (2µ+ dλ)|ε|2 .

We are left to show the coercivity of the inverse of E. We first considerthe auxetic case (−1 < ν < 0). Take σ ∈M2:

Cσ : σ =1 + ν

E|σ|2 − ν

Etr(ε)2 ≥ 1 + ν

E|σ|2 .

37

For non-auxetic materials (0 ≤ ν < 1/2) it holds:

Cσ : σ =1

E

[(1 + ν)|σ|2 − ν tr(σ)2

]]=

1

E

[(1 + ν)

( d∑j=1

σ2jj + 2

∑i<j

σ2ij

)− ν( d∑j=1

σ2jj + 2

∑i<j

σiiσjj

)]

=1

E

[2(1 + ν)

∑i<j

σ2ij +

d∑j=1

σ2jj − 2ν

d∑i<j

σiiσjj

]

≥ 1

E

[2(1 + ν)

∑i<j

σ2ij +

d∑j=1

σ2jj − ν

∑i<j

(σ2ii + σ2

jj)︸ ︷︷ ︸(1−(d−1)ν)

∑dj=1 σ

2jj

]

≥ (1− (d− 1)ν)

E|σ|2 .

It is easy to check that for d = 2, 3 the coercivity constants listed in(3.17) are all strictly positive.

In the region of intermediate composition of material A and B we inter-polate between EA and EB. Different choices could be made, but since theminima of the Ohta-Kawasaki functional show very little regions of inter-mediate density, this choice has nearly no relevance. Nevertheless, in orderto apply the results of homogenization theory, it should be checked that thestiffness tensor belong to the spaceM4

α,β in almost any point of the domain.Then we suppose that in the region of intermediate composition the solidbehaves as a linear elastic isotropic medium, with Lame moduli which area convex combination of those of pure materials. In other words, in eachpoint x, we suppose:

E(x) = 2µ(x) I4 + λ(x) I2 ⊗ I2

µ(x) =µA + µB

2+µA − µB

2ϕ(x)

λ(x) =λA + λB

2+λA − λB

2ϕ(x) .

(3.18)

We stress that, since the admissible set of the parameters (λ, µ) is theconvex set

(λ, µ) ∈ R2 s.t. µ > 0, 2µ + 3λ > 0

, any convex combinations

of admissible parameters is admissible as well.

3.2.2 Two-scales model

Because the two scales involved in the class of problems we are address-ing are very different (nanometres versus centimetres, at least), we look

38

for a two-scales formulation. We recall that diblock copolymers patternsare determined by the couple of parameters m, the difference of relativedensities of materials A and B, and γ, which depends on the propertiesof the monomers. We suppose that a controller has faculty to choose thecomposition of the solution in the different points of the domain, and thuswe introduce the control variable m ∈ L∞(Ω; [m,m]), where m and m de-note the lower and upper limit for m (for instance m = −m2, m = m2).The variable m determines the micro-structure in each point of the domain.Where m ∈ (−m1,m1), for instance, the body exhibits lamellae, while wherem ∈ (m1,m2) B-spots are present. Since once the couple of monomers Aand B are chosen the parameter γ is set, the latter does not account as acontrol variable.

Denoting by ε the length-scale of the micro-structure, we consider thefollowing two-scales formulation:

ϕε(x) = ϕ(x,

x

ε

),

where for each x ∈ Ω the function y 7→ ϕ(x,y) is the periodic micro-structure attaining the minimum of the Ohta-Kawasaki functional for m =m(x). It is remarkable that the periodicity cell may depend on the intervalwhich m(x) belongs to: in the interval (−m1,m1) we have rectangular ele-mentary cell, while in the region (m1,m2) and (−m2,−m1) the elementarycell has hexagonal shape.

Remark 3. In 2D it is possible to choose an elementary cell which is validfor all values of m (see Section 2.5). However, since this is not possible in3D, to preserve generality we will let the cell free to be varied.

Although the pure phases are modelled as isotropic media, the com-posite material is not isotropic in general: the properties of the materialmay depend on its orientation. For this reason the model has to take intoaccount the orientation of the micro-structure. To do so, we introduce an-other control variable, ϑ(x), which describes the local orientation of thediblock copolymer pattern. Generally speaking ϑ belongs to the rotationgroup SO(d), but in two dimensions it can be represented by a point ofthe interval [0, 2π) (the rotation angle), while in three dimensions it can berepresented by the three Euler’s angles.

Therefore, we denote by Ym ⊂ Rd the rescaled periodicity cell associatedwith the monomers proportion m. We then introduce the rotated periodicitycell:

Ym,ϑ = R(ϑ)Ym =

y ∈ Rd s.t. R(ϑ)−1y ∈ Ym,

where R(ϑ) represents the orthogonal transformation associated with ϑ. Intwo dimensions, for instance, we have:

R(ϑ) =

(cos(ϑ) − sin(ϑ)sin(ϑ) cos(ϑ)

).

39

We introduce the space of candidate minima for the Ohta-Kawasaki func-tional for the monomer proportion m and orientation ϑ

Vm,ϑ :=ϕ ∈ H1(Ym,ϑ) , Ym,ϑ-periodic, s.t

Ym,ϑ

ϕ = m,

and the associated Ohta-Kawasaki functional

Hm,ϑ(ϕ) :=

ˆYm,ϑ

( 1

2γ2|∇ϕ|2 + F (ϕ)

)dy

+1

2

ˆYm,ϑ

ˆYm,ϑ

G(y1,y2)(ϕ(y1)−m)(ϕ(y2)−m) dy1 dy2 .

Finally, for each x ∈ Ω we define the function ϕ(x, ·) as follows:

ϕ(x, ·) = argminψ∈Vm(x),ϑ(x)

Hm(x),ϑ(x)(ψ) . (3.19)

Remark 4. Unfortunately the minimum in (3.19) is not unique in general,then the function ϕ(x, ·) is not well-defined in principle. Indeed, on aninfinite domain, the Ohta-Kawasaki functional is invariant with respect totranslations and rotations. However, the periodicity ansatz (ϕ(x, ·) Ym,ϑ-periodic) allows to get rid of rotational-invariance, since the periodic cellforces somehow the micro-structure orientation (in the case of lamellae itis necessary to choose a rectangular cell, otherwise in case of square cellHm,ϑ would be invariant under rotations of π/2). On the other hand, thetranslational-invariance of the Ohta-Kawasaki functional is preserved evenafter the periodicity assumption; however the homogenization process isnot affected by translations of the elementary cell, because the resultingpattern is not varied. Therefore, ϕ(x, ·) can be defined as any minimumof the functional Hm, since the homogenized tensor is not affected by theparticular choice.

3.2.3 Homogenization

To sum up, the elastic properties of the medium are modelled by thefollowing two-scales model:

Eε(x) = 2µε(x) I4 + λε(x) I2 ⊗ I2

µε(x) =µA + µB

2+µA − µB

2ϕε(x)

λε(x) =λA + λB

2+λA − λB

2ϕε(x)

ϕε(x) = ϕ(x,

x

ε

)ϕ(x, ·) = argmin

ψ∈Vm(x),ϑ(x)

Hm(x),ϑ(x)(ψ) .

(3.20)

40

To apply the results of homogenization theory we have to prove thefollowing proposition:

Proposition 2. The tensor Eε defined in (3.20) belongs to L∞(Ω;M4α,β)

for some α, β > 0.

Proof. Thanks to Proposition 1, for almost any x ∈ Ω we have Eε(x) ∈M4

α(x),β(x) for some α(x), β(x) > 0, possibly depending on x. Because theLame coefficients are bounded by

min(λA, λB) < λ(x) < max(λA, λB)

min(µA, µB) < µ(x) < max(µA, µB) ,

and the coercivity constants are continuous functions of λ and µ (see (3.17)),by Weierstrass theorem α(x) and β(x) are uniformly bounded from below,so Eε(x) ∈ M4

α,β for almost any x ∈ Ω for some α, β > 0. Since the space

M4α,β is bounded, Eε belongs to L∞(Ω;M4

α,β).

By Theorem 1, there exists a Hook’s law E∗ ∈ M4α,β such that Eε H-

converges to E∗. Moreover, since the Hook’s law Eε is written in the form(3.1), the tensor E∗ is given by the homogenization formula (3.13):

E∗ijkl(x) =1

|Ym(x),ϑ(x)|

ˆYm(x),ϑ(x)

E(x,y)(eij +∇s

ywij)

:(ekl +∇s

ywkl)

dy ,

(3.21)where the function y 7→ wij(x,y) solves the cell problem:−div y

(E(x,y)∇s

ywij(x,y))

= div y

(E(x,y)eij

)in Ym(x),ϑ(x)

y 7→ wij(x,y) Ym(x),ϑ(x)-periodic.

(3.22)

Remark 5. The homogenization procedures aims at capturing the asymp-totic properties of the material, in the limit ε → 0. In the “real” problemwe are addressing, on the other hand, the length scale ε has a finite value.However in real applications, as we have already stressed, typical values forε range from 10−6 to 10−9, then the approximation ε ' 0 is not far fromreal.

Remark 6. At first glance it may seem that H-convergence is a too weaknotion of convergence to capture the limit behaviour of the medium. As amatter of fact, the displacement field u converges weakly in H1

ΓD(Ω;Rd): in

one dimension this implies uniform convergence, but in higher dimensionsthis topology turns out to be quite weak. However we should not forget thatthe objective functional we intend to minimize, as we anticipated in Section1.1, is the compliance

C(u) =

ˆΩ

b(x) · u(x) dx +

ˆΓN

t(x) · u(x) dx .

41

Then we are not interested in the description of the actual displacementfield, but just on the associated compliance. Since C is a linear functional onH1

ΓD(Ω;Rd), weak convergence of the displacement field yields to the conver-

gence of the compliance. Then, for our purposes, this notion of convergenceis enough.

3.2.4 A simpler formulation

The expression (3.21) may be cumbersome in actual implementations,since it requires the solution of the cell problem (3.22) in each point of thedomain (or in each node of the mesh, in a numerical implementation). Evenworse, the domain itself of the cell problem depends on x. Fortunatelyformula (3.21) can be restated in an equivalent and simpler form.

First of all we notice that E∗(x) depends on x only through the con-trol variables m and ϑ, and thus we can write E∗(x) = E∗m(x),ϑ(x), whereE∗m,ϑ denoted the homogenized Hook’s law associated with the monomersproportion m and the micro-structure orientation ϑ.

Moreover, the effect of ϑ is simply a rotation of the local frame of ref-erence. Then, instead of computing the homogenized tensor on the rotateddomain Ym,ϑ, it is possible to compute the homogenized tensor on the non-rotated domain Ym, and finally to rotate the tensor. As a matter of fact,through a change of variables z = R(ϑ)−1y, it can be proved that

E∗m,ϑ = Q(ϑ)E∗m,0 , (3.23)

where Q(ϑ) is the 8-th order tensor of components

Qijklpqrs(ϑ) = Rip(−ϑ)Rjq(−ϑ)Rkr(−ϑ)Rls(−ϑ) . (3.24)

Therefore, we just need to compute the Hook’s law E∗m := E∗m,0 foreach admissible value of m. Although no analytical formula is available, itis possible to build a database of homogenized tensors for many differentvalues of m, and then interpolate.

Tu sum up, the homogenization theory shows that the sequence of Hook’slaws Eε defined in (3.20) H-converges to the homogenized tensor E∗, thatcan be computed as follows:

E∗(x) = Q(ϑ(x))E∗m(x)

(E∗m)ijkl =1

|Ym|

ˆYm

Em(y)(eij +∇swij

m

):(ekl +∇swkl

m

)dy

Em(y) = 2µm(y) I4 + λm(y) I2 ⊗ I2

µm(y) =µA + µB

2+µA − µB

2ϕm(y)

λm(y) =λA + λB

2+λA − λB

2ϕm(y)

ϕm = argminψ∈Vm,0

Hm,0(ψ) ,

(3.25)

42

where the function wijm(y) solves the cell problem:

−div

(Em(y)∇swij

m(y))

= div(Em(y)eij

)in Ym

wijm Ym-periodic .

(3.26)

3.3 Numerical Results

To construct a database of homogenized tensors, we solve numericallythe cell problems associated with different values of m. The cell problem(3.26) is a linear elasticity problem, whose weak formulation reads as follows:

find wijm ∈ H1

per(Ym;Rd) s.t.´Ω Em∇swij

m : ∇sv = −´

Ω Emeij : ∇sv ∀v ∈ H1per(Ym;Rd) ,

(3.27)where H1

per(Ym;Rd) denotes the space of H1 functions with values in Rd,periodic on the domain Ym.

We consider the finite elements formulation of problem (3.27), imple-mented through the software FreeFem++. The unknown wij

m is discretizedwith P1 finite elements, on the same meshes employed in the simulationsdescribed in Section 2.5.3.

Pure phases Poisson’s coefficients are set to 0.3 for both materials, whileYoung’s moduli are set to different values, in order to investigate the effectof the relative stiffness between pure phases. The homogenized value of thesix independent entries of the stiffness tensor are shown in Figures 3.1 and3.2. In the former the ratio EA/EB equals 10, while in the latter it is equalto 103. As one could expect, since material A is stiffer than material B,the tensor entries increase as the concentration of material A raises (i.e. mincreases).

In the case of lamellae patterns the anisotropic character of the homoge-nized medium is evident. The material is stiffer in direction y, that is to sayin the direction of stripes. This result as well agrees with common sense. Asfar as spots patterns are concerned, any potential anisotropy is not evidentfor the moment, since we have Exxxx = Eyyyy and Exxxy = Eyyxy.

The results reported for spots patterns are those of simulations run onthe hexagonal elementary cell of Figure 2.10. However these figures are veryclose to those obtained with the rectangular cells (Figure 2.9), as we couldexpect, since they describe the same medium.

Finally we observe that, while inside each region of the phase plane theentries of the homogenized tensor behave in a smooth way, at the boundarybetween spots and lamellae the homogenized tensor reports jumps. In thefollowing chapter we will see the implications of this remark.

43

-0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25

m

-100

0

100

200

300

400

500

600

700

800

900

1000

Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(a) Spots A

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

m

-100

0

100

200

300

400

500

600

700

800

900

1000

Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(b) Lamellae

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

m

-100

0

100

200

300

400

500

600

700

800

900

1000

Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(c) Spots B

Figure 3.1: Homogenized stiffness tensor for the ratio EA/EB = 10. Purephases parameters are set as follows: EA = 1000, EB = 100, νA = νB = 0.3.The markers associated with the entries Exxxy are identically null, and arehidden under the markers of Eyyxy. In the charts of spots patterns, moreover,the markers for Exxxx coincide and are hidden under those associated withEyyyy.

44

-0.6 -0.55 -0.5 -0.45 -0.4 -0.35 -0.3 -0.25

m

-100

0

100

200

300

400

500

600

700

800

900

1000

Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(a) Spots A

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

m

-100

0

100

200

300

400

500

600

700

800

900

1000

Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(b) Lamellae

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

m

-100

0

100

200

300

400

500

600

700

800

900

1000

Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(c) Spots B

Figure 3.2: Homogenized stiffness tensor for the ratio EA/EB = 103. Purephases parameters are set as follows: EA = 1000, EB = 1, νA = νB = 0.3.As in Figure 3.1, the markers of Exxxy coincides with those on Eyyxy, andfor spots micro-structures Exxxx coincides with Eyyyy.

45

3.3.1 Tensor rotation

In this section we investigate the behaviour of the homogenized tensorswhen the micro-structure is rotated according to equation (3.23).

Figure 3.3 refers to lamellae structures, and underlines the orthotropicnature of this micro-structure. After a rotation by π/4 the roles of x and yare switched, and after a further rotation by π/4 the tensor coincides withthe non-rotated one. Some entries take negative values for certain rotationangles, which is not a mistake: it is sufficient that the tensor as a wholepreserves is positive definiteness. This fact is guaranteed by the definitionitself of H-convergence, which requires that the H-limit belongs to the spaceM4

α,β.Figure 3.4 reports rotated entries of homogenized tensors for spots micro-

structures. Surprisingly, the homogenized tensors are not affected by rota-tion of the micro-structure. This means that the homogenized tensor isisotropic. Although the microstucture is not rotationally-invariant (it isinvariant just by rotations multiple of π/3), the medium, if we look at itat a sufficiently coarse scale, reports no preferential directions. This prop-erty still holds when the stiffness ratio EA/EB is changed, and in the caseνA 6= νB.

