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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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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.
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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,
Ku and 0 , Kw , ||w-u||
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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
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