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Contents

Introduction vi

1 Mathematical background

1.1 Mathematican Modelling and Numerical Simulation1.1.1 Introduction 1

1.1.2 Mathematical Modelling 1

1.1.3 Some classical models 2

1.1.4 Numerical calculation. 4

1.1.5 The idea of a well-posed problem.. 41.1.6 Classification of PDEs 6

1.2 Variational formulation of elliptic problems1.2.1 Introduction. 7

1.2.2 Classical formulation... 7

1.2.3 Greens formulas. 8

1.2.4 Variational formulation 9

1.2.5 Lax-Milgram theory. 10

1.2.6 System of linearized elasticity. 11

1.3 Finite Element Method1.3.1 Variational approximation 14

1.3.1.1 Introduction... 14

1.3.1.2 General internal approximation 14

1.3.1.3 Finite element method ( general principles ) 16

1.3.2 Finite elements in N=1 dimension 16

1.3.2.1 P1 finite elements. 16

1.3.2.2 Convergence and error estimation 18

2 Optimization

2.1 Introduction. 20

2.2 Definitions and notation.. 21

2.3 Categories of optimization problems2.3.1 Continuous versus discrete optimization 22

2.3.2 Constrained and unconstrainted optimization 22

2.3.3 Nature of objective function and constraints. 22

2.3.4 Global and local optimization. 232.3.5 Stochastic and deterministic optimization.. 23

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2.4 Existence of a minimum2.4.1 Optimization in finite dimensions 23

2.4.2 Optimization in infinite dimensions. 24

2.4.3 Convex analysis 25

2.5 Optimality conditions2.5.1 Introduction. 26

2.5.2 Differentiability 26

2.5.3 Euler inequalities and convex constraints 27

2.5.4 Lagrange multipliers 28

2.6 Numerical algorithms2.6.1 Introduction..... 32

2.6.2 Overview of algorithms 32

2.6.3 Gradient algorithms ( case without constraints ). 33

2.6.4 Gradient algorithms ( case with constraints ).. 352.6.5 Penalization of constraints 36

2.6.6. Newtons method 37

3 Shape Optimization

3.1 Introduction................................................................................................ 39

3.2 Examples

3.2.1 Optimization of the thickness of a membrane. 403.2.2 Some remarks on the criteria of optimization. 42

3.2.3 Optimization of the shape of a membrane.. 43

3.2.4 Shape Optimization in elasticity. 46

3.3 Optimization of distributed systems 47

3.4 Parametrical Optimization3.4.1 Modelling.... 51

3.4.2 Gradient method. 52

3.4.3 Numerical algorithm 54

3.5 Geometrical Optimization3.5.1 Introduction 55

3.5.2 Diffeomorphisms 56

3.5.3 Differentiation with respect to a domain 57

3.5.4 Gradient and optimality condition-Method of the Lagrangian.. 60

3.5.5 Numerical algorithms. 63

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v

4 Applications using FreeFem++

4.1 Introduction to FreeFem++.. 69

4.2 Example of a Cantilever4.2.1 Description of the problem 70

4.2.2 Parametrical Optimization 70

4.2.3 Geometrical Optimization. 94

4.3 Example of an L-shaped structure. 125

5 Conclusions and Perspectives........................................................... 156


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Introduction

Optimization is something more than a pure mathematrical concept. Nature optimizes

in most of its procedures. A structure or body shall deform or displace to a position that

minimizes the total potential energy: this is the stable configuration for equilibrium.People tried to take advantage of optimization procedures many decades ago. Several

optimization techniques have been developed, but the cost in calculation made impossible their

application in engineering problems. Therefore, enineers were mostly using their experience

and intuition, or trial and error techniques in order to design structures with improved

characteristics, always compared with the previous ones and not with some optimal solution,

which remained unknown so far. As it has been expected, this has led to serious failures,

sometimes with great impact on human lifes.

The tremendous evolution of computers during the last decades settled the engineers

able to abandon the former traditional methods of designing, which were both expensive in

time and money and imprecise, and to develop and adopt optimization techniques for a variety

of difficult problems they faced.Combining mechanics and applied mathematics, StructuralOptimization endows Civil Engineers with the capability to design structures with optimal

characteristics or with some desired mechanical behaviour.

The variety of criteria to be minimized (or maximized) that we can settle is huge. We

can use optimization techniques to minimize the cost of a structure, by varying the boundary of

the structure and consequently reducing its volume ( [2], [14], [15], [17], [18], [19] ). Also, we

can achieve maximum rigidity of a structure with predefined volume ( [2] ), which is a very

important topic from the point of view of a Civil Engineer. Moreover, in many applications of

a Civil Engineer, the problem of minimum stress design appears, which can be adequately

treated with optimization techniques ( [4], [7] ). Finaly, a great number of industrial

applications include optimization techniques, such as the design of compliant mechanisms (

[16] ), where the use of other methods can be highly uneffective.Of course, in order for all these problems to obtain a practical significance, engineers

have to set most of the times some constraints, that is, some additional requirements that the

structure should fulfill. These constraints could introduce great difficulties on the solution of

the initial problem and make the problem too complicated.Some of the most regularly

presented constraints are perimeter constraints ( [11] ), volume constraints ( [2] ) and stress

constraints ( [12], [13] ), where we exclude from our structure the possibility to overcome

some stress level, usually linked with a yield criterion (for example a Von Mises bound for the

stress).

In order to treat such problems, various techniques have been developed. Some of the

most popular in Structural Optimization, is the Parametrical Optimization ( [2] ), the

Geometrical Optimization ( [2], [14], [15], [17], [18], [19] ) and the Topological Optimization (

[2], [3], [8], [9] ). All of these methods above present benefits and drawbacks, which make

each of them more or less effective for some kind of problems. As a result, many variants of

these basic methods have been proposed, sometimes presenting significant improvement.For

example,the huge computational cost ( because of remeshing ) of Geometric Optimization and

its tendency to fall into local minima far away from global ones, have led to the development

of a Structural optimization technique, using sensitivity analysis and a level set method.The

knowledge of such techniques constitutes a powerful tool for Civil Engineers that need to keep

up with the contemporary demands of our society and design with safety, precision and

economy.

The objectives of this Thesis are, at first, to familiarize the reader with the idea ofOptimization and to present a new way of designing a variety of structures. Moreover, we want

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to question ourselfes about the practical implementation of such techniques to applications and

problems that cover the whole scientific spectrum of Civil Engineers.

The plan of this Thesis is the following. In the first Chapter we present the

mathematical background needed for the understanding of the examples that we will present.

We explain the notions and the difference between the Mathematical Modelling of a problem

and its Numerical Simulation ( [1] ). After this, we analyze the Variational Formulation ofelliptic problems in general and in the sequence we focus on the system of linearized elasticity,

which is of our interest in this work. We mention that this is done for reasons of simplicity,

since we are already familiar with linear elasticity, and that there is no particular difficulty in

considering another system. At the end of this Chapter we present some details of the Finite

Element Method ( F.E.M. ), which will be the method to use to approximate the solution of our

systems. We have to mention that the knowledge gained in the first few years of university is

the only prerequisite of this Thesis.

Chapter 2 is dedicated to Optimization. After a general introduction about Optimization

methods and especially methods for Shape Optimization, we present the theory of optimality

conditions and of constrained optimization, with equality or inequality constraints. In the last

part of this Chapter, we describe one very popular numerical method used in Optimization, thegradient method and we give some brief description of other methods.

In Chapter 3, we make a full explanation of the two methods used in our examples, the

Parametrical and the Geometrical Optimization. We present the Lagrangian method used to

reveal the necessary equations in order to solve numerically our problem. We also criticize

these two methods and reveal their drawbacks and the possible problems coming from their

implementation in computers.

In Chapter 4, we present the results taken from implementing these two methods in the

free finite element software FreeFem++, developed by F. Hecht and O. Pironneau. FreeFem++

is available on the website: http://www.freefem.org . At the beginning, we give a brief

description of the program, with the help of an example. After this, we give a short explanation

of the Geometrical Optimization code introduced in FreeFem++ ( [10] ). Finaly, we present

two examples. The first one refers to a cantilever and we try to optimize its shape using several

objective functions ( functions to be minimized ) such as the compliance ( the work done by

the loads ), a desired displacementof a boundary and a desired stress distribution in the

structure.The second example is an L-shaped structure, where we try to minimize the stress

concentration, considering as an objective function various norms of the stress tensor.

