DIPLOMOV ´A PR ´ACE R´obert Path´o Tvarov´a optimalizace v kontaktn´ıch ´uloh´ach se tˇren´ım

Univerzita Karlova v Praze Matematicko-fyzik´aln´ı fakulta

DIPLOMOV ´ A PR ´ ACE

R´ obert Path´ o

Tvarov´ a optimalizace v kontaktn´ıch ´ uloh´ ach se tˇ ren´ım

Katedra numerick´e matematiky

Vedouc´ı diplomov´e pr´ace: prof. RNDr. Jaroslav Haslinger, DrSc.

Studijn´ı program: Numerick´a a v´ ypoˇctov´a matematika

Let me express my deepest gratitude to my supervisor, prof. Jaroslav Haslinger for introducing me to the fascinating world of shape optimization and contact problems, his careful guidance, numerous comments and help throughout the work. For all the support, my sincerest thanks belongs to my girlfriend and family, as well.

Prohlaˇsuji, ˇze jsem svou diplomovou pr´aci napsal samostatnˇe a v´yhradnˇe s pouˇzit´ım citovan´ych pramen˚u. Souhlas´ım se zap˚ujˇcov´an´ım pr´ace a jej´ım zveˇrejˇnov´an´ım.

V Praze dne 5.8.2009 R´obert Path´o

Contents

Introduction 5

Notation 7

1 The state problem 9

1.1 The geometry . . . 9 1.2 Formulation of the state problem . . . 11 2 Contact shape optimization problem 14 2.1 Formulation of the problem . . . 14 2.2 Existence of an optimal shape . . . 16

3 Approximation of (P) 25

3.1 Formulation of the discrete problem . . . 25 3.2 Existence of a discrete optimal shape . . . 29 3.3 Convergence analysis . . . 38

Conclusion 49

A Auxiliary tools 50

A.1 Cone property . . . 50 A.2 On Korn’s inequality and the trace mapping . . . 52 A.3 Uniqueness of the solution to the discrete state problem . . . 54

Bibliography 58

N´azev pr´ace: Tvarov´a optimalizace v kontaktn´ıch ´uloh´ach se tˇren´ım Autor: R´obert Path´o

Katedra: Katedra numerick´e matematiky

Vedouc´ı diplomov´e pr´ace: prof. RNDr. Jaroslav Haslinger, DrSc.

e-mail vedouc´ıho: hasling@karlin.mff.cuni.cz

Abstrakt: V diplomov´e pr´aci se zab´yv´ame ´ulohou tvarov´e optimalizace pro 2D Signoriniho ´ulohu s dan´ym tˇren´ım a koeficientem tˇren´ı, kter´y z´avis´ı na ˇreˇsen´ı. C´ılem je nal´ezt optim´aln´ı kontaktn´ı ˇc´ast elastick´eho tˇelesa. V pr´aci navrhneme vhodnou mnoˇzinu pˇr´ıpustn´ych oblast´ı a dok´aˇzeme exis- tenci optim´aln´ı oblasti pro dostateˇcnˇe bohatou tˇr´ıdu cenov´ych funkcion´al˚u.

V dalˇs´ı ˇc´asti se zamˇeˇr´ıme na aproximaci t´eto ´ulohy. Existence diskr´etn´ıch optim´aln´ıch oblast´ı je dok´az´ana a je provedena konvergenˇcn´ı anal´yza.

Kl´ıˇcov´a slova: tvarov´a optimalizace, Signoriniho ´uloha s dan´ym tˇren´ım, ko- eficient tˇren´ı z´avisl´y na ˇreˇsen´ı

Title: Shape optimization in contact problems with friction Author: R´obert Path´o

Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Jaroslav Haslinger, DrSc.

Supervisor’s e-mail address: hasling@karlin.mff.cuni.cz

Abstract: In the present work we formulate a shape optimization problem for the 2D Signorini problem with given friction and a coefficient of friction which depends on the solution. The aim is to find an optimal contact part of an elastic body. A suitable set of admissible domains is given, among which the existence of an optimal one is established for a large class of cost func- tionals. The shape optimization problem is then approximated. Existence of discrete optimal shapes is proven and convergence analysis is done.

Keywords: optimal shape design, Signorini problem with given friction, co- efficient of friction depending on the solution

Introduction

The motivation for the research presented in this work is practical: the aim is to design objects which exhibit a priori given “optimal” properties. To achieve this goal, some of the date of a mathematical model, referred to as the state problem, are considered to be parameters, the so-called control or design variables. Then one tries to adjust the parameters in such a way that the corresponding model enjoys the desired optimality. This is usually done by minimizing a given functional, called thecost functional, which evaluates the state of the system for each admissible value of the control parameter.

As the term indicates, in shape optimizationthe geometry of the model is of primary interest.

In our case the 2D model of the Signorini problem with given friction and a solution-dependent coefficient of friction will be considered as a state problem. It describes the equilibrium state of a deformable body that is uni- laterally supported by a perfectly rigid foundation and is subject to body forces (e.g. gravitation, air pressure) and surface tractions. Besides unilat- eral conditions, friction will be taken into account on the contact part. From the various models available in the literature, we consider one of the simpli- est ones: the model with given friction, which involves a given slip bound multiplied by a coefficient of friction. Nevertheless, as physical experiments suggest, the coefficient of friction may depend not only on the spatial vari- ables, but on the magnitude of the tangential displacement as well. This model of friction is assumed here.

The purpose of this thesis is to extend the shape optimization analysis in [6], where the Signorini problem without friction is considered, to the above class of contact problems. The aim is to find an optimal shape of the contact part. When choosing the cost functional appropriately, this can result e.g.

in even distribution of the normal stresses along the contact zone (stress concentration is undesirable in engineering).

Note, that in the frictionless case the state problem is represented by

a simple elliptic variational inequality of the first kind, while in latter case it becomes an implicit variational inequality, making the analysis more in- volved. Another difference is that we do not assume unique solvability of the state problems. In contrast to the frictionless case, the existence of a solution is now guaranteed for a more regular class of admissible domains.