46

0 π/2 π (3 π)/2 2 π

θ

-200

0

200

400

600

800 Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(a) EA/EB = 10

0 π/2 π (3 π)/2 2 π

θ

-200

0

200

400

600

800E

xxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(b) EA/EB = 1000

Figure 3.3: Rotated homogenized tensor of lamellae patterns for m = 0, fordifferent stiffness ratios between pure phases.

47

0 π/2 π (3 π)/2 2 π

θ

-200

0

200

400

600

800 Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(a) Spots A (m = −0.4)

0 π/2 π (3 π)/2 2 π

θ

-200

0

200

400

600

800 Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(b) Spots B (m = 0.4)

Figure 3.4: Rotated homogenized tensor of spots patterns for EA/EB = 10.

48

Chapter 4

Topology optimizationproblem formulation

In Chapter 2 the mathematical modelling of diblock copolymers patternformation has been described, and the results of numerical simulations hasbeen shown. In Chapter 3 then a two-scales model for diblock copolymerselastic bodies has been proposed, and through homogenization theory amacroscopic description of the body has been derived. Numerical resolutionof the cell problems associated with diblock copolymers micro-structures hasallowed to compute the homogenized Hook’s laws for the typical patternsexhibited by diblock copolymers materials. To summarize, given the controlvariables m(x) and ϑ(x), we are now able to model at a macroscopic levelthe elastic properties of the body, according to (3.25).

In Section 4.1 we formulate a minimum compliance problem for bodiesobtained by self-assembling diblock copolymers. Together with the micro-structure type and its orientation, we want to optimize the body microscopicshape too, with a resource constraint that limits the mass of the body (seeequation (4.9)). In the next two sections two issues arising from the for-mulation (4.9) are addressed: in Section 4.2 a multi-material formulation isproposed to overcome the lack of differentiability of the stiffness tensor withrespect to the design variable m, and in Section 4.3 a length-scale controlstrategy is taken into account to guarantee well-posedness of the problemand consistency of the model. The chapter ends with the proof of Theo-rem 2, which states the existence of solutions of the topology optimizationproblem for diblock copolymers bodies.

4.1 Optimizing the shape of the body

We suppose to be given a reference domain Ω ⊂ Rd, which the bodyshould be contained in. The task of finding the optimal shape of the bodycan be interpreted as that of finding a subset A ⊂ Ω, minimizing a suitable

49

objective functional. The unknown A is identified by its indicator functionz : Ω→ 0, 1 defined as

z(x) = 1A(x) =

1 if x ∈ A0 if x /∈ A .

(4.1)

The variable z determines which point of the space should be filled withmaterial, and which should remain void. In other words we are thinking tothe body as a blank-and-white rendering of an image.

The stiffness tensor in the domain Ω is then given by:

E∗(x) = z(x)Q(ϑ(x))E∗m(x) , (4.2)

where E∗m is the homogenized tensor relative to the micro-stucture associatedwith monomers proportion m, and is given by equation (3.25).

In order to avoid trivial solutions we introduce a resource constraint onthe total mass. Denoting by ρA and ρB the specific weight of pure phases,the specific weight of the composite material for the monomers proportionm is given by

ρ(m) =ρA + ρB

2+ρA − ρB

2m, (4.3)

and the total mass of the body can be computed as

M =

ˆΩz(x)ρ(m(x)) dx . (4.4)

Then we consider the mass constraint M = M , where M is assigned.

4.1.1 Material distribution method

The design variable is the triplet (z,m, ϑ). While m and ϑ are con-tinuous variables, z is discrete, since it takes values in 0, 1. A commonstrategy, known as material distribution method, to switch from a discreteoptimization problem to a continuous one is to replace the discrete variablez : Ω → 0, 1 with an equivalent continuous one z : Ω → [0, 1], which canbe interpreted as the material density. In this way we switch from a black-and-white representation to a grey-scale one. The parametrization of theelasticity tensor (4.2) still makes sense, and interpolates linearly the stiff-ness from void to full material. This enlargement of the design space allowsto use continuous optimization algorithms instead of discrete programming,which would be computationally inefficient because of the huge number ofdesign variables and constraints.

The enlargement of the design space to intermediate values of z has acounter effect, since the physical meaning of those values is not clear. Thusit is desirable that the final design consists mostly of regions of full material

50

or void. In other words, we want little grey regions. We are then led topenalize intermediate densities, to steer the solution to a 0-1 design.

The most common strategy to penalize the intermediate densities is theso-called SIMP method (Solid Isotropic Material with Penalization), origi-nally formulated for isotropic materials [6], but which is here generalized forpossibly anisotropic materials. The SIMP method envisages the substitutionof the linear interpolation (4.2) with the following power-law interpolation,where it is typically set p = 3:

E∗(x) = z(x)pQ(ϑ(x))E∗m(x) . (4.5)

In this way intermediate densities become inconvenient, since they en-tail a consumption of the resource M in exchange of a smaller increase ofstiffness compared to that given, in proportion, by values of z close to 1.Consequently the design is steered towards a discrete 0-1 configuration. Weremark that when z takes values just in the set 0, 1, formulation (4.5)reduces to (4.2).

Remark 7. When formulation (4.5) is employed, singularities in the equi-librium equation can arise where z = 0, since in those point the stiffnesstensor reduces to the null tensor, which of course lacks of coercivity. Toovercome this problem a lower bound for the variable z is introduced, sothat the stiffness tensor never vanishes:

0 < zmin ≤ z(x) ≤ 1 ∀x ∈ Ω ,

where zmin is small (for instance zmin = 10−3). This can be interpreted asfollows: instead of filling the set Ω\A with void, we are filling it with a verycompliant material, which behaves nearly as void but avoids singularities inthe equilibrium.

4.1.2 Minimum compliance problem

We suppose that the boundary of the domain is partitioned into a Dirich-let boundary ΓD, where the body is anchored to the ground, and a Neumannboundary ΓN . We then suppose to be given a bulk load b ∈ H−1(Ω;Rd)and a boundary load t ∈ H−1/2(ΓN ). We want to optimize the mechanicalresponse of the body to the forcing terms b and t. The equilibrium equationis given by:

−div σ = b in Ω

σ(x) = E∗(x)∇su(x) in Ω

σν = t on ΓN

u = 0 on ΓD ,

(4.6)

which reads in weak form:find u ∈ H1

ΓD(Ω;Rd) s.t.

a∗(u,v) = l(v) ∀v ∈ H1ΓD

(Ω;Rd) ,(4.7)

51

where we defines the following bilinear and linear forms (the superscript ∗reminds that the bilinear form depends of the Hook’s law E∗, which in turndepends on the design variables):

a∗(u,v) =

ˆΩE∗∇su : ∇sv dx ,

l(v) = 〈b,v〉H−1,H1ΓD

+ 〈t,v〉H−1/2,H1/2 .(4.8)

A popular way to measure the stiffness of the body with respect tothe load (b, t) is through the compliance l(u), corresponding to the totalwork carried out by the external forces (which coincides with that of internalforces, by the virtual works principle). The compliance is an inverse measureof the stiffness of the body: a high compliance indeed suggests that the bodydeforms a lot under the external forces.

We can now formulate the minimum compliance problem associated withthe load (b, t):

minimizez,m,ϑ

l(u)

subject to: a∗(u,v) = l(v) ∀v ∈ H1ΓD

(Ω;Rd)M = M

zmin ≤ z(x) ≤ 1 ∀x ∈ Ω

m ≤ m(x) ≤ m ∀x ∈ Ω

where: E∗(x) = z(x)pQ(ϑ(x))E∗m(x)

M =´

Ω z(x)ρ(m(x)) dx .

(4.9)

Remark 8. In the context of topology optimization a popular variation of thesingle-load problem is the multi-load problem. In this case one supposes tobe given a finite number of load cases (bi, ti)i=1,...,N , and wants to optimizethe response of the body to all the load cases. The objective functional isthen given by the sum of the compliances computed in the N cases. In thiswork just the single-load problem will be considered, since the generalizationto the multi-load case is straightforward.

4.2 Differentiability issues

In Section 3.3 it has been shown that the map m 7→ E∗m exhibits jumps.Indeed, when m crosses the boundaries between the different regions of thephase plane of Figure 2.5, the micro-structure changes its topology, resultingin a discontinuity of the homogenized tensor. Thus, near the critical values±m1 and ±m2 the map lacks of differentiability, and gradients cannot becomputed. This means that gradient-based algorithms cannot be employed.

Moreover far from the critical values, where derivatives can be computed,gradients information can be misleading in some way. Consider for instance

52

the entry associated with the principle direction x: for m < −0.2 the entryExxxx is monotonically increasing. Imagine an iterative algorithm that,relying on this information makes progressively the variable m raise. At acertain moment the variable would cross the threshold −0.2, and the stiffnessof the structure would fall down, despite the gradient information.

Finally, the formulation (4.9) presents a further complication. Manyoptimization algorithms work progressively improving a candidate solution,moving from a design to a better one. However, looking once again at theentry Exxxx, it is clear that it may happen that moving from A-spots toB-spots implies an improvement of the design, but moving from A-spots tolamellae yields a worse design. Since in order to move from A-spots to B-spots the lamellae configuration is an unavoidable step, the algorithm wouldnever reach the best design of B-spots. Thus the algorithm is likely to betrapped into a local minimum.

We observe that the afore-mentioned problems can be tackled resortingto genetic algorithms. However we now show that an alternative solution,which allows the use of gradient-based algorithms, can be adopted.

We consider an enlargement of the design space, by replacing the vari-ables (z,m, ϑ), with a triplet of design variables for each micro-structuretype (i.e. for each region of the phase plane of the composite material). Inother words, we reformulate problem (4.9) as a multi-material one, lookingat the different micro-structures types as different materials. Therefore weintroduce the following design variables:

zh ∈ L∞(Ω; [zmin, 1]) h = 1, . . . , N

mh ∈ L∞(Ω; [mh,mh]) h = 1, . . . , N

ϑh ∈ L∞(Ω; [0, 2π)) h = 1, . . . , N.

(4.10)

In the case of diblock copolymers we have N = 3 (h = 1 is associatedto A-spots, h = 2 with lamellae and h = 3 with B-spots). The lower andupper bounds mh and mh take the following values (see Section 2.5.1):

m1 = −m2

m1 = m2 = −m1

m2 = m3 = m1

m3 = m2 .

The variables zhh ideally are equal to 1 where the corresponding micro-structure is present, and are equal to zmin elsewhere. Thus we require thatat most one among the variables zhh is equal to 1. In a discrete 0-1formulation, the multi-material formulation should be understood as follows:

E∗(x) =

N∑h=1

zh(x)Q(ϑh(x))E∗mh(x) . (4.11)

53

As in Section 4.1.1, to switch from a discrete problem to a continuousone, intermediate values for zhh are allowed, and then values different from1 and zmin are penalized, in order to have little grey regions in the optimaldesign. Understanding the variable zh as the “density” of the correspondingmicro-structure, we consider the following constraints:

zmin ≤ zh(x) ≤ 1 ∀x ∈ Ω,∀h = 1, . . . , N

N∑h=1

zh(x) ≤ 1 ∀x ∈ Ω .

To penalize intermediate densities we consider the following SIMP-likepower-law interpolation, which makes uneconomical the values different from1:

E∗(x) =N∑h=1

zh(x)pQ(ϑh(x))E∗mh(x) . (4.12)

Remark 9. It is worth noticing that, unlike in formulation (4.5), the expres-sion (4.12) provides a smooth map from the design variables to the Hook’slaw. Indeed inside the intervals [mh,mh] the map m 7→ E∗m does not exhibitjumps (see Section 3.3).

The total mass of the body can be computed by summing the contribu-tion of all the different materials, weighted by their density:

M =N∑h=1

ˆΩzh(x)ρ(mh(x)) dx . (4.13)

Therefore we are led to the following multi-material minimum compli-ance problem:

minimizezh,mh,ϑh

l(u)

subject to: a∗(u,v) = l(v) ∀v ∈ H1ΓD

(Ω;Rd)M = M

zmin ≤ zh(x) ≤ 1 ∀x ∈ Ω,∀h = 1, . . . , N∑Nh=1 zh(x) ≤ 1 ∀x ∈ Ω

mh ≤ mh(x) ≤ mh ∀x ∈ Ω,∀h = 1, . . . , N

where: E∗(x) =∑N

h=1 zh(x)pQ(ϑh(x))E∗mh(x)

M =∑N

h=1

´Ω zh(x)ρ(mh(x)) dx .

(4.14)

4.3 Existence issues and length-scale control

In this section two topics which are linked each other are addressed.The first one deals with the problem of existence of solutions and relatednumerical instabilities; the second one is a gap in the mathematical model.

54

4.3.1 Existence of solutions

It is well known that topology optimization problems lack of existenceof solution in general (in [1] some counterexamples are shown). We now tryto give an intuitive explanation of the reasons behind the ill-posedness ofthis class of problems.

The ill-posedness of topology optimization problems is caused by the pos-sibility of improving the performances of a structure by refining its topologywith the introduction of new holes. In other words, it is possible to builda sequence of finer and finer designs with growing stiffness, without thesequence converging to any admissible design. From Theorem 1 we knowthat the sequence of Hook’s laws H-converges to a limit law (which is themathematical modelling of a micro-structure with a length-scale completelyseparated from to the macro-scale), but it general the limit Hook’s law isnot associated to any design of the kind of (4.12).

From another perspective, the sequence of design variables, being boundedin L∞ (see (4.10)), weakly ∗ converges to some limit design (see Corollary1 in Appendix A). However, since weak∗ limit can be interpreted as a limitin average, the limit densities would exhibit large grey areas (which can beinterpreted with the local density of the micro-structure), and thus wouldbe fairly inefficient because of the SIMP method. This is a consequenceof the lack of lower semi-continuity of the map from the design variables(zh,mh, ϑh)h to the compliance.

The ill-posedness of the problem yields to numerical instabilities whencomputational methods are addressed to solve the optimization problem.The numerical solution typically lacks of mesh-independence, since the length-scale achievable by a discretized design is bounded by that of the mesh.Then, when the mesh is refined, the optimal design is refined as well.

The present work originates exactly from this issue of ill-posedness,and tries to tackle it by formulating a topology optimization problem thatcontemplates micro-structures as admissible designs, through their mod-elling based on the homogenization theory. However, the same kind ofproblem arises: numerical implementations of problem (4.14) report mesh-dependence as well as checkerboard instabilities. This happens because thedensities variables zh are allowed to reproduce a micro-structure on theirown, more precisely a micro-structure of micro-structures: a micro-structurewhose raw material is given by the micro-structure modelled by means ofthe homogenization theory.

The reason why this is possible lies in a gap in the two-scales modelpresented in Chapter 3, which cannot be ignored when it is employed in anoptimization problem.

55

4.3.2 A revision of the two-scales model

In Chapter 3 the two-scales model 3.20, which describes a medium ex-hibiting locally periodic micro-structures, has been considered. The expres-sion locally periodic should be understood in the following way: looking atthe medium at a mesoscopic level, i.e. an intermediate level between the mi-croscopic length-scale ε and the macroscopic one, it is periodic (the functionϕ is indeed periodic in y); However, the periodicity pattern may vary (evenif in a slow way) with x, whence the attribute locally. We emphasize theword slow, since the variation of the pattern should take place at a coarserscale then the micro-scale ε, otherwise the model loses its significance.

This remark was irrelevant in Chapter 3, because in order to take theH-limit there is no need to control the variation of ϕ with respect to thevariable x. As a matter of fact, when homogenization is performed, thefunction ϕ is fixed, and then ε is let go to zero. Instead in the present case,namely when problem (4.14) is addressed, the H-limit has been alreadycomputed, and the variables zh are free to vary at will.

