Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T02:52:13.039Z Has data issue: false hasContentIssue false
  • English
  • Français

On an optimal shape design problem in conduction

Published online by Cambridge University Press:  11 October 2006

José Carlos Bellido
José Carlos Bellido*
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St. Giles', OX1 3LB, Oxford, UK; JoseCarlos.Bellido@uclm.es (On leave from Universidad de Castilla-La Mancha (Spain).)
Get access

Abstract

In this paper we analyze a typical shape optimization problem in two-dimensional conductivity. We study relaxation for this problem itself. We also analyze the question of the approximation of this problem by the two-phase optimal design problems obtained when we fill out the holes that we want to design in the original problem by a very poor conductor, that we make to converge to zero.

Keywords

Optimal shape designrelaxationvariational approachΓ-convergencesemiconvex envelopesquasiconvexity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 4 , October 2006 , pp. 699 - 720
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Allaire, Shape optimization by the homogenization method. Springer (2002).
Allaire, G., Bonnetier, E., Franfort, G. and Jouve, F., Shape optimization by the homogenization method. Numer. Math. 76 (1997) 27–68. CrossRef
Allaire, G. and Kohn, R.V., Optimal bounds on the effective behauvior of a mixture of two well-odered elastic materials. Quat. Appl. Math. 51 (1993) 643–674. CrossRef
Allaire, G. and Kohn, R.V., Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Europ. J. Mech. A/solids 12 (1993) 839–878.
Allaire, G. and Murat, F., Homogenization of the Neumann problem with nonisolated holes. Asymptotic Anal. 7 (1993) 81–95. With an appendix written jointly with A.K. Nandakumar.
Bellido, J.C., Explicit computation of the relaxed density coming from a three-dimensional optimal design prroblem. Nonlinear Analysis TMA 52 (2003) 1709–1726. CrossRef
Bellido, J.C. and Pedregal, P., Optimal design via variational principles: the one-dimensional case. J. Math. Pures Appl. 80 (2000) 245–261. CrossRef
Bellido, J.C. and Pedregal, P., Explicit quasiconvexification for some cost functionals depending on the derivatives of the state in optimal design. DCDS-A 8 (2002) 967–982. CrossRef
Bellido, J.C. and Pedregal, P., Optimal control via variational principles: the three dimensional case. J. Math. Anal. Appl. 287 (2003) 157–176. CrossRef
Bellido, J.C. and Pedregal, P., Existence in optimal control with state equation in divergence form via variational principles. J. Convex Anal. 10 (2003) 365–378.
M.P. Bendsøe and O. Sigmund, Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003).
A. Braides, Γ-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 22 (2002).
Briane, M., Homogenization in some weakly connected domains. Ricerche Mat. 47 (1998) 51–94.
Briane, M., Homogenization in general periodically perforated domains by a spectral approach. Calc. Var. Partial Differ. Equat. 15 (2002) 1–24. CrossRef
A. Cherkaev, Variational methods for structural optimization. Springer (2000).
G. Dal Maso, Introduction to Γ-convergence. Birkhäuser, Boston, 1993.
Fonseca, I., Kinderlehrer, D. and Pedregal, P., Energy functionals depending on elastic strain and chemical composition. Cal. Var. 2 (1994) 283–313. CrossRef
V. Girault and P.A. Raviart, Finite elements methods for Navier-Stokes equations, Theory and Algorithms. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo (1985).
S. Müller and V. Šverák, Convex integration for lipschitz mappings and counterexamples for regularity. Technical Report 26, Max-Planck Institute for Mathematics in the Sciences, Leipzig (1999).
Murat, F., Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients. Ann. Mat Pura Appl. 112 (1977) 49–68. CrossRef
P. Pedregal, Parametrized Measures and Variational Principles. Progress in Nonlinear Partial Differential Equations. Birkhäuser (1997).
Pedregal, P., Optimal design and constrained quasiconvexity. SIAM J. Math. Anal. 32 (2000) 854–869. CrossRef
Pedregal, P., Constrained quasiconvexification of the square of the gradient of the state in optimal design. Quater. Appl. Math. 62 (2004) 459–470. CrossRef
L. Tartar, Remarks on optimal design problems, in Homogenization and continuum mechanics, G. Buttazzo, G. Bouchitte, and P. Suchet Eds, Singapure World Scientific (1994) 279–296.
L. Tartar, An introduction to homogenization method in optimal design. Lect. Notes Math. Springer (2000).
V. Šverák, Lower semicontinuity of variational integrals and compesated compactness, in Proc. ICM, S.D. Chatterji Ed., Birkhäuser 2 (1994) 1153–1158.