Topological Derivatives for Semilinear Elliptic Equations

Mohamed Iguernane; Serguei Nazarov; Jean-Rodolphe Roche; Jan Sokolowski; Katarzyna Szulc

doi:10.2478/v10006-009-0016-4

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Topological Derivatives for Semilinear Elliptic Equations

International Journal of Applied Mathematics and Computer Science

Volume 19 (2009): Issue 2 (June 2009)

By: Mohamed Iguernane, Serguei Nazarov, Jean-Rodolphe Roche, Jan Sokolowski and Katarzyna Szulc

Open Access

|Jul 2009

Abstract

The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L∞ norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

References

Amstutz, S. (2006). Topological sensitivity analysis for some nonlinear PDE system, Journal Mathematiques Pures et Appliquées 85(4): 540-557.10.1016/j.matpur.2005.10.008
Search in Google Scholar
Bucur, D. and Buttazzo, G. (2005). Variational Methods in Shape Optimization Problems, Birkhäuser, Boston, MA.10.1007/b137163
Search in Google Scholar
Casas, E. and Mateos, M. (2002). Uniform convergence of the FEM. Applications to state constrained control problems, Journal of Computational and Applied Mathematics 21(1): 67-100.
Search in Google Scholar
Ciarlet, P. G. (1978). The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.10.1115/1.3424474
Search in Google Scholar
Ciarlet, P. G. and Raviart, P. A. (1972). General Lagrange and Hermite interpolation in with applications to finite element methods, Archive for Rational Mechanics and Analysis 46(3): 177-199.10.1007/BF00252458
Search in Google Scholar
Demlov, A. (2007). Sharply localized pointwise and W^-1∞ estimates for finite element methods for quasilinear problems, Mathematics of Computation 76(260): 1725-1741.10.1090/S0025-5718-07-01983-7
Search in Google Scholar
Frehse, J. and Rannacher, R. (1978). Asymptotic L∞-error estimates for linear finite element approximations of quasilinear boundary value problems, SIAM Journal on Numerical Analysis 15(2): 418-431.10.1137/0715026
Search in Google Scholar
Fulmanski, P., Lauraine, A., Scheid, J.-F. and Sokołowski, J. (2007). A level set method in shape and topology optimization for variational inequalities, International Journal of Applied Mathematics and Computer Science 17(3): 413-430.10.2478/v10006-007-0034-z
Search in Google Scholar
Gilbarg, D. and Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin.10.1007/978-3-642-61798-0
Search in Google Scholar
Haug, E. J. and Céa, J. (1981). Optimization of distributed parameter structures, Proceedings of the NATO Advanced Study Institute on Optimization of Distributed Parameter Structural Systems, Iowa City, IO, USA, NATO Advanced Study Institute Series E: Applied Sciences, Vol. 49, Martinus Nijhoff Publishers, The Hague.
Search in Google Scholar
Il'in, A. M. (1989). Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Nauka, Moscow, (in Russian).
Search in Google Scholar
Jackowska-Strumillo, L., Sokołowski, J., Zochowski, A. and Henrot, A. (2002). On numerical solution of shape inverse problems, Computational Optimization and Applications 23(2): 231-255.10.1023/A:1020528902875
Search in Google Scholar
Kondratiev, V. A. (1967). Boundary problems for elliptic equations in domains with conical or angular points, Trudy Moskovskogo Matematicheskogo Obszhestva 16: 209-292, (in Russian).
Search in Google Scholar
Ladyzhenskaya, O. A. and Ural'tseva, N. N. (1968). Linear and Quasilinear Elliptic Equations, Academic Press, New York, NY London.
Search in Google Scholar
Landkof, N. S. (1966). Fundamentals of Modern Potential Theory, Nauka, Moscow, (in Russian).
Search in Google Scholar
Mazja, V. G., Nazarov, S. A. and Plamenevskii, B. A. (1981). On the asymptotic behavior of solutions of elliptic boundary value problems with irregular perturbations of the domain, Problemy Matematicheskogo Analiza 8: 72-153.
Search in Google Scholar
Mazja, V. G., Nazarov, S. A. and Plamenevskii, B. A. (1991). Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten, Gebieten. Bd. 1, Akademie-Verlag, Berlin; English translation: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. 1, Birkäuser Verlag, Basel, 2000.
Search in Google Scholar
Mazja, V. G. and Plamenevskii, B. A. (1978). Estimates in L_p and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary, Matematische Nachrichten 81(1): 25-82, (in Russian).
Search in Google Scholar
Mazja, V. G. and Plamenevskii, B. A. (1973). On the behavior of solutions to quasilinear elliptic boundary-value problems in a neighborhood of a conical point, Zapiski Nauchnych Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta (LOMI) 38: 91-97.
Search in Google Scholar
Nazarov, S. A. and Plamenevsky, B. A. (1973). Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin.
Search in Google Scholar
Nazarov, S. A. and Sokolowski, J. (2006). Self-adjoint extensions for the Neumann Laplace and applications, Acta Mathematica Sinica 22(3): 879-906.10.1007/s10114-005-0652-z
Search in Google Scholar
Nazarov, S. A. and Sokolowski, J. (2003). Asymptotic analysis of shape functional, Journal de Mathématiques Pures et Appliquées 82(2): 125-196.10.1016/S0021-7824(03)00004-7
Search in Google Scholar
Pólya, G. P. and Szegö, G. (1951). Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, NJ.10.1515/9781400882663
Search in Google Scholar
Raviart, P. A. and Thomas, J.M. (1983). Introduction à l'analyse numérique des équations aux dérives partielles Masson, Paris.
Search in Google Scholar
Sokolowski, J. and Zochowski, A. (1999). Asymptotic analysis of shape functional, Numerische Mathematik 102(1): 145-179.
Search in Google Scholar
Sokolowski, J. and Zochowski, A. (2005). Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer-Verlag, Berlin.
Search in Google Scholar
Sokolowski, J. and Zochowski, A. (1999). On topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272.10.1137/S0363012997323230
Search in Google Scholar
Sokolowski, J. and Zochowski, A. (2003). Optimality conditions for simultaneous topology and shape optimization, SIAM Journal on Control and Optimization 42(4): 1198-1221.10.1137/S0363012901384430
Search in Google Scholar
Sokolowski, J. and Zochowski, A. (2001). Topological derivatives of shape functional for elasticity systems, Mechanics of Structures and Machines 29(3): 331-349.10.1081/SME-100105654
Search in Google Scholar
Sokolowski, J. and Zochowski, A. (1999). On topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272.10.1137/S0363012997323230
Search in Google Scholar
Stampacchia, G. (1965). Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Annales de I'Institut Fourier 15: 189-258.10.5802/aif.204
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10006-009-0016-4 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X

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Language: English

Page range: 191 - 205

Published on: Jul 8, 2009

Published by: University of Zielona Góra

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

shape optimization,

topological derivative,

levelset method,

variational inequality,

asymptotic analysis

Related subjects:

Mathematics,

Applied mathematics

© 2009 Mohamed Iguernane, Serguei Nazarov, Jean-Rodolphe Roche, Jan Sokolowski, Katarzyna Szulc, published by University of Zielona Góra
This work is licensed under the Creative Commons License.

Volume 19 (2009): Issue 2 (June 2009)