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A new approach to solve complex valued systems arising from the solution of Maxwell equations in the frequency domain through real-equivalent formulations. (English) Zbl 1449.65041

Summary: The purpose of this article is to propose algebraic techniques to solve symmetric complex-valued systems. In particular, we focus on the systems which arise from the modeling of electromagnetic fields in the frequency domain. These systems usually present a very ill-conditioned number, so specific preconditioning techniques are supposed to be used. Finally, both analytical and numerical reference tests are proposed on the simple case of the magnetic field produced by a solenoid containing conductive or non-conducting materials.

MSC:

65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
65F08 Preconditioners for iterative methods
78M99 Basic methods for problems in optics and electromagnetic theory

Software:

AztecOO; PETSc
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References:

[1] Salon, Finite element calculation of harmonic losses in ac machine windings, IEEE Trans Magn 29 (2) pp 1442– (1993) · doi:10.1109/20.250674
[2] Hill, A stabilized multilevel vector finite-element solver for time-harmonic electromagnetic waves, IEEE Trans Magn 39 (3) pp 1203– (2003) · doi:10.1109/TMAG.2003.810379
[3] Börner, Fast 3-D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projection, Geophys J Int 173 pp 766– (2008) · doi:10.1111/j.1365-246X.2008.03750.x
[4] Balay S Abhyankar S Adams MF PETSc users manual 2015
[5] Tuminaro RS Heroux MA Hutchinson SA Shadid JN Aztecoo user guide 2007
[6] Day, Solving complex-valued linear systems via equivalent real formulations, SIAM J Sci Comput 23 (2) pp 480– (2006) · Zbl 0992.65020 · doi:10.1137/S1064827500372262
[7] Munankarmy, A comparison of two equivalent real formulations for complex-valued linear systems part 1: introduction and method, Am J Undergraduate Res 1 (3) pp 17– (2002)
[8] Axelsson, A comparison of iterative methods to solve complex valued linear algebraic systems, Numer Algorithms 66 pp 811– (2014) · Zbl 1307.65034 · doi:10.1007/s11075-013-9764-1
[9] Benzi, Block preconditioning of real-valued iterative algorithms for complex linear systems, IMA J Numer Anal 28 pp 598– (2008) · Zbl 1145.65022 · doi:10.1093/imanum/drm039
[10] Axelsson, Real valued iterative methods for solving complex symmetric linear systems, Numer Linear Algebra Appl 7 pp 197– (2000) · Zbl 1051.65025 · doi:10.1002/1099-1506(200005)7:4<197::AID-NLA194>3.0.CO;2-S
[11] Bai, On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer Algorithms 56 pp 297– (2010) · Zbl 1209.65037 · doi:10.1007/s11075-010-9441-6
[12] Bai, Modified HSS iteration methods for a class of complex symmetric linear systems, Comput 87 pp 93– (2010) · Zbl 1210.65074 · doi:10.1007/s00607-010-0077-0
[13] Freund, Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrix, SIAM J Sci Stat Comput 13 pp 425– (1992) · Zbl 0761.65018 · doi:10.1137/0913023
[14] Poirier, Efficient preconditioning scheme for block partitioned matrices with structured sparsity, Numer Linear Algebra Appl 7 pp 715– (2000) · Zbl 1051.65059 · doi:10.1002/1099-1506(200010/12)7:7/8<715::AID-NLA220>3.0.CO;2-R
[15] Nedelec, Mixed finite elements in R3, Numers Math 35 (3) pp 315– (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415
[16] Alves, Modelling and analysis of the electromagnetism in magnetic forming process, IEEE Trans Magn 52 (5) (2016)
[17] Chen, An adaptive finite element method for the eddy current model with circuit/field couplings, SIAM J Sci Comput 32 (2) pp 1020– (2010) · Zbl 1211.78029 · doi:10.1137/080713112
[18] Grayver, Large-scale 3D geoelectromagnetic modeling using parallel adaptive high-order finite element method, Geophys 80 (6) pp E277– (2015) · doi:10.1190/geo2015-0013.1
[19] Haber, Fast finite volume simulation of 3d electromagnetic problems with highly discontinuous coefficients, SIAM J Sci Comput 22 pp 1943– (2001) · Zbl 0992.78033 · doi:10.1137/S1064827599360741
[20] Saad, A comparison of iterative methods to solve complex valued algebric systems, SIAM J Sci Stat Comput 7 (3) pp 856– (1986) · Zbl 0599.65018 · doi:10.1137/0907058
[21] Feriani, Iterative system solvers for the frequency analysis of linear mechanical systems, Comput Meth Appl Mech Eng 190 pp 1719– (2000) · Zbl 0981.70005 · doi:10.1016/S0045-7825(00)00187-0
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