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

Allaire, Grégoire; Bonnetier, Eric; Francfort, Gilles; Jouve, François

Numer. Math. 76, No. 1, 27-68 (1997).

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.

Reviewer: M.Codegone (Torino)

MSC:

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

Keywords:

minimizers; sum of elastic compliance and weight; relaxation method; well-posed problem; microperforated composites; optimal bounds for composite materials; numerical algorithm; penalization technique; topology optimization algorithms