Moreover another issue has to be taken into account. We can considerat least two different ways of manufacturing the diblock copolymer objectsdescribed by the two-scales model (3.20). The first one is to take some self-assembled small units with the desired properties and then assembling themtogether. The other one is to lay down a hot diblock copolymers solution,changing from place to place the composition according to control variablem, and forcing the orientation according to the variable ϑ, and then to let thesolution cool, so that phase separation takes place and produces the desiredpattern. The task of forcing the orientation of the patterns is accomplishedby means of the so-called directed self-assembly (DSA) technologies (see [29,44, 19]).

Despite the focus of this work is on the mathematical aspects and not onthe manufacturing of diblock copolymers bodies, it is important to noticethat in both cases the features of the periodic pattern cannot change toofast from place to place. In other words, a lower bound to the length-scaleof the control variables should be considered.

4.3.3 Length-scale control

We are then led to bound from below the length-scale of the controlvariables. We remark that there are some similarities with the popularstrategies to get rid of the numerical instabilities linked to the non-existenceof solutions. However, in the present case this operation is justified by thephysics of the problem considered, and fits well with the adopted model.

The control on the length-scale can be accomplished in different ways.A first way is to penalize directly the variations of the control variables,supplementing the objective function with a term that measures the total

56

variation (TV) of such variables. Otherwise it is possible to restrict the setof admissible designs to those filtered through a convolution kernel K > 0.A common choice of convolution kernel is the function:

K(x) = max0, ‖x‖ − rmin , (4.15)

where the convolution radius rmin determines the minimum admissible length-scale. Nevertheless, any Lipschitz convolution kernel would do the job.

Given a generic variable ψ we denote by ψ the filtered variable, obtainedthrough a normalized convolution with K.

ψ(x) =(K ∗ ψ)(x)

(K ∗ 1)(x)=

´ΩK(x− z)ψ(z) dz´

ΩK(x− z) dz. (4.16)

To rule out trivial cases (such as domains with a sequence of sharperand sharper cusps), in the following we will suppose that the denominatorin (4.16) is bounded from below:ˆ

ΩK(x− z) dz ≥ Kmin ∀x ∈ Ω . (4.17)

We then replace the original variables with the filtered ones in equations(4.12) and (4.13):

E∗(x) =

N∑h=1

zh(x)pQ(ϑh(x))E∗mh(x)

M =N∑h=1

ˆΩzh(x)ρ(mh(x)) dx .

(4.18)

We remark that the filtered variables satisfy the same constraints of theoriginal ones, thanks to the following lemma:

Lemma 1. Suppose that ψ ∈ L1(Ω) satisfies:

ψ(x) ≤ C a.e. in Ω ,

for some C ∈ R. Then its filtered counterpart satisfies:

ψ(x) ≤ C ∀x ∈ Ω .

Proof. The proof follows easily from the definition of filtered variables, andby linearity of the convolution operator:

ψ(x) =

´ΩK(x− z)ψ(z) dz´

ΩK(x− z) dz

≤´

ΩK(x− z)C dz´ΩK(x− z) dz

= C ∀x ∈ Ω .

57

To sum up, the revised optimization problem reads as follows:

minimizezh,mh,ϑh

l(u)

subject to: a∗(u,v) = l(v) ∀v ∈ H1ΓD

(Ω;Rd)M = M

zmin ≤ zh(x) ≤ 1 ∀x ∈ Ω,∀h = 1, . . . , N∑Nh=1 zh(x) ≤ 1 ∀x ∈ Ω

mh ≤ mh(x) ≤ mh ∀x ∈ Ω,∀h = 1, . . . , N

where: E∗(x) =∑N

h=1 zh(x)pQ(ϑh(x))E∗mh(x)

M =∑N

h=1

´Ω zh(x)ρ(mh(x)) dx .

(4.19)

We conclude this section with some remarks.

Remark 10. In the following we will refer to the filtered variables as “phys-ical variables”, and to the original ones as “control variables”. Indeed thefirst ones are the only ones with physical relevance, while the second onesare mere mathematical tools. Therefore the solution of the optimizationproblem is represented by the filtered variables. This is the reason why thetotal mass is computed with respect to the filtered variables.

Remark 11. Despite the physical variables are represented by the filteredones, the constraints

zmin ≤ zh(x) ≤ 1 ∀x ∈ Ω, ∀h = 1, . . . , N

N∑h=1

zh(x) ≤ 1 ∀x ∈ Ω

mh ≤ mh(x) ≤ mh ∀x ∈ Ω, ∀h = 1, . . . , N

(4.20)

are evaluated on the original variables. The reason for this choice is that thevariables accounting as control variables are the original ones. In any way,thanks to Lemma 1, the constraints on the original values imply the sameconstraints on the filtered ones (the constraint on

∑Nh=1 zh(x) is transferred

to its filtered counterpart thanks to the linearity of the filtering operator).

Remark 12. A counter effect of the filtering of the design variables is that,despite the SIMP penalization, no design without grey areas is admissible.Indeed the interface between void and material cannot be as sharp as in theoriginal formulation, since its slope is bounded from above by the Lipschitzconstant of the convolution kernel (as it will be shown in the followingsection). The grey contours of material regions are the price that we have topay to get a well posed problem, and require a post-processing phase thatconverts the grey-scale design into a black-and-white rendering.

58

4.3.4 Existence result

In this section we show how the length-scale control introduced in theprevious section by means of the filtering of control variables yields to exis-tence of solutions for the minimum compliance problem. We will show thefollowing fundamental result:

Theorem 2. Let Ω ∈ Rd be a bounded Lipschitz domain. Consider a subsetΓD of ∂Ω of positive (d − 1)-dimensional Hausdorff measure. Let K ∈L1(Rd) be a non-negative Lipschitz function satisfying (4.17). Then, forevery forcing term (b, t) ∈ H−1(Ω;Rd) × H−1/2(ΓN ), the problem (4.19)admits at least a solution.

In order to prove Theorem 2 we need the following lemma.

Lemma 2. Let Ω be a domain of Rd. Consider a sequence ϕjj∈N ⊂L∞(Ω), such that:

a ≤ ϕj(x) ≤ b a.e. in Ω

ϕj∗ ϕ in L∞(Ω)

for some a, b ∈ R and ϕ ∈ L∞(Ω). Then:

a ≤ ϕ(x) ≤ b a.e. in Ω.

Proof. Consider x ∈ Ω. By lower semi-continuity of the norm with respectto weak∗ convergence (see [11]), we have:

ϕ(x)− a ≤ ‖ϕ− a‖L∞(Ω) ≤ lim infj‖ϕj − a‖L∞(Ω) ≤ b− a ,

which entails ϕ(x) ≤ b. Arguing as before:

b− ϕ(x) ≤ ‖b− ϕ‖L∞(Ω) ≤ lim infj‖b− ϕj‖L∞(Ω) ≤ b− a ,

whence the second inequality.

We can now prove Theorem 2. The proof exploits the regularizing effectof the filtering operator to recover compactness in the set of admissibledesigns.

Proof of Theorem 2. In the following we will denote by η = (zh,mh, ϑh)h=1,...,N

an admissible design, and by Y the space of admissible designs:

Y =

(zh,mh, ϑh)h=1,...,N :

zh ∈ L∞(Ω; [zmin, 1]) h = 1, . . . , N

mh ∈ L∞(Ω; [mh,mh]) h = 1, . . . , N

ϑh ∈ L∞(Ω; [0, 2π)) h = 1, . . . , N

N∑h=1

zh(x) ≤ 1 ∀x ∈ Ω

N∑h=1

ˆΩzh(x)ρ(mh(x)) dx = M

.

59

We notice that the space Y is a bounded subset of the space L∞(Ω;R3N ).We denote by C(η) the compliance l(u) associated with the design η, andby C? the infimum over all admissible designs:

C? = infη∈Y

C(η) . (4.21)

By definition of infimum, there exists a minimizing sequence, i.e. asequence ηjj∈N ⊂ Y such that:

C(ηj)→jC? . (4.22)

Being the set Y bounded, the sequence ηjj is bounded as well, and byBanach-Alaoglu-Bourbaki theorem (see Appendix A) there exists a subse-quence ηjhh, and a η? ∈ Y such that:

ηjh∗hη? in L∞(Ω;R3N ) .

By definition of weak∗ convergence, since K ∈ L1(Ω) and L∞(Ω) =(L1(Ω)

)∗, the sequence of filtered controls converges pointwise to the filtered

counterpart of η?:

ηjh(x) =(K ∗ ηjh) (x)

(K ∗ 1) (x)=

´Ω ηjh(z)K(x− z) dz

(K ∗ 1) (x)

−→h

´Ω η

?(z)K(x− z) dz

(K ∗ 1) (x)=

(K ∗ η?) (x)

(K ∗ 1) (x)= η?(x) ∀x ∈ Ω .

Thanks to the regularization properties of the convolution operator itis possible to prove more than pointwise convergence. Indeed, consider thefiltered sequence ηjhh. It satisfies the properties of:

• Equiboundedness, since, by Remark 11, ηjh ∈ Y ∀h

• Equicontinuity, since for any x,x′ ∈ Ω and for any h ∈ N, denotingby LK the Lipschitz constant of K:∥∥ηjh(x)− ηjh(x′)

∥∥R3N

=

∥∥∥∥´

Ω ηjh(z)K(x− z) dz´ΩK(x−w) dw

−´

Ω ηjh(z)K(x′ − z) dz´ΩK(x′ −w) dw

∥∥∥∥R3N

=

∥∥∥∥∥´

Ω

´Ω ηjh(z) [K(x− z)K(x′ −w)−K(x′ − z)K(x−w)] dz dw(´

ΩK(x−w) dw) (´

ΩK(x′ −w) dw) ∥∥∥∥∥

R3N

60

≤ 1

K2min

∥∥∥ˆΩηjh(z)

ˆΩ

[K(x− z)

(K(x′ −w)−K(x−w)

)+K(x−w)

(K(x− z)−K(x′ − z)

)]dw dz

∥∥∥R3N

≤ 1

K2min

ˆΩ‖ηjh(z)‖R3N︸ ︷︷ ︸≤‖ηjh‖L∞(Ω;R3N )

[K(x− z)

ˆΩ

∣∣K(x′ −w)−K(x−w)∣∣︸ ︷︷ ︸

≤LK‖x−x′‖

dw

+

ˆΩK(x−w) dw

∣∣K(x′ − z)−K(x− z)∣∣︸ ︷︷ ︸

≤LK‖x−x′‖

]dz

≤2‖ηjh‖L∞(Ω;R3N )‖K‖L1(Rd)|Ω|LK

K2min

‖x− x′‖ .

Then, by Ascoli-Arzela theorem (see Appendix A), there exists a sub-subsequence of ηjhh (which we do not relabel), uniformly convergent tosome limit. Since uniform convergence entails pointwise convergence and byuniqueness of the limit, the limit coincides with η?. Therefore we have:

ηjh →hη? in L∞(Ω;R3N ).

We claim that the design-to-solution map η 7→ u is continuous from thespace Y (endowed with the L∞(Ω;R3N ) topology) to H1

ΓD(Ω;Rd). Once

the claim is shown, thanks to the continuity of the linear form l(·) fromH1

ΓD(Ω;Rd) to R, it follows that:

C(ηjh)→hC(η?) .

Then, by (4.21) and (4.22), the design η? is optimal. Note that η?

is also admissible. Indeed by Lemma 2 the upper and lower bounds of thedesign variables are transferred to their weak∗ limits (the same holds true for∑N

h=1 zh), and by uniform convergence of the filtered variables it is possibleto pass to the limit into the mass constraint. Therefore we have η? ∈ Y.

We are left to show the claim. First of all, we show that the bilinear formsassociated with filtered designs in the space Y are equicoercive. Consider afiltered design η = (zh, mh, ϑh)h=1,...,N ∈ Y, and the associated Hook’s law:

E∗(x) =N∑h=1

zh(x)pQ(ϑh(x))E∗mh(x) .

Thanks to Proposition 2 and by definition of H-convergence, there existsa constant α > 0 such that the tensor E∗m (and its rotated version Q(ϑ)E∗m) isα-coercive for any admissible value of m (and ϑ). Then, thanks to the lowerbound on the variables zh and the Korn’s inequality (A.1), the equicoercivity

61

property is transferred to the bilinear form a(·, ·):

a(u,u) =

ˆΩE∗(x)∇su(x) : ∇su(x) dx

=

ˆΩ

N∑h=1

zh(x)︸ ︷︷ ︸≥zmin

p(Q(ϑh(x))E∗mh(x)

)∇su(x) : ∇su(x)︸ ︷︷ ︸

≥α|∇su(x)|2

dx

≥ Nzpminα ‖∇su‖2L2 ≥ NzpminαCK ‖u‖

2H1 .

where the constant α := NzpminαCK does not depend on η.Consider therefore two filtered designs η1, η2 ∈ Y, and the associated

bilinear forms a1(·, ·) and a2(·, ·). The corresponding states u1 and u2 satisfy:

a1(u1,v) = l(v) ∀v ∈ H1ΓD

a2(u2,v) = l(v) ∀v ∈ H1ΓD

.

By subtracting the previous equations, adding and subtracting a1(u2,v),and choosing v = u2 − u1, we get:

a1(u2 − u1,u2 − u1)︸ ︷︷ ︸≥α‖u2−u1‖2H1

= a1(u2,u2 − u1)− a2(u2,u2 − u1)

=

ˆΩ

(E∗1 − E∗2)∇su2 : ∇s(u2 − u1) dx

≤ ‖E∗1 − E∗2‖L∞ ‖∇su2‖L2 ‖∇s(u2 − u1)‖L2 .

(4.23)

The three factors of the last line of (4.23) can be estimated as follows:

• ‖E∗1 − E∗2‖L∞ ≤ M ‖η1 − η2‖L∞ , for some constant M , since the mapη(x) 7→ E∗(x) is continuous (see Remark 9);

• ‖∇su2‖L2 ≤ 1α ‖l‖H−1 by Lax-Milgram lemma;

• ‖∇s(u2 − u1)‖L2 ≤ ‖u2 − u1‖H1 .

Therefore, plugging the estimates into (4.23) we conclude:

‖u2 − u1‖H1 ≤M ‖l‖H−1

α2‖η1 − η2‖L∞ ,

whence the continuity of the design-to-solution map.

62

Part II

Solution method

63

Chapter 5

Optimization algorithm

In this chapter an algorithm to solve numerically the problem (4.19)is proposed. The algorithm is based on the optimality conditions (OC)method, a fixed-point scheme largely used in the context of topology opti-mization (see [8]). In the first section of this chapter we briefly describe theOC method for the standard topology optimization problem, and we pro-pose a generalization of the OC method to the multi-material case. Then inSection 5.2 an algorithm for the solution of topology optimization problemfor diblock copolymers is described. The two last sections of the chapterdeals respectively with the implementation of the length-scale control in-troduced in Section 4.3, and with the actual evaluation of the homogenizedstiffness tensor.

5.1 The Optimality Conditions method

We review the standard topology optimization problem for the optimaldistribution of an isotropic medium (see [8] for more details). Consider theHook’s law:

E = 2µ I4 + λ I2 ⊗ I2 .

The design of the body is determined by the variable z ∈ L∞(Ω; [zmin, 1]),which is understood as the density of the body. Given a bulk load b ∈H−1(Ω;Rd) and a boundary load t ∈ H−1/2(ΓN ), by employing the SIMPinterpolation for intermediate densities, we consider the state equation:

find u ∈ H1ΓD

(Ω;Rd) s.t.

az(u,v) = l(v) ∀v ∈ H1ΓD

(Ω;Rd) ,

where:

az(u,v) =

ˆΩzpE∇su : ∇sv dx

l(v) = 〈b,v〉H−1,H1ΓD

+ 〈t,v〉H−1/2,H1/2 .

65

We suppose that the volume of the structure is prescribed by the follow-ing constraint: ˆ

Ωz(x) dx = V . (5.1)

The minimum compliance problem reads as follows:minimize

zl(u)

subject to: az(u,v) = l(v) ∀v ∈ H1ΓD

(Ω;Rd)´Ω z = V

zmin ≤ z(x) ≤ 1 ∀x ∈ Ω .

