CardioWave is a modular system for simulating wavefront conduction in the heart. These simulations may be used to investigate the factors that generate and sustain life-threatening arrhythmias such as ventricular fibrillation. The user selects a set of modules which most closely reflects the simulation they are interested in and the simulator is built automatically. Thus, we do not present one monolithic simulator, but rather a simulator-generator which allows the researcher to make the trade-offs of complexity versus performance. The results presented here are from simulations run on an IBM SP parallel computer and a cluster of workstations. The performance numbers show excellent scalability up through 128 processors. With the larger memory of the parallel machines, we have been able to perform highly realistic simulations of the human atria. These simulations include realistic, 3-D geometries with inhomogeneity and anisotropy as we as highly complex membrane dynamics.

An algorithm, called Newton waveform relaxation (NWR), for solving large differential algebraic systems (DASs) that arise from circuit simulation is presented. Analytical and experimental comparisons will be made with waveform relaxation (WR), waveform relaxation Newton (WRN), and direct methods. This approach has been implemented in a circuit simulator called WCAzM and was found to be as much as three times faster than a direct method and always faster than waveform relaxation Newton for the circuits benchmarked. Improvements to the convergence of the linear inner loop of the NWR algorithm are discussed.>

Two-dimensional (2-D) effects are becoming increasingly important in the diffusion of impurities in submicrometer silicon devices. The authors describe a 2-D process simulator, PREDICT2, that handles implant damage effects, annealing, and lateral diffusion. PREDICT2 simulates the diffusion of impurities in silicon by using phenomenological diffusion coefficients. The phenomenological models are verified by comparing simulated and experimental results. This approach is compared with that of point-defect-based simulators. The experimental technique for measuring impurity profiles in two dimensions is outlined. The method for generating diffusion models is illustrated by examining 2-D phosphorous diffusion. Numerical simulations and experimental measurements are compared. The numerical methods used in the simulator are described, and directions for future work are suggested.>

Block iterative methods are applied to hydrodynamic simulations of a one-dimensional submicrometer semiconductor device. It is shown that block successive underrelaxation (SUR) converges with a fixed relaxation factor omega =0.13 for simulations at 300 K and omega =0.04 at 77 K. To demonstrate the robustness of the block iterative method, numerical simulations of a steady-state electron shock wave in Si at 300 K for a 0.1- mu m channel and at 77 K for a 1.0- mu m channel are presented. The block SUR method is parallelizable if each diagonal block solve can be done efficiently in parallel. Using chaotic relaxation and the preconditioned conjugate gradient method for the parallel diagonal block solves, a parallel speed up of approximately 2.5 is obtained on 10 processors of a Butterfly GP-1000.>

A novel algorithm, called Newton waveform relaxation, for solving large differential algebraic systems which arise from circuit simulation is discussed. It will be compared to waveform relaxation, waveform relaxation Newton (WRN), and direct methods. This approach has been implemented as a SPICE level circuit simulator called WCAzM (waveform circuit analyzer with macromodeling) and was found to be as much as three times faster than a direct method and always faster than WRN for the circuits benchmarked. Furthermore it was found that the number of model evaluations required by WCAzM was as much as seven times smaller than a direct method. These numbers indicate that further improvements in speed can be made.>

An introduction to the hydrodynamic model for semiconductor devices is presented. Special attention is paid to classifying the hydrodynamic PDEs (partial differential equations) and analyzing their nonlinear wave structure. Numerical simulations of the ballistic diode using the hydrodynamical device model are presented, as an illustrative elliptic problem. The importance of nonlinear block iterative methods is emphasized. Arguments for existence of solutions and convergence of numerical methods are given for the case of subsonic electron flow.>

In this paper, we present an overview of the physical principles and numerical methods used to solve the coupled system of nonlinear partial differential equations that model the transient behavior of silicon VLSI device structures. We also describe how the same techniques are applicable to circuit simulation. A composite linear multistep formula is introduced as the time-integration scheme. Newton-iterative methods are exploited to solve the nonlinear equations that arise at each time step. We also present a simple data structure for nonsymmetric matrices with symmetric nonzero structures that facilitates iterative or direct methods with substantial efficiency gains over other storage schemes. Several computational examples, including a CMOS latchup problem, are presented and discussed.

We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations. We give a new linear-time algorithm to calculate the fill-in produced by any elimination ordering, and we give two new related algorithms for finding orderings with special properties. One algorithm, based on breadth-first search, finds a perfect elimination ordering, if any exists, in $O(n + e)$ time, if the problem graph has n vertices and e edges. An extension of this algorithm finds a minimal (but not necessarily minimum) ordering in $O(ne)$ time. We conjecture that the problem of finding a minimum ordering is NP-complete

A graph-theoretic elimination process is considered which is related to performing Gaussian elimination on sparse symmetric and unsymmetric systems of linear equations. The authors discuss good algorithms for finding elimination orderings, showing that a generalization of breadth-first search, called lexicographic search, can be used to find perfect orderings in O(n+e) time and minimal orderings in O(ne) time, if the problem graph is undirected and has n vertices and e edges. Also given are efficient (though slower) algorithms for generating such orderings on directed graphs. It is claimed that the minimum ordering problem for directed graphs is NP-complete, and it is conjectured that it is also NP-complete for undirected graphs. The authors include a brief discussion of the relation of elimination to transitive closure and discuss some unresolved, more general, issues.

Perfect elimination digraphs arise in the study of Gaussian elimination on sparse linear systems. With a view toward numerical computational complexity we show four conditions (C1–C4) to be necessary for the perfect elimination property. The sufficiency of C1 is shown in general and the sufficiency of C2–C4 is shown in the symmetric case. The equivalence of C1–C3 is conjectured.

The notions of consistent ordering and zero circulation are shown to be dual notions by relating these notions to the incidence and circuit matrices of an appropriate directed graph.