A moving mesh fictitious domain approach for shape optimization problems

Free access article

Issue		ESAIM: M2AN Volume 34, Number 1, January-February 2000


Page(s)		31 - 45
DOI		10.1051/m2an:2000129

DOI: 10.1051/m2an:2000129
M2AN, Vol. 34, N $^{\rm o}$ 1, 2000, pp. 31-45

A moving mesh fictitious domain approach for shape optimization problems

Raino A.E. Mäkinen
Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35 (MaE), 40351 Jyväskylä, Finland. (Raino.Makinen@mit.jyu.fi)

Tuomo Rossi
Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35 (MaE), 40351 Jyväskylä, Finland. (Tuomo.Rossi@mit.jyu.fi)

Jari Toivanen
Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35 (MaE), 40351 Jyväskylä, Finland. (Jari.Toivanen@mit.jyu.fi)

Received: March 3, 1999.

Abstract: A new numerical method based on fictitious domain methods for shape optimization problems governed by the Poisson equation is proposed. The basic idea is to combine the boundary variation technique, in which the mesh is moving during the optimization, and efficient fictitious domain preconditioning in the solution of the (adjoint) state equations. Neumann boundary value problems are solved using an algebraic fictitious domain method. A mixed formulation based on boundary Lagrange multipliers is used for Dirichlet boundary problems and the resulting saddle-point problems are preconditioned with block diagonal fictitious domain preconditioners. Under given assumptions on the meshes, these preconditioners are shown to be optimal with respect to the condition number. The numerical experiments demonstrate the efficiency of the proposed approaches.

Keywords and phrases: Shape optimization, fictitious domain method, preconditioning, boundary variation technique, sensitivity analysis.

AMS Subject Classification: 49M29, 65F10, 65K10, 65N30, 65N55.

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