Abstract
When the Boundary Element Method (BEM) is used to analyse electromagnetic problems one is able to achieve an almost linear complexity by applying matrix compression techniques. Beyond this, on symmetrical domains the computational costs can be reduced by significant factors. By using several symmetry considerations (geometry, mesh, kernel, excitation) it will be shown how the combination of the Adaptive Cross Approximation (ACA) and the symmetry exploitation allows an efficient solution of electromagnetic problems. This approach will be demonstrated on the scalar BEM formulation for electrostatics and can also be applied to the vectorial eddy current formulations. The symmetry exploting ACA algorithm not only reduces the problem size due to the symmetry but also possesses an almost linear complexity w.r.t. the number of unknowns.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
E. L. Allgower, K. Georg, R. Miranda, J. Tausch: Numerical exploitation of equivariance. Z. Angew. Math. Mech. 78 (1998) 795–806.
M. Bebendorf: Approximation of boundary element matrices. Numer. Math. 86 (2000) 565–589.
M. Bebendorf, S. Rjasanow: Adaptive Low-Rank Approximation of Collocation Matrices. Computing 70 (2003) 1–24.
M. Bonnet. Exploiting partial or complete geometrical symmetry in 3D symmetric Galerkin indirect BEM formulations. Int. J. Num. Meth. Engrg. 57 (2003) 1053–1083.
A. Bossavit: Symmetry, groups and boundary value problems: a progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry. Comp. Meth. in Appl. Mech. Engrg. 56 (1986) 167–215.
H. Cheng, L. Greengard, V. Rokhlin: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155 (1999) 468–498.
W. Hackbusch: A sparse matrix arithmetic based on \( \mathcal{H} \)-matrices. Part I. Computing 62 (1999) 89–108.
W. Hackbusch, B. N. Khoromskij: A sparse \( \mathcal{H} \)-matrix arithmetic. Part II. Application to multi-dimensional problems. Computing (2000) 64 (2000) 21–47.
W. Hackbusch, Z. P. Nowak: On the fast matrix multiplication in the boundary element method by panel clustering. Numer. Math. 54 (1989) 463–491.
S. Küppers, G. Henneberger, I. Ramesohl: The influence of the number of poles on the output performance of a claw-pole alternator. Proc. of the ICEM, pp. 268–272, 1996.
S. Kurz, J. Fetzer, G. Lehner, W. M. Rucker: Numerical analysis of 3D eddy current problems with moving bodies using BEM-FEM coupling. Surv. Math. Industry 9 (1999) 131–150.
S. Kurz, O. Rain, V. Rischmüller, S. Rjasanow: Discretization of boundary integral equations by differential forms on dual grids. IEEE Trans. Magnetics 40 (2004) 826–829.
S. Kurz, O. Rain, S. Rjasanow: Application of the adaptive cross approximation technique for the coupled BE-FE-solution of symmetric electromagnetic problems. Comp. Mech. 32 (2003) 423–429.
T. Nakata, N. Takahashi, K. Fujiwara: Summary of results for benchmark problem 10 (steel plates around a coil). COMPEL 11 (1992) 335–344.
J. Ostrowski, Z. Andjelić, M. Bebendorf, B. Crânganu-Creţu, J. Smajić: Fast BEM-solution of Laplace problems with \( \mathcal{H} \)-matrices and ACA. Proceedings of the IEEE, 2005.
K. Preis, I. Bardi, O. Biro, C. Magele, W. Renhart, K. R. Richter, G. Vrisk: Numerical analysis of 3D magnetostatic fields. IEEE Trans. Magnetics 27 (1991) 3798–3803.
O. Rain: Kantenelementbasierte BEM mit DeRham-Kollokation für Elektromagnetismus. PhD thesis, Universität des Saarlandes, 2004.
S. Rjasanow: Effective algorithms with block circulant matrices. Linear Alg. Appl. 202 (1994) 55–69.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kurz, S., Rain, O., Rjasanow, S. (2007). Fast Boundary Element Methods in Computational Electromagnetism. In: Schanz, M., Steinbach, O. (eds) Boundary Element Analysis. Lecture Notes in Applied and Computational Mechanics, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-47533-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-47533-0_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-47465-4
Online ISBN: 978-3-540-47533-0
eBook Packages: EngineeringEngineering (R0)