Finite Element and Finite Difference Methods in by M.A. Morgan (Eds.)

By M.A. Morgan (Eds.)

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1. 3 F4 PA\ • -F7. — • -P7J The finite element solution thus comes down to inverting a matrix. As was the case for the finite difference example, the global array is mostly filled with zeros. This is in contrast to integral equation methods, which produce full matrix structures. A sparse matrix allows highly economical inversion, for even very large matrix order, by any of a number of different algorithms [15]. In addition, by properly ordering the nodes, as was done here, the matrix can often be made to have a block structure.

As developed in the unimoment method, the exterior region fields are represented by a functional expansion in one of the separable co­ ordinate systems for the vector Helmholtz equation [14: sect. 1]. The spatial interface for coupling the interior numerical solution to the unbounded exterior region is thus a constant coordinate surface of the separable system employed in the outside expansion. Spherical interfaces were utilized in [16] and [22-26] due to the relative ease of generation of exterior region spherical harmonic field expansions.

Such a phenomenon is related to the classical "Rayleigh hypothesis" [31]. As a result, we may not be able to obtain as a good match between the numerical solution at SB and the original truncated analytical expan­ sion, as we were able to do on the spherical surface for the unimoment method. To evaluate the expansion coefficients for the boundary field in the FEBI, we can use a system of two combined field integral equations, as employed in the EBCM [30]. These integral equations are vector field 3-D versions of that in (63); they relate the tangential fields just inside of the boundary SB to that just outside, without making use of a knowledge of the material structure inside of SB · The FEBI method has been shown to work well for scattering cal­ culations involving moderately elongated lossy dielectric objects.

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