(5.2)

Introducing the Lagrange multipliers Λ, λ−(x) and λ−(x) for the con-straints of problem (5.2), and the multiplier u for the equilibrium equation,the necessary conditions of optimality for the design variable z are a subsetof the stationarity conditions of the Lagrangian:

L = l(u)− (az(u, u)− l(u)) + Λ

(ˆΩz(x) dx− V

)+

ˆΩλ+(x) (z(x)− 1) dx +

ˆΩλ−(x) (zmin − z(x)) dx .

(5.3)

By computing the first variation of the Lagrangian with respect to theadjoint variable u we recover the state equation:

∂L∂u

(v) = −az(u,v) + l(v) = 0 ∀v ∈ H1ΓD

(Ω;Rd) .

The first variation with respect to the state variable u reads:

∂L∂u

(v) = l(v)− az(v, u)

= l(v)− az(u,v) = 0 ∀v ∈ H1ΓD

(Ω;Rd) .

thanks to the symmetry of the stiffness tensor. Then it turns out that theproblem is self-adjoint, i.e. u = u.

We are left to compute the first variation with respect to the designvariable z:

∂L∂z

(ξ) = −ˆ

Ω

∂z(x)p

∂zξE∇su : ∇su dx + Λ

ˆΩξ dx

+

ˆΩλ+(x)ξ dx−

ˆΩλ−(x)ξ dx

=

ˆΩ

[−p z(x)p−1E∇su : ∇su + Λ + λ+(x)− λ−(x)

]ξ dx .

(5.4)

66

Since the (5.4) should vanish for any choice of ξ, we are led to thefollowing optimality condition:

p z(x)p−1E∇su : ∇su = Λ + λ+(x)− λ−(x) , (5.5)

with the switching conditions:

λ+(x) ≥ 0, z(x) ≤ 1, λ+(x)(z(x)− 1) = 0 ∀x ∈ Ω

λ−(x) ≥ 0, zmin ≤ z(x), λ−(x)(zmin − z(x)) = 0 ∀x ∈ Ω .(5.6)

Therefore any stationary point of (5.3) should satisfy:

p z(x)p−1E∇su : ∇su

= Λ if zmin < z(x) < 1

≤ Λ if z(x) = zmin

≥ Λ if z(x) = 1.

(5.7)

We are then led to the following fixed-point scheme, where zk and uk

denote the design and the state variable at the iteration step k:

zk+1(x) =

maxzk(x)− ζ, zmin if (Bk(x))ηzk(x) ≤ maxzk(x)− ζ, zminminzk(x) + ζ, 1 if (Bk(x))ηzk(x) ≥ minzk(x) + ζ, 1(Bk(x))ηzk(x) else,

(5.8)where:

Bk(x) =p zk(x)p−1E∇suk : ∇suk

Λk. (5.9)

The Lagrange multiplier Λk, which is involved in the definition of Bk,should be determined in an inner iteration loop in order to satisfy the volumeconstraint (5.1). It is immediate to verify that the volume of the updateddensities is a continuous and decreasing function of Λk, then its value canbe uniquely determined by means of a bisection or Newton method.

The move limit ζ and the tuning parameter η prevent that the sequenceof zk converges too quickly to a local minimum close to the initial guess.Their value should be chosen by experiment, and can be adjusted to improvethe efficiency of the method. Typical values are ζ = 0.2 and η = 0.5.

Note that the (5.8) reaches a (local) optimum when the left-hand sideterm of (5.5), known as specific strain energy, is constant in the region withintermediate density, is higher where z = 1 and lower where z = zmin. Thisis close to the concept of fully stressed design. The fixed-point scheme (5.8)moves the resource z from areas of low to areas of high specific strain energy,until the conditions (5.7) are fulfilled.

The scheme (5.8) represents a really efficient algorithm, thanks to thefact that the design variables are updated independently from each others.Their mutual coupling (due to the mass constraint) is accounted for bymeans of the bisection loop employed to compute Λk.

67

5.1.1 Generalization to the multi-material case

The extension of the OC method to the present case is not straightfor-ward, first of all due to the multi-material nature of the problem. For thisreason in this section we propose a generalization of the scheme (5.8) to themulti-material case.

We consider the simplest minimum compliance problem featuring a multi-material formulation of the kind of (4.14):

minimizezh

l(u)

subject to: a∗(u,v) = l(v) ∀v ∈ H1ΓD

(Ω;Rd)M = M

zmin ≤ zh(x) ≤ 1 ∀x ∈ Ω,∀h = 1, . . . , N∑Nh=1 zh(x) ≤ 1 ∀x ∈ Ω

where: E∗(x) =∑N

h=1 zh(x)pEhM =

∑Nh=1

´Ω zh(x)ρh dx .

(5.10)

Problem (5.10) consists in finding the optimal distribution of N different(isotropic or anisotropic) materials, given the total mass M . With Eh ∈M4

α,β and ρh > 0 the Hook’s law and the specific weight of the materiallabelled with h are respectively denoted.

We suppose that the parameters zmin and M have non-trivial values,namely:

0 < zmin <1

N

|Ω|zminN∑h=1

ρh < M < |Ω|minhρh .

(5.11)

Note that, thanks to the constraint

N∑h=1

zh(x) ≤ 1 ∀x ∈ Ω, (5.12)

the upper bound on zh is redundant, since the variables zh are positive (weare supposing zmin > 0). Therefore we can omit the upper-bound constrainton zh when writing the Lagrangian.

We introduce the multiplier Λ associated with the mass constraint, themultipliers λh(x) associated with the lower-bound of the design variables,and finally a multiplier µ for the constraint (5.12). Therefore the Lagrangian

68

reads:

L = l(u)− (a∗(u, u)− l(u)) + Λ

(N∑h=1

ˆΩzh(x)ρh dx− V

)

+

N∑h=1

ˆΩλh(x) (zmin − zh(x)) dx +

ˆΩµ

(N∑h=1

zh(x)− 1

)dx .

(5.13)

By imposing null first variation of the Lagrangian with respect to theadjoint variable the state equation is recovered. Like in the mono-materialcase, by computing the first variation with respect to the state variable itturns out that the problem is self-adjoint, therefore u = u. Finally, the firstvariation with respect to each design variable zh reads as follows:

∂L∂zh

(ξ) =

ˆΩ

[−∂E

∗

∂zh(x)∇su : ∇su + Λ ρh − λh(x) + µ(x)

]ξ dx , (5.14)

where the sensitivity of the stiffness tensor E∗ with respect to each designvariable is given by the formula:

∂E∗

∂zh(x) = p zh(x)p−1Eh . (5.15)

Therefore the candidate minima should be looked for in the set of designsfulfilling the following set of conditions:

∂E∗

∂zh(x)∇su : ∇su = Λ ρh − λh(x) + µ(x) ∀x, h

λh(x) ≥ 0, zmin ≤ zh(x), λh(x)(zmin − zh(x)) = 0 ∀x, h

µ(x) ≥ 0,N∑h=1

zh(x) ≤ 1, µ(x)

(N∑h=1

zh(x)− 1

)= 0 ∀x

N∑h=1

ˆΩzh(x)ρh dx = M .

(5.16)Because of the presence of the variable µ, it is not possible to write

directly the counterpart of the set of conditions (5.7). Nevertheless, we canrewrite (5.16) in the following equivalent way:

∂E∗∂zh

(x)∇su : ∇su

Λ ρh + µ(x)

= 1 if zh(x) > zmin

≤ 1 if zh(x) = zmin∀x, h

µ(x) ≥ 0,N∑h=1

zh(x) ≤ 1, µ(x)

(N∑h=1

zh(x)− 1

)= 0 ∀x

N∑h=1

ˆΩzh(x)ρh dx = M .

(5.17)

69

Guided by (5.17), we devise the following fixed-point updating scheme:

zk+1h (x) = max

zmin,

∂E∗∂zh

(x)∇suk : ∇suk + ε

Λk ρh + µk(x) + εzkh(x)

, (5.18)

where ε > 0 is a small constant, and Λk and µk are chosen in such a waythat it holds true:

µk(x) ≥ 0,N∑h=1

zk+1h (x) ≤ 1, µk(x)

(N∑h=1

zk+1h (x)− 1

)= 0 ∀x

(5.19a)

Mk+1 =N∑h=1

ˆΩzk+1h (x)ρh dx = M . (5.19b)

One may wonder whether the scheme (5.18) is well defined or not, namelywhether:

1. it is always possible to find a couple (Λ, µ(x)) such that the conditions(5.19a) and (5.19b) hold true;

2. the couple (Λ, µ(x)) fulfilling (5.19a) and (5.19b), if it exists, is unique.

The answer is in the following proposition:

Proposition 3. Consider a provisional design (zh)h=1,...,N ⊂ L∞(Ω; [zmin, 1])and suppose that conditions (5.11) are fulfilled. Consider the following up-date scheme, dependent on the parameters Λ > 0 and µ ∈ L∞(Ω; [0,+∞)):

zh(x) = max

zmin,

∂E∗∂zh

(x)∇su : ∇su + ε

Λ ρh + µ(x) + εzh(x)

, (5.20)

where u solves the state equation associated with the design (zh)h, and ε isa positive constants. Then:

(a) For each Λ > 0, in each point x ∈ Ω the map µ 7→∑N

h=1 zh(x) iscontinuous and non increasing.

(b) For each Λ > 0, in each point x ∈ Ω there exists a unique µ (whichwe denote by µΛ(x)) such that:

µ(x) ≥ 0,

N∑h=1

zh(x) ≤ 1, µ(x)

(N∑h=1

zh(x)− 1

)= 0 .

70

(c) With the choice µ = µΛ, the map Λ 7→ M =∑N

h=1

´Ω zh(x)ρh dx is

continuous and non increasing.

(d) For sufficiently small ε, there exists a unique Λ such that the constraintM = M is fulfilled.

Proof. We denote by Ch the following positive quantity:

Ch(x) =

(∂E∗

∂zh(x)∇su : ∇su + ε

)zh(x)

=

p zh(x)︸ ︷︷ ︸≥zmin

p−1 Eh∇su : ∇su︸ ︷︷ ︸≥α|∇su|2

+ε

zh(x)︸ ︷︷ ︸≥zmin

≥ εzmin > 0 .

Moreover, in each point x ∈ Ω, and given the parameters Λ and µ, wedenote by H (omitting the dependence form x, Λ and µ) the set of indexesh such that the constraint zh(x) ≥ zmin is not active (i.e. the max in (5.20)is attained by the second term).

Finally we denote by Z the total local density:

Z(x; Λ, µ) =N∑h=1

zh(x) .

Fix x ∈ Ω. It is easily seen that the Z(x) is continuous and non increasingin µ(x), being the composition of continuous and non increasing functions,then (a) is proved. Moreover, in a neighbourhood of those µ(x) such thatthe set H is not empty (this happens for µ(x) sufficiently small), Z(x) isstrictly decreasing in µ(x). This happens, for instance, in a neighbourhoodof those values of µ(x) such that Z(x; Λ, µ) ≥ 1 (this remark will be usefulin a while).

To prove (b), we consider the two possibilities:

• Z(x; Λ, 0) < 1. Then µΛ(x) = 0 does the job. This choice is unique,since for any µ(x) > 0 we have Z(x; Λ, µ) ≤ Z(x; Λ, 0) < 1.

• Z(x; Λ, 0) ≥ 1. Since Z(x) → N zmin < 1 for µ(x) → +∞, and beingZ(x) continuous and strictly decreasing in µ(x) in the interval [0,+∞)(the set H cannot be empty in this case), there exists a unique µ(x)such that Z(x) = 1.

Summarizing, denoting by µΛ(x) the only (possibly negative) value inthe interval (−Λ min

hρh − ε,+∞) such that Z(x; Λ, µΛ) = 1 we can write:

µΛ(x) = max 0, µΛ(x) .

71

Note that the function µΛ(x) is well defined since the map is continuousand strictly decreasing in that interval, and

Z(x)→ N zmin < 1 for µ(x)→ +∞Z(x)→ +∞ for µ(x)→ (−Λ min

hρh − ε)+ .

Moreover, since near the point (Λ, µΛ(x)) in the parameters space themap (Λ, µ(x)) 7→ Z(x) is of class C1, by Dini’s theorem the correspondenceΛ 7→ µΛ(x) is C1 as well, and it holds (note that the set H is not empty):

∂

∂ΛµΛ(x) = −

∂∂ΛZ(x)∂∂µZ(x)

= −∑

h∈HCh(x) ρh

(Λρh+µΛ(x)+ε)2∑h∈H

Ch(x)(Λρh+µΛ(x)+ε)2

< 0 , (5.21)

where we have employed the following results:

∂zh∂µ

(x) = − Ch(x)

(Λρh + µΛ(x) + ε)2∀h ∈ H,

∂zh∂Λ

(x) = − Ch(x) ρh(Λρh + µΛ(x) + ε)2

= ρh∂zh∂µ

(x) ∀h ∈ H.(5.22)

Therefore µΛ(x) is monotonically decreasing in Λ. Thus, there exists athreshold Λ∗(x) (possibly +∞) such that:

µΛ(x)

> 0 if Λ < Λ∗(x)

≤ 0 if Λ ≥ Λ∗(x),

µΛ(x) =

µΛ(x) if Λ < Λ∗(x)

0 if Λ ≥ Λ∗(x).

(5.23)

The total mass is given by the function M(Λ, µ) =∑N

h=1

´Ω zh(x)ρh dx.

Consider now the map Λ 7→ M(Λ, µΛ): it is continuous, being the compo-sition of continuous functions, and moreover, denoting by d+ and ∂+ theright total and partial derivatives:

dM

d+Λ=∂M

∂+Λ+

ˆΩ

∂M

∂µ

∂µΛ

∂+Λdx

=

ˆΩ

[∑h∈H

∂zh∂+Λ

ρh +∂M

∂µ

∂µΛ

∂+Λ

]dx .

(5.24)

We claim that the integrand of the last line of (5.24) is non-positive foreach x ∈ Ω. In fact, thanks to (5.23), on the set x : Λ ≥ Λ∗(x) the term∂µΛ∂+Λ vanishes, and thus:

∑h∈H

∂zh∂+Λ

ρh +∂M

∂µ

∂µΛ

∂+Λ= −

∑h∈H

Ch(x) ρh(Λρh + µΛ(x) + ε)2

≤ 0 . (5.25)

72

Note that the equality holds only when the set H is empty.

On the other hand, on the set x : Λ < Λ∗(x) (observe that on this setH cannot be empty), by (5.21) and (5.22) we have:

∑h∈H

∂zh∂+Λ

ρh +∂M

∂µ

∂µΛ

∂+Λ

=∑h∈H

∂zh∂+Λ

ρh −

(∑j∈H

∂zj∂µ ρj

)(∑h∈H

∂zh∂+Λ

)∑

h∈H∂zh∂µ

=1∑

h∈H∂zh∂µ

∑j∈H

∂zj∂+Λ

ρj

(∑h∈H

∂zh∂µ

)−

(∑h∈H

∂zh∂µ

ρh

)(∑h∈H

∂zh∂+Λ

)=

1∑h∈H

∂zh∂µ

∑h,j∈H

[ρ2j

∂zh∂µ

∂zj∂µ− ρjρh

∂zh∂µ

∂zj∂µ

]

=1∑

h∈H∂zh∂µ

∑h<j

ρj(ρj − ρh)∂zh∂µ

∂zj∂µ

+∑j<h

ρj(ρj − ρh)∂zh∂µ

∂zj∂µ

=

1∑h∈H

∂zh∂µ

∑h<j

ρj(ρj − ρh)∂zh∂µ

∂zj∂µ−∑h<j

ρh(ρj − ρh)∂zh∂µ

∂zj∂µ

=

1∑h∈H

∂zh∂µ︸︷︷︸<0

∑h<j

(ρj − ρh)2 ∂zh∂µ︸︷︷︸<0

∂zj∂µ︸︷︷︸<0

< 0 .

(5.26)

Then point (c) is proved. We are left to show point (d). We study Mwhen Λ takes extremal values:

• Consider what happens for Λ → 0. In the limit ε → 0, µ is strictlypositive in all the domain, otherwise the constraint Z(x) ≤ 1 wouldbe violated. Therefore we have Z(x) = 1∀x ∈ Ω. Then, thanks toassumptions (5.11):

M =N∑h=1

ˆΩzh(x)ρh dx ≥ min

hρh

ˆΩZ(x) dx ≥ min

hρh|Ω| > M .

