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2009 | 19 | 2 | 191-205
Tytuł artykułu

Topological derivatives for semilinear elliptic equations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
19
Numer
2
Strony
191-205
Opis fizyczny
Daty
wydano
2009
otrzymano
2008-04-28
poprawiono
2008-10-27
Twórcy
  • Laboratoire de Mathématiques, Institut Elie Cartan, Université Henri Poincaré, UMR 7502 Nancy-Université-CNRS-INRIA, Nancy 1, B.P. 239, 54506 Vandoeuvre lès Nancy Cedex, France
  • Institute of Mechanical Engineering Problems, V.O., Bolshoi pr., 61, 199178, St. Petersburg, Russia
  • Laboratoire de Mathématiques, Institut Elie Cartan, Université Henri Poincaré, UMR 7502 Nancy-Université-CNRS-INRIA, Nancy 1, B.P. 239, 54506 Vandoeuvre lès Nancy Cedex, France
  • Laboratoire de Mathématiques, Institut Elie Cartan, Université Henri Poincaré, UMR 7502 Nancy-Université-CNRS-INRIA, Nancy 1, B.P. 239, 54506 Vandoeuvre lès Nancy Cedex, France
  • Laboratoire de Mathématiques, Institut Elie Cartan, Université Henri Poincaré, UMR 7502 Nancy-Université-CNRS-INRIA, Nancy 1, B.P. 239, 54506 Vandoeuvre lès Nancy Cedex, France
Bibliografia
  • Amstutz, S. (2006). Topological sensitivity analysis for some nonlinear PDE system, Journal Mathematiques Pures et Appliquées 85(4): 540-557.
  • Bucur, D. and Buttazzo, G. (2005). Variational Methods in Shape Optimization Problems, Birkhäuser, Boston, MA.
  • 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.
  • Ciarlet, P. G. (1978). The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam.
  • Ciarlet, P. G. and Raviart, P. A. (1972). General Lagrange and Hermite interpolation in Rn with applications to finite element methods, Archive for Rational Mechanics and Analysis 46(3): 177-199.
  • Demlov, A. (2007). Sharply localized pointwise and $W_∞^{-1}$ estimates for finite element methods for quasilinear problems, Mathematics of Computation 76(260): 1725-1741.
  • 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.
  • 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.
  • Gilbarg, D. and Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin.
  • 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.
  • Il'in, A. M. (1989). Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Nauka, Moscow, (in Russian).
  • 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.
  • 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).
  • Ladyzhenskaya, O. A. and Ural'tseva, N. N. (1968). Linear and Quasilinear Elliptic Equations, Academic Press, New York, NY London.
  • Landkof, N. S. (1966). Fundamentals of Modern Potential Theory, Nauka, Moscow, (in Russian).
  • 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.
  • 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.
  • Mazja, V. G. and Plamenevskii, B. A. (1978). Estimates in Lp 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).
  • 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.
  • Nazarov, S. A. and Plamenevsky, B. A. (1973). Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin.
  • Nazarov, S. A. and Sokolowski, J. (2006). Self-adjoint extensions for the Neumann Laplace and applications, Acta Mathematica Sinica 22(3): 879-906.
  • Nazarov, S. A. and Sokolowski, J. (2003). Asymptotic analysis of shape functional, Journal de Mathématiques Pures et Appliquées 82(2): 125-196.
  • Pólya, G. P. and Szegö, G. (1951). Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, NJ.
  • Raviart, P. A. and Thomas, J. M. (1983). Introduction à l'analyse numérique des équations aux dérives partielles Masson, Paris.
  • Sokolowski, J. and Zochowski, A. (1999). Asymptotic analysis of shape functional, Numerische Mathematik 102(1): 145-179.
  • Sokolowski, J. and Zochowski, A. (2005). Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer-Verlag, Berlin.
  • Sokolowski, J. and Zochowski, A. (1999). On topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272.
  • Sokolowski, J. and Zochowski, A. (2003). Optimality conditions for simultaneous topology and shape optimization, SIAM Journal on Control and Optimization 42(4): 1198-1221.
  • Sokolowski, J. and Zochowski, A. (2001). Topological derivatives of shape functional for elasticity systems, Mechanics of Structures and Machines 29(3): 331-349.
  • Sokolowski, J. and Zochowski, A. (1999). On topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272.
  • 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.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-amcv19i2p191bwm
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