Time-harmonic Eddy Current Models Research Articles

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models

A Local Positive (Semi)Definite Shift-Splitting Preconditioner for Saddle Point Problems with Applications to Time-Harmonic Eddy Current Models

Asymptotic Models and Impedance Conditions for Highly Conductive Sheets in the Time-Harmonic Eddy Current Model

This work is concerned with the time-harmonic eddy current problem for a medium with a highly conductive thin sheet. We present asymptotic models and impedance conditions up to the second order of ...

A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models

Based on a general splitting of the (1,1) leading block matrix, we first construct a general class of shift-splitting (GCSS) preconditioners for non-Hermitian saddle point problems. Convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the GCSS preconditioned saddle point matrices are analyzed. Then the GCSS preconditioner is specifically applied to the non-Hermitian saddle point problems arising from the finite element discretizations of the hybrid formulations of the time-harmonic eddy current models. With suitable choices of the splittings, the new GCSS preconditioners are easier to implement and have faster convergence rates than the existing shift-splitting preconditioner and its modified variant. Two numerical examples are presented to verify the theoretical results and show effectiveness of the new proposed preconditioners.

A new alternating positive semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

Based on the special positive semidefinite splittings of the saddle point matrix, we propose a new alternating positive semidefinite splitting (APSS) iteration method for the saddle point problem arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current problem. We prove that the new APSS iteration method is unconditionally convergent for both cases of the simple topology and the general topology. The new APSS matrix can be used as a preconditioner to accelerate the convergence rate of Krylov subspace methods. Numerical results show that the new APSS preconditioner is superior to the existing preconditioners.

The dimensional splitting iteration methods for solving saddle point problems arising from time-harmonic eddy current models

A dimensional splitting iteration method is proposed for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current models, which is by making use of the special positive semidefinite splittings of the saddle point matrix. It is proved that the proposed iteration method is unconditionally convergent for both cases of simple topology and general topology. Numerical results show that the corresponding preconditioner is superior to the existing preconditioners, when those preconditioners are used to accelerate the convergence rate of Krylov subspace methods.

An alternating positive-semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

An alternating positive-semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

A symmetric BEM–FEM method for an axisymmetric eddy current problem

This paper deals with the numerical solution of the time-harmonic eddy current model in an axisymmetric unbounded domain. To this end, a new symmetric BEM–FEM formulation is derived and also analyzed. Moreover, error estimates for the corresponding discretization are proven.

A Unified Analysis of Transmission Conditions for Thin Conducting Sheets in the Time-Harmonic Eddy Current Model

We introduce tools for a unified analysis and a comparison of impedance transmission conditions (ITCs) for thin conducting sheets within the time-harmonic eddy current model in two dimensions. The first criterion is the robustness with respect to the frequency or skin depth, which means whether they give meaningful results for small and for large frequencies or conductivities. As a second tool we study the accuracy for a range of sheet thicknesses and frequencies for a relevant example and finally analyze their asymptotic order in different asymptotic regimes. For the latter we write all the ITCs in a common form and show how they can be realized within the finite element method. Two new conditions, which we call ITC-2-0 and ITC-2-1, are introduced in this article, which appear in a symmetric form. They are derived by asymptotic expansions in the asymptotic regime of constant ratio between skin depth and thickness like those in [K. Schmidt and A. Chernov, Technical report 1102, Institut for Numerical Simulation, University of Bonn, Bonn, Germany, 2011]. We analyze these ITCs in comparison with the often used perfect electric boundary condition, the conditions by Levi-Civita [T. Levi-Civita, Rend. Lincei (5), 11 (1902), pp. 75--81 (in Italian); K. Schmidt and S. Tordeux, Z. Angew. Math. Phys., 61 (2010), pp. 603--626] the shielding element by Nakata et al. [IEEE Trans Magn., 26 (1990), pp. 2379--2381], the thin layer impedance boundary conditions by Mayergoyz and others [O. Tozoni and I. Mayergoyz, Technika, Kiev, 1974 (in Russian); I. Mayergoyz and G. Bedrosian, IEEE Trans. Magn., 31 (1995), pp. 1319--1324] and a family of ITCs derived by asymptotic expansions in the asymptotic regime of constant shielding by Schmidt and [ESAIM: Math. Model. Numer. Anal., 45 (2011), pp. 1115--1140]. Our analysis shows the superiority of the transmission conditions derived by asymptotic expansions where especially the worst-case error level of the ITC-2-1 is remarkably lower than for all the other conditions.

2D - 1D time-harmonic model for rotating electrical machines

In this paper, a combined two-dimensional - one-dimensional method for analyzing rotating electrical machines is presented. In the method, the cross-sectional AC field and winding equations of a machine are solved together with a one-dimensional time-harmonic eddy current model of electrical steel sheets. The approach is applied to compute a 37 kW cage induction machine, and the results are compared to simulation results received with a two-dimensional finite element model.