Asymptotical compliance optimization for connected networks

Giuseppe Buttazzo; Filippo Santambrogio

doi:10.3934/nhm.2007.2.761

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2007, Volume 2, Issue 4: 761-777. Doi: 10.3934/nhm.2007.2.761

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Asymptotical compliance optimization for connected networks

Giuseppe Buttazzo^1, and
Filippo Santambrogio^2,

1.
Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa
2.
Université Paris Dauphine, Laboratoire CEREMADE, UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16

Received: May 31, 2007

Revised: June 30, 2007

Published: November 2007

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Abstract

We consider the problem of the optimal location of a Dirichlet region in a two-dimensional domain $\Omega$ subject to a force $f$ in order to minimize the compliance of the configuration. The class of admissible Dirichlet regions among which we look for the optimum consists of all one-dimensional connected sets (networks) of a given length $L$. Then we let $L$ tend to infinity and look for the $\Gamma$-limit of suitably rescaled functionals, in order to identify the asymptotical distribution of the optimal networks. The asymptotically optimal shapes are discussed as well and links with average distance problems are provided.

Keywords:

$\Gamma-$convergence,
shape optimization.

Mathematics Subject Classification: Primary: 49J45, Secondary: 49Q10,74P05.

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