Fast integral equation solutions using geometrical-theory-of-diffraction-like matrices

Frequency domain integral equations allow high-accuracy direct solvers, while often requiring excessive computational power for large problems. This contrasts with high-frequency methods such as the geometrical theory of diffraction (GTD), which allow efficient solution of large problems, though often with less accuracy. Use of a special type of expansion functions in integral equations results in the impedance matrix localization method, which localizes the information in the impedance matrix to clumps of large numbers that are analogous to the terms of GTD. Fast numerical methods are developed for calculating these large matrix elements. Then the analogy to GTD is strengthened by developing an integral equation based on sources which radiate only exterior to a closed body. This is a new kind of combined source equation. The peaks (i.e., clumps of large numbers) are now restricted to near the diagonal for a convex body. This both allows a faster matrix solution and further strengthens the analogy to GTD. The local structure of the resulting matrix, its inverse, and the method of finding the inverse are found to be closely analogous to the associated high-frequency method quantities. This suggests the possibility of deriving numerical GTD coefficients.