Optimal Design in Elasticity

This chapter is devoted to the application of the homogenization method in structural optimization. Namely, we generalize the approach introduced in Chapter 3 for conductivity problems to optimal design problems in the setting of linearized elasticity. We first focus on two-phase optimization problems, which amount to finding the optimal distribution of two elastic components in a fixed domain that minimizes an objective function. This objective function depends on the solution of a state equation, which is here the linearized elasticity system. Like in the conductivity case, this type of problem is generically ill-posed, i.e., it admits no optimal solution in the proposed class of admissible designs. Homogenization theory still provides a notion of generalized designs (which are composite materials in the sense of Chapter 2), which makes the problem well-posed and allows one to derive optimality conditions and new numerical algorithms. There is however a serious additional difficulty in the elasticity setting compared to the conductivity one. Since the homogenization method defines generalized designs as composite materials, it is necessary to know the full set of such two-phase composites. Unfortunately, this set is yet unknown for the mixture of two elastic isotropic components. Therefore, in full generality the homogenization method is useless in the elasticity setting since no explicit characterization of the space of composite admissible designs is available.