Fourier-enhanced reduced-order surrogate modeling for uncertainty quantification in electric machine design

TL;DR

The paper tackles torque uncertainty in a PMSM arising from geometric design variations and introduces a Fourier-enhanced reduced-order surrogate that couples discrete Fourier transform-based dimension reduction of torque signals with regression models to efficiently predict torque statistics. It evaluates three regression techniques (PCE, FNN, GP) and finds that Fourier-based dimension reduction combined with Gaussian process regression yields the best accuracy and robustness for Monte Carlo uncertainty quantification, while enabling substantial offline-online computational savings. Key contributions include automatic selection of influential frequency components, demonstration of significant speedups (orders of magnitude) over high-fidelity evaluations, and a framework that can replace the expensive solver in parameter studies for electric machine design. The approach leverages isogeometric analysis for high-fidelity torque generation and produces a practical, scalable pathway for uncertainty-aware PMSM design under geometric variations, with potential applicability to other periodic electromechanical systems.

Abstract

This work proposes a data-driven surrogate modeling framework for cost-effectively inferring the torque of a permanent magnet synchronous machine under geometric design variations. The framework is separated into a reduced-order modeling and an inference part. Given a dataset of torque signals, each corresponding to a different set of design parameters, torque dimension is first reduced by post-processing a discrete Fourier transform and keeping a reduced number of frequency components. This allows to take advantage of torque periodicity and preserve physical information contained in the frequency components. Next, a response surface model is computed by means of machine learning regression, which maps the design parameters to the reduced frequency components. The response surface models of choice are polynomial chaos expansions, feedforward neural networks, and Gaussian processes. Torque inference is performed by evaluating the response surface model for new design parameters and then inverting the dimension reduction. Numerical results show that the resulting surrogate models lead to sufficiently accurate torque predictions for previously unseen design configurations. The framework is found to be significantly advantageous compared to approximating the original (not reduced) torque signal directly, as well as slightly advantageous compared to using principal component analysis for dimension reduction. The combination of discrete Fourier transform-based dimension reduction with Gaussian process-based response surfaces yields the best-in-class surrogate model for this use case. The surrogate models replace the original, high-fidelity model in Monte Carlo-based uncertainty quantification studies, where they provide accurate torque statistics estimates at significantly reduced computational cost.

Figures (13)

• Figure 1: One-fourth of the 's geometry in two dimensions, including material distribution, boundary conditions, and geometric design parameters (see Table \ref{['tab:pmsm-parameters']}). The phase and current direction in the copper coils is also shown. Figure adapted from wiesheu2024combined.
• Figure 2: Torque signals for different geometric design configurations of the .
• Figure 3: Schematic illustration of the surrogate modeling framework, separated into reduced-order modeling and inference parts.
• Figure 4: Reconstruction error versus dimension reduction and worst-case torque signal reconstructions using -based dimension reduction with $R=5$ and $R=11$ reduced frequency components.
• Figure 5: for surrogates computed with different combinations of training dataset size, , and dimension reduction approach.