In the last Chapter, Chapter 5, we end up with some conclusions concerning our results

and our methods, focusing on the difficulties presented in the implementation of Optimization

algorithms in freeFem++. We propose some ways to overcome such difficulties and we

comment their influence in our final results. Furthermore, we propose some other examples to

be studied, which are of great significance from the point of view of a Civil Engineer.As we mentioned before, we just study the system of linearized elasticity in our

examples, but the practical implementations of Optimization techniques in Civil Engineering

problem are numerous. We just have to replace our system with another one, coming from

Fluid Mechanics, Transportation or Management problems, and the global character of

Optimization in Engineering problems is revealed.
http://www.freefem.org/http://www.freefem.org/


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1.Mathematical Background

1.1 Mathematical Modelling and Numerical Simulation ( [1] )

1.1.1 Introduction

In this first chapter, we try to describe two closely linked, although distinct, aspects of

applied mathematics: mathematical modeling and numerical simulation. Using the term

mathematical model, we refer to a representation or an abstract interpretation of physical

reality that is amenable to analysis and calculation. In our work, these models will be partial

differential equations (PDEs), that is, differential equations in several variables. On the other

hand, numerical simulation allows us to calculate the solutions of these models on a

computer, and therefore to simulate physical reality.

We shall see that the numerical calculation of the solutions of several physical models

sometimes has some unpleasant surprises, which can only be explained by a soundunderstanding of their mathematical properties. Therefore, we shall also pay attention to a third

fundamental aspect of applied mathematics, that is, the mathematical analysisof models.

In our work, we confine ourselves to linear problems for simplicity. Likewise, we only

consider deterministic problems, that is, with no random or stochastic components.

Finally, in order for this work to be understandable and applicable from the aspect of

view of a Civil Engineer, we shall often be a little imprecise in our mathematical arguments.

Without underestimating the need for a rigorous mathematical justification of our models, it is

the physical interpretation that dominates and allows engineers to make this knowledge

applicable to their specific problems. However, we shall often face difficulties and results, that

cannot be explained just with our physical intuition, but need a combined knowledge of

mathematics, computer science and engineering.

1.1.2 Mathematical Modelling

Although mathematical modeling is not one of the main purposes of our work, we have to

explain at the beginning the symbols that we will use in order to define our problem. In this

chapter, we present some more general models, but in the sequence we will focus on systems

of linearized elasticity.

Let us consider a domain in N space dimensions (denoted by RN, with in general

N=1,2,or 3) which we assume is occupied by an homogeneous, isotropic, elastic material. We

denote the space variable by x, that is a point of , and the time variable by t. In N=1dimensions, the mass forces applied on the material are represented by a given function f(x,t)

and the external forces by g(x,t), while the deformation is an unknown function u(x,t). In

higher dimensions, the applied forces are represented by vectors in RN, while the displacement

is described by a vector field u: RN RN.In order to calculate the unknown function u, we have to consider some fundamental laws

of mechanics. In the case of a body in balance, we use the equilibrium of forces, but we can

also consider non-balanced situations and use the fundamental laws of Newton. These

equations are usually applied in an elementary volume V contained in and also involve the

boundary of V, denoted V, with surface element ds, and the outward normal from V, denoted

n.

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Moreover, we need to introduce a constitutive law of the material, in order to link the

stress tensor that appears in the equilibrium equation with the strain tensor and so with the

displacements of the body.

The equation coming from our previous considerations is valid in the entire domain and

we must add another relation, called a boundary condition, which describes what happens at

the boundary of the domain, and another relation which describes the initial state of ourunknown function u. By convention, we choose the instant t=0 to be the initial time, and we

impose an initial condition

u(t=0,x)=u0(x),

where u0 is the function giving the initial distribution of the displacement in the domain . The

type of boundary condition depends on the physical context. If the domain is fixed across its

boundary , the displacement satisfies the Dirichletboundary condition

u(t,x)=0 for all xand t>0.

If the domain is assumed to be free across its boundary, then the slope of the deformed shape

across the boundary is zero and the displacement satisfies the Neumannboundary condition

( , ) ( ) ( , ) 0u

t x n x u t xn

for all x and t>0,

where n is the unit outward normal to .

1.1.3. Some classical models

In this section we shall quickly describe some classical models, which mostly involve

applications of Civil Enginnering. Our goal is not to study in depth the specific details of these

examples, but to present the principal classes of PDEs which can appear in our problems. For

simplicity, we shall nondimensionalize all the variables, which will allow us to set the

constants in the models equal to 1.

The wave equation:

The wave equation models propagation of waves or vibration. For example, in two

space dimensions it is a model to study the vibration of a stretched elastic membrane, like the

skin of a drum (see Figure 1.1).

Figure 1.1 Stretched elastic membrane. ( [21] )

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The fact that a mathematical model is a Cauchy problem or a boundary value

problem does not automatically imply that it is a good model. The expression good

modelis not used here in the sense of the physical relevance of the model and of its

results, but in the sense of its mathematical coherence. As we shall see, this

mathematical coherence is a necessary condition before we can consider numerical

simulations and physical interpretations. The mathematician Jacques Hadamard gavea definition of what is a good model, while speaking about well-posed problems

( an ill-posed problem is the opposite of a well-posed problem ). We denote by f the

data ( the right-hand side, the initial conditions, the domain, etc. ), u the solution

sought, and A the operator which acts on u. We are using abstract notation, A

denotes simultaneously the PDE and the type of initial or boundary conditions. The

problem is therefore to find u, the solution of:

A(u)=f . ( 1.4 )

Definition 1.3: We say that problem ( 1.4 ) is well-posed if for all data f it has a

unique solution u, and if this solution u depends continuously on the data f.

Let us examine Hadamards definition in detail: it contains, in fact, three

conditions for the problem to be well-posed. First, a solution must at least exist: this is

the least we can ask of a model supposed to represent reality! Second, the solution

must be unique: this is more delicate since, while it is clear that, if we want to predict

tomorrows weather, it is better to have sun or rain ( with an exclusive or ) but

not both with equal chance, there are other problems which reasonably have several

or an infinity of solutions. For example, problems involving finding the best route

often have several solutions: to travel from the South to the North Pole then any

meridian will do, likewise, to travel by plane from Paris to New York, your travel

agency sometimes makes you go via Brussels or London, rather than directly, because

it can be more economic. Hadamard excludes this type of problem from his definition

since the multiplicity of solutions means that the model is indeterminate: to make the

final choice between all of those that are best, we use another criterion ( which has

been forgotten until now ), for example, the most practical or most comfortable

journey. This is a situation of current interest in applied mathematics: when a model

has many solutions, we must add a selection criterion to obtain the good solution.

Third, and this is the least obvious condition a priori, the solution must depend

continuously on the data. At first sight, this seems a mathematical fantasy, but it is

crucial from the perspective of numerical approximation. Indeed, numerically

calculating an approximate solution of ( 1.4 ) amounts to perturbing the data ( whencontinuous becomes discrete ) and solving ( 1.4 ) for the perturbed data. If small

perturbations of the data lead to large perturbations of the solution, there is no chance

that the numerical approximation will be close to reality ( or at least to the exact

solution ). Consequently, this continuous dependence of the solution on the data is an

absolutely necessary condition for accurate numerical simulations. We note that this

condition is also very important from the physical point of view since measuring

apparatus will not give us absolute precision: if we are unable to distinguish between

two close sets of data which can lead to very different phenomena, the model

represented by ( 1.4 ) has no predictive value, and therefore is of almost no practical

interest.

We finish by acknowledging that, at this level of generality, the definition 1.3is a little fuzzy, and that to give it a precise mathematical sense we should say in

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1.2 Variational formulation of elliptic problems ( [1] )

1.2.1 Introduction

In this chapter we are interested in the mathematical analysis of elliptic partial

differential equations ( PDEs ). In general, these elliptic equations correspond to stationaryphysical models, that is, models which are independent of time. We shall see that boundary

value problems are well-posed for these elliptic PDEs, that is, they have a solution which is

unique and depends continuously on the data. The approach that we shall follow is called the

variational formulation. This approach has a very natural physical or mechanical

interpretation, and it will be crucial for understanding the finite element method that we

explain later.

In this chapter, the prototype example of elliptic PDEs will be the Laplacian for which

we shall study the following boundary value problem,

in ,0 on ,

u fu

( 1.6 )

where we impose the Dirichlet boundary conditions. In ( 1.6 ), is an open set of the space

RN, is its boundary, f is a right-hand side data for the problem, and u is the unknown.

The plan of this chapter is the following. First, we recall some integration by parts

formulas, called Greens formulas, then we define the variational formulation. Finally, we

refer to Lax-Milgram theoremwhich will be the essential tool allowing us to show existence

and uniqueness of the solutions of the variational formulation.

1.2.2 Classical formulation

The classical formulation of ( 1.6 ), which might appear natural at first sight, is to

assume sufficient regularity for the solution u so that equations ( 1.6 ) have a meaning at every

point of or of . First we recall some notation related to spaces of regular functions.

