We develop a new method to precondition the matrix equation resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE). This preconditioning method is based on first applying the loop-tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MoM matrix thus obtained still cannot be solved efficiently by iterative solvers due to the large number of iterations required. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MoM matrix that can be solved efficiently by iterative solvers. Consequently, dramatic reduction in iteration count has been observed. The various steps can be regarded as a rearrangement of the basis functions to arrive at the MoM matrix. Therefore, they are related to the original MoM matrix by matrix transformation, where the transformation requires the inverse of the connection matrix. We have also developed a fast method to invert the connection matrix so that the complexity of the preconditioning procedure is of O(N) and, hence, can be used in fast solvers such as the low-frequency multilevel fast multipole algorithm (LP-MLFMA). This procedure also makes viable the use of fast solvers such as MLFMA to seek the iterative solutions of Maxwell's equations from zero frequency to microwave frequencies.

Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies

We describe a numerically efficient strategy for solving a linear system of equations arising in the Method of Moments for solving electromagnetic scattering problems. This novel approach, termed as the characteristic basis function method (CBFM), is based on utilizing characteristic basis functions (CBFs)-special functions defined on macro domains (blocks)-that include a relatively large number of conventional sub-domains discretized by using triangular or rectangular patches. Use of these basis functions leads to a significant reduction in the number of unknowns, and results in a substantial size reduction of the MoM matrix; this, in turn, enables us to handle the reduced matrix by using a direct solver, without the need to iterate. In addition, the paper shows that the CBFs can be generated by using a sparse representation of the impedance matrix-resulting in lower computational cost-and that, in contrast to the iterative techniques, multiple excitations can be handled with only a small overhead. Another important attribute of the CBFM is that it is readily parallelized. Numerical results that demonstrate the accuracy and time efficiency of the CBFM for several representative scattering problems are included in the paper.

An Iteration-Free MoM Approach Based on Excitation Independent Characteristic Basis Functions for Solving Large Multiscale Electromagnetic Scattering Problems

Techniques based on the method of modal expansions, the Rayleigh-Stevenson expansion in inverse powers of the wavelength, and also the method of moments solution of integral equations are essentially restricted to the analysis of electromagnetic radiating structures which are small in terms of the wavelength. It therefore becomes necessary to employ approximations based on “high-frequency techniques” for performing an efficient analysis of electromagnetic radiating systems that are large in terms of the wavelength.

Techniques for High-Frequency Problems

Heuristic wedge diffraction coefficients for computing propagation path loss over finitely conducting earth are extended to include slope diffraction, with the assumption that propagation of energy through the wedge is negligible. The slope diffraction terms for the lossy wedge are obtained in an analogous manner as for the perfectly conducting case, except that special care must be taken with the factor multiplying the incident field for grazing incidence. Results given show that the slope diffraction term produces continuous results that behave reasonably when compared with results for perfectly conducting wedges. >

A heuristic UTD slope diffraction coefficient for rough lossy wedges

This investigation is concerned with the numerical solution of time-harmonic electromagnetic scattering by axisymmetric penetrable bodies having arbitrary cross-sectional profiles and even continuously inhomogeneous consistency. The initiation of this effort involved the discovery and development of the coupled azimuthal potential (CAP) formulation, which is valid in generally lossy isotropic inhomogeneous rotationally symmetric media. Electromagnetic fields in such regions can be represented, using the CAP formulation, in terms of two continuous potentials which satisfy a self-adjoint system of partial differential equations or, equivalently, a variational criterion. Using an optimized variational finite-element algorithm in conjunction with a triregional unimoment method, a versatile computer program is described that provides scattering solutions for each of multiple incident fields impinging upon an arbitrarily shaped inhomogeneous penetrable body of revolution. An extensive evaluation of the accuracy and convergence of the algorithm is presented, which includes comparison of scattering computations and experimental measurements at X -band for several solid and hollow plexiglas bodies of revolution with maximum interior dimensions of over 4 wavelengths.

R. G. Kouyoumjian

Papers

The geometrical theory of diffraction and its application