On an optimal shape design problem in conduction
ESAIM: Control, Optimisation and Calculus of Variations (2006)
- Volume: 12, Issue: 4, page 699-720
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topBellido, José Carlos. "On an optimal shape design problem in conduction." ESAIM: Control, Optimisation and Calculus of Variations 12.4 (2006): 699-720. <http://eudml.org/doc/249683>.
@article{Bellido2006,
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.
},
author = {Bellido, José Carlos},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal shape design; relaxation;
variational approach; Γ-convergence; semiconvex envelopes; quasiconvexity.; optimal shape design; variational approach; -convergence; quasiconvexity},
language = {eng},
month = {10},
number = {4},
pages = {699-720},
publisher = {EDP Sciences},
title = {On an optimal shape design problem in conduction},
url = {http://eudml.org/doc/249683},
volume = {12},
year = {2006},
}
TY - JOUR
AU - Bellido, José Carlos
TI - On an optimal shape design problem in conduction
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/10//
PB - EDP Sciences
VL - 12
IS - 4
SP - 699
EP - 720
AB -
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.
LA - eng
KW - Optimal shape design; relaxation;
variational approach; Γ-convergence; semiconvex envelopes; quasiconvexity.; optimal shape design; variational approach; -convergence; quasiconvexity
UR - http://eudml.org/doc/249683
ER -
References
top- G. Allaire, Shape optimization by the homogenization method. Springer (2002).
- G. Allaire, E. Bonnetier, G. Franfort and F. Jouve, Shape optimization by the homogenization method. Numer. Math.76 (1997) 27–68.
- G. Allaire and R.V. Kohn, Optimal bounds on the effective behauvior of a mixture of two well-odered elastic materials. Quat. Appl. Math.51 (1993) 643–674.
- G. Allaire and R.V. Kohn, Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Europ. J. Mech. A/solids12 (1993) 839–878.
- G. Allaire and F. Murat, Homogenization of the Neumann problem with nonisolated holes. Asymptotic Anal.7 (1993) 81–95. With an appendix written jointly with A.K. Nandakumar.
- J.C. Bellido, Explicit computation of the relaxed density coming from a three-dimensional optimal design prroblem. Nonlinear Analysis TMA52 (2003) 1709–1726.
- J.C. Bellido and P. Pedregal, Optimal design via variational principles: the one-dimensional case. J. Math. Pures Appl.80 (2000) 245–261.
- J.C. Bellido and P. Pedregal, Explicit quasiconvexification for some cost functionals depending on the derivatives of the state in optimal design. DCDS-A8 (2002) 967–982.
- J.C. Bellido and P. Pedregal, Optimal control via variational principles: the three dimensional case. J. Math. Anal. Appl.287 (2003) 157–176.
- J.C. Bellido and P. Pedregal, 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).
- M. Briane, Homogenization in some weakly connected domains. Ricerche Mat.47 (1998) 51–94.
- M. Briane, Homogenization in general periodically perforated domains by a spectral approach. Calc. Var. Partial Differ. Equat.15 (2002) 1–24.
- A. Cherkaev, Variational methods for structural optimization. Springer (2000).
- G. Dal Maso, Introduction to Γ-convergence. Birkhäuser, Boston, 1993.
- I. Fonseca, D. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition. Cal. Var.2 (1994) 283–313.
- 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).
- F. Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients. Ann. Mat Pura Appl.112 (1977) 49–68.
- P. Pedregal, Parametrized Measures and Variational Principles. Progress in Nonlinear Partial Differential Equations. Birkhäuser (1997).
- P. Pedregal, Optimal design and constrained quasiconvexity. SIAM J. Math. Anal.32 (2000) 854–869.
- P. Pedregal, Constrained quasiconvexification of the square of the gradient of the state in optimal design. Quater. Appl. Math.62 (2004) 459–470.
- 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.
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.