Structure Preserving Model Reduction of Low-Frequency Electromagnetic Problem Based on POD and DEIM

The Proper Orthogonal Decomposition (POD) combined with the (Discrete) Empirical Interpolation Method (DEIM) can be used to reduce the computation time of the solution of a Finite Element (FE) model. However, it can lead to numerical instabilities. To increase the robustness, the POD_DEIM model must be constructed by preserving the structure of the full FE model. In this paper, the structure preserving is applied for different potential formulations used to solve electromagnetic problems.

I. Introduction

To study electrical devices, the Finite Element (FE) method is often used to solve low-frequency electromagnetic problems in the time or frequency domain. The computation time required to solve this kind of problems can be large due to a fine mesh, to an important number of time or frequency steps and to the nonlinear behavior law of the ferromagnetic material. In the literature, to reduce the size of the FE model and the computation time, the Proper Orthogonal Decomposition (POD) is the most popular approach [1], [2]. With a nonlinear behavior law, the POD is not so efficient due to the calculation of nonlinear terms. Then, to reduce this computation cost, interpolation methods have been developed [3]–[7]. The POD combined with the (Discrete) Empirical Interpolation Method (DEIM) has been used to solve a lot of problems in engineering. In electromagnetic modeling, this approach has been already applied to solve a nonlinear magnetostatic problem coupled with an electric circuit [8], [9], a magneto-quasistatic problem including a motion of a subdomain [10] or a nonlinear magnetodynamic problem with a model order reduction of an adaptive subdomain [11].