Learning electromagnetic fields based on finite element basis functions

TL;DR

The paper tackles the need for fast, physically consistent electromagnetic-field predictions under geometric variations in electric machines. It introduces a POD–DNN surrogate that learns spline-basis coefficients from isogeometric discretizations, using a weighted POD basis and a physics-informed neural network to map geometry and operating parameters to reduced coefficients, which are then projected back to the full field. Two PMSM-focused cases are demonstrated: an air-gap field surrogate requiring fewer modes and a full-field surrogate capturing rotor, stator, and air-gap fields, both achieving sub-percent to a few-percent errors in key metrics like torque while delivering orders-of-magnitude speedups over high-fidelity FEM. The results indicate the approach is suitable for rapid design optimization and real-time monitoring in complex geometries where traditional solvers are too costly, with clear guidance on mode counts and network architectures to balance accuracy and efficiency.

Abstract

Parametric surrogate models of electric machines are widely used for efficient design optimization and operational monitoring. Addressing geometry variations, spline-based computer-aided design representations play a pivotal role. In this study, we propose a novel approach that combines isogeometric analysis, proper orthogonal decomposition and deep learning to enable rapid and physically consistent predictions by directly learning spline basis coefficients. The effectiveness of this method is demonstrated using a parametric nonlinear magnetostatic model of a permanent magnet synchronous machine.

Figures (5)

• Figure 1: Parametrization of the PMSM geometry including parameter names, material definitions and boundary conditions, based onKomann_2024aa.
• Figure 2: Illustration of the parameter-dependent mapping from the reference domain $\hat{\Omega}$ to different physical domains $\Omega_1$ and $\Omega_2$.
• Figure 3: This flowchart illustrates the steps that the POD-DNN surrogate model makes to predict the coefficients for an new parametrization $\mathbf{P}.$
• Figure 4: Decay of $\epsilon_\mathrm{rel, POD}$ for air gap and full field snapshots. Reconstruction errors of $0.1\%$ and $0.6\%$ are marked with dashed grey line, selected number of modes in red.
• Figure 5: Reconstructed magnetic field and absolute error for $\mathbf{P}=$[$5.55mm$, $22.90mm$, $6.08mm$, $\ang{8.58}$, $1.39mm$, $2.57mm$, $\ang{19.09}$, $\ang{0.49}$, $10.29A$, $9.01A$, $9.6A$, $\ang{-0.9}$, $\ang{2.95}$, $\ang{-2.67}$] with relative error of $2.83\%$.