The topological sensitivity analysis provides an asymptotic expansion of a shape function with respect to the insertion of a small hole or obstacle inside a domain. This expansion can then be used for shape optimization. In this paper, such an expansion is obtained for the Stokes equations with general shape functions and arbitrarily shaped holes. A numerical example illustrates the use of the topological sensitivity in a shape optimization problem.

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In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain.In this paper we study higher order derivatives. But one can ask the following questions:-are they expensive to calculate? -are they complicated to use? -are they imprecise? -are they useless?At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Mathematics Subject Classification (1991): 49Q10Correspondence to: M. Masmoudi 232 Ph. Guillaume and M. Masmoudi

Abstract. New non-reflecting boundary conditions are introduced for the solution of the Helmholtz equation in a waveguide. These boundary conditions are perfectly transparent for all propagating modes. They do not require the determination of these propagating modes but only their propagation constants. A quasi-local form of these boundary conditions is well suited as terminating boundary condition beyond finite element meshes. Related convergence properties to the exact solution and optimal error estimates are established.

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