Showing papers on "Discretization published in 1983"

Reconstruction algorithms: Transform methods

Abstract: Transform methods for image reconstruction from projections are based on analytic inversion formulas. In this tutorial paper, the inversion formula for the case of two-dimensional (2-D) reconstruction from line integrals is manipulated into a number of different forms, each of which may be discretized to obtain different algorithms for reconstruction from sampled data. For the convolution-backprojection algorithm and the direct Fourier algorithm the emphasis is placed on understanding the relationship between the discrete operations specified by the algorithm and the functional operations expressed by the inversion formula. The performance of the Fourier algorithm may be improved, with negligible extra computation, by interleaving two polar sampling grids in Fourier space. The convolution-backprojection formulas are adapted for the fan-beam geometry, and other reconstruction methods are summarized, including the rho-filtered layergram method, and methods involving expansions in angular harmonics. A standard mathematical process leads to a known formula for iterative reconstruction from projections at a finite number of angles. A new iterative reconstruction algorithm is obtained from this formula by introducing one-dimensional (1-D) and 2-D interpolating functions, applied to sampled projections and images, respectively. These interpolating functions are derived by the same Fourier approach which aids in the development and understanding of the more conventional transform methods.

Method for Generating Discrete Soliton Equation. III

Abstract: As a continuation of previous work, discretization of nonlinear Schrodinger equation and its analogues are discussed as the reduction of 2-component KP hierarchy

A control volume finite-element method for two-dimensional fluid flow and heat transfer

Abstract: The formulation of a general numerical method for two-dimensional incompressible flow and heat transfer in irregular-shaped domains is presented. The calculation domain is first divided into six-node macroelements. Then each macroelement is divided into four three-node triangular subelements. Polygonal control volumes are associated with the nodes of these elements. All dependent variables other than pressure are stored at the nodes of the subelements, and they are interpolated by functions that respond appropriately to an element Peclet number and the direction of an element-averaged velocity vector. The pressure is stored only at the vertices of the macroelements and is interpolated linearly in these elements. The discretization equations are obtained by deriving algebraic approximations to integral conservation equations applied to the polygonal control volumes. An iterative procedure akin to SIMPLER is used to solve the discretization equations.

Numerical methods for semiconductor device simulation

Abstract: This paper describes the numerical techniques used to solve the coupled system of nonlinear partial differential equations which model semiconductor devices. These methods have been encoded into our device simulation package which has successfully simulated complex devices in two and three space dimensions. We focus our discussion on nonlinear operator iteration, discretization and scaling procedures, and the efficient solution of the resulting nonlinear and linear algebraic equations. Our companion paper [13] discusses physical aspects of the model equations and presents results from several actual device simulations.

Numerical evaluation of path-integral solutions to Fokker-Planck equations

Abstract: A numerical method, based on the path-integral formalism, is presented to solve nonlinear Fokker-Planck equations with natural boundary conditions. For one-dimensional stochastic processes, several specific examples possessing exact analytic solutions are evaluated numerically for purposes of comparison. Various discretization prescriptions are investigated and found to be equivalent as expected. The numerical method is shown to give accurate results provided the spatial discretization and the time step satisfy certain relationships determined by the drift and the diffusion functions of the nonlinear Fokker-Planck equations. 26 refs., 5 figs.

A New Convergence Proof for the Multigrid Method Including the V-Cycle

Abstract: For a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem. Hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the “V-cycle”. The presented proof applies to procedures with any number of smoothing iterations and to the V-cycle.

Some new procedures for numerical solution of variably saturated flow problems

Abstract: Persistent difficulties that arise in forming numerical solutions of variably saturated flow problems include controlling the stability of the nonlinear equation solvers and devising a reliable, yet efficient, method for determining the positions of seepage surfaces. New techniques for addressing these problems are applied to a subdomain finite element discretization of the governing flow equations. A series of test problems demonstrates that the techniques are reliable and efficient for a wide variety of problems.

