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A moving mesh fictitious domain approach for shape optimizationproblems

Published online by Cambridge University Press:  15 April 2002

Raino A.E. Mäkinen ,
Tuomo Rossi  and
Jari Toivanen
Raino A.E. Mäkinen
Affiliation:
Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35 (MaE), 40351 Jyväskylä, Finland. ([email protected])
Tuomo Rossi
Affiliation:
Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35 (MaE), 40351 Jyväskylä, Finland. ([email protected])
Jari Toivanen
Affiliation:
Department of Mathematical Information Technology, University of Jyväskylä, P.O. Box 35 (MaE), 40351 Jyväskylä, Finland. ([email protected])
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Abstract

A new numerical method based on fictitious domain methods for shapeoptimization problems governed by the Poisson equation is proposed.The basic idea is to combine the boundary variation technique, in whichthe mesh is moving during the optimization, and efficient fictitiousdomain preconditioning in the solution of the (adjoint) state equations.Neumann boundary value problems are solved using an algebraic fictitiousdomain method. A mixed formulation based on boundary Lagrangemultipliers is used for Dirichlet boundary problems and the resultingsaddle-point problems are preconditioned with block diagonal fictitiousdomain preconditioners. Under given assumptions on the meshes, thesepreconditioners are shown to be optimal with respect to the conditionnumber. The numerical experiments demonstrate the efficiency ofthe proposed approaches.

Keywords

Shape optimizationfictitious domain methodpreconditioningboundary variation techniquesensitivity analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 1 , January 2000 , pp. 31 - 45
Copyright
© EDP Sciences, SMAI, 2000

