Shape optimization by the homogenization method. (English) Zbl 0889.73051

In the framework of shape optimization, the authors look for minimizers of the sum of elastic compliance and weight of a solid structure under specified loading. The resulting formulation is ill-posed, thus a relaxation method is presented to enlarge the space of admissible designs in order to get a well-posed problem. It allows for microperforated composites as admissible designs. A generalization of the relaxed formulation is obtained with the help of the theory of homogenization and optimal bounds for composite materials. First, a relaxation result is proved, valid in any dimensions; secondly, a new numerical algorithm is introduced for computing optimal designs, complemented with a penalization technique which permits to remove composite designs to the final shape. No assumption are given on the number of holes within the domain; the numerical algorithms are viewed as “topology optimization” algorithms, since they are able to compute very fine patterns of the optimal shape on a fixed numerical grid. Numerical results are presented for two- and three-dimensional problem.


74P99 Optimization problems in solid mechanics
74E05 Inhomogeneity in solid mechanics
65K10 Numerical optimization and variational techniques
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