Preconditioning and two-level multigrid methods of arbitrary degree of approximation

Abstract: Let h be a mesh parameter corresponding to a finite element mesh for an elliptic problem. We describe preconditioning methods for two-level meshes which, for most prob- lems solved in practice, behave as methods of optimal order in both storage and computa- tional complexity. Namely, per mesh point, these numbers are bounded above by relatively small constants for all h - ho, where ho is small enough to cover all but excessively fine meshes. We note that, in practice, multigrid methods are actually solved on a finite, often even a fixed number of grid levels, in which case also these methods are not asymptotically optimal as h - 0. Numerical tests indicate that the new methods are about as fast as the best implementations of multigrid methods applied on general problems (variable coefficients, general domains and boundary conditions) for all but excessively fine meshes. Furthermore, most of the latter methods have been implemented only for difference schemes of second order of accuracy, whereas our methods are applicable to higher order approximations. We claim that our scheme could be added fairly easily to many existing finite element codes. 1. Introduction. Consider the numerical solution of elliptic boundary value prob- lems discretized by finite element methods. We assume that the boundary is polygonal or consists of planes. We note that in practical problems one often has a fine enough grid already after the definition of the boundary and the minimal number of vertices needed for a first (coarse) triangulation. Anyhow, if not so, in most cases one makes only a few steps of mesh refinement. Hence the power of multigrid methods-their optimal order of computational complexity-is most often not achieved fully, because optimality requires a large number of recursively defined meshes (for details see, e.g., (4) and for further references see (7)). Hence one might as well consider other methods, perhaps simpler and more effective on a fixed mesh, but which are not asymptotically optimal. Here we shall describe a method which uses only a fixed mesh, but for which one nevertheless achieves a low order of computational complexity and of seemingly optimal order except for, from a practical viewpoint, excessively small meshes. To be more precise, the computational cost per mesh point is bounded by clog N for N < No, where N is the number of mesh points, No is large enough to cover most applications and c is small enough that the method is competitive with multigrid methods. As is well known, the latter need recursion and the usual smoothing followed by corrections of the solutions on the different mesh levels. We claim that the new method is more suitable for implementation in existing finite element packages. In fact most packages for the multigrid methods are only for second order difference methods.

Three-dimensional unsteady Euler equation solutions using flux vector splitting

Abstract: A method for numerically solving the three dimensional unsteady Euler equations using flux vector splitting is developed. The equations are cast in curvilinear coordinates and a finite volume discretization is used. An explicit upwind second-order predictor-corrector scheme is used to solve the discretized equations. The scheme is stable for a CFL number of two and local time stepping is used to accelerate convergence for steady-state problems. Characteristic variable boundary conditions are developed and used in the far field and at surfaces. No additional dissipation terms are included in the scheme. Numerical results are compared with results from an existing three dimensional Euler code and experimental data.

Direct and inverse scattering in the time domain via invariant imbedding equations

Abstract: This paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain, which is applied here to the case of a one‐dimensional lossless medium. The method is based on an integro‐differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques. An example of an exact solution to this equation is given. Numerical schemes for solving the direct and inverse problems are derived by discretizing this equation. The behavior of the resulting algorithms is then tested on several examples. The results suggest that the inversion method is stable in the presence of additive noise.

Device modeling

Abstract: This paper reviews the progress in device modeling with emphasis on numerical modeling approaches. The reason for this is its ever-increasing importance for the design of small-scale devices suited for VLSI applications. First, the basic field equations with their respective boundary conditions are given. Followed by a description of empirical models for the physical device mechanisms, i.e., mobility, avalanche generation, band-gap narrowing. Subsequently, different numerical models, mainly developed in the past decade, are outlined briefly and discretization as well as solution methods are being discussed. Some remarks are given concerning the relations between finite-difference and finite-element methods. Simplified numerical models are also mentioned and their usefulness for certain type of applications is stressed. In order to clearly demonstrate the power of numerical device modeling, a number of representative examples is given. The last sections deal with analytical device modeling. Bipolar transistor models are only briefly reviewed since the evolution has led to some kind of standardization, but the development of MOS transistor models, where the same is not true, is described in more detail. Cross references to numerical results should clarify that with decreasing device dimensions the model parameters of analytical MOST models tend to loose their physical significance and change increasingly into fitting parameters.

