Introduction to Shape optimization

Transcription

1 Introduction to Shape optimization Noureddine Igbida 1 1 Institut de recherche XLIM, UMR-CNRS 6172, Faculté des Sciences et Techniques, Université de Limoges 123, Avenue Albert Thomas Limoges, France. noureddine.igbida@unilim.fr

2 Introduction 1 Shape optimization has received a lot of intention in many application that require a focus on shapes instead of parameters or functions. These issues go back to antiquity, a classic example is : for a given predetermined length of fence, find the shape of the largest possible field that can be surrounded by this fence. The Greek already knew the answer to this problem (isoperimetric) is a disk. The aim of this course is to give an illustrated introduction to the problems of shape optimization. 1. Generalities 1.1 A standard optimization problem Consider V a Banach space K V a subspace of V (the set of admissible element) J : K IR a functional (cost function, objectif function, criterion...)

3 A standard optimization problem is written as Recall that inf is the supremum of all the lower bound inf J(v) or min J(v) v K v K if J has no lower bound, then inf J(v) =. v K By convention, if K = then inf J(v) =. v K Obviously, if there exists u K, such that then There exists a sequence (v n ) n IN, such that 1.2 A shape optimization problem inf J(u) = u K J(u ) inf J(Ω) = min J(u) = u K u K J(u ) lim J(v n n) = inf J(v). v K This is a minimization problem where the unknown variable runs over a class of domains. More precisely, let : V the set of subset of IR N (shapes, forms...) Introduction to Shape optimization N. Igbida 2

4 K V a subspace of V (the set of admissible sets, or admissible shapes) J : K IR a functional A shape optimization problem associated with J and K is (P ) inf Ω K J(Ω) If there exists Ω K, such that inf J(Ω) = min J(Ω) = Ω K Ω K J(Ω ) then Ω is aid to be the optimal shape, optimal form, optimal domain or optimal set. 1.3 Related questions The mathematical questions associated with shape optimization problem are numerous and various, for instance : 1. Existence of a solution : - study of the dependence of J(Ω) with respect to Ω - variation calculus of J(Ω) and/or some intermediate functionnal, with respect to Ω - compactness properties for the admissible forms 2. Necessary condition (sufficient sometimes) for the optimality : - of first order (first order derivative) - of second order (second order derivative) - the derivative here with respect to domains Introduction to Shape optimization N. Igbida 3

5 3. Qualitative and geometrical properties : - connectivity - convexity - symmetry - regularity or singularity 4. Effective calculus of the optimal form : - exhibit explicit solutions (ball, ellipsoids,...) - numerical schemes to produce approximate solutions 1.4 Related difficulties It must be noticed that, in general a shape optimization problem is quite difficult than standard optimization problem. In particular Convex functionals is not adapted : the set of admissible domains does not have any linear or convex structure, so in shape optimization problem it is meaningless to speak about convex functionals and similar notions. Topology : The mathematical proof of existence of a solution for optimization problem always use an adequate topology combined with compactness and continuity arguments to prove the existence of a solution for the optimization problems. Indeed, recall that there exists a sequence (Ω n ) n IN, such that So, if lim J(Ω n n) = inf J(Ω). Ω K Introduction to Shape optimization N. Igbida 4

6 Ω n Ω, in a sense to be precise (Topology on families of domains) J is lower semi-continuous on K, then J(Ω ) lim inf J(Ω n n) and J(Ω ) = inf J(Ω). Ω K Even, if several topologies on families of domains are available, in general there is not a priori choice of a topology in order to apply the direct methods of the calculus of variations, for obtaining the existence of at least an optimal domain. 1.5 Applications Here we give some applications and common problems in shape optimization : Isoperimetric problem (Dido s problem) : This is the most classical problem of a shape optimization. It can be formulated in the following way : find, among all admissible domains with a given perimeter (this explains the term isoperimetric ), the one whose Lebesgue measure is as large as possible. Equivalently, one could minimize the perimeter of a set among all admissible domains whose Lebesgue measure is prescribed. The name of the Queen Dido is often associated with a variant of this problem (the case where a part of the boundary is fixed). Indeed : A long time ago, about three hundred years after the great city of Troy found its ruin, a young woman with the name of Dido had to flee from her homeland, Phoenicia. Her tyrant brother had killed her wealthy husband and was after her riches now. With several boats full of belongings and many people fleeing with her, she sailed away and reached the coast of Africa. The natives there Introduction to Shape optimization N. Igbida 5

7 weren t too happy about the newcomers, but Dido was able to make a deal with their king: she promised him a fair amount of money for as much land as she could mark out with a bull s skin. The king thought he was getting the better end of the deal, but he soon noticed that the woman he was dealing with was smarter than he had expected. Dido took her bull skin and cut it into thin strips which she sewed together into one long string. She then took the seashore as one edge for her piece of land and laid the skin into a half-circle. Like this she got a much bigger piece of land than the king had thought possible, on which she founded the city of Carthage. The Newton s problem of optimal aerodynamical profiles The problem of finding the best aerodynamical profile for a body in a fluid stream under some constraints on its size is another classical question which can be considered as a shape optimization problem. This problem was first considered by Newton, who gave a rather simple variational expression for the aerodynamical resistance of a convex body in a fluid stream, assuming that the competing bodies are radially symmetric, which makes the problem onedimensional. Optimal Dirichlet regions It is Dirichlet problem over an unknown domain, which has to be optimized according to a given cost functional. Optimal mixtures of two conductors An interesting question that can be seen in the form of a shape optimization problem is the determination of the optimal distribution of two given conductors (for instance in the thermostatic model, where the state function is the temperature of the system) into a given set. Mass optimization problem Suppose we are given two density of a given electric charge. We interpret IR N as an insulating medium, into which we place a fixed amount Introduction to Shape optimization N. Igbida 6

8 of some conducting material described by their conductivity (= inverse resistivity). We imagine then the resulting steady current flow within IR N from the positive charge to the negative charge, and ask if we can optimize the placement of the conducting material so as to minimize the heating induced by the flow. Optimal design for structures Shape optimization has received a lot of attention in recent years, particularly in relation to a number of applications in physics and engineering, where the name of optimal design for structures is more common. In general, a Shape optimization is defined by three ingredients : a model (typically a partial differential equation) to evaluate (or analyse) the mechanical behavior of a structure an objective function which has to be minimized or maximized, or sometimes several objectives (also called cost functions or criteria) a set of admissible designs which precisely defines the optimization variables, including possible constraints. In particular Optimal design problems can roughly be classified in three categories from the easiest ones to the most difficult ones: Parametric or sizing optimization : optimization for which designs are parametrized by a few variables (for example, thickness or member sizes), implying that the set of admissible designs is considerably simplified, Geometric optimization : optimization for which all designs are obtained from an initial guess by moving its boundary (without changing its topology, i.e., its number of holes Introduction to Shape optimization N. Igbida 7

9 in 2-d), Topology optimization : optimization where both the shape and the topology of the admissible designs can vary without any explicit or implicit restrictions. 2. Plan of the course Preliminaries on functional analysis Preliminaries on PDE Geometrical optimization : topologies for families of sets Geometrical optimization : isoperimetric problem Geometrical optimization : existence results Parametric optimization Introduction to Shape optimization N. Igbida 8