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Earth, Planets and Space

, Volume 51, Issue 10, pp 1013–1018 | Cite as

Domain decomposition for 3D electromagnetic modeling

  • Zonghou Xiong
Open Access
Article

Abstract

Using the staggered grid full domain 3D modeling schemes of various accuracies have been developed. This study focuses on the second order finite difference method with the 13-point rule for meshes extending into the air. Tests with Krylov space iterative solvers indicate that the restarted Bi-CG Stablised method offers the best convergence for our problems. Because the air and the conductive earth have distinctive physical properties which greatly broaden the spectra of the whole matrix system, the whole mesh with both domains in one system either converges very slowly or fails to converge completely. However, the matrix systems for each domain have much smaller condition numbers. To overcome instability caused by the inclusion of the air in the mesh a domain decomposition method are experimented. Tests show that the adaptive iteration amongst the subdomains converges exponentially, which implies that large models can be solved by using the domain decomposition method.

Keywords

Domain Decomposition Matrix System Domain Decomposition Method Iterative Solver Electromagnetic Induction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Agarwal, A. K. and J. T. Weaver, Comparison of electric fields obtained from 3D magnetic field calculations on fixed staggered grids, in The 14th Workshop on Electromagnetic Induction in the Earth, Sinaia, Romania, Aug. 16–22, 1998.Google Scholar
  2. Árnason, K., A consistent discretization of the electromagnetic field in conducting media and application to the TEM problem, in 3D Electromagnetics, edited by M. Oristaglio and B. Spies, Society of Exploration Geophysicists, 1998.Google Scholar
  3. Bossavit, A., Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism, IEE Proc., 135, 493–499, 1988a.CrossRefGoogle Scholar
  4. Bossavit, A., A rationale for “edge-elements”, IEEE Trans. Magnetics, 24, 74–79, 1988b.CrossRefGoogle Scholar
  5. Druskin, V. and L. Knizhnerman, A spectral semi-discrete method for numerical solution of 3-D non-stationary problems in electrical prospecting, Phys. Sol. Earth, 24, 641–648, 1988.Google Scholar
  6. Druskin, V. and L. Knizhnerman, Spectral approach to solving three-dimensional Maxwell’s diffusion equations in the time and frequency domain, Radio Sci., 29, 937–953, 1994.CrossRefGoogle Scholar
  7. Larsson, E., A domain decomposition method for the Helmholtz equation in a multilayer domain, SIAM J. on Scientific Computing, 20, 1713–1731, 1999.CrossRefGoogle Scholar
  8. Mackie, R. L., J. T. Smith, and T. R. Madden, Three-dimensional electromagnetic modeling using finite difference equations: The magnetotelluric example, Radio Sci., 29, 923–935, 1994.CrossRefGoogle Scholar
  9. Mayergoyz, I. D. and J. D’Angelo, A new point of view on the mathematical structure of Maxwell equations, IEEE Trans. Magnetics, 29, 1315–1320, 1993.CrossRefGoogle Scholar
  10. Mur, G., Edge elements, their advantages and their disadvantages, IEEE Trans. Magnetics, 30, 3552–3557, 1994.CrossRefGoogle Scholar
  11. Shlager, K. L., J. G. Maloney, S. L. Ray, and A. F. Peterson, Relative accuracy of several finite-difference time-domain methods in two and three dimensions, IEEE Trans. Antennas Propagat., 41, 1732–1737, 1993.CrossRefGoogle Scholar
  12. Sleijpen, G. L. G. and D. R. Fokkema, BiCGSTAB(L) for linear matrices involving nonsymmetric matrices with complex spectrum, ETNA, 1, 11–32, 1993.Google Scholar
  13. Smith, J. T., Conservative modeling of 3-D electromagnetic fields; Part I: Properties and error analysis, Geophysics, 61, 1308–1318, 1996a.CrossRefGoogle Scholar
  14. Smith, J. T., Conservative modeling of 3-D electromagnetic fields; Part II: Biconjugate gradient solution and an accelerator, Geophysics, 61, 1319–1324, 1996b.CrossRefGoogle Scholar
  15. Sugeng, F., Modelling the transient responses of complex 3D geological structures using the 3D full-domain hexahedral edge-element finite-element technique, in The 14th Workshop on Electromagnetic Induction in the Earth, Sinaia, Romania, Aug. 16–22, 1998.Google Scholar
  16. Wang, T. and G. W. Hohmann, A finite-difference, time-domain solution for three-dimensional electromagnetic modeling, Geophysics, 58, 797–809, 1993.CrossRefGoogle Scholar
  17. Weaver, J. T., Agarwal, A. K., and X. H. Pu, Recent developments in three-dimensional finite difference modelling of the magnetic field in geoelectromagnetic induction, in International Symposium on Threedimensional Electromagnetics, Schlumberger-Doll Research, Ridgefield, Connecticut, USA, Oct. 4–6, 1995.Google Scholar
  18. Weidelt, P., Three-dimensional conductivity models: implications of electrical anisotropy, in 3D Electromagnetics, edited by M. Oristaglio and B. Spies, Society of Exploration Geophysicists, 1998.Google Scholar
  19. Xiong, Z., A study of high accuracy methods for full-domain 3D electromagnetic modelling, in The 14th Workshop on Electromagnetic Induction in the Earth, Sinaia, Romania, Aug. 16–22, 1998.Google Scholar
  20. Xiong, Z., A. Raiche, and F. Sugeng, A high accuracy staggered grid Galerkin method for 3D electromagnetic modelling, in The 13th Workshop on Electromagnetic Induction in the Earth, Onuma, Japan, July-12–18, 1996.Google Scholar
  21. Yee, K. S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media, IEEE Trans. Antennas Propagat., AP-14, 302–307, 1966.Google Scholar

Copyright information

© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences. 1999

Authors and Affiliations

  1. 1.CRC AMET, Earth ScienceMacquarie UniversitySydneyAustralia

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