The text is organized as follows: in Chapter 1 we introduce the state problem and the set of admissible domains, which are used in the shape optimization problem. Since the mathematical analysis of the state problem has already been studied in [9], the existence and uniqueness theorems from [9] are only cited. The shape optimization problem is formulated in Chapter 2 and existence of an optimal shape is established. Results obtained here form the groundstones for proving the existence of a solution to the discretized shape optimization problems, which is the content of Chapter 3. We conclude this chapter by showing that the discrete optimal shapes converge to an optimal shape in an appropriate sense. Finally, auxiliary results are presented in the Appendix.

Notation

Let us list some of the widely used and common symbols that will appear in the text.

Domains and sets

N set of all positive integers;

R^d Euclidian space of dimension d∈N;

R⁺0 set of non-negative real numbers [0,∞);

Ω bounded domain in R^d with the Lipschitz boundary;

Ω closure of Ω;

∂Ω boundary of Ω;

Function spaces

C^k(D) functions whose derivatives up to orderk are continuous inD, k ∈N∪ {0} ∪ {∞};

C^0,1(D) Lipschitz continuous functions in D;

L^p(D) Lebesgue integrable functions inD, p∈[1,∞);

L^∞(D) essentially bounded, measurable functions inD;

H^k(D) Sobolev space W^k,2(D), k ∈N∪ {0};

P1(D) space of linear functions in D;

X :=X×X Cartesian product of a spaceX, e.g. L²(Ω);

Since it will be always clear from the context, we shall use the same symbol for norms and scalar products in scalar- and vector-valued function spaces. In the sequel k ∈N∪ {0}.

Norms and scalar products

(·,·) scalar product in R^d;

(·,·)_k,D scalar product in H^k(D) and H^k(D), respectively;

k · k euclidian norm in R^d;

k · k_k,D norm in H^k(D) and H^k(D), respectively;

k · k_0,∞,D norm in L^∞(D);

k · kX norm in the space X;

| · |_k,D seminorm in the space H^k(D) andH^k(D), respectively;

Convergences

→ inX strong convergence in a space X;

⇀ in X weak convergence in X;

⇉ in [a, b] uniform convergence in [a, b];

−→M convergence of domains in the sense of uniform convergence of boundaries;

−→O convergence of domains in the sense of C¹-convergence of boundaries;

Chapter 1

The state problem

The thesis starts with the classical and variational formulation of the Sig- norini problem with given friction and a coefficient of friction which depends on the solution. The mathematical analysis of this problem has already been done and the reader may find more details in [8], [12] and [9]. After becom- ing familiar with the state problem, we introduce a class of contact shape optimization problems, which will be treated thoroughly in the next chapter.

1.1 The geometry

Let a planar deformable body made of an elastic material be represented by a domain Ω ⊂ R², which is unilaterally supported by the half-space S = {(x₁, x2)|x2 ≤ 0}. The boundary of Ω will be decomposed into three non-empty, open, non-overlapping parts: ∂Ω = Γ_u ∪Γ_P ∪Γ_c. Next we will consider the case when the contact part Γc along which Ω comes into contact withS can be described by the graph of a continuous, non-negative function α: [a, b]→R:

Γc(α) = {(x1, x2)|a < x1 < b, x2 =α(x1)} and Ω := Ω(α) where

Ω(α) = {(x1, x2)∈R² |x1 ∈(a, b), α(x1)< x2 < γ}, γ >0 given Figure 1.1).

ByUadwe denote the set of all admissibleα’s, also calleddesign variables.

A suitable choice of U_ad is given by:

U_ad :={α ∈C^1,1([a, b])|0≤α(x)≤C0 ∀x∈[a, b],

|α(x)−α(y)| ≤C1|x−y| ∀x, y ∈[a, b],

|α^′(x)−α^′(y)| ≤C2|x−y| ∀x, y ∈[a, b], meas Ω(α) = C3}.

(1.1)

i.e. U_ad contains all functions which are together with their first derivatives Lipschitz equicontinuous in [a, b] and preserve the constant area of Ω(α).

In the sequel we shall suppose that the positive constants C0, C1, C2, C3 are chosen in a way that C0 < γ and U_ad 6=∅.

b

C

γ

Ω(α)

Γ (α) Γ

Γ

P

c u

a

Γ

Figure 1.1: Deformable body Ω(α)

As most of the results of the subsequent chapters apply to a larger class of admissible functions (domains), let us further define:

Qad :={α∈C^0,1([a, b])|0≤α ≤C0 in [a, b],

|α(x)−α(y)| ≤C1|x−y| ∀x, y ∈[a, b], meas Ω(α) =C3}.

(1.2)

Obviously, U_ad ⊂Q_ad.

1.2 Formulation of the state problem

Throughout this section we shall suppose that α ∈Q_ad is fixed. Consider a deformable body Ω := Ω(α), which is subject to body forces of density F ∈ L²(Ω) and surface tractions of density P ∈ L²(ΓP) acting on ΓP. Further, the body is fixed on Γ_u and friction is taken into account on the contact part Γc. We shall use the model with given friction, in which the coefficient of friction F depends on the displacement itself and the slip bound g is given.

The aim is to find a displacement vector u:=u(α) : Ω→R² characterizing the equilibrium state of our system. The classical formulation leads to the following system of differential equations and boundary conditions:

(equilibrium equations)

∂τ_ij

∂xj

+F_i = 0 in Ω, i= 1,2;¹ (1.3) (kinematic boundary conditions)

ui = 0 on Γu, i= 1,2; (1.4) (static boundary conditions)

T_i(u) =P_i on Γ_P, i= 1,2; (1.5) (non-penetration conditions)

u2(x1, α(x1)) ≥ −α(x1) T2(u)(x1, α(x1)) ≥ 0

¡u2(x1, α(x1)) +α(x1)¢

T2(u)(x1, α(x1)) = 0



 (1.6)

for x1 ∈(a, b), (friction conditions)

u1 = 0 ⇒ |T₁(u)| ≤ F(0)g

u₁ 6= 0 ⇒ T₁(u) = −sgn(u₁)F(|u₁|)g

¾

on Γc (1.7) First of all, let us recall some notions from the linear elasticity and define the symbols used above.