• For Λ→ +∞, thanks again to assumptions (5.11) we have:

M → |Ω|zminN∑h=1

ρh < M .

73

Therefore, since M is continuous with respect to Λ, there exists a valuesuch that the mass constraint is fulfilled. By (5.25) and (5.26) it is clear thatM is strictly decreasing in Λ, unless the set H is empty. But for the valuesof Λ for which H is empty, we have M = |Ω|zmin

∑Nh=1 ρh < M , then we

can conclude the values of Λ fulfilling the mass constraint lie in the regionof strict monotonicity of the mass. Therefore the choice of Λ is unique.

Remark 13. The proof of Proposition 3 may appear quite convoluted, and itis likely to distract from the reasons behind the well-definition of the scheme,which are more plain than it might appear. Everything works thanks tomonotonicity of mass and density with respect to the Lagrange multipliers.The only critical aspect is what happens when the constraint

∑Nh=1 ρh ≤ 1

is active, and Λ is raised, since if Λ increases then µ decreases. The localresource Z(x) is fully employed, and an increase of Λ has the only effect ofmoving the resources from heaviest materials to the lightest. In other words,the share of Z(x) owned by materials with low specific weight increases, tothe detriment of that of heavy materials. This is the meaning of (5.26).

Moreover the fixed-point scheme is compliant with the optimality con-ditions (5.17), in a sense that is clarified by the following proposition.

Proposition 4. A design (zh)h=1,...,N fulfils the optimality conditions (5.17)if and only if it is a fixed-point for the update scheme (5.18).

Proof. Suppose that the design (zh)h=1,...,N fulfils (5.17). Then it is easilyseen that plugging into (5.18)(5.19a)(5.19b) zk+1

h = zk+1h = zh, Λk = Λ

and µk = µ, all conditions are satisfied. By uniqueness of Λk and µk (seeProposition 3), (zh)h is a fixed point for the update scheme (5.18).

On the other hand, suppose that a design (zkh)h=1,...,N is a fixed pointfor the scheme (5.18) (i.e. zk+1

h = zkh). Then it is easy to see that zkh, Λk,µk fulfil the optimality conditions (5.17).

Finally we note that Proposition 3 suggests that the value for the mul-tipliers Λk and µk can be found by means of a bisection loop. Thereforewe devise Algorithm 1: at each iteration an external bisection loop detectsthe value of the multiplier Λk, and in each point of the domain an inneriteration loop determines the local value of the multiplier µk. A good initialguess for the Lagrange multipliers is given by the corresponding value at theprevious iteration. Algorithm 1 is written in a rather general way, since thestrategy to follow in the iteration loops is not specified. A good strategycomprises two phases: in a first bracketing phase an interval containing thedesired value is detected, and then the exact value is computed (with thedesired accuracy) by bisection of the interval.

The efficiency of the fixed-point scheme (5.18) can be enhanced, as forthe mono-material scheme (5.8), by introducing a move limit ζ and a tuning

74

Algorithm 1 Multi-material Topology Optimization

for h← 1, N doInitialize zh

end forwhile Not converged do

Solve equilibrium (FEM) . see equation (5.10)Compute sensitivities . see equation (5.14)Initial guess for Λloop

for all x ∈ Ω doInitial guess for µ(x)loop

for h← 1, N doUpdate zh(x) . see equation (5.18)

end forZ(x)←

∑Nh=1 zh(x)

if (Z(x) < 1 AND µ(x) = 0) OR |Z(x)− 1| < εtol thenbreak loop

else if Z(x) < 1 thenIncrease µ(x)

else if Z(x) > 1 thenDecrease µ(x) (not below 0)

end ifend loop

end forUpdate M . see equation (4.13)if |M −M | < εtol then

break loopelse if M > M then

Increase Λelse if M < M then

Decrease Λend if

end loopend whilePost-processing

75

parameter η:

zk+1h (x) =

maxzkh(x)− ζ, zmin if (Bk

h(x))ηzkh(x) ≤ maxzkh(x)− ζ, zminzkh(x) + ζ if (Bk

h(x))ηzkh(x) ≥ zkh(x) + ζ

(Bkh(x))ηzkh(x) else

Bkh(x) =

p zkh(x)p−1E∇suk : ∇suk + ε

Λk ρh + µk(x) + ε.

(5.27)

5.2 Optimization algorithm for diblock copolymersbodies

To extend the OC method to the present case we derive the optimal-ity conditions of the two-scales topology optimization problem for diblockcopolymers. For the sake of generality, we will consider the non filteredversion (4.14), and we will later move to the filtered one.

5.2.1 Optimality conditions

As for the problem (5.10), the constraints on zh are redundant, thereforewe can omit the upper bound constraint zh < 1 in the Lagrangian. Moreover,despite in Theorem 2 we supposed ϑh ∈ L∞(Ω; [0, 2π)), since 0 and 2π canbe identified we are allowed to treat ϑh as an unconstrained variable.

Then we introduce the multiplier Λ for the mass constraint, the multi-pliers λh(x) for the lower bound of zh, a multiplier µ(x) for the constraint(5.12), and the multipliers γ+

h (x) and γ−h (x) for the upper and lower boundsof the variables mh:

L = l(u)− (a∗(u, u)− l(u)) + Λ

(N∑h=1

ˆΩzh(x)ρ(mh(x)) dx−M

)

+N∑h=1

ˆΩλh(x) (zmin − zh(x)) dx +

ˆΩµ

(N∑h=1

zh(x)− 1

)dx

+

N∑h=1

ˆΩ

[γ+h (x) (mh(x)−mh) + γ−h (x) (mh −mh(x))

]dx .

(5.28)

Computing the first variation of the Lagrangian with respect to the ad-joint variable u the state equation is recovered. By imposing null first vari-ation of the Lagrangian with respect to the state variable it turns out thatthe adjoint variable satisfies the same equation as the state variable (theproblem is self-adjoint). To find the condition of stationarity with respect

76

to the design variables, we compute the following derivatives:

∂L∂zh

(ξ) =

ˆΩ

[−∂E

∗

∂zh(x)∇su : ∇su + Λ ρ(mh(x))− λh(x) + µ(x)

]ξ dx

∂L∂mh

(ξ) =

ˆΩ

[− ∂E

∗

∂mh(x)∇su : ∇su + Λ zh(x)

ρA − ρB

2+ γ+

h (x)− γ−h (x)

]ξ dx

∂L∂ϑh

(ξ) =

ˆΩ

[−∂E

∗

∂ϑh(x)∇su : ∇su

]ξ dx ,

(5.29)

where:

∂E∗

∂zh(x) = p zh(x)p−1Q(ϑh(x))E∗mh(x)

∂E∗

∂mh(x) = zh(x)pQ(ϑh(x))

∂E∗m∂m

(mh(x))

∂E∗

∂ϑh(x) = zh(x)pQ′(ϑh(x))E∗mh(x) .

(5.30)

In (5.30) the 8-th order tensor Q′(ϑ) denotes the element-wise derivative

of the rotation tensor Q (see (3.24)). How to evaluate the derivative ∂E∗m∂m

will be treated in Section 5.4.

By imposing null first variation of the Lagrangian (5.28) with respect toevery design variables we get the following set of conditions:

∂E∗

∂zh(x)∇su : ∇su = Λ ρ(mh(x))− λh(x) + µ(x) ∀x, h

∂E∗

∂mh(x)∇su : ∇su = Λ zh(x)

ρA − ρB

2+ γ+

h (x)− γ−h (x) ∀x, h

∂E∗

∂ϑh(x)∇su : ∇su = 0 ∀x, h

λh(x) ≥ 0, zmin ≤ zh(x), λh(x)(zmin − zh(x)) = 0 ∀x, h

µ(x) ≥ 0,N∑h=1

zh(x) ≤ 1, µ(x)

(N∑h=1

zh(x)− 1

)= 0 ∀x

γ+h (x) ≥ 0, mh(x) ≤ mh, γ+

h (x)(mh(x)−mh) = 0 ∀x, h

γ−h (x) ≥ 0, mh ≤ mh(x), γ−h (x)(mh −mh(x)) = 0 ∀x, h

N∑h=1

ˆΩzh(x)ρ(mh(x)) dx = M .

(5.31)

Exploiting the positivity of the Lagrange multipliers, it is possible to

77

rewrite (5.31) in an equivalent way:

∂E∗∂zh

(x)∇su : ∇su

Λ ρ(mh(x)) + µ(x)

= 1 if zh(x) > zmin

≤ 1 if zh(x) = zmin∀x, h

∂E∗∂mh

(x)∇su : ∇su

Λ zh(x)ρA−ρB

2

≥ 1 if mh(x) = mh

= 1 if mh < mh(x) < mh

≤ 1 if mh(x) = mh

∀x, h

∂E∗

∂ϑh(x)∇su : ∇su = 0 ∀x, h

µ(x) ≥ 0,

N∑h=1

zh(x) ≤ 1, µ(x)

(N∑h=1

zh(x)− 1

)= 0 ∀x

N∑h=1

ˆΩzh(x)ρ(mh(x)) dx = M .

(5.32)Observe that the role of the variables zh and mh in (5.32) differs a

lot from that of ϑh. Indeed the variables in the first set are constrainedones, and are coupled each other being involved in the mass constraints aswell as in the constraint (5.12). On the other hand, the variables ϑh areunconstrained, and are uncoupled from the others. Therefore the variableszh and mh need to be updated simultaneously, while the update of ϑh can becarried out on its own. Moreover, the features of the first set of variables fitwell with the optimality conditions method, while the unconstrained natureof ϑh suggests the use of gradient-descent methods.

5.2.2 Optimization of zh and mh

On the base of the optimality conditions (5.32), we generalize the fixed-point scheme (5.27) to the following one:

zk+1h (x) =

maxzkh(x)− ζz, zmin if (Bk

h(x))ηzkh(x) ≤ maxzkh(x)− ζz, zminzkh(x) + ζz if (Bk

h(x))ηzkh(x) ≥ zkh(x) + ζz

(Bkh(x))ηzkh(x) else

mk+1h (x) =

maxmkh(x)− ζm,mh

if mh − δ + (Dkh(x))η(mk

h(x)−mh + δ) ≤ maxmkh(x)− ζm,mh

minmkh(x) + ζm,mh

if mh − δ + (Dkh(x))η(mk

h(x)−mh + δ) ≥ minmkh(x) + ζm,mh

mh − δ + (Dkh(x))η(mk

h(x)−mh + δ)

else,

(5.33)

78

where

Bkh(x) =

∂E∗∂zh

(x)∇suk : ∇suk + ε

Λk ρ(mkh(x)) + µk(x) + ε

Dkh(x) =

∂E∗∂mh

(x)∇suk : ∇suk + ε

Λk zkh(x)ρA−ρB

2 + ε,

(5.34)

and the Lagrange multipliers Λk and µk are chosen in such a way that itholds true:

µk(x) ≥ 0,N∑h=1

zk+1h (x) ≤ 1, µk(x)

(N∑h=1

zk+1h (x)− 1

)= 0 ∀x

(5.35a)

Mk+1 =N∑h=1

ˆΩzk+1h (x)ρ(mk+1

h (x)) dx = M . (5.35b)

The parameters ζz and ζm are move limits, and η is a tuning parameter.The purpose of the parameter δ, which is a small positive number, is toprevent that the design variable mh gets stuck to its minimal value mh.

The value of the Lagrange multipliers Λk and µk can be determinedby means of bisection loops. Indeed the scheme (5.33) features the samemonotonicity properties as the scheme (5.18) (see Remark 13). As a mat-ter of fact, the introduction of the variables mh does not undermine themonotonicity of the mass with respect to the multiplier Λ: an increase ofΛ translates into a decrease of the variable mh, and consequentially in areduction of the total mass. The pseudo-code for the update of (zh)h and(mh)h is shown in Algorithm 2.

5.2.3 Optimization of the micro-structure orientation

As far as the update of the unconstrained variables (ϑh)h is concerned,we employ a gradient descent method. Denoting by (zkh,m

kh, ϑ

kh)h=1,...,N the

design at the iteration step k, we consider the problem of minimizing theobjective function with respect to the variable Θ := (ϑ1, . . . , ϑN ). Therefore,we introduce the objective functional

f(Θ) = l(u) ,

where u is the state variable associated with the design (zkh,mkh, ϑh)h=1,...,N .

We consider the following update scheme:

Θk+1 = Θk + αkpk ,

79

Algorithm 2 Update of (zh)h and (mh)h

Solve equilibrium (FEM) . see equation (4.14)Compute sensitivities . see equation (5.29)Initial guess for Λloop

for h← 1, N doUpdate mh(x) . see equation (5.33)

end forfor all x ∈ Ω do

Initial guess for µ(x)loop

for h← 1, N doUpdate zh(x) . see equation (5.33)

end forZ(x)←

∑Nh=1 zh(x)

if (Z(x) < 1 AND µ(x) = 0) OR |Z(x)− 1| < εtol thenbreak loop

else if Z(x) < 1 thenIncrease µ(x)

else if Z(x) > 1 thenDecrease µ(x) (not below 0)

end ifend loop

end forUpdate M . see equation (4.13)if |M −M | < εtol then

break loopelse if M > M then

Increase Λelse if M < M then

Decrease Λend if

end loop

80

where the search direction pk is the negative of the gradient computed atthe current point:

pk = −∇Θf(Θk) .

By components, the gradient is given by

∇Θf = (∇ϑ1f, . . . ,∇ϑN f) ,

where, using (5.29), the L2 Riesz representative of the gradient of the ob-jective functional with respect to each variable ϑh can be computed as

[∇ϑhf(Θ)] (x) = −∂E∗

∂ϑh(x)∇su(x) : ∇su(x)

= −zkh(x)pQ′(ϑh(x))E∗mkh(x)

∇su(x) : ∇su(x) .(5.36)

As to the choice of the step length αk, the optimal choice would be theminimizer over α > 0 of the univariate function ϕ defined as

ϕ(α) = f(Θk + αpk) . (5.37)

However the computation of the global minimizer, even with a moderateprecision, is too expensive, since it requires a large number of resolutionsof the state equation and possibly of computations of the gradient (5.36).Therefore the accuracy in the computation of αk is traded off for the effi-ciency of the research, performing an inexact line search.

Typically, inexact line search algorithms compute a sequence of can-didate values and stop when certain conditions are satisfied. A popularstopping criterion is given by the strong Wolfe conditions [34]:

f(Θk + αkpk) ≤ f(Θk) + c1αk(∇Θf(Θk), pk

)(5.38a)∣∣∣(∇Θf(Θk + αkpk), pk

)∣∣∣ ≤ c2

∣∣∣(∇Θf(Θk), pk)∣∣∣ , (5.38b)

with 0 < c1 < c2 < 1, where (·, ·) denotes the L2(Ω;RN ) scalar product.Typical values for c1 and c2 are 10−4 and 0.9 respectively.

In order to give a graphical interpretation to the conditions (5.38a)-(5.38b), observe that the derivative of ϕ can be computed as:

ϕ′(α) =(∇Θf(Θk + αpk), pk

)= −

N∑h=1

(∇ϑhf(Θk + αpk),∇ϑhf(Θk)

).

(5.39)

Therefore we can reformulate the strong Wolfe conditions as follows:

ϕ(αk) ≤ ϕ(0) + c1αkϕ′(0) (5.40a)∣∣ϕ′(α)

∣∣ ≤ c2

∣∣ϕ′(0)∣∣ , (5.40b)

81

The condition (5.38a) is illustrated in Figure 5.1a. For small positive α,the graph of ϕ(α) lies below the graph of the right-hand side of (5.38a), sincec1 < 1. Therefore the first Wolfe condition, known as sufficient decreasecondition, requires that the step αkpk yields a decrease of the objectivefunction not too low compared with that of the first order approximationϕ(0) + αϕ′(0).

