A Novel Dirichlet-Neumann Random-Walk Algorithm for the Solution of Time-Harmonic Helmholtz Equation at Multiple Wavelength Length Scales: 1D and 2D Verification

Abstract

The electrical properties of IC interconnects at multi-GHz frequencies must be described with Maxwell’s equations. We have created an entirely new floating random-walk (RW) algorithm to solve the timeharmonic Maxwell-Helmholtz equations. Traditional RW algorithms for Maxwell-Helmholtz equations are constrained to length scales that are less than a quarterwavelength. This is because of the problem of resonance in finite-domain Green’s function for Helmholtz equation at multiple quarter-wavelength length scales. In this paper, we report the major discovery of extending our floating RW algorithm beyond a quarter-wavelength. The problem of Green’s function resonance has been eliminated by the use of an infinite-domain Green’s function. In this work, we formulate this algorithm and describe its successful application to homogeneous and heterogeneous 1D problems and homogeneous 2D problems. We believe, that with additional work, this RW algorithm will prove useful in the development of CAD tools for electromagnetic analysis of IC interconnect systems. It can be noted that the algorithm exhibits full parallelism, requiring minimal interprocessor communication. Thus, significant performance enhancement can be expected in any future parallel software or hardware implementation.