1Throughout the thesis the summation convention is adopted.

The linearized strain tensor ε(u) ={εij(u)}²_i,j=1 is defined by εij(u) = 1

2

¡∂ui

∂x_j + ∂uj

∂x_i

¢, i, j = 1,2 (1.8)

and thesymmetric stress tensor τ(u) = {τ_ij(u)}²_i,j=1related toε(u) by means of linear Hook’s law:

τij :=τij(u) = cijklεkl(u),

where the linear elasticity coefficients cijkl ∈ L^∞(Ω) satisfy the following symmetry and ellipticity conditions:

cijkl(x) = cjikl(x) =cklij(x) a.a. x∈Ω, ∀i, j, k, l= 1,2

∃Cell>0 : cijkl(x)ξijξkl ≥Cellξijξij a.a. x∈Ω, ∀ξij =ξji ∈R.

) (1.9) Further, Ti(u) = τij(u)nj (i = 1,2) stands for the i-th component of the stress vector T(u) on∂Ω. Heren = (n1, n2) denotes the unit outward normal vector to ∂Ω. Due to the special geometry of Ω(α), the first component T1(u)|_Γc will play the role of the tangential contact stress, while T2(u)|_Γc will be the normal contact stress on Γc. The same holds for the trace of u= (u1, u2) on Γc, i.e. u1|Γc, u2|Γc will represent the tangential and normal contact displacement, respectively.

The function F : R⁺0 → R⁺0 in (1.7) is the coefficient of friction and is assumed to be continuous and bounded in R⁺₀. The function g ∈ L²(Γc), g ≥0 a.e. on Γc is the slip bound.

Finally, let us explain the non-penetration and friction conditions in a few words. Inequality (1.6)1 ensures that the body stays above or on the foundation S, while (1.6)3 states that the normal contact stress can occur only at points of contact. Relations (1.7)1 and (1.7)2 say that the tangential contact stress is bounded by the product F(|u1|)g and the slip occurs only when this bound is attained. In the latter case the direction of the slip is opposite to the acting force.

In order to give the weak formulation of our problem, we introduce the following function spaces:

V :=V(α) ={v ∈H¹(Ω)|v = 0 a.e. on Γu}, V :=V(α) =V ×V

and the closed, convex set K ⊂V of all kinematically admissible displace- ments:

K :=K(α) ={v ∈V |v2(x1, α(x1))≥ −α(x1) a.e. in (a, b)}.

Definition 1.1. By a weak solution to the Signorini problem with a solution- dependent coefficient of frictionF we mean anyu∈K solving the following implicit variational inequality of the second kind:

(P(α))





Findu∈K such that for all v ∈K : a(u, v−u) +

Z

Γc

F(|u₁|)g¡

|v₁| − |u₁|¢

ds≥L(v−u) where

a(u, v) = Z

Ω

τij(u)εij(v)dx, (1.10) L(v) =

Z

Ω

Fividx+ Z

ΓP

Pivids. (1.11) From Green’s formula it easily follows that (P(α)) is formally equivalent to the classical formulation given by (1.3)–(1.7).

The existence of at least one solution to (P(α)) is shown in [9].

Theorem 1.1. For each α∈Qad there exists a solution to (P(α)).

Under the additional assumptions onF andg, uniqueness of the solution to (P(α)) can be guaranteed, as follows from the next theorem.

Theorem 1.2. Suppose that g ∈L^∞(Γc), g ≥0 and F is Lipschitz contin- uous in R⁺₀:

|F(x)− F(x)| ≤CL|x−x| ∀x, x∈R⁺₀. If

0< CLkgk_0,∞,Γc < CellCK

C_tr² , (1.12)

where Cell is the ellipticity constant from (1.9), CK is the constant in Korn’s inequality andCtr is the norm of the trace mappingH¹(Ω)→L²(Γc), then (P(α)) has a unique solution.

Proof. It is identical with the one of Theorem 2.2. in [9] with the only modification: instead ofg ≡1 on Γc we now takeg ∈L^∞(Γc), g ≥0.

Chapter 2

Contact shape optimization problem

In the previous chapter we introduced problem (P(α)) for one elastic body Ω := Ω(α) being in unilateral contact with a perfectly rigid foundation. Now we shall consider the family of bodies {Ω(α)|α ∈ U_ad}, i.e. shapes of Ω(α) will be characterized by their bottom part representing the contact zone Γc(α). Now we will try to find “an optimal” contact part of our elastic body.

The formulation of this problem is given in the next section.

2.1 Formulation of the problem

Let the symbol O denote the set of all admissible shapes, i.e.

O :={Ω(α)|α ∈ U_ad}, (2.1)

where U_ad is defined by (1.1). InO we introduce following convergence:

Ωn

−→O Ω ⇐⇒^def αn→α inC¹([a, b]), (2.2) where αn, α ∈ Uad and Ωn := Ω(αn), Ω := Ω(α). ¹

Similarly, let

M:={Ω(α)|α ∈Qad}, (2.3)

1In order to simplify notation, the abbreviation Ωn := Ω(α_n) and Ω := Ω(α) will be used further on.

with Qad defined by (1.2). We define Ωn

−→M Ω ⇐⇒^def αn⇉α in [a, b], αn, α∈Qad. (2.4) Furthermore, we shall need another bounded domain Ω such that: Ωb ⊂ Ωb ∀Ω∈ M. For the forthcoming analysis we set:

Ω := (a, b)b ×(0, γ), where γ is the same as in the definition of Ω(α).

