On optimal microstructures for a plane shape optimization problem

{ erw = Ae(uw), e(uw) = 21-(VUw + V u T ) , div ~rw = 0 in co, erw.n = f on 0f2, O'w.n = 0 on Ow\OQ, (1) where uw is the displacement vector, e(uw) is the strain tensor, and o-w is the stress tensor. The compliance of the structure is defined by 1 I n t r o d u c t i o n Solving structural optimization problems by the homogenization method, amounts to find extremal microstructures which maximize the rigidity of a structure or equivalently which minimize its compliance (the work done by the load the structure is submitted to). These microstructures are called extremal in the sense that they achieve optimality in the well-known Hashin-Shtrikman bounds on the effective properties of composite materials. For more details on the homogenization method in structural design, we refer to A1laire et al. (1997), Allaire and Kohn (1-993b), Bendsee (1995), Bendsee and Kikuchi (1988), Gibianski and Cherkaev (1997), Jog et al. (1994), Kohn and Strang (1986), Murat and Tartar (1997), and references therein. There are several examples of optimal microstructures in the literature. Mainly, they are the sequential laminates (see e.g. Francfort and Murat 1986), the concentric sphere assemblages of IIashin (1963), the confocal ellipsoid assemblages of Tartar [Tartar (1985), and Grabovsky and Kohn (1995a) in the elasticity setting], the Vigdergauz periodic constructions (Vigdergauz 1994; Grabovsky and Kohn 1995b). Before discussing the properties of these extremal microstructures, we introduce the shape optimization problem considered in this paper. We restrict ourselves to plane problems, corresponding to the generalized shape optimization problem for perforated plates in plane stress [according to the terminology of Rozvany et al. (1995)]. We seek the optimal shape of a linearly elastic structure that minimizes a c(co) = /f.uw=/Ae(uw).e(uw) =/A-lerw' O'c~ 9 0s w w Our structural optimization problem is to minimize, over all subsets w C ~2, the objective function E(co) equal to the weighted sum of the compliance and weight of co E(co) = c(co) + lcol 9 It can be written as