Waveform Relaxation Technique Research Articles

Decoupling technique for CMOS gate with strong-coupled components

A novel decoupling technique for a CMOS gate with strong-coupled components is presented. The feedback structure is reduced to a unidirectional one. The hysteresis phenomena are characterised. Using the waveform relaxation technique, the decoupling principle and procedures are discussed in detail. The model predictions are verified by SPICE simulation.

Dynamic Iteration Using Reduced Order Models: A Method for Simulation of Large Scale Modular Systems

We describe a new iterative method, dynamic iteration using reduced order models (DIRM), for simulation of large scale modular systems using reduced order models that preserve the interconnection structure. This method may be compared to the waveform relaxation technique; however, unlike DIRM, waveform relaxation does not take advantage of model reduction techniques. The DIRM method involves simulating in turn each subsystem connected to model reduced versions of the other subsystems. The data from this simulation is then used to update the reduced model for that particular subsystem. We provide analytical results on convergence and accuracy of the DIRM method as well as numerical examples that demonstrate the success of DIRM and verify the analysis.

A Method for Global Approximation of the Initial Value Problem

For the numerical solution of the initial value problem a parallel, global integration method is derived and studied. It is a collocation method. If f(x,y)≡f(x) the method coincides with the Filippi's modified Clenshaw–Curtis quadrature [11]. Two numerical algorithms are considered and implemented, one of which is the application of the new method to Picard iterations, so it is a waveform relaxation technique [3]. Numerical experiments are favourably compared with the ones given by the known GAM [2], GBS [14] and Sarafyan [18] methods.

Chebyshev pseudospectral method and waveform relaxation for differential and differential–functional parabolic equations

The waveform relaxation technique for linear parabolic differential and differential–functional equations is studied. We use the second order finite difference method and the Chebyshev pseudospectral method for spatial discretization and apply a Gauss–Seidel waveform relaxation scheme to the resulting systems of ordinary differential and differential–functional equations. Waveform relaxation error bounds are presented for the two semi-discretization schemes in both functional and non-functional cases. Sharp error bounds are obtained after application of an inequalities technique with time-dependent coefficients and logarithmic norm. Convergence of the schemes is studied analytically and compared by means of extensive numerical data obtained for four parabolic equations with different coefficients. Our conclusion is that waveform relaxation error bounds and waveform relaxation convergence are better after Chebyshev pseudospectral semi-discretization than after finite difference method. Moreover, for the same accuracy of semi-discretization, a single Gauss–Seidel waveform iteration after Chebyshev pseudospectral method is less expensive than after finite difference method.

Waveform Relaxation for Functional-Differential Equations

The convergence of waveform relaxation techniques for solving functional-differential equations is studied. New error estimates are derived that hold under linear and nonlinear conditions for the right-hand side of the equation. Sharp error bounds are obtained under generalized time-dependent Lipschitz conditions. The convergence of the waveform method and the quality of the a priori error bounds are illustrated by means of extensive numerical data obtained by applying the method of lines to three partial functional-differential equations.

A waveform relaxation technique for steady state initialization of circuits with nonlinear elements and ideal diodes

This paper presents an efficient and robust technique for finding steady-state harmonic solutions in power circuits with nonlinear elements and ideal diode-type devices. The problem is formulated as a two-point boundary value (TPBV) problem using the waveform relaxation technique. Ideal diodes are modelled using a multiple area Thevenin equivalent technique. The overall solution is very robust and convergence is achieved within a few iterations even for very low damping and stiff systems. Examples are presented comparing the proposed technique with a previous technique and with direct "brute-force" integration.

ACCELERATION OF CONVERGENCE OF WAVEFORM RELAXATION ITERATIONS FOR SECOND ORDER LINEAR SYSTEM

The solution of second order linear systems of differential equations by waveform relaxation techniques is investigated. It is demonstrated that overlapping of components of the system improves significantly the speed of convergence of the resulting waveform relaxation iterations.

The performance of preconditioned waveform relaxation techniques for pseudospectral methods

Waveform relaxation techniques for the pseudospectral solution of the heat conduction problem are discussed. The pseudospectral operator occurring in the equation is preconditioned by transformations in both space and time. In the spatial domain, domain stretching is used to more equally distribute the grid points across the domain, and hence improve the conditioning of the differential operator. Preconditioning in time is achieved either by an exponential or a polynomial transformation. Block Jacobi solutions of the systems are obtained and compared. The preconditioning in space by domain stretching determines the effectiveness of waveform relaxation in this case. Preconditioning in time is also effective in reducing the number of iterations required for convergence. The polynomial transformation is preferred, because it removes the requirement of a matrix exponential calculation in the time-stepping schemes and, at the same time, is no less effective than the direct exponential preconditioning.

