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Projection-based model order reduction (MOR) is a key technique in digital twins, enabling the rapid generation of large-scale, high-fidelity, parametric simulation data through numerical methods. However, a major challenge in projection-based MOR is the evaluation of nonlinear terms, which depend on the size of the full-order model during the iterative process. This reliance significantly degrades the efficiency of MOR techniques. While various hyper-reduction technique, such as the discrete empirical interpolation method (DEIM) discussed in this paper, have been introduced to mitigate this issue by employing low-dimensional representation to approximate nonlinear terms and accelerate computations, classical DEIM faces notable limitations in practical applications. Specifically, when a system’s nonlinear characteristics vary significantly with parameters, it becomes difficult to create a universal low-dimensional representation capable of capturing nonlinear behavior across the entire parameter space. Additionally, as the number of system parameters increases, the low-dimensional representation grows in size, reducing its computational efficiency for nonlinear term evaluations. To address these challenges, we build upon the two-stage model reduction approach that leverages tensor structures, originally proposed by Mamonov and Olshanskii for linear systems (Comput. Methods Appl. Mech. Engrg., 397, 115122, 2022) and later further developed for nonlinear dynamical systems with DEIM in (SIAM J. Sci. Comput., 46(3), A1850–A1878, 2024). Inspired by these contributions, we adapt and extend the methodology to address time-independent parametric problems, with a particular focus on electromagnetic applications. This approach dynamically generates problem-dependent and parameter-specific low-dimensional representation for the nonlinear terms. Its performance is demonstrated through various numerical examples, including an EI transformer with varying excitation, an electric motor under operational variations, and a voice coil actuator (VCA) with non-uniform demagnetization. Results show that this approach significantly outperforms classical DEIM in terms of efficiency and accuracy.

Digital twin is considered the key technique for real-time monitoring and life-cycle management of electric equipment. To construct the digital twin model of electric equipment, a multi-parameter electromagnetic analysis is needed to generate a large amount of high fidelity data under various working condition. However, repeated solving such multi-parameter electromagnetic problems based on full order finite element method may lead to extreme scale calculations. To address this issue, a hybrid approach that combines tensor decomposition and proper orthogonal decomposition (POD) is introduced, which can effectively establish a reduced order model for multi-parameter electromagnetic field problems. The performance of the proposed approach is illustrated through three numerical examples, namely an electrical motor, a transformer and a voice coil actuator including parameter variations of operating conditions, geometric parameters, and material parameters. The numerical results show that the proposed hybrid approach has significant advantage compared to conventional reduced order model in situation where the solution changes dramatically within the parameter variation range and even more apparent when the parameter dimension is high.