Optimal design by homogenization

Prof. Eric Bonnetier

CNRS, Centre de math'ematiques appliqu\'ees Ecole Polytechnique, 91128 Palaiseau Cedex, France

In the context of shape optimization, we seek minimizers of the sum of the elastic compliance and of the weight of a solid structure under specified loading. This problem is known not to be well posed and a relaxed formulation is introduced to allow for microperforated composites as admissible designs. Using the theory of homogenization and optimal bounds, we obtain an explicit form for the relaxed formulation. Based on this formulation, we introduce a numerical algorithm for computing optimal designs. We present numerical results for two and three dimensional problems.