Let I : ∆ → R, where ∆ = {(α, y) | α ∈ Qad, y ∈ V(α)}, be a cost functional and denote:

G:={(α, u)|α ∈ U_ad, u solves (P(α))} ⊂∆,

i.e. G is the graph of the control-to-state mapping S :α7→u. Note, that at this moment we do not require the unicity of the solution to (P(α)), i.e. S is multivalued, in general. On the other hand, to ensure that (P(α)) has at least one solution for eachα ∈ U_ad, we have to impose appropriate conditions on the data which do not depend on α. In what follows we assume that the elasticity coefficients cijkl∈L^∞(Ω) satisfy:b

cijkl(x) = cjikl(x) =cklij(x) a.a. x∈Ω,b ∀i, j, k, l= 1,2

∃C_ell>0 : cijkl(x)ξijξkl ≥Cellξijξij a.a. x∈Ω,b ∀ξ_ij =ξji ∈R, )

(2.5) and that

F ∈L²(Ω),b P ∈L²(∂Ω)b and g ∈H¹(Ω), gb ≥0 a.e. in Ω.b The shape optimization problem is then defined as the minimization of I onG.

Definition 2.1. The domain Ω(α^∗) ∈ O is said to be optimal, iff a pair (α^∗, u^∗) solves the following problem:

(P)

½ Find (α^∗, u^∗)∈ G such that I(α^∗, u^∗)≤I(α, u) ∀(α, u)∈ G

The existence of at least one solution to (P) will be proven for cost functionalsI satisfying the following lower semicontinuity assumption:

Ωn

−→O Ω, Ωn,Ω∈ O,

yn⇀ y inH¹(Ω), yb n, y ∈H¹(Ω)b )

=⇒

=⇒lim inf

n→∞ I(αn, yn|Ωn)≥I(α, y|Ω).

(2.6)

2.2 Existence of an optimal shape

First we present two basic lemmas concerning the familyO defined by (2.1).

Proposition 2.1. O is compact in the following sense: every sequence inO contains a convergent subsequence, i.e.

∀{Ω_n} ⊂ O ∃{Ω_nk} ⊂ {Ω_n} ∃Ω∈ O : Ω_nk−→^O Ω, k → ∞.

In addition,

Ωn

−→O Ω =⇒ Ωn

−→M Ω.

Proof. It follows from the Arzel`a–Ascoli theorem.

The next result is a consequence of Lipschitz equicontinuity of the func- tions belonging to Qad.

Proposition 2.2. The family M possesses the uniform cone property.

For the definition of the uniform cone property, as well as some other important results used later in this chapter, we refer to Sections A.1 and A.2 of the Appendix.

Notation. If v ∈ H¹(Ω(α)) then (v ◦α)(x1) := v(x1, α(x1)) ∈ L²(a, b), where x₁ ∈(a, b).

Lemma 2.1. Letαn, α∈Qad and vn, v ∈H¹(Ω),b n= 1,2, . . . be such that αn⇉α in [a, b] and vn ⇀ v in H¹(Ω), nb → ∞.

Then

vn◦αn →v◦α in L²(a, b), n→ ∞.

Proof. Due to the triangle inequality

kv_n◦αn−v◦αk_L²_(a,b) ≤ kv_n◦αn−vn◦αk_L²_(a,b)+kv_n◦α−v ◦αk_L²_(a,b), it is sufficient to show that both integrals on the right hand side converge to zero. It is well-known that (possibly after changing vn on a set of Lebesgue measure zero) vn is absolutely continuous in Ω on a.a. lines parallel to theb coordinate axes (see Theorem 2.2 in [13]) so that

v_n(x₁, α_n(x₁))−v_n(x₁, α(x₁)) =

Z αn(x1)

α(x1)

∂v_n

∂x2

(x₁, x₂)dx₂ (2.7)

holds for a.a. x1 ∈(a, b). From (2.7) and H¨older’s inequality, the first term can be estimated as follows:

Z b

a

|v_n◦αn−vn◦α|² dx1 = Z b

a

¯¯

¯

Z αn(x1)

α(x1)

∂v_n

∂x2

dx2

¯¯

¯

2

dx1

≤ kα_n−αk_C([a,b])kv_nk²_1,bΩ →0, because {vn} is bounded in H¹(Ω) andb αn ⇉α in [a, b].

For the second term we get immediately:

Z b

a

|v_n◦α−v◦α|²dx₁ ≤ Z b

a

|v_n◦α−v◦α|²p

1 + (α^′)²dx₁

=kv_n−vk²_L2(Γc(α)) →0, due to the compact embedding H¹(Ω(α))֒→^c L²(Γc(α)).

The next lemma ensures that the non-penetration condition (1.6)1 is preserved under weak convergence.

Notation. If v ∈ H¹(Ω(α)),α ∈ Qad then ˜v ∈ H¹(Ω) stands for the uni-b form extension of v from Ω(α) ontoΩ defined in Theorem A.2 in Appendixb A.

Lemma 2.2. Suppose that αn, α ∈ Qad and vn ∈ H¹(Ω(αn)), v ∈ H¹(Ω),b n = 1,2, . . . are such that:

(i) αn ⇉α in [a, b];

(ii) ˜vn ⇀ v in H¹(Ω);b

(iii) v_n◦α_n≥ −α_n a.e. in (a, b)∀n.

Then also v◦α≥ −α a.e. in (a, b).

Proof. Note that for any β ∈Qad and any w∈H¹(Ω(β)) the following two conditions are equivalent:

w◦β≥ −β a.e. in (a, b), (2.8)

and

(w◦β+β)⁻ = 0 a.e. in (a, b). (2.9)

Due to (iii) the functions vn satisfy (2.9) with w :=vn and β :=αn for alln = 1,2, . . . Next we prove that

n→∞lim Z b

a

(vn◦αn+αn)⁻ξ dx1 = Z b

a

(v◦α+α)⁻ξ dx1 ∀ξ ∈C₀^∞(a, b). (2.10) If it is so then the integral on the right of (2.10) is equal to zero implying that (v◦α+α)⁻ = 0 a.e. in (a, b).

