Application of the complete multiple reciprocity method for 3D impedance extraction with multiple frequency points

A complete multiple reciprocity method (CMRM), usually employed for the eigenvalue analysis of Helmholtz equation, is applied to impedance calculation of 3D electric structures for multiple frequency points. Based on a recently proposed boundary integral formulations for impedance calculation, the CMRM is used to separate the boundary integrals into the frequency-dependent and frequency-independent portions so as to accelerate the computation for multiple frequency points. A set of approaches is proposed to handle the severe numerical problems induced by the large varieties of distance r and frequency-dependent k, when applying the CMRM to impedance calculation. As a result, the near-field integrals are calculated with the inner product of a frequency-independent sequence and a frequency-dependent sequence, while the far-field integrals are calculated with an efficient approximate formula. Since the majority of the calculation for generating the overall linear equation system becomes reusable, the impedance extraction with multiple frequency points is greatly accelerated. Several typical structures of interconnects are calculated with the boundary element method combined with CMRM. Numerical results verify the accuracy and efficiency of the proposed methods.