Definition 1.6: Let be an open set of RN, and its closure. We denote by C()( respectively, C( ) ) the space of continuous functions in ( respectively, in ). Let k0 bean integer. We denote by Ck() ( respectively, Ck( ) ) the space of functions k timescontinuously differentiable in ( respectively, in ).

A classical solution ( we also say strong solution ) of ( 1.6 ) is a solution

u )()(2 CC , which implies that the right-hand side f must be in C().This classical

formulation, unfortunately, has a number of problems. Without going into detail, we note that,

under the single hypothesis f (C ), there is not in general a solution of class C2for ( 1.6 ) ifthe dimension of the space is greater than two ( N2). In fact, a solution does exist, as we shall

see later, but it is not of class C2( it is a little less regular except if the data f is more regular

than C( ) ).

In what follows, to study ( 1.6 ), we shall replace its classical formulation by a so-called

variational formulation, which is much more advantageous. The principle of the variational

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approach for the solution of PDEs is to replace the equation by an equivalent so-called

variational formulation obtained by integrating the equation multiplied by an arbitrary

function, called a test function. As we need to carry out integration by parts when establishing

the variational formulation, we start by giving some essential results on this subject.

1.2.3 Greens formulas

In this section is an open set of the space RN( which may be bounded or unbounded),

whose boundary is denoted by . We also assume that is a regularopen set of class C1.It is

not necessary to understand absolutely the precise definition of a regular open set, to follow the

rest of this course. It is enough to know that an open regular set is roughly speaking an open set

whose boundary is a regular hypersurface ( a manifold of dimension N-1 ), and this open set is

locally situated on one side of its boundary. We then define the outward normal at the

boundary as being the unit vector n=( n i)1iN normal at every point to the tangent plane of

and pointing to the exterior of . In RN we denote by dx the volume measure, orLebesque measure of dimension N. On , we denote by dsthe surface measure, or Lebesquemeasure of dimension N-1 on the manifold . The principal result of this section is the

following theorem.

Theorem 1 ( Greens formula ): Let be a regular open set of class C1. Let w be a C1( )function with bounded support in the closure . Then w satisfies Greens formula,

( )( ) ( ) ,

i

i

w xdx w x n x ds

x

( 1.7 )

where niis the ith component of the unit outward normal to .

Theorem 1 has many corollaries which are all immediate consequences of Greens

formula ( 1.7 ). We present here some of them, which are useful for our examples.

Corollary 1 ( Integration by parts formula ):Let be a regular open set of class C1. Let u

and y be two C1( ) functions with bounded support in the closed set . Then they satisfy theintegration by parts formula,

( ) ( )( ) ( ) ( ) ( ) ( )ii i

y x u xu x dx y x dx u x y x n x dsx x

. ( 1.8 )

Proof. It is enough to take w=yu in Theorem 1.

Corollary 2: Let be a regular open set of class C1. Let u be a function of C2( ) and y afunction of C1( ), both with bounded support in the closed set . Then they satisfy theintegration by parts formula,

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( )( ) ( ) ( ) ( ) ( ) ,

u xu x y x dx u x y x dx y x ds

n

( 1.9 )

where

1i i N

uu

x

is the gradient vector of u, and

uu n

n

.

Proof.We apply Corollary 1 to y andi

u

x

and we sum in i.

1.2.4 Variational formulation

To simplify the presentation, we assume that the open set is bounded and regular,

and that the right-hand side f of ( 1.6 ) is continuous on . The principal result of this sectionis the following proposition.

Proposition 1: Let u be a function of C2( ). Let X be the space defined by,

X = { C1( ) such that = 0 on }.

Then u is a solution of the boundary value problem ( 1.6 ) if and only if u belongs to X and

satisfies the equation,

dx = dx for every y X. ( 1.10 ) )()( xyxu )()( xyxf

Equation ( 1.10 ) is called the variational formulation of the boundary value problem ( 1.6 ).

Proof. If u is a solution of the boundary value problem ( 1.6 ), we multiply the equation by

yX and we use the integration by parts formula of Corollary 2.

)()( xyxu dx = dx + )()( xyxu( )

( ) (u x

)y xn

ds ,

where y = 0 on since yX, therefore,

dx = dx )()( xyxu )()( xyxf

which is nothing other than the formula ( 1.10 ). Conversely, if uX satisfies ( 1.10 ), by usingthe integration by parts formula in reverse we obtain,

dx = 0 for every y )())()(( xyxfxu X.

As (u+f) is a continuous function, we conclude that -u(x) = f(x) for all x. In addition,

since uX, we recover the boundary condition u=0 on , that is, u is a solution of theboundary value problem ( 1.6 ).

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Remark 1: An immediate consequence of the variational formulation ( 1.10 ) is that it is

meaningful if the solution u is only a function of C1( ), as opposed to the classicalformulation ( 1.6 ) which requires u to belong to C2( ). We therefore already suspect that it iseasier to solve ( 1.10 ) than ( 1.6 ) since it is less demanding on the regularity of the solution.

In the variational formulation ( 1.10 ), the function y is called the test function. The

variational formulation is also sometimes called the weak form of the boundary value problem

( 1.6 ). In mechanics, the variational formulation is known as the principle of virtual work.

Remark 2:We can rewrite the variational formulation ( 1.10 ) in compact notation: find u Xsuch that,

a(u,y) = L(y) for every yX,with

a(u,y) = dx )()( xyxuand

L(y) = dx, )()( xyxf

where a(,) is a bilinear form on X and L() is a linear form on X. It is in this abstract form that

we solve ( under some hypotheses ) the variational formulation in the next section.

The principal idea of the variational approach is to show the existence and

uniqueness of the solution of the variational formulation ( 1.10 ), which implies the same result

for the equation ( 1.6 ) because of Proposition 1. Indeed, we shall see that there is a theory,

both simple and powerful, for analyzing variational formulations. Nonetheless, this theory only

works if the space in which we look for the solution and in which we take the test functions ( inthe preceding notation, the space X ) is a Hilbert space, which is not the case for X = { y C1( ) such that y = 0 on } equipped with the natural scalar product for this problem. Themain difficulty in the application of the variational approach will therefore be that we must use

a space other than X, that is the Sobolev space H 1 () which is indeed a Hilbert space.0

At this point, we mention once more that the objective of this work is not a rigorous

mathematical justification of the validity of the equations used, but instead the practical interest

that these new formulations give to our problems. However, as we shall see later, neglecting

details that are relevant with functional spaces could prove to be dangerous sometimes.

1.2.5 Lax-Milgram theory

We describe an abstract theory to obtain the existence and the uniqueness of the

solution of a variational formulation in a Hilbert space. We denote by V a real Hilbert space

with scalar product and norm || ||. Following Remark 2 we consider a variational

formulation of the type,

,

find uV such that a(u,y) = L(y) for every yV. ( 1.11 )

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The hypotheses on a and L are:

(1) L() is a continuous linear form on V, that is, yL(y) is linear from V into R and there

exists C>0 such that:

|L(y)| C ||y|| for all yV,

(2)

a(,) is a bilinear form on V, that is, ua(u,y) is a linear form from V into R for all

yV, and ya(u,y) is a linear form from V into R for all u V,

(3) a(,) is continuous, that is, there exists M>0 such that:

|a(u,y)| M ||u|| ||y|| for all u,yV, ( 1.12 )

(4)

a(,) is coercive ( or elliptic ), that is, there exists v>0 such that:

a(u,u) v ||u||2 for all uV. ( 1.13 )

Theorem 2 ( Lax-Milgram ):Let V be a real Hilbert space, L() a continuous linear form on

V, a(,) a continuous coercive bilinear form on V. Then the variational formulation ( 1.11 )

has a unique solution. Further, this solution depends continuously on the linear form L.

Remark 3: The need to replace the space C1( ) with H 1 (), in order to apply the Lax-

Milgram theorem for the Laplacian is non-trivial and need the use of Sobolev spaces.

Moreover, the equivalence between the classical and the variational formulation is also out of

the purposes of this Thesis, and can be found in [1]. The main reason that we presented thevariational formulation above is the use of the Finite Element Method, that we present in the

sequel.

0

1.2.6 System of linearized elasticity

We apply the variational formulation to the solution of the system of linearized

elasticity equations. We start by describing the mechanical model. These equations model the

deformations of a solid under the hypothesis of small deformations and small displacements

( this hypothesis allows us to obtain linear equations, from which we have linear elasticity ).

