Longitudinal Partitioning Waveform Relaxation Methods For The Analysis of Transmission Line Circuits

Three research projects are presented in this manuscript. Projects one and two describe two waveform relaxation algorithms (WR) with longitudinal partitioning for the time-domain analysis of transmission line circuits. Project three presents theoretical results about the convergence of WR for chains of general circuits. The first WR algorithm uses a assignment-partition procedure that relies on inserting external series combinations of positive and negative resistances into the circuit to control the speed of convergence of the algorithm. The convergence of the subsequent WR method is examined, and fast convergence is cast as a generic optimization problem in the frequency-domain. An automatic suboptimal numerical solution of the min-max problem is presented and a procedure to construct its objective function is suggested. Numerical examples illustrate the parallelizability and good scaling of the WR algorithm and point out to the limitation of resistive coupling. In the second WR algorithm, resistances from the previous insertion are replaced with dissipative impedances to address the slow convergence of standard resistive coupling of the first algorithm for low-loss highly reactive circuits. The pertinence and feasibility of impedance coupling are demonstrated and the properties of the subsequent WR method are studied. A new coupling strategy proposes judicious approximations of the optimal convergence conditions for faster speed of convergence. The proposed strategy avoids the difficult problem of optimisation and uses coarse macromodeling of the transmission line to construct approximations with delay under circuit form. Numerical examples confirm a superior speed of convergence which leads to further runtime saving. Finally, new results concerning the nilpotent WR algorithm are presented for chains of circuits when dissipative coupling is used. It is shown that optimal local convergence is necessary to achieve the optimal WR algorithm. However, the converse is not correct: the WR algorithm with optimal local convergences factors can be nilpotent yet not optimal or even be non-nilpotent at all. The second analysis concerns resistive coupling. It is demonstrated that WR always converges for chains circuits. More precisely, it is shown that WR will converge independently of the length of the chain when this late is made of identical symmetric circuits