Recasting modified nodal analysis to improve reliability in numerical circuit Simulation

A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed.

Transient simulation becomes a bottleneck for modern IC designs due to large numbers of transistors, interconnects and tight design margins. For modified nodal analysis (MNA) formulation, we could have differential algebraic equations (DAEs) which consist ordinary differential equations (ODEs) and algebraic equations. Study of solving DAEs with conventional multi-step integration methods has been a research topic in the last few decades. We adopt matrix exponential based integration method for circuit transient analysis, its stability and accuracy with DAEs remain an open problem. We identify that potential stability issues in the calculation of matrix exponential and vector product (MEVP) with rational Krylov method are originated from the singular system matrix in DAEs. We then devise a robust algorithm to implicitly regularize the system matrix while maintaining its sparsity. With the new approach, $\varphi$ functions are applied for MEVP to improve the accuracy of results. Moreover our framework no longer suffers from the limitation on step sizes thus a large leap step is adopted to skip many simulation steps in between. Features of the algorithm are validated on large-scale power delivery networks which achieve high efficiency and accuracy.

The model SWIF (Soil Water in Forested ecosystems) is presented. SWIF is a model for the simulation of water flow in the unsaturated soil zone, including water extraction by roots and lateral saturated drainage. the model gives a halfimplicit finite difference solution to the flow equation. for minimizing computing time, the model chooses its integration time steps based on the rate of change of soil water conditions. the model is not a conceptually new model. However, it meets the requirements of an entirely modular programme in which new theories can easily be implemented. the hydrology of the canopy and the forest floor is described in independent modules.Simulation results are evaluated in relation to temporal and vertical discretization, using a simulation of soil water dynamics in a forest over a period of two growing seasons. Both the site description and the simulation results are given in an accompanying paper (Bouten et al., 1992). in another paper a sensitivity analysis of the model is given (Bouten and Witter, 1992).In order to evaluate the effects of temporal discretization of boundary condition's, the length of the interval of constant averaged boundary conditions was varied from one hour to one month. the simulated annual transpiration is found to be dependent on this interval for a wet year, but not for a dry year. When the boundary conditions are taken constant for intervals shorter than one day, simulated soil water contents show a highly dynamic behaviour during and directly after heavy rainfall. Then small integration time steps are required and computing time increases strongly.For the evaluation of the effects of vertical discretization, the number of soil layers was varied from 5 to 40. the yearly transpiration appears to be almost independent of the layer thickness as long as the horizon boundaries are not altered, but the temporal distribution of transpiration and water contents are affected. the required computing time for the simulation of the water balance of a full growing season increases from 2.5 minutes for a simulation of five layers to 18 minutes for a simulation of 40 layers.When model results are evaluated on a yearly basis, they appear to be not very sensitive to the temporal and vertical discretization. Only if research aims require more detailed model results on a shorter timescale, higher computing times are to be accepted.

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations is analyzed using the theory of dynamical systems. With the aid of parallel Connection Machines (CM-2 and CM-5), the associated bifurcation diagrams as a function of the time step, and the complex behavior of the associated 'numerical basins of attraction' of these iterative implicit schemes are revealed and compared. Studies showed that all of the four implicit LMMs exhibit a drastic distortion and segmentation but less shrinkage of the basin of attraction of the true solution than standard explicit methods. The numerical basins of attraction of a noniterative implicit procedure mimic more closely the basins of attraction of the differential equations than the iterative implicit procedures for the four implicit LMMs.

A novel numerical method based on the differential transformation is proposed for solving nonlinear optimal control problems in this paper. The differential transformation is a linear operator that transforms a function from the original time and/or space domain into another domain in order to simplify the differential calculations. The optimality conditions for the optimal control problems can be represented by algebraic and differential equations. Using the differential transformation, these algebraic and differential equations with their boundary conditions are first converted into a system of nonlinear algebraic equations. Then the numerical optimal solutions are obtained in the form of finite-term Taylor series by solving the system of nonlinear algebraic equations. The differential transformation algorithm is similar to the spectral element methods in that the computational region splits into several subregions but it uses polynomials of high degrees by keeping a small number of subregions. The differential transformation algorithm could solve the finite- (or infinite-) time horizon optimal control problems formulated as either the algebraic and ordinary differential equations using Pontryagin’s minimum principle or the Hamilton–Jacobi–Bellman partial differential equation using dynamic programming in one unified framework. In addition, the differential transformation algorithm can efficiently solve optimal control problems with the piecewise continuous dynamics and/or nonsmooth control. The performance is demonstrated through illustrative examples.

