Solution of the Elasticity PDE

This chapter is concerned with the solution technique of the elasticity PDE (1.7). Its weak formulation (1.11) constitutes a starting point for the development of a finite element method via the Galerkin method (cf. [Bra03]). One could principally employ the standard finite element method which is described for instance in Braess [Bra03]. However, we aim to solve shape optimization problems using a steepest descent method. In particular, this implies that our optimization variable — which means more precisely in our context the underlying domain the PDE (1.7) needs to be solved on — varies from iteration to iteration. Evidently, this might lead to rather complicated structures in the course of optimization, and would require various grids that frequently adapt to the new shape. Roughly speaking, the more complicated the shapes get, the more triangles we need to resolve the boundary, leading to a high number of degrees of freedom — which is linked directly to the size of the system of linear equations that needs to be solved.