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From solutions of finite element (FE) simulations depending on parameters, a surrogate model of the FE solution can be built based on the proper orthogonal decomposition (POD) combined with the radial basis functions (RBFs) interpolation. Then, a fast approximation of an FE solution can be computed for any parameter set. In order to optimize the number of FE solutions required for the construction of the surrogate model and to control the error between the FE and surrogate models, a hierarchical multilevel approach based on an iterative algorithm is investigated. In this article, a hierarchical multilevel (H-POD-RBF) approach is developed for a nonlinear magnetostatic problem. In terms of application, a 3-D three-phase inductance is considered. The results obtained from the surrogate model are compared with those given from the FE model for a current or voltage supply. [ABSTRACT FROM AUTHOR]

The proper orthogonal decomposition (POD) approach is applied to a linear finite element (FE) model of a squirrel cage induction machine. In order to obtain a reduced model valid on the whole operating range, snapshots are extracted from the simulation of typical tests, such as at the locked rotor and the synchronous speed. Then, the reduced model of the induction machine is used to simulate different operating points with variable rotation speed, and the results are compared to the full FE model to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper presents a methodology that allows taking advantage of the geometrical periodicity of electrical machines together with the modeling of rotor motion. It enables by means of the discrete Fourier transform (DFT) to reduce the large-scale system obtained from the finite-element model to several smaller independent subsystems, allowing a shortening of the computational time. Due to DFT properties, the computational time can be more reduced especially when we consider the inter-dependence of the spectral components under either balanced or unbalanced supply condition. In addition, a further reduction is possible in the case of balanced regimes where the distribution of the eventual numerical solution is governed by a limited number of prevailing harmonics. [ABSTRACT FROM AUTHOR]

Among the model order reduction techniques, the proper generalized decomposition (PGD) has shown its efficiency to solve a large number of engineering problems. In this paper, the PGD approach is applied to solve a multi-physics problem based on a magnetoelectric device. A reduced model is developed to study the device in its environment based on an offline/online approach. In the offline step, two specific simulations are performed in order to build a PGD reduced model. Then, we obtain a model very well fitted to study in the online stage the influence of parameters like the frequency or the load. The reduced model of the device is coupled with an electric load ( $R$ – $L$ ) to illustrate the possibility offered by the PGD. [ABSTRACT FROM AUTHOR]

Among the model-order reduction techniques, the proper orthogonal decomposition (POD) has shown its efficiency to solve magnetostatic and magnetoquasistatic problems in the time domain. However, the POD is intrusive in the sense that it requires the extraction of the matrix system of the full model to build the reduced model. To avoid this extraction, nonintrusive approaches like the data-driven (DD) methods enable to approximate the reduced model without the access to the full matrix system. In this paper, the DD-POD method is applied to build a low-dimensional system to solve a magnetostatic problem coupled with electric circuit equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Proper orthogonal decomposition (POD) has been successfully used to reduce the size of linear finite-element (FE) problems, and thus the computational time associated with. When considering a nonlinear behavior law of the ferromagnetic materials, the POD is not so efficient due to the high computational cost associated with the nonlinear entries of the full FE model. Then, the POD approach must be combined with an interpolation method to efficiently deal with the nonlinear terms, thus obtaining an efficient reduced model. An interpolation method consists in computing a small number of nonlinear entries and interpolating the other terms. Different methods have been presented to select the set of nonlinear entries to be calculated. Then, the (discrete) empirical interpolation method [(D)EIM] and the best points interpolation method (BPIM) have been developed. In this paper, we propose to compare two reduced models based on the POD-(D)EIM and on the POD-BPIM in the case of nonlinear magnetostatics coupled with electric equation. [ABSTRACT FROM AUTHOR]

To solve a parametric model in computational electromagnetics, the finite-element (FE) method is often used. To reduce the computational time and the memory requirement, the FE method can be combined with the model-order reduction technique like the proper orthogonal decomposition and (discrete) empirical interpolation methods. These three numerical methods introduce the errors of discretization, reduction, and interpolation, respectively. The solution of the parametric model will be efficient if the three errors are of the same order and so they need to be evaluated and compared. In this paper, we propose an a posteriori error estimator based on the verification of the constitutive law, which estimates the three different errors. An example of application in magnetostatics with 11 parameters is treated where it is shown how the error estimator can be used to control and to improve the accuracy of the solution of the reduced model. [ABSTRACT FROM PUBLISHER]

Among the model order reduction techniques, the proper generalized decomposition (PGD) has shown its efficiency to solve static and quasistatic problems in the time domain. However, the introduction of non-linearity due to ferromagnetic materials, for example, has never been addressed. In this paper, the PGD technique combined with the discrete empirical interpolation method (DEIM) is applied to solve a non-linear magnetostatic problem coupled with the circuit equations. To evaluate the reduction technique, the transient state of a three-phase transformer at no load is studied using the full finite-element model and the PGD–DEIM model. [ABSTRACT FROM AUTHOR]

Model order reduction methods enable reduction of the computation time when dealing with parametrized numerical models. Among these methods, the proper orthogonal decomposition method seems to be a good candidate because of its simplicity and its accuracy. In the literature, the offline/online approach is generally applied but is not always required especially if the study focuses on the device without any coupling with others. In this paper, we propose a method to adaptively construct the reduced model while it limits the evaluation of the full model when appropriate. A stochastic magnetostatic example with 14 uncertain parameters is studied by applying the Monte Carlo simulation method to illustrate the proposed procedure. In that case, it appears that the complexity of this method does not depend on the number of input parameters and so is not affected by the curse of dimensionality. [ABSTRACT FROM AUTHOR]

To solve stochastic static field problems, a discretization by the finite-element method can be used. A system of equations is obtained with the unknowns (scalar potential at nodes, for example) being random variables. To solve this stochastic system, the random variables can be approximated in a finite-dimension functional space—a truncated polynomial chaos expansion. The error between the exact solution and the approximated one depends not only on the spatial mesh, but also on the discretization along the stochastic dimension. In this paper, we propose an a posteriori estimation of the error due to the discretization along the stochastic dimension. [ABSTRACT FROM AUTHOR]

Improvement of the magnetic performances of soft magnetic composites (SMC) materials requires to link the microstructure to the macroscopic magnetic behavior law. This can be achieved with the finite-element method using the geometry reconstruction from images of the microstructure. Nevertheless, it can lead to large computational times. In that context, the proper generalized decomposition (PGD), that is an approximation method originally developed in mechanics, and based on a finite sum of separable functions, can be of interest in our case. In this work, we propose to apply the PGD method to the SMC microstructure magnetic simulation. A nonlinear magnetostatic problem with the scalar potential formulation is then solved. [ABSTRACT FROM PUBLISHER]

The numerical solution of partial differential equations onto random domains can be done by using a mapping transforming this random domain into a deterministic domain. The issue is then to determine this one to one random mapping. In this paper, we present two methods—one based on the resolution of the Laplace equations, one based on a geometric transformation—to determine the random mapping. A stochastic magnetostatic example is treated to compare these methods. [ABSTRACT FROM PUBLISHER]

Deals with a study which presented a 3 dimensional error estimator based on the hypercircle theorem in the case of the finite element method scalar potential formulation in magnetostatics. Ways to calculate the magnetostatic field; Presentation of solutions; Application of the estimator.

Provides information on a study which presents an error estimator that enables control of the quality of finite element solution in nonlinear magnetostatic 2-dimensional problems. Methodology of the study; Results of the study; Conclusions.