The sufficient decrease condition does not guarantee that the iterationproduces a reasonable progress, since, as it is shown in Figure 5.1a, it issatisfied for any sufficiently small values of α. Therefore condition (5.38a) issupplemented with condition (5.38b), which states that the slope of ϕ nearthe new point is sufficiently small compared with the slope at α = 0. Thisrequirement rules out the regions which are too far form stationary points(see Figure 5.1b).

In [34] a search procedure to find in a rapid way a value of α satisfyingthe strong Wolfe conditions is reported. The algorithm is made up of twophases. In the first stage, reported in Algorithm 3, an initial guess of α israised until it reaches an acceptable step length or an interval that bracketsthe desired value. In this case, a second function (Algorithm 4) is invokedto decrease the length of the interval until a value of α satisfying bothconditions is detected.

Clearly the algorithm is run only for the indexes h associated withanisotropic micro-structures (which is the case of lamellae). As a matterof fact, the orientation of spots micro-structures does not account as a de-sign variable, since it does not affect the state variable u.

5.3 Implementation of length-scale control

Starting from the algorithm described is sections 5.2.2 and 5.2.3, it ispossible to derive an algorithm for the filtered version of the topology opti-mization problem 4.19.

Suppose to know the differential of a given quantity Ψ with respect to ageneric design variable ϕ, and denote by ∇ϕΨ its L2 Riesz representative:

∂Ψ

∂ϕ(ξ) =

ˆΩ∇ϕΨ(x) ξ(x) dx .

Then the differential of Ψ with respect to the filtered design variable

ϕ(x) =(K ∗ ϕ)(x)

(K ∗ 1)(x)

82

= (Θ + )

0 + ′ 0

0 + ′ 0

(a)

= (Θ + )

0 + ′ 0

0 + ′ 0

(b)

= (Θ + )

(c)

Figure 5.1: Graphical representation of the strong Wolfe conditions. Theadmissible regions are highlighted with a coloured background. (a) Suffi-cient decrease condition (5.38a) (b) Curvature condition (5.38b) (c) Bothconditions (5.38a)-(5.38b)

.

83

Algorithm 3 Line search algorithm

α0 ← 0α1 ← α (initial guess)i← 1loop

Evaluate ϕ(αi) . see equation (5.37)if ϕ(αi) > ϕ(0) + c1αiϕ

′(0) OR [ ϕ(αi) ≥ ϕ(αi−1) AND i > 1] thenreturn zoom(αi−1,αi)

end ifEvaluate ϕ′(αi) . see equation (5.39)if |ϕ′(αi)| ≤ −c2ϕ

′(0) thenreturn αi

end ifif ϕ′(αi) ≥ 0 then

return zoom(αi,αi−1)end ifαi+1 ← 2 ∗ αii← i+ 1

end loop

Algorithm 4 zoom function

function zoom(αlo, αhi)j ← 1loop

αj ← (αlo + αhi)/2Evaluate ϕ(αj) . see equation (5.37)if ϕ(αj) > ϕ(0) + c1αjϕ

′(0) OR ϕ(αj) ≥ ϕ(αlo) thenαhi ← αj

elseEvaluate ϕ′(αj) . see equation (5.39)if |ϕ′(αj)| ≤ −c2ϕ

′(0) thenreturn αj

end ifif ϕ′(αj)(αhi − αlo) ≥ 0 then

αhi ← αloend ifαlo ← αj

end ifend loop

end function

84

can be computed as:

∂Ψ

∂ϕ(ξ) =

∂Ψ

∂ϕ

(∂ϕ

∂ϕ(ξ)

)=

ˆΩ∇ϕΨ(x)

´ΩK(x− y)ξ(y) dy´

ΩK(x− y) dydx

=

ˆΩ

ˆΩ

K(x− y)∇ϕΨ(x)´ΩK(x−w) dw

dxξ(y) dy

, (5.41)

which entails:

∇ϕΨ(x) =

ˆΩ

K(y − x)∇ϕΨ(y)´ΩK(y −w) dw

dy . (5.42)

Therefore, repeating the same arguments used in Section 5.2.1 and Sec-tion 5.2.2, we recover a scheme analogous to (5.33), where the definitions ofBk and Dk are replaced by

Bkh(x) =

´ΩK(y − x)∂E

∗

∂zh(y)∇suk(y) : ∇suk(y)

(´ΩK(y −w) dw

)−1dy + ε

Λk´

ΩK(y − x)ρ(mkh(y))

(´ΩK(y −w) dw

)−1dy + µk(x) + ε

Dkh(x) =

´ΩK(y − x) ∂E

∗

∂mh(y)∇suk(y) : ∇suk(y)

(´ΩK(y −w) dw

)−1dy + ε

Λk ρA−ρB

2

´ΩK(y − x)zkh(y)

(´ΩK(y −w) dw

)−1dy + ε

,

(5.43)

keeping in mind that the constraints (4.20) are evaluated on the non-filteredvariables. The Lagrange multiplier Λk is chosen in such a way that themass constraint is satisfied when the mass is computed with respect to thefiltered variables (see (4.18)). Notice that equations (5.43) are similar totheir non-filtered counterparts (5.34), with the difference that the update ineach point is determined not only by the sensitivity computed in that point,but by its neighbourhood. Nevertheless, the convolution operator does notinvolve the multiplier µ(x), and thus it is possible to apply Algorithm 2 evenfor the filtered version of the problem, provided that (5.34) is replaced by(5.43).

5.3.1 Filtering of ϑh

As far as the update of the variables ϑh is concerned, applying the chainrule (5.41) to compute the gradient of the objective functional with respectto ϑh, thanks to equation (5.36) it turns out that:[

∇ϑhf(Θ)]

(x) = −ˆ

Ω

K(y − x)∂E∗

∂ϑh(y)∇suk(y) : ∇suk(y)´

ΩK(y −w) dwdy . (5.44)

Also in this case all the quantities involved in the expression (5.44) areknown, therefore it is possible to apply the line search method described inSection 5.2.3.

85

However, the filtering of the variables ϑh may lead to undesired effects.Indeed the filtering operator (4.16) carries out a weighted mean, but thisoperation is not suited for circular quantities. Think for instance to thearithmetic mean of the two angles ϑ1 = 5 and ϑ2 = 355: their arithmeticmean is ϑ = (ϑ1 + ϑ2)/2 = 180, but if we write ϑ2 as −5, the arithmeticmean becomes 0 (which is much closer to our intuition). The reason is that,even if in two dimensions they can be identified with the interval [0, 2π), thespace of rotations is not a vectorial space, then the usual mean operator isnot defined.

As a matter of fact, the space of two dimensional rotations topologicallyspeaking is a circumference, and it is possible to identify each rotation withthe complex number eiϑ. The usual arithmetic mean of the angles ϑ1 andϑ2 is equivalent to a geometric mean in the complex plane:

eiϑ =√eiϑ1eiϑ1 .

However, in the complex plane the square root has two branches, whencethe two different results 180 and 0. Things get even worse if we computethe mean of n angles, since we would get n different solutions. This remarksuggests to perform the arithmetic mean in the complex plane instead of inthe interval [0, 2π). The result typically lies inside the unitary circumference,so we take the argument of the result as “average” angle. Of course, whenthe arithmetic mean gives as result the origin of the complex plane, theargument is not defined. In this case one should choose arbitrarily theresult of the operation (for instance 0).

In other words, given te angles (ϑj)j=1,...,n we define the circular meanoperator:

meancirc(ϑ1, . . . , ϑn) = atan2

1

n

n∑j=1

sin(ϑj),1

n

n∑j=1

cos(ϑj)

, (5.45)

where atan2 denotes the variant of the arctangent function which accountsfor the positioning of the angle in the appropriate quadrant:

atan2(y, x) =

arctan( yx) if x > 0

arctan( yx) + π if x < 0 and y ≥ 0

arctan( yx)− π if x < 0 and y < 0

+π2 if x = 0 and y > 0

−π2 if x = 0 and y < 0

0 if x = 0 and y = 0 .

The operator (5.45) is a generalization of the arithmetic mean operatorfor 2π-periodic functions. In the case of orthotropic materials, the orienta-tion ϑ = π can be identified with the orientation ϑ = 0. In other words, the

86

variable ϑh associated with orthotropic micro-structures, such as lamellae, isπ-periodic. To obtain a mean operator suited for functions of generic periodT , it is possible to map the interval [0, T ] into [0, 2π], apply the operator(5.45) and finally the inverse map from [0, 2π] to [0, T ]:

meancirc,T(ϑ1, . . . , ϑn) =T

2πatan2

1

n

n∑j=1

sin(2π

Tϑj),

1

n

n∑j=1

cos(2π

Tϑj)

,

(5.46)

To sum up, in the case of the variables ϑh, the filtering operator (4.16)can be replaced by

ϑh(x) =Th2π

atan2

((K ∗ sin(2πϑh

Th))(x)

(K ∗ 1)(x),(K ∗ cos(2πϑh

Th))(x)

(K ∗ 1)(x)

)

=Th2π

atan2

((K ∗ sin

(2πϑhTh

))(x), (K ∗ cos

(2πϑhTh

))(x)

).

(5.47)

The gradient of the objective functional with respect to the design vari-able ϑh is found, as usual, by means of the chain rule:

∂f

∂ϑh(ξ)

=∂f

∂ϑh

(∂ϑh∂ϑh

(ξ)

)

=

ˆΩ

∇ϑhfTh2π1 +

(K∗sin(

2πϑhTh

)

K∗cos(2πϑhTh

)

)2(K ∗ cos(2πϑh

Th))2

[(K ∗ cos(

2πϑhTh

)

)(K ∗

(cos(

2πϑhTh

)2π

Thξ

))+

(K ∗ sin(

2πϑhTh

)

)(K ∗

(sin(

2πϑhTh

)2π

Thξ

))]dx

=

ˆΩ

ˆΩ

∇ϑhf(x)K((x)− (y))(K ∗ sin(2πϑh

Th(x))

)2+(K ∗ cos(2πϑh

Th(x))

)2

[(K ∗ cos(

2πϑhTh

))(x) cos(2πϑhTh

(y))

+(K ∗ sin(2πϑhTh

))(x) sin(2πϑhTh

(y))

]ξ(y) dy dx

87

=

ˆΩ

[cos(

2πϑhTh

)K ∗

(K ∗ cos(2πϑh

Th))∇ϑhf(

K ∗ sin(2πϑhTh

))2

+(K ∗ cos(2πϑh

Th))2

+ sin(2πϑhTh

)K ∗

(K ∗ sin

(2πϑhTh

))∇ϑhf(

K ∗ sin(

2πϑhTh

))2+(K ∗ cos

(2πϑhTh

))2

]ξ(y) dy ,

where K(x) = K(−x). Note that in most cases (like in (4.16)), the convolu-tion kernel satisfies K(−x) = K(x). Finally, the gradient of f with respectto ϑh reads:

∇ϑhf = − cos

(2πϑhTh

)K ∗

(K ∗ cos

(2πϑhTh

))∂E∗∂ϑh∇suk : ∇suk(

K ∗ sin(

2πϑhTh

))2+(K ∗ cos

(2πϑhTh

))2

− sin

(2πϑhTh

)K ∗

(K ∗ sin

(2πϑhTh

))∂E∗∂ϑh∇suk : ∇suk(

K ∗ sin(

2πϑhTh

))2+(K ∗ cos

(2πϑhTh

))2 .

(5.48)

Equation (5.48) provides an explicit formula to compute the gradient ofthe objective functional with respect to the design variables. Therefore, alsoin the case the filter (5.47) is employed, it is possible to apply the line searchalgorithm of Section 5.2.3.

Remark 14. The experience shows that the filtering of the variables ϑh canbe avoided in most cases. As a matter of fact, even if the length-scale ofthe variables describing the orientation of the micro-structure is not keptunder control, the solution does not exhibit the formation af finer and finerscales when the mesh is refined (unlike what happens for the variables mh

and zh). The reason is that designs with an orientation that varies rapidlydo not report the same structural advantages that designs with a rapidlyvarying density, since in an optimal design the orientation of the micro-structure tends to align with the principal directions of stress.

5.4 Evaluation of the homogenized tensor and itsderivatives

A still open point is the effective evaluation of the homogenized tensorE∗m and of its derivative ∂E∗m

∂m . In Chapter 3 the numerical computation ofE∗m has been addressed (see (3.25)-(3.26)). In this section a way to computeanalytically the sensitivity of the homogenized tensor with respect to thecontrol variable m is exposed, and then a more efficient method is presented.

88

5.4.1 Exact derivative of the homogenized tensor

Note that inside the intervals (mh,mh) the periodicity cell Ym does notvary. Therefore, denoting by

Ψ(m,wijm,w

klm) =

ˆYm

Em(y)(eij +∇swij

m

):(ekl +∇swkl

m

)dy ,

it holds:

∂(E∗m)ijkl∂m

=∂

∂m

[1

|Ym|

ˆYm

Em(y)(eij +∇swij

m

):(ekl +∇swkl

m

)dy

]=

1

|Ym|

ˆYm

∂Em∂m

(eij +∇swij

m

):(ekl +∇swkl

m

)dy

+1

|Ym|∂Ψ

∂wijm

[∂wij

m

∂m

]+

1

|Ym|∂Ψ

∂wklm

[∂wkl

m

∂m

].

(5.49)

The derivative of Ψ with respect to the cell function wijm reads

∂Ψ

∂wijm

[v] =

ˆYm

Em(y)∇sv :(ekl +∇swkl

m

)dy ,

which is null thanks to the cell problem itself (see (3.27)). Since the sameholds for the derivative with respect to wkl

m, equation (5.49) reduces to

∂(E∗m)ijkl∂m

=1

|Ym|

ˆYm

∂Em∂m

(eij +∇swij

m

):(ekl +∇swkl

m

)dy .

By (3.25), the derivative of Em(y) can be computed as follows:

∂Em∂m

(y) = 2∂µm∂m

(y) I4 +∂λm∂m

(y) I2 ⊗ I2

=

[2µA − µB

2I4 +

λA + λB

2I2 ⊗ I2

]∂ϕm∂m

(y) .

We are left to compute ∂ϕm∂m (y). We know that any minimum of the Ohta-

Kawasaki functional satisfies the stationary Cahn-Hilliard-Oono equation:

g(ϕ,m) = ∆(−δ2∆ϕ+ F ′(ϕ)

)− τ(ϕ−m) = 0 .

Therefore the derivative of ϕ with respect to m can be found by solvingthe following system:

∂g

∂ϕ

[∂ϕ

∂m

]+∂g

∂m= 0 . (5.50)

89

5.4.2 Evaluation by interpolation

The procedure to compute the sensitivity of the homogenized tensor withrespect to m described in the previous section is very expansive form thenumerical viewpoint. A much simpler way to estimate this quantities aswell as the tensor itself is by interpolation of a set of pre-computed cases,since in Section 3.3 it has been shown that inside the intervals (mh,mh) theentries of E∗m are smooth in m (see Figures 3.1 and 3.2).

In the cases we considered, a three-points polynomial interpolation hasbeen enough to cover with sufficient precision the whole intervals. Theresults of the interpolation on the cases examined in Section 3.3 are shownin Figure 5.2.

Writing the entries of the homogenized tensor E∗m as polynomials in m,the computation of their derivatives with respect to the same variable m isstraightforward.

90

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

m

0

100

200

300

400

500

600

700

800

900

1000

Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(a) EA/EB = 10

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

m

-100

0

100

200

300

400

500

600

700

800

900

Exxxx

Exxyy

Exxxy

Eyyyy

Eyyxy

Exyxy

(b) EA/EB = 1000

Figure 5.2: Entries of the homogenized stiffness tensor for different ratiosEA/EB. Circles represent the values computed numerically, while lines arethe polynomial interpolators, computed in the extreme end middle points ofeach interval.

91

92

Chapter 6

Numerical implementation

In this chapter the numerical implementation of the algorithms describedin Chapter 5 is addressed. In the first section a discrete version of thetopology optimization problem is presented, and the discrete counterpart ofthe optimization algorithm devised for the continuous case is derived. Thensome details about the FreeFem++ implementations are given.