MIMD COMPUTERS FOR SCIENTIFIC APPLICATIONS

The purpose of this talk is twofold: to give an overview of the major parallel processor projects at IBM T.J. Watson research and then to discuss in some detail the VICTOR [1] message-passing parallel machine. Finally, concepts of a possible massively-parallel machine for very high performance are briefly discussed. In the first part of the talk the RP3 [2], the ACE [3], the GFII [4], the M/370, the Hybrid Dataflow [5], and the VLIW [6] projects will be discussed at a level sufficient to explain the key features of each of these projects, but without going into technical details. In summary, the RP3 is a 64-node shared memory machine, the ACE a 8-node shared bus workstation, the GF11 a switch-based SIMD machine with up to 576 nodes, the M/370 a project to study commercial applications of message-passing machines. The Hybrid Data Flow and the VLIW projects intend to explore very fine grained parallelism. The VICTOR project is a family of transputer based machines which were designed and build at T.J. Watson. It includes V32 and V256 with 32 and 256 nodes, respectively, and a set of 16-node workstations. All of these are presently operational. All machines except V32 are T800 based, with 4 MB of memory/node. V256 incorporates a distributed 16-node fileserver with 10 GB capacity. Special hardware supports multiple-users via spatial partitioning and non-intrusive processor monitoring. The operating environment is based on a mixture of vendor compiler and loader technology and a runtime system developed in the VICTOR group. In particular, this includes a set of routing routines and support for the filesystem. Languages used are OCCAM, C, Pascal and Fortran 77. Presently, the Trollius [7] operating system which has been developed at Cornell University is being ported onto VICTOR. Applications programs which have been written for VICTOR include presently fractals, ray-tracing, a nuclear physics Monte Carlo code, a computer pipeline model and a neural network code. Under development presently are two codes for VLSI design, dealing with parallel fault simulation and with circuit simulation. The latter one uses a parallel version of waveform relaxation techniques and is expected to be able to handle circuits with up to 106 transistors. Other applications being developed are graphical process monitors, database research, 3D-seismic code and multi-robot simulation. Work in the Mathematical Sciences department is focussed on developing a parallel language with emphasis on dynamic process scheduling. In the last part of the talk issues of scaling message-passing architectures into the Teraflop performance range are discussed.

Applying fast walsh transform technique to analyze high-speed digital integral transmission lines

A new approach, based on the waveform relaxation technique and fast Walsh transform, is presented to analyze the coupled loosy transmission lines (CLTL) with arbitrary terminal networks. The simulation accuracy of the new method can be greatly improved, the disadvantage which always exists in previous methods can be avoided and a considerable saving in time and memory of CPU is obtained.

Circuit simulation by hierarchical waveform relaxation

A circuit analysis method, the hierarchical waveform relaxation technique, that uses multilevel decomposition and relaxation techniques is presented. Sufficient conditions for the convergence of the process are discussed, and a dynamic circuit restructuring technique that uses the concept of sensitivity based subcircuit merging and repartitioning within the hierarchical framework to improve the robustness of the simulation algorithm as well as the speed of convergence is presented. The algorithm is implemented in the circuit simulation program PYRAMID, and test results are compared with those obtained by other methods of analysis. The comparison indicates that this method produces better results than had previously been possible for a large range of circuits of practical interest, particularly strongly coupled digital bipolar and FET circuits containing bidirectionality and feedback, and it is demonstrated that the relative improvement of the method scales with the size of the circuit. >

Time-domain analysis of lossy coupled transmission lines

A new approach, based on the waveform relaxation technique and fast Walsh transform, is first presented for the analysis of lossy coupled transmission lines (LCTL) with arbitrary terminal networks. The simulation accuracy of the new method can be greatly improved, the disadvantage which always exists in previous methods [1]–[7] can be avoided and a considerable saving in time and memory of CPU is obtained.

Waveform relaxation techniques for linear and nonlinear diffusion equations

A recent class of multirate numerical algorithms, collectively referred to as waveform relaxation methods, is applied to the one-dimensional diffusion equation. The methods decouple different parts or blocks of the system in the time domain, effectively allowing each block to take the largest time-step consistent with its accuracy requirements. Significant speedup is obtained over the results using a composite Crank-Nicholson/ second-order backward Euler time-stepping scheme. Possible implementation strategies for the waveform relaxation schemes to the diffusion equation in two dimensions are considered briefly.

Newton waveform relaxation techniques for tightly coupled systems

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. >

Transient analysis of lossy transmission lines with arbitrary initial potential and current distributions

The method of characteristics has been generalized for the transient analysis of lossy transmission lines with arbitrary initial conditions. It is shown that traveling waves generated by the initial potential and current distributions can be simulated by voltage and current generators connecting to the terminals of the lossy transmission-line model consisting of ideal delay-lines and waveshaping networks synthesized by the modified Pade approximation. The discrete-time transient simulation requires neither the FFT nor convolution integral and is as efficient as the iterative waveform relaxation technique. Transient responses of lossy transmission lines with steady-state and transient initial potential and current distributions are simulated for illustration.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Transient simulation of nonuniform coupled lossy transmission lines characterized with frequency-dependent parameters. I. Waveform relaxation analysis

Presents a waveform relaxation technique for simulating the transient response of nonuniform coupled transmission lines which are characterized with frequency-dependent parameters. The method consists of iterative waveform relaxation analysis of asymmetric disjoint resistive networks constructed with voltage-dependent voltage sources generated by applying the fast Fourier transform (FFT). The method requires neither convolution integration nor synthesis of lumped equivalent circuits for the simulation of the frequency-dependence of transmission-line parameters. Transient responses of uniform and nonuniform coupled transmission lines with and without skin-effect parameters and terminated with linear and nonlinear loads are simulated for illustration. The accuracy and efficiency of the relaxation technique are substantiated with exact analytical solutions and experimental data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Relaxation simulation of transverse electromagnetic wave propagation in coupled transmission lines

An efficient method for computing the transient response of an n-conductor coupled transmission line system that supports transverse electromagnetic waves propagating at different velocities is presented. The method consists of iterative generations of travelling voltage waveforms by the method of characteristics and the waveform relaxation technique. It is shown that the spatial and temporal variations of conductor voltages and currents can be derived from the linear combinations of traveling waves and initial potential and current distributions. Closed-form analytical solutions and experimental data are given to substantiate the accuracy and efficiency of the relaxation technique. >