A novel mixed multiscale finite-element method for the eddy-current problem is presented to avoid the necessity of modeling each laminate of the core of electrical devices. The method is based on a current vector potential T and a reduced magnetic scalar potential (RMSP) and copes with the 3-D problems. The edge effect is considered. Material properties are assumed to be linear. Hence, the method is developed for the frequency domain. External currents are represented by the Biot-Savart field serving as excitation. The planes of symmetry are exploited. Numerical simulations are presented, showing excellent accuracy at minimal computational costs.
The multiscale finite-element method (MSFEM) reduces the computational costs for the simulation of eddy currents (ECs) in laminated iron cores compared with the standard finite-element method (SFEM) essentially. Nevertheless, the complexity of the resulting problem is still too large to solve it conveniently. The idea is to additionally exploit model order reduction (MOR). Snapshots (SNSs) for a reduced basis are cheaply calculated by the MSFEM. Numerical simulations of a small transformer show exceptional performance. This is well demonstrated by the overall EC losses and by the distribution of the magnetic-flux density, both with respect to those obtained by the MSFEM.
Eddy currents (ECs) are simulated in a single laminate, representing the whole core of an electrical machine. Despite this drastic reduction in the complexity of the problem, a 3-D finite-element model turns out to be still too expensive for simulations. To overcome this difficulty, 2-D/1-D methods are used. This article presents a solution to consider both air gap and edge effect (EE) based on the multiscale finite-element method (MSFEM) using the magnetic vector potential (MVP) A. Linear material properties are assumed; therefore, this article is carried out in the frequency domain. The new 2-D/1-D MSFEM is discussed, and various simulation results are presented. Index Terms-2-D/1-D method, eddy-current (EC) problems, edge effect (EE), iron core, lamination, magnetic vector potential (MVP) A, multiscale finite-element method (MSFEM).
The multiscale finite element method is a valuable tool to solve the eddy current problem in laminated materials consisting of many iron sheets, which would be prohibitively expensive to resolve in a finite element mesh. It allows to use a coarse mesh which does not resolve each sheet and constructs the local fields using predefined micro-shape functions. This paper presents for the first time an a-posteriori error estimator for the multiscale finite element method which considers the error with respect to the exact solution. It is based on flux equilibration and a modification of the theorem of Prager and Synge and provides an upper bound for the error that does not include generic constants. Numerical examples show a good performance in both the linear and the nonlinear case.
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