It remains to verify (2.10):

¯¯

¯ Z b

a

(vn◦αn+αn)⁻ξ dx1− Z b

a

(v◦α+α)⁻ξ dx1

¯¯

¯

≤ Z b

a

¯¯(vn◦αn+αn)⁻−(v◦α+α)⁻¯

¯|ξ|dx1

≤ Z b

a

|vn◦αn−v◦α||ξ|dx1 + Z b

a

|αn−α||ξ|dx1

≤ kvn◦αn−v ◦αk_L²_(a,b)kξk_L²_(a,b)+kαn−αk_C([a,b])kξk_L¹_(a,b) →0, making use of Lemma 2.1, (ii) and the inequality:|a⁻−b⁻| ≤ |a−b| ∀a, b∈ R.

Corollary 2.1. Letαn, α∈Qad andvn ∈K(αn),v ∈H¹(Ω) (nb = 1,2, . . .) be such that:

αn⇉α in [a, b] and ˜vn ⇀ v in H¹(Ω), nb → ∞.

Then v|_Ω(α) ∈K(α).

An important approximation result is presented in the next lemma.

Lemma 2.3. Let αn ⇉ α in [a, b], αn, α ∈ Qad and ξ ∈ K(α) be given.

Then there exists a sequence {ξl} ⊂H¹(Ω)b such that

ξl→ξ˜ in H¹(Ω), lb → ∞, (2.11) and for every l∈N there exists n0(l)∈N such that

ξl|_Ωn ∈K(αn) ∀n ≥n0(l). (2.12) Proof. For a possible construction of {ξl} see [6] .

Lemma 2.4. G is compact in the following sense:

∀{(α_n, un)} ⊂ G ∃{(α_nj, unj)} ⊂ {(α_n, un)} ∃(α, u)∈ G: αnj →α in C¹([a, b]), u˜nj ⇀u˜ in H¹(Ω), jb → ∞.

Proof. Due to Proposition 2.1 we may assume that Ωn

−→O Ω, where Ω :=

Ω(α) for someα ∈ U_ad.

Inserting v := 0 ∈K(αn)∀n ∈N into (P(αn)) we get:

aΩn(un, un)≤aΩn(un, un) + Z

Γc(αn)

F(|un1|)g|un1|ds≤LΩn(un), where a_Ωn, L_Ωn denote the bilinear form (1.10) and linear form (1.11), respectively, evaluated on Ωn, and un1 stands for the first component of un = (un1, un2) ∈ H¹(Ωn). Thus one can find a constant C > 0, indepen- dent of α_n such that:

kunk1,Ωn ≤C ∀n,

making use of the ellipticity conditions and Corollary A.1 in Appendix A.

Since the family O possesses the uniform cone property, the norms of the extensions ˜un of un are also bounded independently of n (see Lemma A.2):

k˜u_nk_1,Ωb ≤C ∀n.

Therefore the sequence{˜un}contains a weakly convergent subsequence{˜unj}, i.e. ˜unj ⇀ χinH¹(Ω) for someb χ∈H¹(Ω). It remains to verify thatb u:=χ|Ω

solves (P(α)).

First of all, Lemma 2.2 yields that u∈K(α). Next, choose an arbitrary test function ξ ∈ K(α) and approximate it by a sequence {ξl} satisfying (2.11) and (2.12). Let l ∈ N be fixed. Since ξl|_Ωnj ∈ K(αnj) for j large enough, it can be used as a test function in problems (P(αnj)):

¡τ(unj), ε(ξl−unj)¢

0,Ω_nj + Z

Γc(α_nj)

F(|unj1|)g(|ξl1| − |unj1|)ds

≥(F, ξl−unj)0,Ω_nj + (P, ξl−unj)0,ΓP. (2.13) We pass to the limit first with j → ∞ and then with l → ∞ in (2.13). In order to do so, let us investigate each term separately.

Fix an integer m ∈Nand define the set Gm :=Gm(α) = n

(x1, x2)∈R²|x1 ∈(a, b), α(x1) + 1

m < x2 < γo

. (2.14) Due to uniform convergence α_nj ⇉ α in [a, b], Ω_nj ⊃G_m(α) for sufficiently large j and Ωnj can be decomposed as (see Figure 2.2):

Ωnj =Gm∪(Ωnj\Ω)∪¡

(Ω\Gm)∩Ωnj

¢.

G

Ω \ Ω

(Ω \ G ) Ω

Figure 2.1: Partitioning of the domain Ωnj

Hence, for j large enough:

¡τ(unj), ε(ξl−unj)¢

0,Ω_nj ≤¡

τ(unj), ε(ξl−unj)¢

0,Gm

+¡

τ(unj), ε(ξl)¢

0,Ω_nj\Ω+¡

τ(unj), ε(ξl)¢

0,(Ω\Gm)∩Ω_nj. From weak convergence of {unj} to u inH¹(Gm(α)), one arrives at:

lim sup

j→∞

¡τ(u_nj), ε(ξ_l−u_nj)¢

0,Ω_nj

≤¡

τ(u), ε(ξl−u)¢

0,Gm +Ckξlk1,Ω\Gm, (2.15) where the constant C >0 is independent of m, l ∈N and α∈ U_ad.

Similarly:

(F, ξl−unj)0,Ω_nj =(F, ξl−unj)0,Gm+ (F, ξl−unj)0,Ω_nj\Ω

+ (F, ξl−unj)0,(Ω\Gm)∩Ω_nj, whence

lim inf

j→∞ (F, ξl−unj)0,Ω_nj

≥(F, ξl−u)0,Gm−C¡

kFk0,Ω\Gm +kξlk0,Ω\Gm

¢, (2.16) again with a constant C which does not depend onm, l and α.

When dealing with the curvilinear integral on the right hand side of (2.13), one has to consider also the case, when ΓP depends on the design variable α. In this case we define

Mm ={a} ׳

α(a), α(a) + 1 m

´∪ {b} ׳

α(b), α(b) + 1 m

´

, (2.17) and write

(P, ξ_l−u_nj)_0,ΓP = (P, ξ_l−u_nj)_0,ΓP\Mm+ (P, ξ_l−u_nj)_0,Mm.

