Multigrid–Based Optimal Shape and Topology Design in Magnetostatics

Abstract

The paper deals with an efficient solution technique to large–scale discretized shape and topology optimization problems. The efficiency relies on multigrid preconditioning. In case of shape optimization, we apply a geometric multigrid preconditioner to eliminate the underlying state equation while the outer optimization loop is the sequential quadratic programming, which is done in the multilevel fashion as well. In case of topology optimization, we can only use the steepest–descent optimization method, since the topology Hessian is dense and large–scale. We also discuss a Newton–Lagrange technique, which leads to a sequential solution of large–scale, but sparse saddle–point systems, that are solved by an augmented Lagrangian method with a multigrid preconditioning. At the end, we present a sequential coupling of the topology and shape optimization. Numerical results are given for a geometry optimization in 2–dimensional nonlinear magnetostatics.

This research has been supported by the Czech Ministry of Education under the grant AVČR 1ET400300415, by the Czech Grant Agency under the grant GAČR 201/05/P008 and by the Austrian Science Fund FWF within the SFB “Numerical and Symbolic Scientific Computing” under the grant SFB F013, subproject F1309.