We consider the stationary elasticity equations , that is, independent of time. Let be an openbounded set of RN. Let a force f(x) be a function from into RN. The unknown u ( the

displacement ) is also a function from into RN. The mechanical modeling uses the

deformation tensor, denoted by e(u), which is a function with values in the set of symmetric

matrices,

e(u) = Njii

j

j

it

x

u

x

uuu

,1)(

2

1))((

2

1,

as well as the stress tensor ( another function with values in the set of symmetric matrices )

which is related to e(u) by Hookes law,= 2e(u) + tr(e(u))I,

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where and are the Lame coefficients of the homogeneous isotropic material which occupies

. For thermodynamic reasons the Lame coefficients satisfy,

> 0 and 2+N> 0.

We add, to this constitutive law, the balance of forces in the solid,

-div= f in

where, by definition, the divergence of is the vector of components,

N

j j

ij

xdiv

1

.

Using the fact that tr(e(u)) = divu, we deduce the equations for 1 i N,

N

j

iij

i

j

j

i

j

fdivux

u

x

u

x1)( in , ( 1.14 )

with fiand u i, for 1 i N, the components of f and u in the canonical basis of RN. By adding

a Dirichlet boundary condition, and by using vector notation, the boundary value problem is,

fIuetruediv )))(()(2( in

u=0 on . ( 1.15 )

To find the variational formulation we multiply each equation ( 1.14 ) by a test function w i

( which is zero at the boundary to take account of the Dirichlet boundary conditions ) and

we integrate by parts to obtain,

dxwfdxx

wdivudx

x

w

x

u

x

uN

j

ii

i

i

j

i

i

j

j

i

1

.

We sum these equations, for i going from 1 to N, in order to obtain the divergence of the

function w=(w1, , wN) and to simplify the first integral as,

N

ji i

j

j

i

i

j

j

iN

ji j

i

i

j

j

i weuex

w

x

w

x

u

x

u

x

w

x

u

x

u

1,1,

)()(22

1.

Choosing H 1 ()0Nas the Hilbert space, we obtain the variational formulation:find uH 1 ()0

N

such that,

wdxfdivudivwdxdxweue )()(2 w H1

0 ()N.

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Theorem 3: Let be an open bounded set of RN. Let fL2()N. There exists a unique (weak)solution u H 1 ()0

Nof ( 1.15 ).

In order to prove this Theorem, we need to use the Lax-Milgram Theorem for the

variational formulation of ( 1.15 ). The proof, and especially the verification of the coercivity

of the bilinear form, is delicate and we refer to [1] (pg 137-138) for a detailed proof. In effect,

to introduce other boundary conditions ( for example, Neumann ) on part of the boundary, the

proof of the coercivity of the variational formulation becomes much more difficult.

In practice, all the boundary is not fixed and often a part of the boundary is free to

move, or surface forces are applied to another part. These two cases are modeled by Neumann

boundary conditions which are written here:

n=g on , ( 1.16 )

where g is a vector valued function. The Neumann condition ( 1.16 ) is interpreted by saying

that g is a force applied on the boundary. If g=0, we say that no force is applied and theboundary can move without restriction: we say that the boundary is free.

We shall now consider the elasticity system with mixed boundary conditions ( a

mixture of Dirichlet and of Neumann ), that is,

-div(2e(u)+tr(e(u))I)=f in

u=0 on D ( 1.17 )

n=g on N,

where (D,N) is a particion of such that the surface measures of D and N are

nonzero. The analysis of this new boundary value problem is more complicated than in the

case of Dirichlet boundary condition. We just cite a Theorem that guarantees the existence anduniqueness of solution of ( 1.17), and we refer to [1] (pg 140-141) for a full proof.

Theorem 4:Let be a regular open bounded connected set of class C1of RN. Let fL2()Nand gL2()N. We define the space:

V={u such that u=0 on NH )(1 D}. ( 1.18 )

There exists a unique (weak) solution uV of ( 1.17 ) which depends linearly and continuouslyon the data f and g.

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1.3 Finite Element Method ( [1],[2] )

1.3.1 Variational approximation

1.3.1.1

Introduction

In this chapter, we present the method of finite elements which is the numerical

method of choice for the calculation of solutions of elliptic boundary value problems, but is

also used for parabolic or hyperbolic problems. The principle of this method comes directly

from the variational approachthat we have studied in detail in the preceding chapters.

The idea at the base of the finite element method is to replace the Hilbert space V on

which we pose the variational formulation by a subspace Vh of finite dimension. The

approximate problem posed over Vhreduces to the simple solution of a linear system, whose

matrix is called the stiffness matrix. In addition, we can choose the construction of Vhin such

a way that the subspace Vhis a good approximation of V and that the solution uhin Vhof the

variational formulation is close to the exact solution u in V.

1.3.1.2 General internal approximation

We again consider the general framework of the variational formalism introduced in

Chapter 1.2. Given a Hilbert space V, a continuous and coercive bilinear form a(u,w), and a

continuous linear form L(w), we consider the variational formulation,

find uV such that a(u,w)=L(w) wV, ( 1.19 )

which we know has a unique solution by the Lax-Milgram Theorem. The internal

approximationof ( 1.19 ) consists of replacing the Hilbert space V by a finite dimensional

subspace Vh, that is to look for the solution of,

find uhVhsuch that a(uh,wh)=L(wh) whVh. ( 1.20 )

The solution of the internal approximation ( 1.20 ) is easy as we show in the following lemma.

Lemma 1:Let V be a real Hilbert space, and Vha finite dimensional subspace. Let a(u,w) be a

continuous and coercive bilinear form over V, and L(w) a continuous linear form over V. Then

the internal approximation ( 1.20 ) has a unique solution. In addition, this solution can beobtained by solving a linear system with a positive definite matrix ( and symmetric if a(u,w) is

symmetric ).

Proof.The existence and uniqueness of uhVh, the solution of ( 1.20 ), follows from the Lax-Milgram Theorem 2 applied to Vh. To put the problem in a simpler form, we introduce a basis

(j) of VhNj1 h

. If uh= , we set U

hN

j

jju1

h=(u1,,uh) the vector in R h of coordinates of u .

Problem ( 1.20 ) is equivalent to,

N

N

N

h

find such that ahh RU )(,1

i

N

j

ijj Luh

h

Ni1 ,

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which can be written in the form of a linear system

KhUh=bh, ( 1.21 )

with, forhNji ,1 ,

),(ijijh

aK , iih Lb )( .

The coercivity of the bilinear form a(u,w) implies the positive definite character of the matrix

Kh, and therefore its invertibility. In engineering applications the matrix Kh is called the

stiffness matrix.

We shall now compare the error caused by replacing the space V by its subspace V h.

More precisely, we shall bound the difference ||u-uh|| where u is the solution in V of ( 1.19 )

and uhthat in Vhof ( 1.20 ). We denote by v>0 the coercivity constant and M>0 the continuity

constant of the bilinear form a(u,w). The following lemma shows that the distance between theexact solution u and the approximate solution u h is bounded uniformly with respect to the

subspace Vhby the distance between u and Vh.

Lemma 2: (Cea) We use the hypotheses of Lemma 1. Let u be the solution of ( 1.19 ) and u h

that of ( 1.20 ). We have,

hh Vwv

M

inf ||u-wh||. ( 1.22 )||u-uh||

Finally, to prove the convergence of this variational approximation, we give a last

general lemma. Recall that in the notation Vh the parameter h>0 does not have a practical

meaning. Nevertheless, we shall assume that it is in the limit h0 that the internal

approximation ( 1.20 ) converges to the variational formulation ( 1.19 ).

Lemma 3:We use the hypotheses of Lemma 1. We assume that there exists a subspace V V

which is dense in V and a mapping r

hfrom Vinto Vh(called an interpolation operator) such

that,

||w-r0

limh

h(w)||=0 w V. ( 1.23 )

Then the method of internal variational approximation converges, that is,

||u-u0

limh

h||=0. ( 1.24 )

The strategy indicated by Lemmas 1, 2, and 3 above is now clear. To obtain a

numerical approximation of the exact solution of the variational problem ( 1.19 ), we

must introduce a finite dimensional space Vh, then solve a simple linear system associated

with the internal variational approximation ( 1.20 ).Nevertheless, the choice of Vh is notobvious. It must satisfy two criteria:

1. We must construct an interpolation operator rh from Vinto Vhsatisfying ( 1.23 ).

2. The solution of the linear system KhUh=bhmust be economical.

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1.3.1.3 Finite element method (general principles)

The principle of the finite element method is to construct internal approximation spaces

Vh from the usual functional spaces , whose definition is based on the

geometrical concept of a mesh of the domain . A mesh is a tessellation of the space by very

simple elementary volumes: triangles, tetrahedral, parallelopipeds. We shall later give a precisedefinition of a mesh in the framework of the finite element method.

