Fast Green Function Evaluation for Method of Moment

TL;DR

The paper tackles the bottleneck of impedance-matrix filling in the method of moments (MOM) caused by excessive evaluations of $Exp(-jkr)$ and $Exp(-jkr)/r$. It introduces an adaptive, one-dimensional interpolation framework with distance-based sampling tables and local refinement, using linear interpolation for $Exp(-jkr)$ and a Lagrange polynomial for $Exp(-jkr)/r$, plus a hash-based lookup for fast sampling-point retrieval. The authors provide rigorous error bounds for both interpolation schemes and demonstrate substantial runtime savings (over 20% in reported cases) across multiple MOM formulations (EFIE/MFIE) and a practical IC-inductor example, with minimal impact on accuracy. The approach is easy to integrate into existing MOM codes, offering a robust, scalable acceleration for electrically large and multiscale problems.

Abstract

In this letter, an approach to accelerate the matrix filling in method of moment (MOM) is presented. Based on the fact that the Green function is dependent on the Euclidean distance between the source and the observation points, we constructed an efficient adaptive one-dimensional interpolation approach to fast calculate the $Exp$ type function values. In the proposed method, several adaptive interpolation tables are constructed based on the maximum and minimum distance between any two integration points with local refinement near zero function values to minimize the relative error. An efficient approach to obtain the sampling points used in the interpolation phase is carefully designed. Then, any function values can be efficiently calculated through a linear interpolation method for Exp and a Lagrange polynomial interpolation method for the Green function. In addition, the error bound of the proposed method is rigorously investigated. The proposed method can be quite easily integrated into the available MOM codes for different integration equation (IE) formulations with few efforts. Comprehensive numerical experiments validate its accuracy and efficiency through several IE formulations. Results show that over 20% efficiency improvement can be achieved without sacrificing the accuracy.

Figures (3)

• Figure 1: Real and imaginary parts of $Exp(-jkr)$ and $Exp(-jkr)/r$.
• Figure 2: (a) and (b) denote the maximum relative error of real and imaginary part of $Exp(-jkr)$ and $Exp(-jkr)/r$ obtained from linear interpolation method, (c) and (d) from Lagrange polynomial method.
• Figure 3: The relative error of real and imaginary part of elements in impedance matrix through EFIE and MFIE obtained from the proposed method. Note: the relative errorrs are in log scale.