Numerical Methods for Semiconductor Device Simulation

Abstract: This paper describes the numerical techniques used to solve the coupled system of nonlinear partial differential equations which model semiconductor devices. These methods have been encoded into our device simulation package which has successfully simulated complex devices in two and three space dimensions. We focus our discussion on nonlinear operator iteration, discretization and scaling procedures, and the efficient solution of the resulting nonlinear and linear algebraic equations. Our companion paper [13] discusses physical aspects of the model equations and presents results from several actual device simulations.

Characteristics Method Using Time‐Line Interpolations

Abstract: The use of interpolations in time, rather than the more widely used spatial interpolations, demonstrates several benefits in the application of the method of characteristics to wave problems in hydraulics. Two such time-line schemes are presented and analyzed in the solution of a linearized water hammer problem. Reachback time-line interpolations, where the characteristic lines are projected back before the current time step, demonstrate less damping than the corresponding spatial interpolation scheme at the same discretization. Implicit time-line interpolations, where the characteristic line is projected into the current time step, permit relaxation of the restrictive time step required by the Courant condition. An error analysis and numerical experiments demonstrate the degree of damping and dispersion introduced by both reachback and implicit time-line interpolation methods. The error analyses and their verification may encourage further application of these methods in appropriate problem domains.

Decomposition in Reliability Analysis of Fault-Tolerant Systems

Abstract: Two important problems which arise in modeling fault-tolerant systems with ultra-high reliability requirements are discussed. 1) Any analytic model of such a system has a large number of states, making the solution computationally intractable. This leads to the need for decomposition techniques. 2) The common assumption of exponential holding times in the states is intolerable while modeling such systems. Approaches to solving this problem are reviewed. A major notion described in the attempt to deal with reliability models with a large number of states is that of behavioral decomposition followed by aggregation. Models of the fault-handling processes are either semi-Markov or simulative in nature, thus removing the usual restrictions of exponential holding times within the coverage model. The aggregate fault-occurrence model is a non-homogeneous Markov chain, thus allowing the times to failure to possess Weibull-like distributions. There are several potential sources of error in this approach to reliability modeling. The decomposition/aggregation process involves the error in estimating the transition parameters. The numerical integration involves discretization and round-off errors. Analysis of these errors and questions of sensitivity of the output (R(t)) to the inputs (failure rates and recovery model parameters) and to the initial system state acquire extreme importance when dealing with ultra-high reliability requirements.

Finite boxes&#8212;A generalization of the finite-difference method suitable for semiconductor device simulation

Abstract: A two-dimensional numerical device-simulation system is presented. A novel discretization scheme, called "finite boxes," allows an optimal grid-point allocation and can be applied to nonrectangular devices. The grid is generated automatically according to the specified device geometry. It is adapted automatically during the solution process by equidistributing a weight function which describes the local discretization error. A modified Newton method is used for solving the discretized nonlinear system. To achieve high flexibility the physical parameters can be defined by user-supplied models. This approach requires numerical calculation of parts of the coefficients of the Jacobian. Supplementary algorithms speed up convergence and inhibit the commonly known Newton overshoot. The advantages and computer resource savings of the new method are described by the simulation of a 100-V diode. We also present results for thyristor and GaAs MESFET simulations.

A comprehensive two-dimensional VLSI process simulation program, BICEPS

Abstract: Bell Integrated Circuit Engineering Process Simulator (BICEPS) is a comprehensive VLSI process-simulation program developed at Bell Laboratories BICEPS incorporates the most up-to-date physical models and efficient numerical algorithms to make it a highly robust and general-purpose program BICEPS can calculate doping profiles resulting from ion implantation, predeposition, oxidation, and epitaxy in one or two spatial dimensions as well as etching and deposition of oxide, nitride, and photoresist In this paper, the physics of IC process simulation will be reviewed with an emphasis on the various physical models implemented in BICEPS Calculation of the impurity profiles in VLSI devices involves the solution of a coupled set of nonlinear time-dependent partial differential equations, with moving boundaries and in more than one spatial dimension The numerical techniques in obtaining a solution to this problem, namely, spatial discretization, time discretization, and the treatment of moving boundaries are also described in this paper The capabilities of BICEPS are illustrated by the results of simulation of the fabrication of a typical NMOS transistor

Approximation of the signorini problem with friction, obeying the coulomb law

Abstract: The approximation of the Signorini problem with friction by mixed finite element method is studied. The relation between the continuous case and its finite dimensional discretization is analyzed.