6.1 Finite Elements discretization

To solve numerically problem (4.14) or (4.19), a discretization strategyshould be addressed. A first way to find the discrete version of the algo-rithm described in Chapter 5 is the optimize-then-discretize approach, thatis to take the update schemes in continuous form and to discretize them.However, the solution of an optimal design problem is better achieved usingthe discretize-then-optimize approach, which consists in deriving an opti-mization algorithm starting from a discretization of the problem (see [37, p.533]). In this section the latter approach is presented.

6.1.1 Discrete problem formulation

In this section a finite elements discretization of the topology optimiza-tion problem (4.14) is introduced. The state variable u is discretized throughcontinuous finite elements (P1 for instance), while for the design variableszh, mh and ϑh piecewise constant elements (P0) are employed.

We consider a triangulation Th = ell=1,...,Ne in Ne elements of thereference domain Ω. We introduce a finite elements subspace of the spaceH1

ΓD(Ω;Rd), which we denote by Vh, defined over the mesh Th, and we denote

by ϕjj=1,...,NV a basis of the space Vh. The finite elements discretizationof the state variable

u(x) =

NV∑j=1

ujϕj(x) ,

93

is associated to the vector

U = (u1, . . . , uNV ) ∈ RNV .

On the other hand, the design variables are discretized by the piecewiseconstant functions:

zh(x) =

Ne∑l=1

zh,l1el(x) h = 1, . . . , N

mh(x) =

Ne∑l=1

mh,l1el(x) h = 1, . . . , N

ϑh(x) =

Ne∑l=1

ϑh,l1el(x) h = 1, . . . , N.

To derive the Galerkin formulation of the state equation we write thebilinear form a∗(·, ·) applied to two finite elements functions:

a∗(u,v) =

NV∑i,j=1

uivj

ˆΩE∗(x)∇sϕi : ∇sϕj dx

=

NV∑i,j=1

uivj

Ne∑l=1

ˆel

E∗l∇sϕi : ∇sϕj dx ,

where E∗l denotes the stiffness tensor evaluated in the element el (note thatit is constant over each element of the mesh):

E∗l =N∑h=1

zph,lQ(ϑh,l)E∗mh,l

Then, introducing the stiffness matrix K ∈ RNV ×NV defined as

Kij =

Ne∑l=1

ˆel

E∗l∇sϕi : ∇sϕj dx ,

and the right-hand side f ∈ RNV , defined as

fi = l(ϕi) = 〈b,ϕi〉H−1,H1ΓD

+ 〈t,ϕi〉H−1/2,H1/2 ,

the Galerkin approximation of the state equation simply reads

KU = f .

94

The total mass associated with the design (zh,l,mh,l, ϑh,l)h,l is given bythe formula:

M =N∑h=1

ˆΩzh(x)ρ(mh(x)) dx =

N∑h=1

Ne∑l=1

|el|zh,lρ(mh,l) .

To sum up, problem (4.14) can be formulated in discrete version asfollows:

minimizezh,l,mh,l,ϑh,l

fTU

subject to: KU = f

M = M

zmin ≤ zh,l ≤ 1 ∀ l = 1, . . . , Ne,∀h = 1, . . . , N∑Nh=1 zh,l ≤ 1 ∀ l = 1, . . . , Ne

mh ≤ mh,l ≤ mh ∀ l = 1, . . . , Ne ∀h = 1, . . . , N

where: Kij =∑Ne

l=1

´el

∑Nh=1 z

ph,lQ(ϑh,l)E∗mh,l∇

sϕi : ∇sϕj dx

M =∑N

h=1

∑Nel=1 |el|zh,lρ(mh,l) .

(6.1)

6.1.2 Discrete optimization algorithm

To derive the discrete counterpart of the algorithm described in Chapter5, we write the Lagrangian function associated to the problem (6.1):

L = fTU− UT

(KU− f)) + Λ

(N∑h=1

Ne∑l=1

|el|zh,lρ(mh,l)−M

)

+N∑h=1

Ne∑l=1

λh,l (zmin − zh,l) +

Ne∑l=1

µl

(N∑h=1

zh,l − 1

)

+N∑h=1

Ne∑l=1

[γ+h,l (mh,l −mh) + γ−h,l (mh −mh,l)

].

(6.2)

Thanks to the symmetry of the matrix K, also the discrete problemturns out to be self-adjoint. By deriving the Lagrangian with respect toeach design variable, following the same argument as in the continuous case,

95

we get the following set of optimality conditions:

UT ∂K∂zh,l

U

Λ|el|ρ(mh,l) + µl

= 1 if zh,l > zmin

≤ 1 if zh,l = zmin∀ l, h

UT ∂K∂mh,l

U

Λ|el|zh,l ρA−ρB

2

≥ 1 if mh,l = mh

= 1 if mh < mh,l < mh

≤ 1 if mh,l = mh

∀ l, h

UT ∂K

∂ϑh,lU = 0 ∀ l, h

µl ≥ 0,

N∑h=1

zh,l ≤ 1, µl

(N∑h=1

zh,l − 1

)= 0 ∀ l

N∑h=1

Ne∑l=1

|el|zh,lρ(mh,l) = M ,

(6.3)

where (∂K

∂zh,l

)i,j

=

ˆel

p zp−1h,l Q(ϑh,l)E∗mh,l∇

sϕi : ∇sϕj dx(∂K

∂mh,l

)i,j

=

ˆel

zph,lQ(ϑh,l)∂E∗m∂m

(mh,l)∇sϕi : ∇sϕj dx(∂K

∂ϑh,l

)i,j

=

ˆel

zph,lQ′(ϑh,l)E∗mh,l∇

sϕi : ∇sϕj dx .

(6.4)

Note that if the state variable is discretized through P1 elements, thequantity ∇sϕi is constant over each element el, therefore the integrals in(6.4) can be evaluated point-wise and then multiplied by the element area|el|.

Finally, arguing as in the continuous case, we devise the following fixed-point scheme:

zk+1h,l =

maxzkh,l − ζz, zmin if (Bk

h,l)ηzkh,l ≤ maxzkh,l − ζz, zmin

zkh,l + ζz if (Bkh,l)

ηzkh,l ≥ zkh,l + ζz

(Bkh,l)

ηzkh,l else

mk+1h,l =

maxmkh,l − ζm,mh

if mh − δ + (Dkh,l)

η(mkh,l −mh + δ) ≤ maxmk

h,l − ζm,mhminmk

h,l + ζm,mhif mh − δ + (Dk

h,l)η(mk

h,l −mh + δ) ≥ minmkh,l + ζm,mh

mh − δ + (Dkh,l)

η(mkh,l −mh + δ)

else,

(6.5)

96

where

Bkh,l =

(Uk)T ∂K

∂zh,lUk + ε

Λk|el|ρ(mkh,l) + µkl + ε

Dkh,l =

(Uk)T ∂K

∂mh,lUk + ε

Λk|el|zkh,lρA−ρB

2 + ε,

(6.6)

with Uk denoting the state variable at the iteration k, and the Lagrangemultipliers Λk and µkl chosen in such a way that it holds true:

µkl ≥ 0,

N∑h=1

zk+1h,l ≤ 1, µkl

(N∑h=1

zk+1h,l − 1

)= 0 ∀ l (6.7a)

Mk+1 =N∑h=1

ˆΩzk+1h,l ρ(mk+1

h,l ) dx = M . (6.7b)

To update the variables ϑh,l the line-search algorithm described in section5.2.3 can be applied, knowing that the gradient of the objective functionalis given by

∂f

∂ϑh,l= −UT ∂K

∂ϑh,lU . (6.8)

6.1.3 Discrete length-scale control

Consider a generic discretized design variable ψ, defined as

ψ(x) =

Ne∑l=1

ψl1el(x) .

We are interested in finding the discretization of its filtered counterpart

ψ(x) =(K ∗ ψ)(x)

(K ∗ 1)(x),

that is to say an Ne-dimensional vector ψ = (ψ1, . . . , ψNe) such that:

ψ(x) 'Ne∑l=1

ψl1el(x) .

A practical way is to compute its L2 projection on the finite elements

97

P0 subspace, through the formula:

ψl =

(ψ,1el

)L2

‖1el‖L2

=1

|el|

ˆel

ψ(x) dx

=1

|el|

ˆel

´ΩK(x− y)ψ(y) dy´

ΩK(x− y) dydx

=

Ne∑r=1

ψr1

|el|

ˆel

´erK(x− y) dy´

ΩK(x− y) dydx .

Therefore, defining the filtering matrix H ∈ RNe×Ne by

Hlr =1

|el|

ˆel

´erK(x− y) dy´

ΩK(x− y) dydx . (6.9)

we have that the discrete counterpart of the filtering operator can be definedby the following matrix-vector multiplication:

ψl =

Ne∑r=1

Hlrψr . (6.10)

Remark 15. Observe that denoting by xl the barycenter of the element el,and approximating x ' xl over el, we get a simpler formula to compute thematrix H:

Hlr =1

|el|

ˆel

´erK(x− y) dy∑Ne

s=1

´esK(x− y) dy

dx ' K(xl − xr)|er|∑Nes=1K(xl − xs)|es|

dx . (6.11)

It is easy to check that, with the approximation (6.11), the filtering op-erator (6.10) coincides with what it is known in literature as density filtering(see [40] for instance).

To filter the variables ϑh,l, we consider the discrete counterpart of theoperator (5.47):

ˆϑh,l =Th2π

atan2

(Ne∑r=1

Hlr sin

(2πϑh,rTh

),

Ne∑r=1

Hlr cos

(2πϑh,rTh

))(6.12)

Finally we introduce the following filtered version of the discretized

98

topology optimization problem:

minimizezh,l,mh,l,ϑh,l

fTU

subject to: KU = f

M = M

zmin ≤ zh,l ≤ 1 ∀ l = 1, . . . , Ne, ∀h = 1, . . . , N∑Nh=1 zh,l ≤ 1 ∀ l = 1, . . . , Ne

mh ≤ mh,l ≤ mh ∀ l = 1, . . . , Ne ∀h = 1, . . . , N

where: Kij =∑Ne

l=1

´el

∑Nh=1 z

ph,lQ(ϑh,l)E∗mh,l∇

sϕi : ∇sϕj dx

M =∑N

h=1

∑Nel=1 |el|zh,lρ(mh,l) .

(6.13)To derive an algorithm for the solution of the problem (6.13) starting

from the scheme proposed in Section 6.1.2, it is possible to exploit the chainrule, like in the continuous case. Omitting the passages, we report the finalresult. The update of the variables zh,l and mh,l is performed by the samefixed-point scheme (6.5), provided that the definitions of the quantities Bk

h,l

and Dkh,l are replaced by:

Bkh,l =

(Uk)T (∑Ne

r=1 Hrl∂K∂zh,r

)Uk + ε

Λk∑Ne

r=1 Hrl|er|ρ(mkh,r) + µkl + ε

Dkh,l =

(Uk)T (∑Ne

r=1 Hrl∂K∂mh,r

)Uk + ε

Λk ρA−ρB

2

∑Ner=1 Hrl|er|zkh,r + ε

.

(6.14)

On the other hand, the gradient of the objective functional with re-spect to the design variables ϑh,l can be computed as follows (compare with(5.48)):

∂f

∂ϑh,l

= − cos

(2πϑh,lTh

) Ne∑s=1

Hsl

(∑r Hsr cos

(2πϑh,rTh

))UT ∂K

∂ϑh,sU(∑

r Hsr sin(

2πϑh,rTh

))2+(∑

r Hsr cos(

2πϑh,rTh

))2

− sin

(2πϑh,lTh

) Ne∑s=1

Hsl

(∑r Hsr sin

(2πϑh,rTh

))UT ∂K

∂ϑh,sU(∑

r Hsr sin(

2πϑh,rTh

))2+(∑

r Hsr cos(

2πϑh,rTh

))2

(6.15)

6.2 Implementation details

The numerical results reported in Chapter 7 are obtained through aFreeFem++ implementation of the algorithm described in Section 6.1.

99

The state variable is discretized through piecewise linear (P1) finite ele-ments, and the state equation is solved by the UMFPACK sparse solver.

In two dimensions the tensors belonging to the setM4 have six indepen-dent entries. Therefore it is possible to represent the fourth-order tensor Eby a six-elements vector. With this notation the 8th-order tensor Q(ϑ) (see(3.24)) is represented by a six-by-six matrix, which was determined throughthe MATLAB Symbolic Toolbox. The tensor Q′(ϑ) as well was computedby the same toolbox.

6.2.1 Dealing with local minima

It is well known that topology optimization problems are highly non-convex and may feature many local minima, thus small variations in theinitial design may result in drastic changes in the result of the optimizationalgorithm [41, 8]. In the case considered in this work, the problem of localminima is sharpened by the introduction of the variables ϑh. As a matterof fact, the numerical experiments show that the final orientation of theanisotropic micro-structures depends strongly on the initial configuration.In turn, the orientation of the candidate micro-structures influences thechoice of one rather than the other.

According to [41], “Most global optimization methods seem to be unableto handle problems of the size of a typical topology optimization problem”.The most common strategies to deal with the problem of local minima relyon continuation methods. This approach consists in modifying the originalnon-convex problem in a convex one (or, at lest, “more convex” that theoriginal one); after the algorithm has reached convergence, the problem isgradually changed to recover the original formulation. Even if this cannotbe regarded as a global optimization strategy, it takes into account globalinformation, since in the first stage of the procedure a global minimum ofthe convexified version of the problem is reached.

In the FreeFem++ implementation of the algorithm the following strate-gies are employed:

• In a first stage the penalization factor is set to p = 1, and then it isgradually raised to the target value. This is a popular approach toconvexify the problem when a SIMP-like interpolation is employed.

• An effectual method to avoid that the initial guess for ϑh affects ina determinant way the outcome of the optimization algorithm is toenlarge the design space, including each anisotropic micro-structuresmore then once in the list of the N candidate materials, each time witha different initial guess for the orientation (for instance ϑ0

h1= 0, ϑ0

h2=

π/2). Then, in the post-processing stage, the variables associated withthe same micro-structure should be merged into a single one: denoting

100

by (hi)i=1,...,Nh the indexes associated with the same micro-structureh, the overall design variables are given by the following equations:

zh,l =

Nh∑i=1

zhi,l

mh,l =1

zh,l

Nh∑i=1

mhi,lzhi,l

ϑh,l =Th2π

atan2

∑Nhi=1 sin

(2πThϑhi,lzhi,l

)zh,l

,

∑Nhi=1 cos

(2πThϑhi,lzhi,l

)zh,l

.

Note that this approach is analogous to that employed in Section 4.2to avoid that the variables mh get stuck into local minima.

• Numerical experience has shown that a good strategy to avoid theorientation variables ϑh to be trapped into local minima is to employ asinitial design for the density variables zh the optimal solution obtainedwith a single isotropic material. In this way, in the first iterations ofthe algorithm, the orientation of the micro-structure tends to alignwith the principal direction of stresses of the educated guess. In fact,the principal direction of stresses of a uniform design may differ a lotfrom those of the optimal design, which is, on the other hand, likely toresemble (at least at the topological level) to that of a single isotropicmaterial. Anyway, to avoid the initial guess to be too much influentialon the final design, the initial guess should be computed with a higherzmin and a lower penalization parameter (for instance zmin = 0.2 andp = 2).

101

102

Chapter 7

Numerical results

In this chapter we present and discuss some numerical results obtainedwith the FreeFem++ implementation described in Chapter 6. For all thecases that will be presented, the output of the algorithm is shown throughfour types of charts:

• Total density. This chart shows the macroscopic shape of the body,visualizing in a grey-scale colormap the quantity

∑Nh=1 zh. White re-

gions correspond to void, black regions to full material. Sometimes thischart is integrated with some blow-ups revealing the micro-structurespresent in certain points.

• Allocation of micro-structure (m.s.). This chart shows how the dif-ferent kinds of micro-structures (A-spots, lamellae, B-spots) are al-located in the design region. A different colour is assigned to eachmicro-structure h, and it is shown on the chart with transparencythat is inversely proportional to the associated density zh.

• Variable m. In this chart the equivalent of the variable m of the mono-material original formulation (4.9) is shown. In other words, in eachpoint the variable mh corresponding to the micro-structure present inthat point is plotted. In regions of intermediate density, the mean ofthe different mh, weighted by the associated zh, is plotted.