),(),( 0 HH11

In this context the parameter h of Vhcorresponds to the maximum size of the mesh or

the cells which comprise the mesh. Typically, a basis of V h will be composed of functions

whose support is localizedin one or few elements. This will have two important consequences:

on the one hand, in the limit h0, the space Vh will be more and more large and will

approach little by little the entire space V, and on the other hand, the stiffness matrix Khof the

linear system ( 1.21 ) will be sparse, that is, most of its coefficients will be zero ( which will

limit the cost of the numerical solution ).

The finite element method is one of the most effective and most popular methods of

numerically solving boundary value problems. It is the basis of innumerable industrial software

packages.

1.3.2 Finite elements in N=1 dimension

To simplify the exposition, we will present the finite element method in one space

dimension. Without loss of generality we choose the domain =(0,1). In one dimension a mesh

is simply composed of a collection of points (xj)0jn+1such that:

x0= 0 < x1< < xn< xn+1 = 1.

The mesh will be called uniformif the points xjare equidistant, that is,

xj=jh with h=1

1

n, 10 nj .

The points xjare also called the verticesor nodes of the mesh. In all that follows, we denote by

Pkthe set of polynomials, with real coefficients, of one real variable with degree less than or

equal to k.

1.3.2.1

P1finite elements

The P1finite element method uses the discrete space of globally continuous functions

which are affine on each element,

Vh= { u such that u|])1,0([C [xj,xj+1] 1P for all nj0 }, ( 1.24 )

and on its subspace,

V0h= { u V hsuch that u(0) = u(1) =0 }. ( 1.25 )

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The P1finite element method is then simply the method of internal variational approximation

applied to spaces Vhor V0hdefined by ( 1.24 ) or ( 1.25 ).

We can represent the functions of Vhor V0h, which are piecewise affine, with the help

of very simple basis functions. We introduce the hat function defined by,

(x) = 1-|x| if |x| 1 ,0 if |x| > 1.

If the mesh is uniform, for 10 nj we define the basis functions (see Figure 3),

j(x) =

h

xx j. ( 1.26 )

x0=0 x1 x2 xj xn xn+1=1

vh

j

Figure 3: Mesh of =(0,1) and P1finite element basis functions.

Lemma 4: The space Vh, defined by ( 1.24 ), is a subspace of H1 (0,1) of dimension n+2, and

every function uhVhis defined uniquely by its values at the vertices (xj) 1 ,0 nj

uh(x) =

1

0

)()(n

j

jjh xxu 1,0x .

Likewise, V0h, defined by ( 1.25 ), is a subspace of H (0,1) of dimension n, and

every function

1

0

hh Vu 0 is defined uniquely by its values at the vertices (xj) ,nj1

uh(x) =

n

j

jjh xxu

1

)()( 1,0x .

The basis (j), defined by ( 1.26 ), allows us to characterize a function of Vh by its

values at the nodes of the mesh. In this case we talk of Lagrange finite elements.

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As we mentioned before, using the Finite Element Method to approximate the solution

of a PDE ends up with the solution of a linear system of the form,

KhUh= bh.

As the basis functions jhave a small support, the intersection of supports of jand

i, which form in fact the stiffness matrix (Kh)is often empty and most of the coefficients of

Kh are zero. For example, in N=1 dimension, using the P 1 F.E.M. results in a tridiagonal

stiffness matrix.

The exact evaluation of the right-hand side bh can be difficult or impossible if the

function f is complicated. In practice, we use quadrature formulas ( or numerical integration

formulas ) which give an approximation of the integrals b h. For example, we can use the

midpoint formula:

1

2)(1 1

1

i

i

x

x

ii

ii

xx

dxxxx .

1.3.2.2 Convergence and error estimation

We first of all define an interpolation operatorrh:

Definition 1.7: The P1 interpolation operator is the linear mapping from H1(0,1) into Vh

defined, for all uH1(0,1), by:

(rhu)(x) = .

1

0 ( ) ( )

n

j jj u x x

The interpolant rhu of a function u is simply the function which is piecewise affine and

coincides with u on the vertices of the mesh xj.

Lemma 5: Let rhbe the P1interpolation operator. For every u ), it satisfies:1,0(1

H

.0||||lim)1,0(0

1 H

hh

uru

Moreover, if , then there exists a constant C independent of h such that:)1,0(2

Hu

.||||||||)1,0()1,0( 21 LHh

uChuru

In our opinion, a full presentation of the Finite Element Method is out of the purposes

of this Thesis. For a more complete presentation and understanding of the FEM, we refer to

[1]. The main idea to remember is that FEM approximatesa solution of our equation and uses

several methods, according to the polynomials which will substitute our initial function. The

inevitable use of different basis functions for each method, results in methods with differentbenefits and drawbacks. In our examples, we use the P1and P2method.

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19

Considering again that we work in N=1 dimension, the principal advantage of P 2finite

elements is that, if the solution is regular, then the convergence of the method is quadratic

( the rate of convergence is proportional to h2) while the convergence for P1finite elements is

only linear ( proportional to h ).Of course, this advantage has a price: there are twice as many

unknowns ( exactly 2n+1 instead of n for P 1finite elements ) therefore the matrix is twice as

large, also the matrix has five nonzero diagonals instead of three in the P 1case. Let us remarkthat if the solution is not regular, there is no theoretical ( or practical ) advantage in the use of

P2finite elements rather than P1. The fact that the unknowns are twice more is due to the fact

that we also need the values of the function at the midpoints, in order to decompose a second

degree polynomial on a basis.


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2.2 Definitions and notation ( [1] )

Optimization has a particular vocabulary: let us introduce some classical notation and

definitions. We consider principally some minimization problems ( knowing that it is enoughto change the sign to obtain a maximization problem).First of all, the space in which the problem lies, denoted as V, is assumed to be a

normed vector space, that is to say equipped with a norm denoted by ||u||. The space V can be

the space RN, or a real Hilbert space, or even some other space. We also have a subset

where we will look for the solution: we say that K is the set of admissible elements of the

problem, or that K defines the constraints imposed on the problem. Finally, the criterion, orthe cost function, or the objective function, to be minimized, denoted by J, is a function

defined over K with values in R. The problem studied will therefore be denoted as,

VK

. ( 2.1 ))(inf uJVKu

For the maximization problems, the notation max replaces inf. Let us specify some basic

notations.

Definition 2.1: We say that u is a local minimum ( or minimum point ) of J over K if and only

if,

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2.3 Categories of optimization problems ( [1],[20] )

2.3.1 Continuous versus discrete optimization

As we mentioned earlier, in some optimization problems the variables make sense onlyif they take on integer values. The obvious strategy of ignoring the integrality requirement,

solving the problem with real variables, and then rounding all the components to the nearest

integer is by no means guaranteed to give solutions that are close to optimal. The mathematical

model is changed by adding the constraint Zxij for all i and j, to the existing constraints.

The problem is then known as an integer programming problem.

The generic term discrete optimization usually refers to problems in which the

solution we seek is one of a number of objects in a finite set.

Continuous optimization problems are normally easier to solve, because the

smoothness of the functions makes it possible to use objective and constraint information at a

particular point x to deduce information about the functions behaviour at all points close to x.

The same statement cannot be made about discrete problems, where points that are close insome sense may have markedly different function values.

Some models contain variables that are allowed to vary continuously and others that

can attain only integer values. We refer to these as mixed integer programming problems.

2.3.2

Constrained and unconstrained optimization

The most important distinction between optimization algorithms includes the existence

or not of constraints. Sometimes it is safe to disregard some natural constraints on the

variables, and assume that they have no effect on the optimal solution. Unconstrained

problems arise also as reformulations of constrained optimization problems, in which the

constraints are replaced by penalization terms in the objective function that have the effect of

discouraging constraint violations.

Constrained problems arise from models that include explicit constraints on the

variables. These constraints may be simple bounds, more general linear constraints, or

nonlinear inequalities that represent complex relationships among the variables.

2.3.3 Nature of objective function and constraints

According to the nature of the objective function and constraints, we classifyoptimization problems to linear, nonlinear and convex.

In linear programming problems, the objective as well as the constraints are all linear

functions. Such kind of problems appear mostly in management and transportation problems.

The term nonlinear characterizes problems, where at least some of the constraints or

the objective are nonlinear functions. Such problems tend to arise naturally in the physical

sciences and engineering.

Finaly, the term convex programming is used to describe a special case of

constrained optimization problems, in which:

The objective function is convex. The equality constraint functions are linear.

The inequality constraint functions are concave.

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2.3.4 Global and local optimization

The fastest optimization algorithms seek only a local solution, a point at which the

objective function is smaller than at all other feasible points in its vicinity.

A special case is convex programming, in which all local solutions are also global

solutions. Linear programming problems fall in the category of convex programming.

2.3.5 Stochastic and deterministic optimization

In some optimization problems, the model cannot be fully specified because it depends

on quantities that are unknown at the time of formulation.

