ESAIM: Control, Optimisation and Calculus of Variations

Research Article

Design-dependent loads in topology optimization

Bourdin, Blaisea1 and Chambolle, Antonina2

a1 Courant Institute of Mathematical Science, New York University. : Department of Mathematics, Louisiana State University, Baton Rouge LA 70803-4918, USA;

a2 CEREMADE, UMR 7534 du CNRS, Université de Paris-Dauphine, 75775 Paris Cedex 16, France;


We present, analyze, and implement a new method for the design of the stiffest structure subject to a pressure load or a given field of internal forces. Our structure is represented as a subset S of a reference domain, and the complement of S is made of two other “phases”, the “void” and a fictitious “liquid” that exerts a pressure force on its interface with the solid structure. The problem we consider is to minimize the compliance of the structure S, which is the total work of the pressure and internal forces at the equilibrium displacement. In order to prevent from homogenization we add a penalization on the perimeter of S. We propose an approximation of our problem in the framework of Γ-convergence, based on an approximation of our three phases by a smooth phase-field. We detail the numerical implementation of the approximate energies and show a few experiments.

(Received November 20 2001)

(Revised May 13 2002)

(Online publication September 15 2003)

Key Words:

  • Topology optimization;
  • optimal design;
  • design-dependent loads;
  • Γ-convergence;
  • diffuse interface method.

Mathematics Subject Classification:

  • 49Q20;
  • 74P05;
  • 74P15