Then, using the compactness of the embedding of H¹(Ω) intob L²(∂Ω) web have:

lim inf

j→∞ (P, ξl−unj)0,ΓP

≥(P, ξl−u)0,ΓP\Mm−C¡

kPk0,Mm +kξlk0,Mm

¢, (2.18) with C having the same property as before.

Finally, we show that in the frictional term one can pass to the limit:

j→∞lim Z

Γc(α_nj)

F(|u_nj1|)g¡

|ξ_l1| − |u_nj1|¢ ds=

Z

Γc(α)

F(|u₁|)g¡

|ξ_l1| − |u₁|¢ ds.

(2.19)

Indeed, the integral on the left of (2.19) can be written as I^(j) :=

Z b

a

F(|unj1◦αnj|)g◦αnj

¡|ξl1◦αnj| − |unj1◦αnj|¢q

1 + (α^′_nj)²dx1

= Z b

a

F(|unj1 ◦αnj|)g◦α¡

|ξl1◦α| − |u1◦α|¢q

1 + (α^′_nj)²dx1

+ Z b

a

F(|u_nj1◦α_nj|)¡

g◦α_nj|ξ_l1◦α_nj| −g◦α|ξ_l1◦α|¢q

1 + (α^′_nj)²dx₁

− Z b

a

F(|u_nj1◦αnj|)¡

g◦αnj|u_nj1◦αnj| −g◦α|u₁ ◦α|¢q

1 + (α^′_nj)²dx1

=:I₁^(j)+I₂^(j)−I₃^(j)

Next we show thatI₃^(j)→0 forj → ∞, whereas the relationI₂^(j) →0 follows analoguosly. Exploiting the boundedness ofF and the uniform boundedness of {α^′_nj}we can write:

|I₃^(j)| ≤ F q

1 +C₁² Z b

a

¯¯g◦αnj|unj1◦αnj| −g◦α|u1◦α|¯

¯dx1,

where|F(x)| ≤ F ∀x∈R⁺₀. Adding and subtracting the termg◦α_nj|u₁◦α|

and applying H¨older’s inequality we obtain

|I₃^(j)| ≤F q

1 +C₁²kg◦αnjk_L²_(a,b)ku_nj◦αnj −u1◦αk_L²_(a,b) +F

q

1 +C₁²ku₁◦αk_L²_(a,b)kg◦αnj −g◦αk_L²_(a,b). The term kg◦αnjk_L²_(a,b) can be estimated using Theorem A.4:

|I₃^(j)| ≤F q

1 +C₁²Ctrkgk_1,Ωbku_nj1◦αnj −u1◦αk_L²_(a,b) +F

q

1 +C₁²ku₁ ◦αk_L²_(a,b)kg◦αnj−g◦αk_L²_(a,b).

Since the constants F, C1, Ctr are independent ofj, the fact that I₃^(j) →0 follows from Lemma 2.1.

Recall that convergence of a sequence in L²(a, b) implies convergence almost everywhere in (a, b) for an appropriate subsequence. Therefore we may assume without loss of generality that

unj1◦αnj →u1◦α a.e. in (a, b),

for the whole sequence {unj1◦αnj}, otherwise we pass to a subsequence.

Due to the continuity of F we have:

F(|unj1◦αnj|)→ F(|u1◦α|) a.e. in (a, b), so that

j→∞lim I₁^(j) = Z b

a

F(|u1◦α|)g◦α¡

|ξl1◦α| − |u1◦α|¢p

1 + (α^′)²dx1 (2.20) as follows from Lebesgue’s dominated convergence theorem and the fact that αnj →α inC¹([a, b]). Hence (2.19) holds.

Combining (2.15) – (2.19) and letting m→ ∞ one gets:

¡τ(u), ε(ξl−u)¢

0,Ω(α)+ Z

Γc(α)

F(|u₁|)g¡

|ξ_l1| − |u₁|¢ ds

≥(F, ξl−u)0,Ω(α)+ (P, ξl−u)0,ΓP ∀l∈N.

Finally, passing to the limit with l → ∞ we obtain the statement of the theorem.

The proof of the main theorem of this section is now straightforward.

Theorem 2.1. Suppose that the cost functionalI satisfies (2.6). Then there exists at least one solution to shape optimization problem (P).

Proof. Denote

q:= inf

(α,u)∈G I(α, u)

and let {(αn, un)} ⊂ G be a minimizing sequence of I:

q = lim

n→∞I(αn, un).

By Proposition 2.1 one can find a subsequence{Ωnj} ⊂ {Ωn}and a domain Ω(α^∗) ∈ O such that Ωnj

−→O Ω(α^∗). Lemma 2.4 states that the sequence {u_nj} of the corresponding solutions to (P(αnj)) contains a weakly conver- gent subsequence:

˜

un_jk ⇀ χ inH¹(Ω),b ask → ∞,

whereχ∈H¹(Ω) andb u^∗ :=χ|Ω(α^∗)is a solution to (P(α^∗)), i.e. (α^∗, u^∗)∈ G.

The lower semicontinuity assumption (2.6) onI yields:

q= lim

n→∞I(αn, un) = lim inf

k→∞ I(αn_jk,u˜n_jk|Ω_nj

k)

≥I(α^∗, χ|_Ω(α^∗₎) =I(α^∗, u^∗)≥q.

Let us conclude this chapter with the case when the state problem is uniquely solvable.

Taking into account the results of Section A.2 we see that the constants Ctr and CK in (1.12) can be made independent of α ∈ Uad. Thus we can establish sufficient conditions on the slip bound g and the coefficient of friction F guaranteeing uniqueness of the solution to (P(α)) for each α ∈ Uad.

Theorem 2.2. Let g ∈ C(Ω),b g ≥ 0 and F be Lipschitz continuous with modulus CL>0. If

CLkgk

C(Ω)b < CellCK

C_tr² , (2.21)

then (P(α)) has a unique solution for each α∈Qad.

