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Optimal Shape Design for Elliptic Systems (Scientific Computation): Olivier Pironneau

The study of optimal shape design can be arrived at by asking the following question: "What is the best shape for a physical system?" This book is an applications-oriented study of such physical systems; in particular, those which can be described by an elliptic partial differential equation and where the shape is found by the minimum of a single criterion function. There are many problems of this type in high-technology industries. In fact, most numerical simulations of physical systems are solved not to gain better understanding of the phenomena but to obtain better control and design. Problems of this type are described in Chapter 2.

Traditionally, optimal shape design has been treated as a branch of the calculus of variations and more specifically of optimal control. This subject interfaces with no less than four fields: optimization, optimal control, partial differential equations (PDEs), and their numerical solutions-this is the most difficult aspect of the subject. Each of these fields is reviewed briefly: PDEs (Chapter 1), optimization (Chapter 4), optimal control (Chapter 5), and numerical methods (Chapters 1 and 4).

If a computer program is used to yield the numerical solution of a PDE describing an optimal shape design problem, an optimization algorithm will have to be written (usually a gradient algorithm is used). Optimal control theory provides the basic techniques for computing the derivatives of the criteria functions with respect to the boundary. So in essence, optimal control and optimization may be applied when the "control" becomes associated with the shape of the domain (Chapter 6). However, problems are encountered with numerical discretization; thus two chapters are devoted to applications with finite elements (Chapter 7) and in a finite difference, or boundary element context (Chapter 8). Finally, two industrial applications are included (Chapter 9) for a practical illustration of the theory studied. Chapter 2 deals with the problem of the existence of solutions to PDEs. A study of this more theoretical question sheds some light on certain difficulties of convergence that are encountered in practice.

Optimal shape design has been studied in great depth by the French School of Applied Mathematics (at the Universities of Paris and Nice, in particular), and this book presents this approach. It represents a special blend of mathematics and engineering which some engineers may find too theoretical and applied mathematicians too computational. The best compromise between these two opposing points of view is difficult to find.

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