This chapter is devoted to algorithmic and numerical issues in the application of the homogenization method to optimal design problems. We describe how the results of Chapters 3 and 4 can be used to build new numerical algorithms. Although homogenization theory has been devised as a tool for shape optimization by some of its first contributors (see the works of Murat and Tartar , , , ), the first significant numerical applications appeared well after these theoretical breakthroughs. There were some early contributions by Gibiansky and Cherkaev , in the context of plate optimization (equivalent to two-dimensional elasticity), Glowinski , Goodman, Kohn, and Reyna , and Lavrov, Lurie, and Cherkaev  in the conductivity setting (equivalent to a torsion problem for an elastic bar). Note that these works consider only two-phase optimization problems with nondegenerate components. The first study of shape optimization by the homogenization method in a general elasticity setting (including the fact that one phase was degenerating to holes) is due to Bendsoe and Kikuchi . Their work was really pioneering in the sense that they treated convincing real life examples educating the whole community of structural optimization to the homogenization approach. They have been followed by many others, notably , , , , , , , , , .
KeywordsStructural Optimization Numerical Algorithm Shape Optimization Volume Constraint Homogenization Method
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