Stacked-Element Connectivity Parameterization for Topology Optimization of Nonlinear Magnetic Systems

To design a magnetic system under a high-strength magnetic field, a nonlinear B-H relation must be considered. However, topology optimization using the nonlinear relation is rare. If the standard topology optimization formulation-varying the densities of design domain-discretizing magnetic finite elements-is used for nonlinear magnetic systems, the nonlinear material behavior will have to be interpolated. Because no consistent interpolating scheme is available, it is difficult to obtain converged distinct air-nonlinear material distributions without several trial-and-error attempts. Furthermore, the sensitivity analysis is very complicated because of the interpolated material nonlinearity. We have devised an alternative topology optimization formulation, called stacked-element connectivity parameterization (S-ECP), for the topology optimization of magnetic systems involving nonlinear B-H relations. The key idea is to discretize the design domain in terms of linear air elements, stack modified nonlinear magnetic material elements on the air elements, and control the element connectivity by varying the permeability of linearly behaving zero-length one-dimensional magnetic links. The links are introduced mainly to connect the air and nonlinear elements. Because a link is modeled by a linear material, the sensitivity analysis does not require complicated analysis involving the differentiation of nonlinear B-H relations, unlike the standard density method. Furthermore, it is much easier to find interpolation functions yielding converged distinct air-magnetic material states. We present numerical examples of C-shaped core design to demonstrate the effectiveness of the formulation.