fictitious domain method for solving optimal shape design

RAIRO M ODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

J. H ASLINGER K.-H. H OFFMANN ˇ M. KO CVARA Control/fictitious domain method for solving optimal shape design problems RAIRO – Modélisation mathématique et analyse numérique, tome 27, no 2 (1993), p. 157-182.

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T T P l MATHEMATICALMOOELUKGAHDNUMERICALANAIYSIS \ } \ ! \ 11 MODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE

(Vol. 27, n 2, 1993, p. 157 à 182)

CONTROL/FICTITIOUS DOMAIN METHOD FOR SOLVING OPTIMAL SHAPE DESIGN PROBLEMS (*) by J. HASLINGER (•), K.-H. HOFFMANN (2) and M. KOCVARA (3) Communicated by O. PIRONNEAU

Abstract. — Combining a fictitious domain and an optimal control approach, we present a new method for the numerical realization of optimal shape design problems. This approach enables us to perform ail calculations on a fixed domain with a fixed grid. Finite element approximation is studied. Résumé. — On présente une méthode nouvelle pour la résolution numérique des problèmes de Voptimisation de la forme. Cette méthode est basée sur la combinaison d'une méthode des domaines fictifs avec une méthode de contrôle optimal. L'avantage de cette approche est le fait qu'on peut réaliser tous les calculs sur un domaine et un maillage fixe.

Shapc opiimization is a branch of the optimal control theory, in which the control variable is connectée with the geometry of the problem. Mathematical analysis, including the approximation theory and the numerical realization, has been widely discussed in [4, 7]. The numerical realization of optimal shape design problems has spécifie features. One of them is the fact that the state problem is solved many times on the domain, changing during the computation. For domains with complicated shapes this requires the use of mesh generators. Moreover, data, defining the finite dimensional approximation (stiffness matrix, etc.) have to be recomputed again and again. As a resuit, the whole procedure is time consuming and hence expensive. To overcome this difficulty, the control/fictitious domain method is proposed. The method has been used in [1] for the numerical solution of the Helmholtz équation. Nevertheless, the same approach can be used in the (;:) Received for publication April 29, 1991. (]) Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 12000 Praha 2, C/cchoslovakia. l-)

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tram ol the shape optimization. Introducing the new control variable on the right-hand side of the state problem, one can transform the original optimal shape design problem into a new one, in which the state problem is formulated and solved on a given, fixed domain Û with the same differential operators. The advantage of this approach is obvious : choosing the simple shape of Û, the construction of triangulation of Û is elementary and what is more important, ail computations are realized with the same stiffness matrix A. Using the factorization of A, one can solve very efficiently the corresponding System of algebraic équations. An alternative approach, based on the control on the boundary is suggested in [9]. The present paper deals with the mathematical analysis and the approximation of the control/fictitious method for the numerical solution of the optimal shape design problems, with the state given by the homogeneous Dirichlet problem in the first part. The second part of the paper deals with the numerical realization of the method. For the sake of simplicity, numerical experiments are performed for the state, given by an ordinary differential équation. I ne experiments tor partial ditterential équations, as well as the mathematical analysis of the control/fictitious method for the state, given by variational inequalities, will be presented in subséquent papers. 1. INTRODUCTION

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