Asymptotics of an optimal compliance-location problem

Giuseppe Buttazzo; Filippo Santambrogio; Nicolas Varchon

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 4, page 752-769
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Buttazzo, Giuseppe, Santambrogio, Filippo, and Varchon, Nicolas. "Asymptotics of an optimal compliance-location problem." ESAIM: Control, Optimisation and Calculus of Variations 12.4 (2006): 752-769. <http://eudml.org/doc/249616>.

@article{Buttazzo2006,
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. },
author = {Buttazzo, Giuseppe, Santambrogio, Filippo, Varchon, Nicolas},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Compliance; optimal location; shape optimization; Γ-convergence.; compliance; -convergence},
language = {eng},
month = {10},
number = {4},
pages = {752-769},
publisher = {EDP Sciences},
title = {Asymptotics of an optimal compliance-location problem},
url = {http://eudml.org/doc/249616},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Buttazzo, Giuseppe
AU - Santambrogio, Filippo
AU - Varchon, Nicolas
TI - Asymptotics of an optimal compliance-location problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/10//
PB - EDP Sciences
VL - 12
IS - 4
SP - 752
EP - 769
AB - 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.
LA - eng
KW - Compliance; optimal location; shape optimization; Γ-convergence.; compliance; -convergence
UR - http://eudml.org/doc/249616
ER -

References

top
  1. G. Allaire, Shape optimization by the homogenization method. Springer-Verlag, New York (2002).  Zbl0990.35001
  2. M. Bendsoe and O. Sigmund, Topology Optimization. Theory, Methods, and Applications. Springer-Verlag, New York (2003).  Zbl1059.74001
  3. G. Bouchitté and G. Buttazzo, Integral representation of nonconvex functionals defined on measures. Ann. Inst. H. Poincaré Anal. Non Linéaire9 (1992) 101–117.  Zbl0757.49012
  4. G. Bouchitté, C. Jimenez and M. Rajesh, Asymptotique d'un problème de positionnement optimal. C.R. Acad. Sci. Paris Ser. I335 (2002) 1–6.  
  5. D. Bucur and G. Buttazzo, Variational Methods in Shape Optimization Problems. Birkäuser, Boston, Progress in Nonlinear Differential Equations and their Applications 65 (2005).  Zbl1117.49001
  6. G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems: relaxed solutions and optimality conditions. Bull. Amer. Math. Soc.23 (1990) 531–535.  Zbl0718.49017
  7. G. Buttazzo and G. Dal Maso, Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim.23 (1991) 17–49.  Zbl0762.49017
  8. G. Buttazzo and G. Dal Maso, An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal.122 (1993) 183–195.  Zbl0811.49028
  9. G. Buttazzo, G. Dal Maso, A. Garroni and A. Malusa, On the relaxed formulation of Some Shape Optimization Problems. Adv. Math. Sci. Appl.7 (1997) 1–24.  Zbl0971.49024
  10. 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.  Zbl0496.35030
  11. G. Dal Maso, An Introduction to Γ-convergence. Birkhauser, Basel (1992).  
  12. L. Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Die Grundlehren der Math. Wiss., Vol. 65, Springer-Verlag, Berlin (1953).  Zbl0052.18401
  13. A. Henrot and M. Pierre, Variation et Optimisation de Forme. Une analyse géométrique. Springer-Verlag, Berlin, Mathématiques et Applications 48 (2005).  
  14. F. Morgan and R. Bolton, Hexagonal Economic Regions Solve the Location Problem. Amer. Math. Monthly109 (2002) 165–172.  Zbl1026.90059
  15. S. Mosconi and P. Tilli, Γ-Convergence for the Irrigation Problem, 2003. J. Conv. Anal.12 (2005) 145–158.  Zbl1076.49024
  16. J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization. Shape sensitivity analysis. Springer-Verlag, Berlin (1992).  Zbl0761.73003

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.