Mixed Spectral Element Method for Electromagnetic Secondary Fields in Stratified Inhomogeneous Anisotropic Media

Abstract

The spectral element method (SEM) is widely used for analyzing high frequency electromagnetic wave propagation and scattering in complex structures. At low frequencies, however, the primary (direct) fields are usually much larger than the secondary fields caused by scattering, thus making it much more difficult to accurately simulate the secondary (scattered) fields because of the source singularity in the primary fields. To more effectively and accurately simulate the low-frequency secondary fields in subsurface structures, in this work the vector Helmholtz equation for the scattered fields is used to directly obtain the secondary field data in inhomogeneous anisotropic media. The computational method for the secondary electromagnetic fields is based on the mixed spectral element method (mixed SEM) for anisotropic objects in a layered anisotropic background medium. The electromagnetic field is separated into the background fields that are evaluated analytically using the dyadic Green's functions (DGFs) of a layered uniaxially anisotropic medium, and the secondary fields caused by anomalous bodies with arbitrary shapes are numerically computed by the mixed SEM. This avoids the source singularity, and allows the transmitters to be located outside the computational domain. By enforcing Gauss' law as the constraint condition, the mixed SEM efficiently eliminates the low frequency breakdown problem. Numerical results verify the proposed method over the traditional SEM in low-frequency scattering from objects in subsurface layered anisotropic media.INDEX TERMS Mixed spectral element method, secondary field, anisotropic media, source singularity.

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“…For solving the scattered field problem in three-dimensional electromagnetic fields, Equation (3) can be employed as the governing equation for the SEM method [28]. This approach enables the direct derivation of the scattered field within the computational region when the incident field is known.…”

“…where S is the stiffness matrix, M is the mass matrix, K is the constraint matrix, K T is the transpose of K, vectors x and y denote the unknowns of H s and the Lagrange multiplier p, respectively, and b and c a are the source terms corresponding to the right-hand sides of Equation (3). The exact derivation and expression of the formula can be found in [28]. Consequently, as the order of the basis function increases, the interpolation error of SEM decreases exponentially, resulting in exponential convergence and a significantly reduced computational effort.…”