A shape optimization method for nonlinear axisymmetric magnetostatics using a coupling of finite and boundary elements

In this paper we propose a method for constrained shape optimization governed with a nonlinear axisymmetric magnetostatic state problem and we apply it to an optimal shape design of an electromagnet. The state problem is solved via Hiptmair’s symmetric coupling of finite elements employed in the interior ferromagnetic domain and boundary elements modelling the exterior air domain as well as current excitations. The boundary element integrals are regularized by a Duffy transform and evaluated then using a tensor–product Gaussian quadrature. Nonlinear ferromagnetic behaviour is resolved by Newton iterations. The optimization method under both linear and nonlinear constraints relies on the steepest–descent search, projections onto the set of linearized constraints, and an adjoint method of shape sensitivity analysis. Shape perturbations influence grid deformation via a solution to an auxiliary torsion–free linear elasticity problem. Finally, numerical results are presented.