This project concerns a very rich and interesting topic in mathematics, the optimization of shapes. Shape optimization is the field where we look for the best shape (or at least a better one) for some given criterion. It requires competencies in geometry, partial differential equations, calculus of variations, mechanics, scientific computing, and optimization. Along with the exponential increase of the industrial demand for more mathematical and numerical modelization, interest for shape optimization has, in particular, increased tremendously over the last thirty years. On the industrial side, this activity ranges from classical problems in the aeronautics industry (find the shape of a wing or of the whole plane which ensures better drag), in structural mechanics (try to minimize the weight) to what might seem simpler problems such as the design of glassware, but which also pose quite a number of interesting and difficult problems.
Several research teams all over the world have put much effort to understand shape optimization and serious progress has indeed been made on the mathematical analysis, numerical approximation and simulation and applications to a large range of shape problems.
Despite (or because) their false simplicity, several problems are still open, although formulated hundreds year ago (see for instance the recent books by Bucur and Buttazzo [BB05], Henrot [He06], Henrot and Pierre [HP05]) and give rise to important debates within the international scientific community.
The main scientific challenge is the qualitative study of optimal shapes for some classes of shape functional involving partial differential equations and/or geometric quantities associated naturally with shapes: measure, perimeter, curvature. We are particularly interested to handle non-smooth and singular shapes (often with constraints, among them the convexity constraint turns out to be very important and difficult), to understand the minimal regularity under which one can extract information from the optimality conditions, to prove this regularity to obtain qualitative information about the optimal shape and to compute them.
Precisely, in our project we will concentrate on the following topics:
- Spectral problems
- Critical shapes and optimality conditions
- Inverse problems in Fluids
- Optimal shapes with convexity constraints
- Computation of optimal shapes
Moreover, as a transverse theme to these five topics, we will also concentrate on the analysis of the regularity of optimal shapes. Active research is done both to understand analytical aspects and to develop efficient numerical algorithms. Since the first existence results due to D. Chenais in the seventies
[Ch75], there was a major step forward in the study of both the existence question and the much more difficult question of regularity of optimal shapes (free boundaries). Alt and Caffarelli [AC81] were the first to prove in 1981 that for a class of energy minimizing free boundary problems, the solution has smoothness properties in relationship with the regularity of minimal surfaces. Concerning Dirichlet boundary conditions, general existence of solution was proved by Buttazzo and Dal Maso in 1993 [BD93], who exhibited a class of shape functionals for which the relaxation does not occur, and for which an optimal shape (a quasi-open set) is a minimizer. Finally, a most surprising result was due to Sverak in 1993 [Sv93] who proved in two space dimensions that a topological constraint, on the number of holes, is crucial for the existence. This last result opened the way to the intensive use of potential theory and of fine regularity properties of solutions of partial differential equations on non-smooth domains, in the study of shape optimization problems. Homogeneous Neumann conditions on the free part of the boundary are studied in a different framework, which is not directly related to our project (see for instance the book of Allaire [Al02]).
Concerning numerical methods, together with the classical descent algorithms by shape gradient and the relaxation/homogenization methods, several new tools were developed in the last decade for the computation of optimal shapes, among which we refer to fictitious materials, phase field, global stochastic optimization by using genetic algorithms, the bubble method, or the topological gradient, the level set method, material distribution optimization where the mass is allowed to concentrate on lower dimensional structures , etc. (see for instance [Al02,HP05,SZ92]).
[Al02] Allaire, G. Shape optimization by the homogenization method. Applied Mathematical Sciences, 146. Springer-Verlag, New York, 2002.
[AC81] Alt, H. W.; Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math. 325 (1981), 105-144.
[BB05] Bucur, D., Buttazzo, G. Variational methods in shape optimisation problems. Progress in Nonlinear Differential Equations and their Applications, 65. Birkhauser Boston, Inc., Boston, MA, 2005.
[BD93] Buttazzo, Giuseppe; Dal Maso, Gianni An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993), no. 2, 183-195.
[Ch75] Chenais, Denise On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52 (1975), no. 2, 189-219.
[He06] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhauser Verlag, Basel, 2006.
[HP05] A. Henrot, M. Pierre, Variation et optimisation de formes, Mathématiques et Applications 48, Springer, 2005.
[SZ92] Sokolowski, Jan; Zolésio, Jean-Paul Introduction to shape optimization. Shape sensitivity analysis. Springer Series in Computational Mathematics, 16. Springer-Verlag, Berlin, 1992.
[Sv93] Sverak, V. On optimal shape design. J. Math. Pures Appl. (9) 72 (1993), no. 6, 537--551.