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An algorithm for the identification of fronts prescribed by Signorini type boundary conditions. (Un algorithme d’identification de frontières soumises à des conditions aux limites de Signorini.) (French) Zbl 0954.65050
The purpose of this paper is to study a question concerning the geometrical inverse problem, which consists in finding the shape of an unknown part \(\gamma\) of the boundary \(\partial\Omega\) of a two-dimensional body \(\Omega\), by using thermal measurements of some part \(M\) of the boundary. The two extremal points of the unknown boundary \(\gamma\) are supposed to be fixed, while Signorini type boundary conditions are prescribed on \(\gamma\). The problem is turned into an optimal shape one, by constructing a Kohn-Vogelius-like cost function, the only minimum of which is proved to be the unknown boundary.
Main result: The authors prove that the derivative of this cost functional with respect to a direction \(\theta\) depends only on the state function \(u^0\) and not on its Lagrangian derivative \(u^1(\theta)\). Expressions which allow to implement a gradient algorithm to solve the inverse problem are proposed. The authors obtain very good numerical results by linearization (from Signorini to mixed obundary conditions on the auxiliary Dirichlet problem) and by the use of gradient expressions, which do not depend on the derivatives of the solution.
The main interest of this work lies in the way used to overcome the difficulties arising from the nonlinear boundary conditions. It appears that this approach may be extended to the thermoelastic coupled problem, which is a more relevant problem from a physical point of view.

MSC:
65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49N45 Inverse problems in optimal control
34H05 Control problems involving ordinary differential equations
35R30 Inverse problems for PDEs
49Q12 Sensitivity analysis for optimization problems on manifolds
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References:
[1] S. Andrieux, A. Ben Abda et M. Jaoua, Identifiabilité de frontiètres inaccessibles par une mesure unique de surface. Annales Maghrébines de l’Ingénieur, 7 (1993) 5-24.
[2] A. Ben Abda, S. Chaabane, F. El Dabaghi et M. Jaoua, On a non linear geometrical inverse problem of Signorini type: identifiability and stability. Math. Meth. in the Appl. Sci.21 (1998) 1379-1398. · Zbl 0936.35189
[3] F. Ben Belgacem, Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element method. Sinum (à paraître). Zbl0974.74055 · Zbl 0974.74055
[4] F. Brezzi, W.W. Hager et P.A. Raviart Error estimates for the finite element solution of variational inequalities. Numer. Math.28 (1977) 431-443. · Zbl 0369.65030
[5] S. Chaabane et M. Jaoua, Identification of Robin coefficients by the means of boundary measurements. Inverse Problems15 (1999) 1425-1438. · Zbl 0943.35100
[6] F. Hettlich et W. Rundell Iterative methods for the reconstraction of an inverse potential problem. Inverse Problems12 (1996) 251-266. Zbl0858.35134 · Zbl 0858.35134
[7] K. Khodja et M. Moussaoui, Régularité des solutions d’un problème mêlé Dirichlet-Signorini dans un domaine polygonal plan. Comm. Partial Diff. Eq.17 (1992) 805-826. Zbl0806.35049 · Zbl 0806.35049
[8] R.V. Kohn et A. McKenney Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems6 (1990) 389-414. Zbl0718.65089 · Zbl 0718.65089
[9] R.V. Kohn et M. Vogelius, Determinig conductivity by boundary measurements; interior results. Comm. Pure Appl. Math.38 (1985) 644-667. · Zbl 0595.35092
[10] R.V. Kohn et M. Vogelius, Relaxation of a variational method for impedance computed tomography. Comm. Pure Appl. Math.40 (1987) 745-777. Zbl0659.49009 · Zbl 0659.49009
[11] K. Kunisch et X. Pan, Estimation of interfaces from boundary measurements. SIAM J. Cont. Opt.32 (1994) 867-894. Zbl0807.35162 · Zbl 0807.35162
[12] J.L.M. Lions, Quelques méthodes de résolution de problèmes aux limites non linéaires. Dunod, Paris (1969). Zbl0189.40603 · Zbl 0189.40603
[13] J.L. Lions et E. Magenes, Problèmes aux limites non homogènes et applications, tome 1. Dunod, Paris (1968). Zbl0165.10801 · Zbl 0165.10801
[14] J.R. Roche et J. Sokolowski, Numerical methods for shape identification problems. Control and Cybernetics25 (1996) 867-894. · Zbl 0876.65048
[15] J. Simon, Differentiation with respect to the domaine in boundary value problems. Num. Func. Anal. Opt.2 (1980) 649-687. Zbl0471.35077 · Zbl 0471.35077
[16] J. Sokolowski et J.P. Zolesio, Introduction to shape optimization; shape sensitivity analysis. Springer Verlag (1992). · Zbl 0761.73003
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