Frequently, however, modelers can predict or estimate the unknown quantities with

some degree of confidence. They may, for instance, come up with a number of possible

scenarios for the values of the unknown quantities and even assign a probability to each

scenario. Stochastic optimization algorithms use these quantifications of the uncertainty to

produce solutions that optimize the expected performance of the model.Here, we focus on deterministic optimization problems, in which the model is fully

specified.

2.4 Existence of a minimum ( [1] )

Generally, the question of the existence of a minimum point is out of the purpose of this

Thesis. However, it is interesting to highlight the difference on this topic between finite and

infinite dimensions. Moreover, referring to convex analysis will allow us to understand better

some results in the sequel.

2.4.1

Optimization in finite dimensions

Let us interest ourselves now in the question of the existence of minima for

optimization problems posed in finite dimensions. We shall assume in this section ( without

loss of generality ) that V=RNprovided with the usual scalar product and with

the Euclidean norm

1

N

i iiu w u w

|| ||u u u

n

. A general result guaranteeing the existence of a minimum is

the following.

Theorem 2.1: (existence of a minimum in finite dimensions)Let K be a closed nonempty set

of RN, and J a continuous function over K with values in R satisfying the property, called

infinite at infinity,

. ( 2.2 )0

(u ) a sequence in K, lim || || lim ( )n nnn n

u J u

Then there exists at least one minimum point of J over K. Further, from every minimizing

sequence of J over K we can extract a subsequence converging to a minimum point over K.

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Remark 2.1: The property ( 2.2 ), which assures that every minimizing sequence of J over K is

bounded, is automatically satisfied if K is bounded. When the set K is not bounded, this

condition means that, in K, J is infinite at infinity.

2.4.2 Optimization in infinite dimensions

We give an example showing that the existence of a minimum in infinite dimensions is

not absolutely guaranteedby conditions like those used in the statement of Theorem 2.1. This

difficulty is closely linked to the fact that in infinite dimensions the closed bounded sets are not

compact!

Example 2.2:Take the Hilbert space ( of infinite dimensions ) of square summable sequences

in R,

2

2 1

1( ) { ( ) such that },i i i

il R x x x

equipped with the scalar product1

,i ii

x y

x y . We consider the function J defined over

by,2( )l R

2

2 2

1

( ) (|| || 1) .i

i

xJ x x

i

Taking K= , we consider the problem,2( )l R

( 2.3 )

2 ( )inf ( ),

x l RJ x

For which we shall show that there does not exist a minimum point. We verify first of all that,

2 ( )

inf ( ) 0.x l R

J x

( 2.4 )

Let us introduce the sequence xnin defined by2( )l R nix

n

i for all i1. We verify easily that,

1( ) 0 when n + .nJ x

n

As J is positive, we deduce that xn is a minimizing sequence and that the minimum value is

zero. However, it is evident that there does not exist any2( )x l R such that ( ) 0J x .

Consequently, there does not exist a minimum point of ( 2.4 ). We see in this example that the

minimizing sequence xnis not compact in ( although it is bounded ).2( )l R

In order to obtain an existence result in infinite dimension, we need to add some

additional condition. Unfortunately, even these conditions are not verifiable in general.

However, we can verify it for a particular class of problems, which are very important in

theory and practice: convexminimization problems.

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2.5 Optimality conditions ( [1] )

2.5.1 Introduction

In this part, we shall obtain necessary and sometimes sufficient conditions for

minimality. The objective is in a certain way much more practical, since these optimalityconditions will be more often used to try to calculate a minimum( sometimes even without

having shown its existence! ). The general idea of optimality conditions is the same as writing

that the derivative must be zero, when we calculate the extremum of a function over R.

These conditions will therefore be expressed with the help of the first derivative

( conditions of order 1 ) or second derivative ( conditions of order 2 ). Above all we will obtain

necessaryconditions for optimality, but the use of the second derivative or the introduction of

convexity hypotheses will also allow us to obtain sufficient conditions, and to distinguish

between minima and maxima.

These optimality conditions generalize the following elementary remark: if x 0is a local

minimum point of J on the interval [a,b] R ( J being a differentiable function on [a,b] ), then

we have:

0 0 0 0 0 0( ) 0 if x , J (x ) 0 if x , , J (x ) 0 if x .J x a a b b The strategy to obtain and to prove the minimality conditions is therefore clear: we take

account of constraints to test the minimality of0

x in particular directions which respect the

constraints: we shall talk of admissible directions. We then use the definition of the derivative

( and the second order Taylor formulas ) to conclude.

2.5.2 Differentiability

From now on, we assume that V is a real Hilbert space, and that J is a continuous

function with values in R. The scalar product in V is always denoted ,u w and the associated

norm ||u||.

We start by introducing the idea of a first derivative of J, since we shall need this to

write optimality conditions. When there are several variables ( that is to say if the space V is

not R ), the good theoretical idea of differentiability, called differentiability in the sense of

Frechet, is given by the following definition.

Definition 2.5: We say that the function J, defined on a neighbourhood of u with values in

R, is differentiable in the sense of Frechet at u if there exists a continuous linear form on V,, such that,

V

L V

0

| ( ) |( ) ( ) ( ) ( ), with lim 0.

|| ||w

o wJ u w J u L w o w

w ( 2.8 )

We call L the derivative ( or the differential, or the gradient ) of J at u and we denote

( ).L J u

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Proposition 2.2 ( second order optimality condition ): We assume that K=V and that J is

0 and

twice differentiable at u. If u is a local minimum point of J, then,

( )J u ( )( , ) 0 w V.J u w w ( 2.12 )

he converse of Proposition 2.2 is also true.

.5.4 Lagrange multipliers

We shall now try to write the minimality conditions when the set K is not convex. More

precise

efinition 2.7: At every point

T

2

ly, we shall study sets K defined by equality constraintsor inequality constraints( or

both at the same time ). We start with a general remark about admissible directions.

D w K , the set:

called the cone of admissible directions at the point w.

In more visual terms, we can also say that K(w) is the set of all the vectors which are

tangent

he interest in the cone of admissible directions lies in the following result, which gives

a neces

roposition 2.3 (Euler inequality, general case): Let u be a local minimum of J over K. If J

( ) { , ( ) , ( ) ( ) ,

lim , lim 0, lim ( ) / }

n N n N

n n n n

n n n

y V w K R

w w w w y

K w

is

s at w with a curve contained in K and passing through w. In other words, K(w) is the

set of all the possible directions of variations starting from w which remain infinitesimally in

K.

T

saryoptimality condition.

P

is differentiable at u, we have,

( ), 0 K(w)J u y y .

We shall now make precise the necessary condition of Proposition 2.3 in the case where

K is giv

quality constraints

en by equalityand inequality constraints.

E

this first case we assume that K is given by,

}

In

{ , F(y)=0K y V , ( 2.13 )

here 1( ) ( ),..., ( )MF y F y F y is a mapping from V intoMRw , with M1. The necessary

optimality condition then takes the following form.

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Theorem 2.5:Take where K is given by ( 2.13 ). We assume that J is differentiable atu Ku K and that the functions (1 )

iF i M are continuously differentiable in a neighbourhood

e further assume that t F uof u. W he vectors 1( ( ))i i M

M

are linearly independent. Then, if u is a

local minimum of J over K, there exist1,..., R , called Lagrange multipliers, such that,

1

( ) ( ) 0M

i i

i

J u F u

. ( 2.14 )

roof. See [1] ( pg 307 ).

emark 2.4:It is useful to introduce the function Ldefined over

P

M

V RR by,

at we call the Lagrangian of the minimization problem of J over K. If is a local

1

( , ) ( ) ( ) ( ) ( )M

i i

i

L y J y F y J y F y

,

th u K

s aminimum of J over K, Theorem 2.5 tells us that, in the regular case, there exist MR suchthat:

( , ) 0, ( , ) 0,L L

u uy

( , ) ( ) 0L

u F u

if andu K ( , ) ( ) ( ) 0

Lu J u F u

y

since from ( 2.14 ). We can

therefore write the constraint and the optimality condition as the annihilation of the gradient

We now give a necessarysecond order optimality condition.

roposition 2.4:We take the hypotheses of Theorem 2.5 and we assume that the functions J

( the stationarity ) of the Lagrangian.

P

and ,...,1 M

F F are twice continuously differentiable and that the vectors ( ( ))F u 1i i M are

linearly independent. Let MR be the Lagrange multiplier defined by The hen

every local minimum of J over K satisfies,

orem 2.5. T

. ( 2.15 )

roof.See [1] ( pg 310 ).

