P. Gangl

M. H. Gfrerer, P. Gangl

In this paper we investigate the influence of the discretization of PDE constraints on shape and topological derivatives. To this end, we study a tracking-type functional and a two-material Poisson problem in one spatial dimension. We consider the discretization by a standard method and an enriched method. In the standard method we use splines of degree $p$ such that we can control the smoothness of the basis functions easily, but do not take any interface location into consideration. This includes for p=1 the usual hat basis functions. In the enriched method we additionally capture the interface locations in the ansatz space by enrichment functions. For both discretization methods shape and topological sensitivity analysis is performed. It turns out that the regularity of the shape derivative depends on the regularity of the basis functions. Furthermore, for point-wise convergence of the shape derivative the interface has to be considered in the ansatz space. For the topological derivative we show that only the enriched method converges.

A. Cesarano, B. Endtmayer, P. Gangl

First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local convergence properties, necessitating a sufficiently close initial guess. In this work, we present an unregularized shape-Newton method and combine shape optimization with homotopy (or continuation) methods in order to allow for the use of higher order methods even if the initial design is far from a solution. The idea of homotopy methods is to continuously connect the problem of interest with a simpler problem and to follow the corresponding solution path by a predictor-corrector scheme. We use a shape-Newton method as a corrector and arbitrary order shape derivatives for the predictor. Moreover, we apply homotopy methods also to the case of multi-objective shape optimization to efficiently obtain well-distributed points on a Pareto front. Finally, our results are substantiated with a set of numerical experiments.