Design-dependent loads in topology optimization

Bourdin, Blaise; Chambolle, Antonin

ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 19-48.

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 $\Gamma$-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.

Classification : 49Q20, 74P05, 74P15
Mots clés : topology optimization, optimal design, design-dependent loads, $\Gamma $-convergence, diffuse interface method

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