Multiple‐parameter reduced basis technique for bifurcation and post‐buckling analyses of composite plates

Abstract: A multiple-parameter reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analyses of composite plates subjected to combined loadings. The computational algorithm can be conveniently divided into three distinct stages. The first stage is that of determining the stability boundary. The plate is discretized by using displacement finite element models and the analysis region is reduced by exploiting the special symmetries exhibited by the response of the plate. The vector of unknown nodal displacements is expressed as a linear combination of a small number of path derivatives (derivatives of the nodal displacements with respect to path parameters), and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of algebraic equations. The reduced equations are used to determine the stability boundary of the plate. In the second stage, a nonllnear solution in the vicinity of the stability boundary is obtained by using a bifurcation buckling mode as a predictor, and a set of reduced equations is generated. In the third stage, the reduced equations are used to trace post-buckling paths corresponding to various combinations of the load parameters. The potential of the proposed approach is discussed and its effectiveness is demonstrated by means of a numerical example of laminated composite plate subjected to combined compressive and shear loadings.

On the importance of the discrete maximum principle in transient analysis using finite element methods

Abstract: In transient analysis it is generally thought that small time steps can only improve the accuracy, because standard stability theorems limit the maximum time step for a given mesh size. In finite element approximations, however, small time steps may cause stability problems which lead to physically unreasonable results. It is shown that this is due to the violation of a discrete maximum principle. The influence of lumped and consistent mass matrices on a stable discretization of time and space is presented.

On multi-grid methods for variational inequalities

Abstract: We consider here a general class of algorithms for the numerical solution of variational inequalities. A convergence proof is given and in particular a multi-grid method is described. Numerical results are presented for the finite-difference discretization of an obstacle problem for minimal surfaces

An eigenvalue numerical technique for solving unsteady linear groundwater models continuously in time

Abstract: A procedure for solving the differential groundwater flow equation is presented herein. Using a finite difference or finite element discretization scheme, a set of simultaneous linear equations is obtained. The eigenvalues and eigenvectors of a matrix, which is a function of the coefficients of the set, are the key to the solution. A vector L is obtained straightforwardly by combining the eigenvector matrix A, the eigenvalue vector α, the pumping vector P, and the initial head vector H. Vector L, which depends on time, can be expressed simply and explicitly as a function of the eigenvalues. Piezometric heads can be obtained by combining A and L. L is the only vector that needs to be computed as P changes with time. In this way, influence functions of a piezometric head, flow velocity and flow depletion of a stream connected with the aquifer under a unit stress, can be obtained explicitly and continuously in time. The method can be applied to confined as well as to leaky aquifers and to one-, two-, or three- dimensional linear models. Its main advantage lies in the fact that it is unnecessary to repeatedly solve a matrix for every time increment. The method is particularly useful for groundwater management problems in which a large number of alternatives have to be evaluated.

A proposed upstream weight numerical method for simulating pollutant transport in groundwater

Abstract: An alternative upstream weighting numerical method is proposed to simulate the transport process in groundwater under a variety of conditions. The discretized system of the solute transport equations is derived from the integral form that expresses the balance of mass in each element. An upstream weight that depends on the local Peclet number is directly added to the basis functions. When convective transport dominates the dispersive transport, this technique can eliminate the oscillation of solutions efficiently, with only a nominal increase of computation effort over that of the general linear triangular Galerkin finite element method. The accuracy of the proposed numerical model is tested against two classical problems for which analytical solutions exist. The agreement is nearly exact. An additional numerical example is used to illustrate the applicability of this model.

On deterministic control problems: An approximation procedure for the optimal cost

Abstract: In optimal control problems with continous and impulse controls, it is necessary to solve a quasi-variational inequality of Hamilton-Jacobi type. For the non-stationnary problem, we present a discretization method to solve it numerically and we show an application to the management of energy production systems.