Integrated and Explicit Boundary Conditions of Electromagnetic Fields at Arbitrary Interfaces between Two Anisotropic Media

Bing Zhou^1*, Graham Heinson², Aixa Rivera-Rios²

¹Petroleum Geosciences, The Petroleum Institute, Abu Dhabi, UAE.
²Geology and Geophysics, The University of Adelaide, Adelaide, Australia.

DOI: 10.4236/jemaa.2015.73009                

Abstract

This paper derives two new integrated and explicit boundary conditions, named the “explicit normal version” and “explicit tangential versions” respectively for electromagnetic fields at an arbitrary interface between two anisotropic media. The new versions combine two implicit boundary equations into a single explicit matrix formula and reveal the boundary values linked by a 3 × 3 matrix, which depends on the interface topography and model property tensors. We analytically demonstrate the new versions equivalent to the common implicit boundary conditions and their application to transformation of the boundary values in the boundary integral equations. We also give two synthetic examples that show recovery of the boundary values on a hill and a ridge, and highlight the advantage of the new versions of being a simpler and more straightforward method to compute the electromagnetic boundary values.

Keywords

Electromagnetic Theory, Electromagnetism, Boubdary Condition, Electric Anisotropy

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Weaver, J.T. (1995) Mathematic Methods for Geo-Electromagnetic Induction. Research Studies Press Ltd., Taunton, Somerset.
[2]	Key, K. and Weiss, C. (2006) Adaptive Finite-Element Modeling Using Unstructured Grids: The 2D Magnetotelluirc Example. Geophysics, 71, G291-G299. http://dx.doi.org/10.1190/1.2348091
[3]	Mukheriee, S. and Everett, M. (2011) 3D Controlled-Source Electromagnetic Edge-Based Finite Element Modeling of Conductive and Permeable Heterogeneities. Geophysics, 76, F215-F226. http://dx.doi.org/10.1190/1.3571045
[4]	Axia, R. (2014) Multi-Order Hexahedral Vector Finite Element Method for 3-D MT Modeling, Including Anisotropy and Complex Geometry. PhD Thesis, Adelaide University, Adelaide.
[5]	Thide, B. (2004) Electromagnetic Field Theory. Upsilon Books, Communa AB, Uppsala.
[6]	Zhdanov, M.S., Varentsov, I.M., Waever, J.T., Golubev, N.G. and Krylov, V.A. (1997) Methods for Modeling Electromagnetic Fields Results from COMMEMI—The International Project on the Comparison of Modeling Methods for Electromagnetic Induction. Journal of Applied Geophysics, 37, 133-271. http://dx.doi.org/10.1016/S0926-9851(97)00013-X
[7]	Helmuth, S. (1995) Two Dimensional Spline Interpolation Algorithms. A. K. Peter Ltd, Wellesley.
[8]	Brebbia, C.A. and Dominguez, J. (1992) Boundary Elements: An Introductory Course. Computational Mechanics Publications, Boston.
[9]	Beer, G., Smith, I.M. and Duenser, C. (2008) The Boundary Element Method with Programming for Engineering and Scientists. Springer Wien, New York.
[10]	Everett, M.E. and Constable, S. (1999) Electric Dipole Fields over an Anisotropic Seafloor: Theory and Application to the Structure of 40Ma Pacific Ocean Lithosphere. Geophysical Journal International, 136, 41-56. http://dx.doi.org/10.1046/j.1365-246X.1999.00725.x