Iterative-solver convergence for loop-star and loop-tree decompositions in method-of-moments solutions of the electric-field integral equation

Abstract

Method-of-moments (MoM) solutions of the electric-field integral equation using Rao-Wilton-Glisson (RWG) basis functions suffer from the so-called low-frequency breakdown. Introduction of loop-tree or loop-star decompositions of the basis functions can effectively solve this problem, and a number of papers have been published discussing various aspects with respect to these techniques. Several papers imply that loop-tree or loop-star decompositions may help to improve iterative-solver convergence for the solution of the resulting linear-equation systems. Since only a few results with respect to this issue are available, a study of the frequency-dependent iterative-solver convergence for RWG, loop-tree, and loop-star basis functions was performed. Two metallic scattering objects, with meshes comprising up to 21060 unknowns, were considered. RWG functions were found to provide the best convergence behavior, as long as the frequency considered was high enough to prevent the low-frequency breakdown. Among the loop-tree and loop-star bases, the loop-tree functions were found to be superior to the loop-star functions. The loop-tree functions resulted in good and stable convergence behavior if the number of subdivisions per wavelength was larger than a few hundred. Moreover, it is shown that the so-called loop-tree decomposition can also be viewed as a loop-cotree decomposition if an alternative tree of edges connecting the free vertices of the mesh is constructed

A fast three-dimensional deterministic ray tracing coverage simulator for a 24 GHz anti-collision radar

“…For this purpose, we used the calculation method of Eibert (2004) to simulate the fields in 2 series of points in Fig. 8 Comparing in both cases shows that the result of ray tracing stays in a good agreement with MoM-IE technique in the illumination and transition regions.…”

“…Hence, the IE method is used as a standard method to compare and validate the results of the ray tracer as it is still a big problem for every-day optimization and design processes. For this purpose, an in-house IE-MoM code is used Eibert (2004).…”

Applying Divergence-Free Condition in Solving the Volume Integral Equation

“…The loop basis generated from surface mesh has been studied and successfully applied to solve low frequency problems [7][8][9][10][11]. For volume tetrahedral mesh, Mendes and Carvalho proposed the volume solenoidal loop basis function [12,13], which is an extension of the surface loop basis.…”

On a Well-Conditioned Electric Field Integral Operator for Multiply Connected Geometries

“…As a result, when k approaches zero, the EFIE becomes increasingly difficult and sometimes impossible to solve. This so-called low frequency breakdown phenomenon traditionally has been remedied by using Loop-Star/Tree (quasi-Helmholtz, or Hodge) decompositions [5], [6], [7], [8], [9]. When using these decompositions with the EFIO and after appropriate matrix scaling with suitably chosen powers of (kh), the low frequency breakdown is solved; that is, in the limit of k going to zero, the matrix condition number is constant.…”

“…Since Loop and Star bases are very well-known [5], [6], [7], [8], [9], the treatment will be minimal and solely meant to introduce notation.…”