If the assumptions of Theorem 2.2 are satisfied, then the control-to-state mapping S :α7→u(α) is single valued and the shape optimization problem (P) takes the following form:

½ Find α^∗ ∈ Uad such that

I(α^∗, u(α^∗))≤I(α, u(α)) ∀α∈ Uad.

Chapter 3

Approximation of (P)

This chapter is concerned with the approximation of (P). We introduce a suitable finite element discretization of the state problem (P(α)) and a discretization of the set of admissible shapes U_ad. We show that the discrete shape optimization problem (P_h) has a solution. Convergence analysis is studied in the last section.

3.1 Formulation of the discrete problem

In shape optimization problems both the admissible setUad and the state problem have to be discretized. We will consider piecewise linear approxima- tions ofU_ad: they have the advantage that the discrete admissible shapes be- come polygonal, so that the state problem may be discretized by a standard finite element method. In our analysis linear finite elements on a triangular mesh will be used.

Letd∈Nbe given and seth:= (b−a)/d. Byδ_h we denote the equidistant partition of [a, b]:

δh : a ≡a0 < a1 <· · ·< ad(h) ≡b, where

aj =a+jh ∀j = 0,1, . . . , d.

The set of discrete admissible shapes U_ad^h consists of continuous, piecewise linear functions onδh which satisfy analoguous constraints to those imposed

in (1.1):

U_ad^h :={α_h ∈C([a, b])|α_h|_[ai−1,ai]∈P₁([a_i−1, a_i]) ∀i= 1, . . . , d, 0≤αh(ai)≤C0 ∀i= 0, . . . , d,

|α_h(ai)−αh(ai−1)| ≤C1h ∀i= 1, . . . , d,

|α_h(ai+1)−2αh(ai) +αh(ai−1)| ≤C2h², ∀i= 1, . . . , d−1, meas Ω(αh) = C3 }.

(3.1)

The positive constants C0, C1, C2, C3, γ are the same as in (1.1) and the symbols Γu, ΓP, Γc(αh), Ω(αh), etc. have the same meaning as in Chapter 1.

We denote the set of discrete admissible shapesby

O_h :={Ω(αh)|αh ∈ U_ad^h }. (3.2) Remark 3.1. Notice, that U_ad^h ⊂ Qad, but U_ad^h * Uad, i.e. we speak about anexternal approximation of U_ad.

Due to the special geometry of the polygonal domain Ω(α_h), one can construct its triangulation T(h, αh) such that the nodes of T(h, αh) lie on the lines {a_i} ×R,i= 0, . . . , d.

Let h > 0 be fixed. Next we shall consider the system{T(h, α_h)}, α_h ∈ U_ad^h which consists of topologically equivalent triangulations, i.e.:

(T1) the number of the nodes inT(h, αh) as well as the neighbours of each triangle from T(h, αh) are the same for all αh ∈ U_ad^h;

(T2) the position of the nodes ofT(h, αh) depends continuously on changes of αh ∈ U_ad^h;

(T3) the triangulations T(h, α_h) are compatible with the decomposition of

∂Ω(αh) into Γc, ΓP and Γu for any h >0 and any αh ∈ U_ad^h . In order to establish convergence results we shall also need:

(T4) the system {T(h, αh)} is uniformly regular with respect to h > 0 as well as αh ∈ U_ad^h , i.e. there exists a constantθ0 >0 such that

θ(h, αh)≥θ0 ∀h >0∀αh ∈ U_ad^h ,

whereθ(h, αh) denotes the minimal interior angle of all triangles from T(h, αh).

Example. Let Γu(α) :={a} ×(α(a), γ) for every α ∈Qad. Then one can introduce a triangulation of Ω(αh) satisfying (T1)–(T4) as in [10]:

Chooseα₀ ∈(C₀, γ) and consider a uniform triangulation of the rectangle [a, b]×[α0, γ], which will be independent of αh ∈ U_ad^h for h fixed. Further, divide the line segments{a_i} ×[αh(ai), α0] (i= 0, . . . , d) into M equal parts, where

M = 1 + [α0d],

and [·] denotes the integer part of a real number. Finally, connect the newly created nodes as shown in Figure 3.1. It is readily seen that (T1), (T2) and

Figure 3.1: One possible triangulation

(T3) are automatically satisfied for this system {T(h, αh) | h > 0, αh ∈ U_ad^h}, while for justification of (T4) we refer to [10].

Notation. The domain Ω(α_h) with the triangulation T(h, α_h) will be de- noted by Ωh(αh), or just shortly Ωh in what follows.

In order to give a finite element discretization of the state problem, we approximate V(αh) by piecewise linear functions on Ωh:

Vh(αh) :={vh ∈C(Ωh)|vh|T ∈P1(T) ∀T ∈ T(h, αh), vh = 0 on Γu}, V_h(αh) :=Vh(αh)×Vh(αh)

and

K_h(αh) :={vh = (vh1, vh2)∈V_h(αh)|vh2(ai, αh(ai))≥ −αh(ai)∀ai ∈ Nh},

where Nh is the set of all contact nodes, i.e. ai ∈ Nh iff (ai, αh(ai)) ∈ Γc(αh)\Γu. Observe, that K_h(αh)⊂K(αh) ∀h >0∀α_h ∈ U_ad^h.

Now the discrete state problem reads as follows:

(Ph(αh))













Findu_h :=u_h(α_h)∈K_h(α_h) such that:

aΩh(uh, vh−uh) +

Z b

a

F(rh|u_h1◦αh|)g◦αh(|v_h1◦αh| − |u_h1◦αh|)q

1 + (α_h^′)²dx1

≥LΩh(vh−uh) ∀v_h ∈K_h(αh).

where rh : C([a, b]) → C([a, b]) stands for the piecewise linear Lagrange interpolation operator on δh, i.e. for every v ∈ C([a, b]) the function rhv is continuous, piecewise linear on the partition δh and

(rhv)(ai) =v(ai) ∀i= 0, . . . , d.