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References

Astrakhantsev, G.P., Method of fictitious domains for a second-order elliptic equation with natural boundary conditions. USSR Comput. Math. Math. Phys. 18 (1978) 114-121. CrossRef
Atamian, C., Dinh, G.V., Glowinski, R., He, J. and Périaux, J., On some imbedding methods applied to fluid dynamics and electro-magnetics. Comput. Methods Appl. Mech. Engrg. 91 (1991) 1271-1299. CrossRef
Babuska, I., The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. CrossRef
Bespalov, A., Kuznetsov, Yu.A., Pironneau, O. and Vallet, M.-G., Fictitious domain with separable preconditioners versus unstructured adapted meshes. Impact Comput. Sci. Eng. 4 (1992) 217-249. CrossRef
Börgers, C., A triangulation algorithm for fast elliptic solvers based on domain imbedding. SIAM J. Numer. Anal. 27 (1990) 1187-1196. CrossRef
Börgers, C. and Widlund, O.B., On finite element domain imbedding methods. SIAM J. Numer. Anal. 27 (1990) 963-978. CrossRef
Braibant, V. and Fleury, C., Shape optimal design using B-splines. Comput. Methods Appl. Mech. Engrg. 44 (1984) 247-267. CrossRef
Bramble, J.H., The Lagrangian multiplier method for Dirichlet's problem. Math. Comp. 37 (1981) 1-11.
Bramble, J.H., Pasciak, J.E. and Schatz, A.H., The construction of preconditioners for elliptic problems by substructuring, I. Math. Comp. 47 (1986) 103-134. CrossRef
Brockman, R.A., Geometric sensitivity analysis with isoparametric finite elements. Comm. Appl. Numer. Math. 3 (1987) 495-499. CrossRef
Chan, T.F., Analysis of preconditioners for domain decomposition. SIAM J. Numer. Anal. 24 (1987) 382-390. CrossRef
J. Danková and J. Haslinger, Fictitious domain approach used in shape optimization: Neumann boudary condition, in Control of Partial Differential Equations and Applications (Laredo, 1994), Lecture Notes in Pure and Appl. Math., Dekker, New York 174 (1996) 43-49.
Danková, J. and Haslinger, J., Numerical realization of a fictitious domain approach used in shape optimization. I. Distributed controls. Appl. Math. 41 (1996) 123-147.
P. Duysinx, W.H. Zhang and C. Fleury, Sensitivity analysis with unstructured free mesh generators in 2-D and 3-D shape optimization, in Structural Optimization 93, Vol. 2, Rio de Janeiro (1993) 205-212.
P.E. Gill, W. Murray and M.H. Wright, Practical Optimization. Academic Press, New York (1981).
R. Glowinski, T. Hesla, D.D. Joseph, T.-W. Pan and J. Périaux, Distributed Lagrange multiplier methods for particulate flows, in Computational Science for the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler Eds., Wiley, Chichester (1997) 270-279.
Glowinski, R. and Kuznetsov, Yu.A., On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrande multiplier method. C.R. Acad. Sci. Paris Sér. I Math. 327 (1998) 693-698. CrossRef
Glowinski, R., Pan, T.-W., Kearsley, A.J. and Périaux, J., Numerical simulation and optimal shape for viscous flow by a fictitious domain method. Internat. J. Numer. Methods Fluids 20 (1995) 695-711. CrossRef
Glowinski, R., Pan, T.-W. and Périaux, J., A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Engrg. 111 (1994) 283-303. CrossRef
A. Greenbaum, Iterative Methods for Solving Linear Systems. Frontiers in Applied Mathematics, SIAM, Philadelphia, PA, USA 17 (1997).
Haslinger, J., Imbedding/control approach for solving optimal shape design problems. East-West J. Numer. Math. 1 (1993) 111-119.
J. Haslinger, Comparison of different fictitious domain approaches used in shape optimization. Tech. Rep. 15, Laboratory of Scientific Computing, University of Jyväskylä (1996).
Haslinger, J., Hoffmann, K.H. and Kocvara, M., Control/fictitious domain method for solving optimal shape design problems. RAIRO Modél. Math. Anal. Numér. 27 (1993) 157-182. CrossRef
Haslinger, J. and Jedelský, D., Genetic algorithms and fictitious domain based approaches in shape optimization. Structural Optimization 12 (1996) 257-264. CrossRef
Haslinger, J. and Klarbring, A., Fictitious domain/mixed finite element approach for a class of optimal shape design problems. RAIRO Modél. Math. Anal. Numér. 29 (1995) 435-450. CrossRef
J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd ed., Wiley, Chichester (1996).
J. He, Méthodes de domaines fictifs en méchanique des fluides applications aux écoulements potentiels instationnaires autour d'obstacles mobiles. Ph.D. thesis, Université Paris VI (1994).
E. Heikkola, Y. Kuznetsov, T. Rossi and P. Tarvainen, Efficient preconditioners based on fictitious domains for elliptic FE-problems with Lagrange multipliers, in ENUMATH 97 - Proceedings of the 2nd European Conference on Numerical Mathematics and Advanced Applications, H.G. Bock, G. Kanschat, R. Rannacher, F. Brezzi, R. Glowinski, Yu.A. Kuznetsov and J. Périaux Eds., World Scientific Publishing Co., Inc., River Edge, NJ (1998) 646-661.
Kunisch, K. and Peichl, G., Shape optimization for mixed boundary value problems based on an embedding method. Dynam. Contin. Discrete Impuls. Systems 4 (1998) 439-478.
Kuznetsov, Yu.A., Efficient iterative solvers for elliptic finite element problems on nonmatching grids. Russian J. Numer. Anal. Math. Modelling 10 (1995) 187-211.
Yu.A. Kuznetsov, Iterative analysis of finite element problems with Lagrange multipliers, in Computational Science for the 21st Century, M.-O. Bristeau, G. Etgen, W. Fitzgibbon, J.L. Lions, J. Périaux and M.F. Wheeler Eds., Wiley, Chichester (1997) 170-178.
Yu.A. Kuznetsov, M.F. Wheeler, Optimal order substructuring preconditioners for mixed finite element methods on nonmaching grids. East-West J. Numer. Math. 3 (1995) 127-143.
Mäkinen, R., Finite-element design sensitivity analysis for non-linear potential problems. Comm. Appl. Numer. Math. 6 (1990) 343-350. CrossRef
Marchuk, G.I., Kuznetsov, Yu.A. and Matsokin, A.M., Fictitious domain and domain decomposition methods. Soviet J. Numer. Anal. Math. Modelling 1 (1986) 3-35. CrossRef
NAG, The NAG Fortran Library Manual: Mark 18. NAG Ltd, Oxford (1997).
Paige, C.C. and Saunders, M.A., Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12 (1975) 617-629. CrossRef
O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer-Verlag, New York (1984).
Proskurowski, W. and Vassilevski, P.S., Preconditioning capacitance matrix problems in domain imbedding. SIAM J. Sci. Comput. 15 (1994) 77-88. CrossRef
T. Rossi, Fictitious Domain Methods with Separable Preconditioners. Ph.D. thesis, Department of Mathematics, University of Jyväskylä (1995).
Rossi, T. and Toivanen, J., A parallel fast direct solver for block tridiagonal systems with separable matrices of arbitrary dimension. SIAM J. Sci. Comput. 20 (1999) 1778-1793. CrossRef
J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer-Verlag, Berlin (1992).
Swarztrauber, P.N., The methods of cyclic reduction and Fourier analysis and the FACR algorithm for the discrete solution of Poisson's equation on a rectangle. SIAM Rev. 19 (1977) 490-501. CrossRef
J. Toivanen, Fictitious Domain Method Applied to Shape Optimization. Ph.D. thesis, Department of Mathematics, University of Jyväskylä (1997).
L. Tomas, Optimisation de Forme et Domaines Fictifs: Analyse de Nouvelles Formulations et Aspects Algorithmiques. Ph.D. thesis, École Centrale de Lyon (1997).