1 1

( ) ( ) ( , ) 0 w K(u)= ( )MM

i i i

i i

J u F u w w F u

P

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Inequality constraints

In this second case we assume that K is given by,

{ , F ( ) 0 for 1 i M}iK y V y , ( 2.16 )

where1,...,

MF F

)w

are always functions from V into R. When we want to determine the cone of

admissible directions K(y), the situation is a little more complicated than before as all the

constraints in ( 2.16 ) do not play the same role depending on the point y where we calculate

K(y). In effect, if , it is clear that, for sufficiently small, we will also have( ) 0iF y

(iF y 0 ( we say that the constraint i is inactive at y ). If for certain indices I,

it is not clear that we can find a vector w

( ) 0iF y

V such that, for >0 sufficiently small, (y+w)satisfies all the constraints in ( 2.16 ). It will therefore be necessary to impose supplementary

conditions on the constraints, called constraint qualifications. Roughly speaking, these

conditions will guarantee that we can make variations around a point y in order to test its

optimality. There exist different types of constraint qualification ( more or less sophisticated

and general ). We shall give a definition whose principle is to look at the linearizedproblem if

it is possible to make variations respecting the linearized constraints. These calculus of

variations considerations motivate the following definitions.

Definition 2.8:Take . The setu K ( ) { {1,..., }, F ( ) 0}iI u i M u is called the set of active

constraints at u.

Definition 2.9:We say that the constraints ( 2.16 ) are qualifiedat u if and only if there

exists a direction

K

w V such that we have for all ( )i I u ,

either ( ), 0iF u w ,

or ( ), 0iF u w and is affine. ( 2.17 )iF

The direction w is in some way a re-entrant direction since we deduce from ( 2.17 )

that u w K for all 0 sufficiently small. Of course, if all the functions are affine, we

can take

iF

0w and the constraints are automatically qualified.

We can then state the necessaryoptimality conditions over the set ( 2.17 ).

Theorem 2.6:We assume that K is given by ( 2.16 ), that the functions J and1,..., MF F are

differentiable at u and that the constraints are qualified at u. Then, if u is a local minimum of J

over K, there exist1,..., 0M , called Lagrange multipliers, such that,

1

( ) ( ) 0, 0, 0 if F ( ) 0 i {1,...,M}.M

i i i i i

i

J u F u u

( 2.18 )

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Remark 2.5:We can rewrite the condition ( 2.18 ) in the following form,

,1

( ) ( ) 0, 0, F( ) 0M

i i

i

J u F u u

where 0 means that each of the components of the vector 1,..., is positive, since,

for every index {1,..., }i M , we have either ( ) 0iF u , or 0i . The fact that ( ) 0F u is

called the condition of complementary variations.

Proof.See [1] ( pg 313 ).

Equality and inequality constraints

We can of course mix the two types of constraints. We therefore assume that K is given

by,

, ( 2.19 ){ , G(y)=0, F(y) 0}K y V

where and 1( ) ( ),..., ( )NG y G y G y 1( ) ( ),..., ( )MF y F y F y are two mappings from V intoN

R andM

R . In this new context, we must give an adequate definition of the qualification of

constraints. We always denote by ( ) { {1, ..., }, ( )iI u i M F u 0}

N

the set of active inequality

constraints at .u K

Definition 2.10:We say that the constraints ( 2.19 ) are qualifiedat u if and only if the

vectors are linearly independent and there exists a direction

K

1( ( ))i iG u 1

( )N

ii

w G u

such that we have for all ,( )i I u

( ), 0iF u w . ( 2.20 )

We can then state the necessaryoptimality conditions over the set ( 2.19 ).

Theorem 2.7: Take where K is given by ( 2.19 ). We assume that J and F are

differentiable at u, that G is differentiable in a neighbourhood of u, and that the constraints are

qualified at u ( in the sense of Definition 2.10 ). Then, if u is a local minimum of J over K,

there exist lagrange multipliers

u K

1,..., and 1,..., 0 , such that,

1 1

( ) ( ) ( ) 0, 0, F(u) 0, F(u)=0.N M

i i i i

i i

J u G u F u

( 2.21 )

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2.6 Numerical algorithms ( [1],[20] )

2.6.1 Introduction

In this section we present and analyse some gradient algorithms, which will allow us tocalculate, or more exactly to approximate the solution of the optimization problems studiedabove. We also refer to some Newton algorithms, although we have not used them in our

examples. All the algorithms studied here are used effectively in practice to solve concreteoptimization problems by computer.

These algorithms are also of an iterative nature: starting from a given initial point ,

each method constructs a sequence

0u

( )n

n Nu which converges, under certain hypotheses, to the

solution u of the optimization problem considered. In this Thesis, we sometimes refer to theconvergence of these algorithms, but the proofs and the rate of convergence are out of thepurpose of this work.

In all of this section we assume that the objective function J to be minimized isa-convex and differentiable. The application of the algorithms presented here for the

minimization of convex functions which are not strongly convex can present some small

difficulties, without mentioning the great difficulties which appear when we want toapproximate the minimum of a non-convex function! Typically, these algorithms cannot

converge and oscillate between several minimum points, or worse they converge to a localminimum, very far from a global minimum.

Remark 2.6: We limit ourselves to deterministic algorithms and we say nothing about

stochastic algorithms. Besides the fact that their analysis calls on probability theory ( which wedo not discuss in this Thesis ), their use is very different. To put it simply, let us say that

deterministic algorithms are the most efficient for the minimization of convex functions, whilestochastic algorithms allow us to approximate global ( not only local ) minima of non-convexfunctions.

2.6.2 Overview of algorithms

In this section, we refer to unconstrained optimization problems, since algorithms for

constrained problems are just combinations of these basic methods and of techniques used to

treat with constraints. All algorithms for unconstrained minimization problems require the user

to supply a starting point, which we usually denote by . Beginning at , optimizationalgorithms generate a sequence of points

0

u

0

u( )

n

n Nu that terminate when either no more progress

can be made or when it seems that a solution point has been approximated with sufficient

accuracy. In deciding how to move from one iterate to the next, the algorithms use

information about the function J at , and possibly also information from earlier iterates

. They use this information to find a new iterate

nu

nu

0 1 1, ,..., nu u u 1n

u with a lower function value

than . In this work, we use monotone algorithms, although there are also non-monotone

algorithmsthat do not insist on a decrease in J at every step, but even these algorithms require

J to be decreased after some prescribed number of iterations.

nu

There are two fundamental strategies for moving from the current point to a new

iterate , the line search and the trust region method. In the line search algorithmic

strategy, the distance to move along the direction w can be found by approximately solving

nu

1nu

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the following minimization problem to find a step length n : ( 2.12 ). The

exact solution of ( 2.12 ) is expensive and usually unnecessary. Instead, the line search

algorithm generates a limited number of trial step lengths until it finds one that looselyapproximates the minimum of ( 2.12 ).

0min ( )n n nJ u w

nu

nm

In the trust region algorithmic strategy, the information gathered about J is used toconstruct a model function whose behaviour near the current point is similar to that of

the actual objective function J. Because the model may not be a good approximationof J

when u is far from , we restrict the search for a minimizer of to some region around .

In other words, we find the candidate step by approximately solving the following sub-

problem: , where

nm

nm

nu

n

nu

nw

min ( )n

n

nw

m u w n

u wn lies inside the trust region. If the candidate solution

does not produce a sufficient decrease in J, we conclude that the trust region is too large, and

we shrink it and re-solve our problem. Usually, the trust region is a ball defined by || ||n

w ,

where is called the trust-region radius.0

Each time we decrease the size of the trust region after failure of a candidate iterate, thestep from to the new candidate will be shorter, and it usually points in a different direction

from the previous candidate, unlike the line-search method.

nu

In a sense, the line-search and the trust-region approaches differ in the order in which

they choose the direction and distanceof the move to the next iterate. In trust-region, we first

choose a maximum distance ( the trust region radius n ) and then seek a direction and step

that attain the best improvement possible subject to this distance constraint. In line-search, wefollow the opposite procedure.

In this Thesis, we occupy ourselves with line-search methods and especially withgradient algorithms, which we present in the sequel of this Chapter.

2.6.3 Gradient algorithms ( case without constraints )

Let us start by studying the practical solution of optimization problems in the absenceof constraints. Let J be a function which is a-convex ( or even convex ) and differentiable

defined over the real Hilbert space V. We consider the problem without constraints,

inf . ( 2.23 )( )y V

J y

From Theorem 2.2 there exists a unique solution u, characterized by the Euler equation,

( )J u 0 .

Gradient algorithm with optimal step

The gradient algorithm consists of moving from an iterate by following the line of

greatest slope associated with the cost function J(y). The direction of descent corresponding to

this line of greatest slope from is given by the gradient . In effect, if we look for

in the form,

nu

( )n

J unu1n

u

, ( 2.24 )1n n n

u u w n

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with small and a unit vector in V, it is with the choice of direction

that we can hope to find the smallest value of ( in the absence

of other information such as higher derivatives or previous iterates).