• Variable ϑ. This chart shows the variable ϑh in the regions wherean anisotropic micro-structure is present. Regions where an isotropicmicro-structure is present are marked with grey. Since in orthotropicmaterials the orientation ϑ = π can be identified with ϑ = 0, a circularcolormap which identifies the over-mentioned angles is employed.

103

7.1 Case study traction : convergence history

To evaluate the effectiveness of the algorithm we are interested in theconvergence history of the objective functional. However, when the continu-ation strategies described in Section 6.2.1 are employed, in the first stages ofthe algorithm a different problem is being solved, so a comparison betweenthe initial compliance and the final one would not make sense. Therefore weconsider a simple case study, which does not require the strategies of Section6.2.1: the algorithm in applied directly to the final problem, starting froma uniform initial guess for each design variable. The initial guess for thelamellae configuration is oriented horizontally.

The geometry of the case study is shown in Figure 7.1. A rectangularreference domain is anchored on its left-hand side, and a traction force isapplied in the mid-point of the opposite side. The domain is discretizedthrough a uniform mesh composed of 1600 triangular elements. The prop-erties of the monomers A and B are set to the following values:

EA

EB= 1000,

ρA

ρB= 1000.

Figure 7.2 shows the result of the optimization. Close to the point of ap-plication of the traction load a B-spots region is present, while the remainingpart of the body is occupied by lamellae of near horizontal orientation (onlya small torsion can be observed far from horizontal symmetry axis).

Note that in the final design no A-spots areas can be observed. Thisis a common feature of optimal designs, since in the competition betweenthe two isotropic micro-structures, namely A-spots and B-spots, the latterresult to be more convenient, since they provide the structure with morestiffness the mass being equal.

Figure 7.3 shows the evolution of the compliance during the optimizationhistory. Odd steps correspond to the update of the variables ϑh (see Section5.2.3), while in even steps the variables zh and mh are updated (see Section5.2.2). It is evident that the compliance decreases monotonically throughthe whole convergence history, but the steps associated with the update ofzh and mh are responsible of a much higher improvement than the steps inwhich the orientation in optimized. It should be stressed that in the presentcase the initial guess for the orientation is much closer to the final designthan the initial guesses for the variables zh and mh, therefore there is littleroom, in comparison, for improvements in the orientation. Nevertheless theconvergence of the variables zh and mh is faster than that of ϑh even whenthe latter is set initially to a configuration which is far from the optimal one.This is an advantage of the fixed-point scheme proposed in Chapter 5 withrespect to gradient-descent method.

104

Figure 7.1: Geometry and boundary conditions of the test traction.

(a) Total density (b) Allocation of m.s.

(c) Variable m (d) Variable ϑ

Figure 7.2: Result of the test traction.

105

0 10 20 30 40 50 60 70 80 90 100

# iterations

0

0.02

0.04

0.06

0.08

com

plia

nce

100 101 102

# iterations

10-3

10-2

10-1

com

plia

nce

Figure 7.3: Convergence history of the objective function in natural scale(top) and log-log scale (bottom).

7.2 Case study cantilever : mesh independence

In this section the case study cantilever, depicted in Figure 7.4, is con-sidered. The reference domain is a rectangle of measures 4 × 8, anchoredon its left-hand side. A downward force is acting on the bottom right-handcorner of the triangle. The simulations are performed with the followingsettings:

EA

EB= 10,

ρA

ρB= 10.

To check mesh-independence several simulation are run on uniform meshesof increasing resolution (from 1600 to 10000 elements), keeping the filter ra-dius constant. The variables zh and mh are filtered with a radius equalto rmin = 0.5, while for the variables ϑh a smaller radius is employed(rmin = 0.2). The choice is conceived to let the micro-structure free torotate inside the beams which are produced by the algorithm. Figure 7.5shows that switching from a coarser mesh to a finer one, the topology of thesolution is not affected.

On the other hand, Figure 7.6 shows the solution of the same case studywith same mesh as the last column of Figure 7.5, but with a smaller filterradius (rmin = 0.12 for zh and mh, rmin = 0.1 for ϑh). It is evident thatdecreasing the filter radius the length-scale of the solution is refined as well.

106

Figure 7.4: Geometry and boundary conditions of the test cantilever.

(a) Total density (b) Total density (c) Total density

(d) Allocation of m.s. (e) Allocation of m.s. (f) Allocation of m.s.

(g) Variable m (h) Variable m (i) Variable m

(j) Variable ϑ (k) Variable ϑ (l) Variable ϑ

Figure 7.5: Results of the test cantilever with increasing mesh resolutions.First column: 1600 elements; Second column: 3600 elements; Third column:10000 elements.

107

(a) Total density (b) Allocation of m.s.

(c) Variable m (d) Variable ϑ

Figure 7.6: Result of the test cantilever with fine filter radius.

7.3 Case study bridge: algorithm complexity

In this section the computational complexity of the algorithm is inves-tigated. The considered case study is shown in Figure 7.8a: a rectangulardomain is supported at its lower vertices, and it is loaded in the middlepoint of the lower side. The test is performed on different mesh sizes, usingan Intel Core i5-4570 CPU (3.20 Ghz). Figure 7.9 shows the result of theoptimization for the finer mesh (7200 elements).

The execution times are reported in Figure 7.7. The chart shows thatthe initialization time of the algorithm scales quadratically with the numberof elements Ne. On the other hand, the average execution times of updatestep of the variables ϑh and of the update step of the variables zh and mh

scale linearly with Ne.

Comparing the computational effort of the update of ϑh and of (zh,mh),it is remarkable that the update of the orientation requires roughly tentimes as much. This datum highlights a further advantage of the fixed-pointscheme proposed in Chapter 5 with respect to gradient-descent methods.It would be interesting to look for more efficient update schemes for theorientation of the micro-structure, in order to enhance the overall algorithmefficiency.

108

103 104

# mesh elements

10-2

10-1

100

101

102

103

104

time

[sec

]

initializationstep θstep z/mstep z/m: sensitivities computationstep z/m: bisection looplinear scalingquadratic scaling

Figure 7.7: Execution times for the test bridge with different mesh sizes.

(a) (b)

Figure 7.8: Geometry and boundary conditions of the case study bridge (a)and square (b)

109

(a) Total density (b) Allocation of m.s.

(c) Variable m (d) Variable ϑ

Figure 7.9: Results of test bridge.

7.4 Case study square: effect of the ratio EA/EB

The reference domain of the case study shown in Figure 7.8b is a square,anchored in the mid-points of its vertical sides. A downward force appliedin the middle point of the square acts on the body. The outcome of twosimulations, with different ratio EA/EB are shown side by side in Figure7.10. In both cases a uniform mesh of 7200 elements is employed. Thevariables zh and mh are filtered with a radius accounting for 1, 8% of theside-length of the domain, while the filter radius for the micro-structureorientation is 1, 5% of the side-length. We can conclude that the B-spotsmicro-structure is more competitive in the case of high ratio EA/EB.

7.5 Case study L-shape: a popular test case

In Figure 7.11 the geometry of the L-shape test, a popular case studyin the topology optimization community, is shown. In this test the stiffnessratio EA/EB = 103 is considered, and the same filtering radii of the testsquare are employed. The outcome of the algorithm on a 4900 elementsmesh is shown in Figure 7.12. Like in the previous test cases it can beobserved that the final design is a truss structure. Inside the beams, wherea clear preferential direction is present, anisotropic micro-structures alignedwith the beam itself result to be the most convenient; next to junction pointof beams, for high values of the ratio EA/EB, anisotropic micro-structuresare present.

110

(a) Total density (b) Total density

(c) Allocation of m.s. (d) Allocation of m.s.

(e) Variable m (f) Variable m

(g) Variable ϑ (h) Variable ϑ

Figure 7.10: Results of test square. Left column: EA/EB = 10; rightcolumn: EA/EB = 103.

111

Figure 7.11: Geometry and boundary conditions of the test L-shape.

(a) Total density (b) Variable m

(c) Allocation of m.s. (d) Variable ϑ

Figure 7.12: Results of test L-shape.

112

Chapter 8

Conclusions

In this work we investigated the possibility of realizing bodies with highstructural performances by means of a class of self-assembling materialsexhibiting micro-structures, namely diblock copolymers. We implemented atwo-dimensional finite elements solver for the Cahn-Hilliard-Oono equation,the parabolic PDE that models the phase separation phenomenon of thisclass of materials. Then, guided by the works available in the literature, wereconstructed the phase plane of the different micro-structures exhibited bydiblock copolymers.

By means of the homogenization theory we formulated a two-scalesmodel to describe at a macroscopic level the elastic properties of a body ob-tained by self-assembly of diblock copolymers. The model allows to computethe effective macroscopic stiffness tensor of a body exhibiting locally-periodicmicro-structures, and overcomes the infeasibility of a direct simulation of allthe characteristic scales of the equilibrium equation. The actual computa-tion of the homogenized tensor requires the solution of an elliptic secondorder PDE, known as cell-problem, which we solved again by means of thefinite elements method. In this way, a database of homogenized tensors forthe admissible values of the parameters governing the diblock copolymerspattern formation has been built.

The afore-mentioned two-scales model allowed to formulate an optimiza-tion problem for diblock copolymers bodies. The local properties of themicro-structure and the macroscopic shape and topology of the body are si-multaneously optimized in order to maximize the stiffness of the body itself(i.e. to minimize its compliance). The problem relies on a multi-materialformulation of the state equation, in which the different classes of micro-structures exhibited by diblock copolymers (in the two dimensional case,A-spots, lamellae end B-spots) are treated as different materials competingeach others in the optimization problem.

Because of both modellistic arguments (namely the need of controllingthe length-scale of the design variables) and pure mathematical reasons

113

(namely the non-existence of solution of the non-filtered problem), the spaceof admissible design is restricted to that of filtered ones, i.e. the designs ob-tained by convolution with a regularizing kernel. In this way the length-scaleof the design variables is bounded from below and, moreover, the regulariz-ing effect of the convolution equation provides with compactness the set ofadmissible designs, by Ascoli-Arzela theorem. Thanks to this fact we provedan existence result for the topology optimization problem of bodies obtainedby self-assembly of diblock copolymers.

Subsequently we proposed an algorithm for the topology optimizationproblem considered in this work. First of all we generalized a popularmethod in the context of topology optimization, namely the optimality con-ditions method, to the multi-material case. The proposed algorithm requiresthe computation of two Lagrange multipliers satisfying certain conditions; aresult proved in this work states the existence and uniqueness of the valuesof the multipliers, and suggests a bisection method for their actual compu-tation. Then we applied the multi-material update scheme to the topologyoptimization problem of diblock copolymers bodies. Since the optimalityconditions method cannot be applied to the optimization of the orientationof the micro-structure, we employed a linear search method to update thesevariables.

Finally we considered a finite elements approximation of the equilibriumequation of diblock copolymers bodies, and we derived a discrete versionof the algorithm, which we implemented restricting ourselves to the two-dimensional case. Numerical results displaying the capabilities of the pro-posed method have been reported and commented.

We observe that the pertinence of the algorithm proposed in the presentwork goes beyond the application to diblock copolymers bodies. Indeed thealgorithm can be applied to the problem of finding the optimal distribu-tion in a reference domain of N different candidate micro-structures, whosefeatures are parametrized by a number of design parameters.

Moreover, the multi-material optimality conditions method presented inthis work may be interesting in and of itself, since it provides an efficientand easy to implement alternative to non-linear programming methods suchas the method of moving asymptotes (MMA).

Future developments

Despite in this work only a two-dimensional implementation has beencarried out, the extension to three dimensions should be quite straightfor-ward. In fact the whole theoretical analysis, included the application of thehomogenization theory to derive the two-scales formulation and the proofof existence of solution of the optimization problem, can be applied to anyspace dimension. In the actual implementation the only aspect that requiressome care is the representation of the orientation of the micro-structures,

114

which in 3D can be carried out by means of the three Euler’s angles. Theextension to three dimensions would be interesting since the phase plane ofdiblock copolymers in 3D is richer that in 2D (lamellae, gyroids, cylindersand spheres patterns are present), and thus micro-structures with differentanisotropy features would compete each other.

A possible improvement to the algorithm proposed in this work is in theoptimization of the rotation of the micro-structure. Other schemes can beconsidered to improve the efficiency of the algorithm.

115

116

Appendix A

Useful results

In this appendix some useful results are reported. References where moredetails and proofs can be found are reported in the headings. In the followingthe word “domain” denotes an open connected subset of the reference spaceRd.

Theorem 3 (Banach-Alaoglu-Bourbaki). Let X be a separable Banach space.Then for every bounded sequence in its dual space X∗ we can extract a weakly∗ converging subsequence.

Corollary 1. Let ujj∈N be a bounded sequence in L∞(Ω), where Ω isa domain of Rd. Then there exists a subsequence ujhh∈N and a limitu ∈ L∞(Ω) such that

ujh∗ u in L∞(Ω).

Theorem 4. Let X be a reflexive Banach space. Then for every boundedsequence in X we can extract a weakly converging subsequence.

Theorem 5 (Ascoli-Arzela). Let K ⊂ Rd be a compact set, and E be anormed vector space. Consider a sequence ujj∈N of functions uj : K → Esuch that:

• ujj is equibounded:

∃M < +∞ ∀ j ∈ N supx∈Ω‖uj(x)‖ ≤M ;

• ujj is equicontinuous:

∀ ε > 0 ∃ δ > 0 ∀x,y ∈ Ω s.t. ‖x− y‖ < δ ∀ j ∈ N‖uh(x)− uh(y)‖ < ε .

Then we can extract a subsequence ujhh∈N uniformly converging tosome u ∈ C0(K;E).

117

Theorem 6 (Poincare inequality). Let Ω ∈ Rd be a bounded Lipschitz do-main. Consider a subset ΓD of ∂Ω of positive (d − 1)-dimensional Haus-dorff measure. Then there exists a constant CP > 0 such that for anyu ∈ H1

ΓD(Ω;Rn)

‖u‖L2(Ω;Rn) ≤ CP ‖∇u‖L2(Ω;Rn×d) .

Theorem 7 (Korn’s inequality (1/3) [36]). Let Ω ∈ Rd be a bounded Lip-schitz domain. Then there exists a constant CK > 0 such that for anyu ∈ H1(Ω;Rd) (we stress that domain and co-domain must have the samedimensionality)

‖u‖2H1(Ω;Rd) ≤ CK(‖u‖2L2(Ω;Rd) + ‖∇su‖2L2(Ω;Rd×d)

).

Theorem 8 (Korn’s inequality (2/3) [36]). Let Ω ∈ Rd be a bounded Lips-chitz domain. Let V be a closed subspace of H1(Ω;Rd), such that V ∩R = ∅,where R denotes the space of rigid displacements:

R =η = a +Ax : a ∈ Rd, A ∈ Rd×d, A skew-symmetric

.

Then there exists a constant CK > 0 such that every u ∈ V satisfies thefollowing inequality:

‖u‖2H1(Ω;Rd) ≤ CK‖∇su‖2L2(Ω;Rd×d) . (A.1)

Corollary 2 (Korn’s inequality (3/3) [36]). Let Ω ∈ Rd be a bounded Lip-schitz domain. Consider a subset ΓD of ∂Ω of positive (d− 1)-dimensionalHausdorff measure. Then there exists a constant CK > 0 such that everyu ∈ H1

ΓD(Ω;Rd) satisfies the inequality (A.1).

Theorem 9 (Lax-Milgram Lemma). Let V be a real Hilbert space. Let Fbe an element of its dual space V ∗, and a(·, ·) be a bilinear form on V suchthat:

• a is continuous:

∃M < +∞ |a(u, v)| ≤M‖u‖V ‖v‖V ∀u, v ∈ V ;

• a is V -coercive:

∃α > 0 a(u, u) ≥ α‖u‖2V ∀u ∈ V.

Then there exists a unique solution of the variational problemfind u ∈ V s.t.

a(u, v) = 〈F, v〉∗ ∀v ∈ V.

Moreover the solution u satisfies the following stability estimate:

‖u‖V ≤1

α‖F‖V ∗ .

118

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