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

A New Multistep Technique Based on the Nonuniform Rational Basis Spline Curves for Nonlinear Transient Heat Transfer Analysis of Functionally Graded Truncated Cone

In this study, the analysis of fixed capacitor thyristor controlled reactor (FC-TCR), one of Static VAR compensation techniques, is performed by modified nodal analysis (MNA). In the analysis, semiconductor elements are modelled with two- valued resistor concept. The resulting equations are solved by Trapezoidal method, used in numerical analysis. All of the simulation results are obtained in Matlab M-File environment.

This paper presents a simulation technique and a computer program for fast DC analysis of MOS transistor based VLSI circuits. Time domain analysis of a VLSI circuit is evaluated by a piecewise constant waveform approximation. This approximation is realized by repeatedly applying the developed DC analysis engine with initial conditions. All proposed methods such as device modeling, circuit partitioning and event-driven simulation were implemented and combined with such well known algorithms as modified nodal analysis (MNA), Katzenelson algorithm and Gaussian elimination in the form of a circuit simulation program: SAMOC. Time domain waveforms and benchmark circuit simulation results comparison of SPICE and SAMOC are presented.

Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type

This article presents an efficient numerical solution for specific integro‐differential equations of all fractional orders in the Caputo sense within the interval . We propose a new numerical scheme to evaluate the approximate solution of a nonlinear system of integro‐fractional differential equations of the Volterra–Hammerstein type by utilizing a quadratic B‐spline curve, while a linear B‐spline curve is employed as a predictor at the initial step to initiate the scheme. First, we establish the Caputo‐type fractional‐order derivative of the quadratic B‐spline curve and apply it to transform the considered problem into a system of nonlinear algebraic equations, where the Gauss–Legendre quadrature formula is applied for integral evaluations. The resulting system is then solved using Newton's method. Additionally, the linear B‐spline curve is utilized as a predictor to determine the values of the unknown functions at a single node and to compute one of the unknown control points. The presented method enhances the accuracy and efficiency of numerical computations, making it a useful technique for addressing complex fractional‐order models in applied mathematics and engineering. Some illustrative examples are presented, with results displayed in graphs and tables, to validate the method's effectiveness and robustness. MATLAB 9.2 was used to execute and perform the computations for the proposed algorithms.

A hybrid approximation method for solving Hutchinson’s equation

This document is intended to contain a detailed description of the mathematical formulation of Xyce, a massively parallel SPICE-style circuit simulator developed at Sandia National Laboratories. The target audience of this document are people in the role of 'service provider'. An example of such a person would be a linear solver expert who is spending a small fraction of his time developing solver algorithms for Xyce. Such a person probably is not an expert in circuit simulation, and would benefit from an description of the equations solved by Xyce. In this document, modified nodal analysis (MNA) is described in detail, with a number of examples. Issues that are unique to circuit simulation, such as voltage limiting, are also described in detail.

The melting process with moving boundary for mixture consisting of two fluids

The paper considers a hemispherical resonator gyroscope. It describes the structure of a hemispherical resonator gyroscope as a measuring and computing system, as well as the problems and main factors involved in increasing its accuracy. We studied a measuring component utilising alternating voltage, its circuit layout and mathematical model. We stated the problem of synthesising a computational component taking into account possible computational resource limitations in the device in the form of a system of nonlinear algebraic equations. We considered the specifics and implementation requirements concerning the computational algorithm, as well as the specifics of utilising redundant information about the wave pattern state in the gyroscope. The equation system proposed is overdetermined. We analysed the issues peculiar to the iterative method of solving this nonlinear system, which we propose to use as the basis for the mathematical model of the computational component. The method described may be generalised as the Gauss --- Newton method. The paper provides alternative iterative methods for designing the computational component. We investigated convergence issues for the computational component model obtained via the Gauss --- Newton method. The paper presents computational component simulation results and concludes that using the computational component model obtained is possible and feasible