0n

( )/n n

u

nw

|| ( ) |n

w J J u | )

n

1( n

J u

This simple remark leads us, among the methods ( 2.24 ) which are called methods of

descent, to the gradient algorithm with optimal step, in which we solve a succession ofminimization problems with one real variable ( even if V is not finite dimensional ). Starting

from an arbitrary in V, we construct a sequence ( defined by,0u )nu

, ( 2.25 )1 ( )n n nu u J u

where n R is chosen at each step such that,

. ( 2.26 )1( ) inf ( ( )n nR

J u J u J u

)n

Gradient algorithm with fixed step

The gradient algorithm with fixed step consists simply of the construction of a sequence

defined by,n

u

, ( 2.27 )1 ( )n n

u u J u n

where is a fixed positive parameter. This method is therefore simpler than the gradientalgorithm with optimal step, since at each step we save the cost of calculating the solution of

( 2.26 ).

Remark 2.7:In order to choose between the two methods that we already presented, we haveto take under consideration the benefits and drawbacks of each one, and the specific problem to

which we apply the method. Generally, the advantage of the gradient algorithm with optimalstep lies at the rate of convergence. This can prove to be very important, especially in problem

with a large amount of data. On the other hand, the gradient algorithm with fixed step is

simpler ( as we mentioned above ) and costs less at each iteration. In our examples, we use agradient algorithm with fixed step, especially in favor of simplicity.

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2.6.4 Gradient algorithms (case with constraints )

We now study the solution of optimization problems with constraints,

inf , ( 2.28 )( )y K

J y

where J is a function which is a-convex ( or even convex ) and differentiable defined over K, a

convex closed nonempty subset of the real Hilbert space V. Theorem 2.2 ensures the existenceand uniqueness of the solution u of ( 2.28 ), characterized by the condition,

( ), 0 y K J u y u . ( 2.29 )

According to the algorithms studied below, we will sometimes need to state supplementaryhypotheses over the set K.

Gradient algorithm with fixed step and projection

Theorem 2.8 ( projection over a convex set ): Let V be a Hilbert space. Let be a

convex closed nonempty subset. For all

K Vx V , there exists a unique kx K such that,

|| || min || ||ky K

x x x

y .

Equivalently, kx is characterized by the property,

kx K , , 0 y K k kx x x y . ( 2.30 )

We call kx the orthogonal projection over K of x.

The gradient algorithm with fixed step adapts to problem ( 2.28 ) with constraints

starting from the following remark. For all real >0, ( 2.29 ) is written,

( ( )), 0 yu u J u y u K . ( 2.31 )

Let us denote by KP the projection operator over the convex set K, defined in Theorem

2.8. Then, ( 2.31 ) is none other than the characterization of u as the orthogonal projection of

( )u J u over K. In other words,

( ( )) >0Ku P u J u . ( 2.32 )

It is easy to see that ( 2.32 ) is in fact equivalent to ( 2.29 ), and therefore characterizes thesolution u of ( 2.28 ). The gradient algorithm with fixed step and projectionalgorithm ( or

more simply projected gradient ) is then defined by the iteration,

, ( 2.33 )1

( (n n

K

u P u J u ))n

where is a fixed positive parameter.

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2.6.5 Penalization of constraints

We conclude this subsection by briefly describing another way to approximate aminimization problem with constraints, by a sequence of minimization problems without

constraints: this is the procedure of penalization of constraints. We avoid talking here of a

method or an algorithm since penalization of the constraints is not, properly speaking, amethod. The solution of the problems without constraints that we shall construct must be donewith the help of the algorithms of section 2.6.3. This solution can raise some difficulties, since

the penalized problem ( 2.35 ) is often ill conditioned, as for example the new objective

function may be less smooth than the initial objective and the constraints.

For simplicity, we shall take the case where , and we again consider the convex

minimization problem,

NV R

, ( 2.34 )( ) 0inf ( )

F yJ y

where J is a continuous convex function from NR into R and F is a continuous convexfunction from NR into MR .

For 0 , we then introduce the problem without constraints,

2

1

1inf ( ) [max( ( ),0]

N

M

iy R

i

J y F y

, ( 2.35 )

where the constraints are penalized. We can then state the following result, which

shows that, for small, the problem ( 2.35 ) approximates well the problem ( 2.34 ).

( ) 0iF y

Proposition 2.5:We assume that J is continuous, strictly convex, and is infinite at infinity, that

the functions are convex and continuous foriF 1 i M , and that the set,

{ , F ( ) 0 i {1,...,M}}Ni

K y R y

is nonempty. Denoting by u the unique solution of ( 2.34 ) and, for >0, uthe unique solution

of ( 2.35 ), we then have,

0limu u

.

Proof.See [1] ( pg 341-342 ).

Remark 2.8:There are also other methods of penalization, such as the barrier-methods, in

which we introduce barrier functions in order to replace the constraints. The most well-known method from this category is the log-barrier method, which can prove to be extremely

sensitive when we try to combine it with a gradient method. For this reason, we avoid the use

of this method in our examples.

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method to the minimization of a function J as explained above, it may be that the method

converges to a maximum or a saddle point of J, and does not tend to a minimum, since it only

looks for the zeros of .J

Remark 2.10: A major drawback of Newtons method is the need to know the Hessian ( )J y

( or the derivative matrix ).When the problem is very large or if J is not easily twice

differentiable, we can modify Newtons method to avoid the calculation of this matrix

= . The methods, called quasi-Newton, propose iteratively calculating an

approximation of . We replace the formula ( 2.34 ) by:

( )F y

1))

n

( )J y ( )F yn

S ( (

S

F u

.1 ( ) for n 0n n n n

u u S F u

In general, we calculate by a recurrence formula of the type:n

1n n

S S C

n

1)

where is a matrix of rank 1 which depends on , chosen so that

converges to 0.

nC

( (

1, , ( ), (n n n n

u u F u F u

1))n nS F u

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3. Shape Optimization ( [2] )

3.1 Introduction

A problem of optimal design of structures, or shape optimization in mechanics, is

defined by three data:

1.a model ( typically an equation with partial derivatives ) which allows us to evaluate

and analyse the mechanical behaviour of a structure,

2. a criterionof whose we should search for a minimum or maximum and eventually

several criteria

3. an admissible direction of the variables of optimization that take account of theeventual constraints that are imposed on the variables.

A problem of shape optimization is a problem where the variables of optimization are

the shapes of the structure themselves. The shape optimization is evidentially essential in many

applications, but it is clearly more complicated than the traditional optimization, in which the

variables are, for example, the mechanical properties of the materials.

Among the problems of shape optimization, we can distinguish three great categories,

from the simplest to the most difficult.

1. the parametric shape optimization, in which the shapes are parametrized by a

reduced number of variables ( for example the thickness, a diameter, the dimensions ), whoselimit takes under consideration the variety of admissible shapes,

2. the geometrical shape optimization, in which, starting from an initial shape, we

authorize the variations of the boundaries of the shape ( without however changing the

topology of the shape, that is, in the 2-dimensional space, the number of components that

connect the edges or , more simply, the number of the holes in the shape),

3. the topological shape optimization, in which we search, without no restriction

explicit or implicit, the best possible shape, free to change the topology.

The last type of optimization is, for sure, the most general but also the most difficult.

Note that, if the definition of the topology of ones shape is simple enough in the 2-

dimensional space, it is clearly more complicated in 3-d, where the interpretation is not only

the number of components that connect the edges of the shape ( which allows, for example, to

differentiate a flat ball into a hollow crown ) but also, between others, its number of handles

or the loops ( think the difference between a swell, a tore, a bretzel, etc. ). In this Thesis we

do not refer to topological optimization, and so we escape the technical complications that are

necessary in order to define precisely this notion of topology.

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The questions that can be posed to the problems of shape optimization are the following

ones:

1.theoretical questions for the existence, the uniqueness, or the qualitative properties of

the solutions. We will only speak very little about this here, except when that has direct

implications to the questions of numerical calculation,

2. optimality conditions, which are very important not only from the point of

theoretical sight, but also from the point of numerical sight ( they are often at the base of the

numerical algorithms or the gradient method type),

3.numerical calculation of the approximate optimal shape.

We should principally concentrate on the target of understanding the numerical

algorithms of the shape optimization ( that is necessary for a good comprehension of the

subjacent theory ). We notice that we will be privileged from a continue approach of these

problems, with detriment of the discrete approach which has its partisans but whichcamouflages a little the true stakes and the crucial points of the analysis. By the continue

approach, we want to say that we will suppose that the mechanical model is indeed an equation

with partial derivatives, that will oblige us to work with spaces of functions. In contrast, the

discrete appr

michail id is

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