An optimal shape design approach for the numerical realization of Nonhomogeneous Dam Problems
by
Abdeljalil Nachaoui
CNRS UMR6629/Université de Nantes, BP 92208, 44322, Nantes, France.
Coauthors: A. Chakib (Université Mohamed V, Faculté des Sciences, Département de Mathématiques et Informatique Rabat, Maroc), T. Ghemires (Université Mohamed V, Faculté des Sciences, Département de Mathématiques et Informatique Rabat, Maroc)

The stationary flow of fluid through an nonhomogeneous porous medium (say dam), leads to a problem posed in a domain with a partially unknown boundary separating the wet and dry part of the dam. The first idea used to solve it is the so-called Baiocchi transformation which reformulates the problem in a variational inequality solved in a known domain. This idea gives arise to many study from several points of view (see [1, 3, 5] and the references there).

This work deals with an optimal shape design approach for the numerical realization of dam problems. The Dirichlet condition is included into a least square cost functional and the unknown part of the boundary is adjusted in such a way that corresponding cost functional attains its minimum for such a shape. The idea is note new. It was used by Begis and Glowinski [2] to solve a simple problem on filtration in a rectangular dam, but without theoretical justification. Later, Ha-slinger et al. [4] used this idea in its dual form. Their approach requires the use of the primal and the dual formulation of the state problem. Also, the divergence free finite elements have to be used in the numerical realization. Our approach is based on the free boundary regularity results developed in [3] and [5]. Under some regularity assumptions about the permeability coefficient, we prove that this formulation is equivalent to the classical one. We prove the existence of a solution of the new optimal shape design formulation for a class of inhomogeneous dam problems. The standard finite element method is used to give numerical results which show the efficiency of the approach proposed.


Keywords : Dam problems, Inhomogeneous porous medium, Optimal shape design, Finite element.

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C. Baiocchi, V. Comincioli, E. Magenes, G.A. Pozzi, Free boundary problems in the theory of fluid flow through porous media Existence and Uniqueness Theorems. Ann. Mat. Pura Appl. 96, 1-82 (1973).

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D. Begis and R. Glowinski, application de la méthode des éléments finis à l'approximation d'un problème de domaine optimal. Méthodes de résolution des problèmes approchés. Appl. Math. Optimization 2, 130-169 (1975).

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A. Friedman and S.-Y. Huang, The inhomogeneous dam problem with discontinuous permeability. Ann. Scu. Norm. Sup. Pisa vol. 14, N^o4, 49-77 (1987).

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J. Haslinger, K.-H. Hoffmann and R.A.E. Makinen, Optimal control/dual approach for the numerical solution of the dam problem. Adv. Math. sci. Appl. 2, N^o1, 189-213 (1993).

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A. Lyaghfouri, The inhomogeneous dam with linear Darcy's law and Dirichlet boundary conditions, Math. Models Methods Appl. Sci. 6, N^o8 : 1051 - 1077 (1996).

Date received: July 18, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadk-32.