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Asymptotics of an optimal compliance-location problem

Published online by Cambridge University Press:  11 October 2006

Giuseppe Buttazzo ,
Filippo Santambrogio  and
Nicolas Varchon
Giuseppe Buttazzo
Affiliation:
 Università di Pisa, Dip. di Matematica, Largo B. Pontecorvo, 5, 56127 Pisa, Italy; [email protected]
Filippo Santambrogio
Affiliation:
 Scuola Normale Superiore, Classe di Scienze, Piazza dei Cavalieri, 7, 56126 Pisa, Italy; [email protected]
Nicolas Varchon
Affiliation:
 Collège Condorcet de Bresles, 60510 Bresles, France; [email protected]
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Abstract

We consider the problem of placing a Dirichlet region made by n small balls of given radius in a given domain subject to a force f in order to minimize the compliance of the configuration. Then we let n tend to infinity and look for the Γ-limit of suitably scaled functionals, in order to get informations on the asymptotical distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.

Keywords

Complianceoptimal locationshape optimizationΓ-convergence.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 4 , October 2006 , pp. 752 - 769
Copyright
© EDP Sciences, SMAI, 2006

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References

G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002).
M. Bendsoe and O. Sigmund, Topology Optimization. Theory, Methods, and Applications. Springer-Verlag, New York (2003).
Bouchitté, G. and Buttazzo, G., Integral representation of nonconvex functionals defined on measures. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992) 101–117. CrossRef
Bouchitté, G., Jimenez, C. and Rajesh, M., Asymptotique d'un problème de positionnement optimal. C.R. Acad. Sci. Paris Ser. I 335 (2002) 1–6. CrossRef
D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems. Birkäuser, Boston, Progress in Nonlinear Differential Equations and their Applications 65 (2005).
Buttazzo, G. and Dal Maso, G., Shape optimization for Dirichlet problems: relaxed solutions and optimality conditions. Bull. Amer. Math. Soc. 23 (1990) 531–535. CrossRef
Buttazzo, G. and Dal Maso, G., Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991) 17–49. CrossRef
Buttazzo, G. and Dal Maso, G., An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122 (1993) 183–195. CrossRef
Buttazzo, G., Dal Maso, G., Garroni, A. and Malusa, A., On the relaxed formulation of Some Shape Optimization Problems. Adv. Math. Sci. Appl. 7 (1997) 1–24.
D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs. Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. II (1982), 98–138 and Vol. III (1982) 154–178.
G. Dal Maso, An Introduction to Γ-convergence. Birkhauser, Basel (1992).
L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Die Grundlehren der Math. Wiss., Vol. 65, Springer-Verlag, Berlin (1953).
A. Henrot and M. Pierre, Variation et Optimisation de Forme. Une analyse géométrique. Springer-Verlag, Berlin, Mathématiques et Applications 48 (2005).
Morgan, F. and Bolton, R., Hexagonal Economic Regions Solve the Location Problem. Amer. Math. Monthly 109 (2002) 165–172. CrossRef
Mosconi, S. and Tilli, P., Γ-Convergence for the Irrigation Problem, 2003. J. Conv. Anal. 12 (2005) 145–158.
J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization. Shape sensitivity analysis. Springer-Verlag, Berlin (1992).