Remark 3.2. It is easy to see that the frictional term in (P_h(α_h)) is equiv-

alent to Z

Γc(αh)

F¡ R_h^Γc¯

¯(uh1)|Γc

¯¯¢g¡

|vh1| − |uh1|¢ ds,

where the operatorR^Γ_h^c is the piecewise linear Lagrange operator defined on C(Γ_c(α_h)), i.e.

(R^Γ_h^cvh)◦αh =rh(vh◦αh) ∀vh ∈C(Γc(αh))∀αh ∈ U_ad^h,

but it will be more convenient to work with the first expression involvingrh. Under the same conditions onF one can show that also the discrete state problems are solvable, as follows from [9].

Theorem 3.1. (P_h(αh)) has at least one solution for each h >0 and αh ∈ U_ad^h.

As in the continuous setting, the discrete shape optimiziation problem is defined as the minimization of the cost functional I on the graph of the discrete, generally multivalued control-to-state mapping:

G_h :={(α_h, u_h)|α_h ∈ U_ad^h, u_h solves (P_h(α_h))}.

Thus, for each h >0 thediscrete shape optimization problem reads as:

(P_h)

½ Find (α_h^∗, u^∗_h)∈ G_h such that

I(α^∗_h, u^∗_h)≤I(αh, uh) ∀(αh, uh)∈ Gh.

In the next section we shall establish the existence of a solution to (Ph).

3.2 Existence of a discrete optimal shape

Firstly, we have to introduce suitable convergence in the set of discrete ad- missible shapes. Unfortunately, the set U_ad^h contains less regular functions than U_ad does, therefore (2.2) is inappropriate. Nevertheless, as one will see later, uniform convergence of the boundaries in the sense of (2.4) will be sufficient to ensure the existence of discrete solutions and their convergence to a solution of (P).

Our aim will be to prove the following existence result.

Theorem 3.2. Suppose that the cost functional I is lower semicontinuous in the following sense:

Ω⁽ⁿ⁾_h −→^M Ω_h, Ω⁽ⁿ⁾_h ,Ω_h ∈ O_h,

y⁽ⁿ⁾ ⇀ y in H¹(Ω), yb ⁽ⁿ⁾, y ∈H¹(Ω), nb → ∞ )

=⇒

=⇒lim inf

n→∞ I(α⁽ⁿ⁾_h , y⁽ⁿ⁾|_Ω(n)

h )≥I(α, y|_Ωh).

(3.3)

Then (P_h) has at least one solution.

As soon as we prove the compactness ofG_h(an analogy of Lemma 2.4), the proof of Theorem 3.2 becomes identical with that of Theorem 2.1, therefore will be omitted.

In order to prove the compactness of G_h, we will need some auxiliary results.

Recall that the symbolestands for the extension of functions from their domain of definition onto Ω (see Theorem A.2 in Appendix A).b

Lemma 3.1. Let α⁽ⁿ⁾_h , αh ∈ U_ad^h, v⁽ⁿ⁾_h ∈ Vh(α⁽ⁿ⁾_h ), v ∈ H¹(Ω) (nb = 1,2, . . .) be such that

α⁽ⁿ⁾_h ⇉αh in [a, b] and ˜v_h⁽ⁿ⁾ ⇀ v in H¹(Ω), nb → ∞.

Then vh :=v|_Ωh ∈Vh(αh).

Proof. From weak convergence of {˜v_h⁽ⁿ⁾} in H¹(Ω) it follows thatb vh = 0 on Γu, hence we only have to verify that vh is continuous and piecewise linear over Ωh.

Choose an arbitrary triangle T ∈ T(h, αh) and a point z ∈ intT. Let U(z)⊂T denote an open neighbourhood of z. Due to (T1) and (T2) there exist triangles Tn ∈ T(h, α⁽ⁿ⁾_h ) and n0 ∈Nsuch that

U(z)⊂T_n ∀n≥n₀.

Denote v⁽ⁿ⁾ :=v⁽ⁿ⁾_h |_U(x) and their gradients c⁽ⁿ⁾ = (c⁽ⁿ⁾₁ , c⁽ⁿ⁾₂ ) := ∇v⁽ⁿ⁾ ∈ R² inU(z). Weak convergence of {˜v_h⁽ⁿ⁾} tov inH¹(Ω) yields weak convergenceb of {v⁽ⁿ⁾}to v inH¹(U(z)), from which:

∇v⁽ⁿ⁾ ⇀∇v in L²(U(z)), n → ∞,

i.e. Z

U(z)

q· ∇v⁽ⁿ⁾dx→ Z

U(z)

q· ∇v dx ∀q ∈L²(U(z)). (3.4) Inserting q:= (1,0) and q:= (0,1) into (3.4), we obtain:

n→∞lim c⁽ⁿ⁾_i = 1 measU(z)

Z

U(z)

∂v

∂xi

dx=:ci for i= 1,2. (3.5) Observe that the numbers c⁽ⁿ⁾_i do not depend on U(z) at all, therefore one gets the same limit c_i in (3.5) for every z ∈ intT and U(z) ⊂ T. Conse- quently ∇v = (c1, c2) in T, so that v|T is linear ∀T ∈ T(h, αh). In addi- tion,v ∈H¹(Ω), ensuring thatb vh is also continuous on Ωh. Thus the proof is complete.

Corollary 3.1. Let α_h⁽ⁿ⁾, αh ∈ U_ad^h , v_h⁽ⁿ⁾ ∈ K_h(α⁽ⁿ⁾_h ), v ∈ H¹(Ω) (nb = 1,2, . . .) be such that

α⁽ⁿ⁾_h ⇉αh in [a, b] and v˜_h⁽ⁿ⁾⇀ v in H¹(Ω), nb → ∞.

Then vh :=v|_Ωh ∈K_h(αh).

Proof. Indeed, v_h ∈ K(α_h) follows from Corollary 2.1 and the fact that K_h(α⁽ⁿ⁾_h ) ⊂ K(α_h⁽ⁿ⁾) ∀n. Now it is sufficient to use Lemma 3.1 to each component to show that vh ∈V_h(αh).