Topology optimization method applied to the design of electromagnetic devices: Focus on convexity issues

To perform parameter and shape optimization, an initial topology is required which affects the final solution. Topology optimization methods have the advantage to release this constraint. They are based on a splitting of the design space into cells, in which they attempt to distribute optimally some given materials. In this paper, the optimization is based on a discrete formulation of the Maxwell equations obtained thanks to a finite element model. The gradient of the objective function is computed immediately from this discrete form, which is more convenient than the adjoint variable method. A line-search method (steepest descent direction) is then applied. The step size is computed by a simple and quick algorithm, to achieve good and fast convergence. Several convexity issues were already highlighted in topology optimization. We explain how we can face some of them by performing the optimization on a variable linked to the permeability by a given mapping. This mapping actually affects the value of intermediate materials and must be selected carefully to avoid the algorithm to be trapped in some local minimizers. In order to illustrate the concepts presented along the paper, the method is applied for the topology optimization of a linear reluctant actuator.