A Waveform Relaxation Solver for Transient Simulation of Large-Scale Nonlinearly Loaded Shielding Structures

This article introduces an algorithm for transient simulation of electromagnetic structures loaded by lumped nonlinear devices. The reference application is energy-selective shielding, which adopts clipping devices uniformly spread along shield apertures to achieve a shielding effectiveness that increases with the power of the incident field, thereby blocking high-power interference while allowing low-power communication. Transient simulation of such structures poses a number of challenges, related to their large-scale and low-loss nature. In this work, we propose a waveform relaxation (WR) scheme based on decoupling the linear electromagnetic structure from its nonlinear terminations. In a preprocessing stage, the electromagnetic subsystem is characterized in the frequency domain and converted into a behavioral rational macromodel. Transient simulation is performed by refining estimates of the port signals through iterations. The proposed scheme combines a time partitioning approach with an inexact Newton–Krylov solver. This combination provides fast convergence also in those cases where standard WR schemes fail due to a strong mismatch at the decoupling sections. Numerical results on several test cases of increasing complexity with up to 1024 ports show that the proposed approach proves as reliable as HSPICE in terms of accuracy, with a speedup ranging from one to three orders of magnitude.