Decomposition and regularization of nonlinear anisotropic curl‐curl DAEs

Markus Clemens (Bergische Universität Wuppertal, Wuppertal, Germany)
Sebastian Scho¨ps (Bergische Universität Wuppertal, Wuppertal, Germany and Wave Propagation and Signal Processing Research Group, Katholieke Universiteit Leuven, Kortrijk, Belgium)
Herbert De Gersem (Wave Propagation and Signal Processing Research Group, Katholieke Universiteit Leuven, Kortrijk, Belgium)
Andreas Bartel (Bergische Universität Wuppertal, Wuppertal, Germany)

Abstract

Purpose

The space discretization of eddy‐current problems in the magnetic vector potential formulation leads to a system of differential‐algebraic equations. They are typically time discretized by an implicit method. This requires the solution of large linear systems in the Newton iterations. The authors seek to speed up this procedure. In most relevant applications, several materials are non‐conducting and behave linearly, e.g. air and insulation materials. The corresponding matrix system parts remain constant but are repeatedly solved during Newton iterations and time‐stepping routines. The paper aims to exploit invariant matrix parts to accelerate the system solution.

Design/methodology/approach

Following the principle “reduce, reuse, recycle”, the paper proposes a Schur complement method to precompute a factorization of the linear parts. In 3D models this decomposition requires a regularization in non‐conductive regions. Therefore, the grad‐div regularization is revisited and tailored such that it takes anisotropies into account.

Findings

The reduced problem exhibits a decreased effective condition number. Thus, fewer preconditioned conjugate gradient iterations are necessary. Numerical examples show a decrease of the overall simulation time, if the step size is small enough. 3D simulations with large time step sizes might not benefit from this approach, because the better condition does not compensate for the computational costs of the direct solvers used for the Schur complement. The combination of the Schur approach with other more sophisticated preconditioners or multigrid solvers is subject to current research.

Originality/value

The Schur complement method is adapted for the eddy‐current problem. Therefore, a new partitioning approach into linear/non‐linear and static/dynamic domains is proposed. Furthermore, a new variant of the grad‐div gauging is introduced that allows for anisotropies and enables the Schur complement method in 3D.

Keywords

Citation

Clemens, M., Scho¨ps, S., De Gersem, H. and Bartel, A. (2011), "Decomposition and regularization of nonlinear anisotropic curl‐curl DAEs", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 30 No. 6, pp. 1701-1714. https://doi.org/10.1108/03321641111168039

Publisher

:

Emerald Group Publishing Limited

Copyright © 2011, Emerald Group Publishing Limited

To read the full version of this content please select one of the options below

You may be able to access this content by logging in via Shibboleth, Open Athens or with your Emerald account.
To rent this content from Deepdyve, please click the button.
If you think you should have access to this content, click the button to contact our support team.