Showing papers on "Finite difference published in 1984"

Analysis and simulation of semiconductor devices

Abstract: 1. Introduction.- 1.1 The Goal of Modeling.- 1.2 The History of Numerical Device Modeling.- 1.3 References.- 2. Some Fundamental Properties.- 2.1 Poisson's Equation.- 2.2 Continuity Equations.- 2.3 Carrier Transport Equations.- 2.4 Carrier Concentrations.- 2.5 Heat Flow Equation.- 2.6 The Basic Semiconductor Equations.- 2.7 References.- 3. Proeess Modeling.- 3.1 Ion Implantation.- 3.2 Diffusion.- 3.3 Oxidation.- 3.4 References.- 4. The Physical Parameters.- 4.1 Carrier Mobility Modeling.- 4.2 Carrier Generation-Recombination Modeling.- 4.3 Thermal Conductivity Modeling.- 4.4 Thermal Generation Modeling.- 4.5 References.- 5. Analytical Investigations About the Basic Semiconductor Equations.- 5.1 Domain and Boundary Conditions.- 5.2 Dependent Variables.- 5.3 The Existence of Solutions.- 5.4 Uniqueness or Non-Uniqueness of Solutions.- 5.5 Sealing.- 5.6 The Singular Perturbation Approach.- 5.7 Referenees.- 6. The Diseretization of the Basic Semiconductor Equations.- 6.1 Finite Differences.- 6.2 Finite Boxes.- 6.3 Finite Elements.- 6.4 The Transient Problem.- 6.5 Designing a Mesh.- 6.6 Referenees.- 7. The Solution of Systems of Nonlinear Algebraic Equations.- 7.1 Newton's Method and Extensions.- 7.2 Iterative Methods.- 7.3 Referenees.- 8. The Solution of Sparse Systems of Linear Equations.- 8.1 Direct Methods.- 8.2 Ordering Methods.- 8.3 Relaxation Methods.- 8.4 Alternating Direction Methods.- 8.5 Strongly Implicit Methods.- 8.6 Convergence Acceleration of Iterative Methods.- 8.7 Referenees.- 9. A Glimpse on Results.- 9.1 Breakdown Phenomena in MOSFET's.- 9.2 The Rate Effect in Thyristors.- 9.3 Referenees.- Author Index.- Table Index.

Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations

Abstract: Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media, and they promise to be invaluable in solving the full inverse problem. This paper quantitatively compares finite-difference and finite-element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time-domain and frequency-domain techniques. It is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy. To be of value, a solution technique must be able to minimize (1) numerical attenuation or amplification, (2) polarization errors, (3) numerical anisotropy, (4) errors in phase and group velocities, (5) extraneous numerical (parasitic) modes, (6) numerical diffraction and scattering, and (7) errors in reflection and transmission coefficients. This paper shows that in homogeneous media the explicit finite-element and finite-difference schemes are comparable when solving the scalar wave equation and when solving the elastic wave equations with Poisson's ratio less than 0.3. Finite-elements are superior to finite-differences when modeling elastic media with Poisson's ratio between 0.3 and 0.45. For both the scalar and elastic equations, the more costly implicit time integration schemes such as the Newmark scheme are inferior to the explicit central-differences scheme, since time steps surpassing the Courant condition yield stable but highly inaccurate results. Frequency-domain finite-element solutions employing a weighted average of consistent and lumped masses yield the most accurate resuls, and they promise to be the most cost-effective method for CDP, well log, and interactive modeling.--Modified journal abstract.

A transmitting boundary for transient wave analyses

Abstract: An artificial boundary condition at the edge of finite computational grids is devised. Itcan simulate the transmitting process of clastic surface waves and body waves incident atarbitrary angles under any accuracy required. It may be used for two- or three-dimensionaltransient wave analyses in laterally heterogeneous media and easily incorporated into exist-ing finite element or finite difference computational codes.

DIF3D: A Code to Solve One-, Two-, and Three-Dimensional Finite-Difference Diffusion Theory Problems

Abstract: The mathematical development and numerical solution of the finite-difference equations are summarized. The report provides a guide for user application and details the programming structure of DIF3D. Guidelines are included for implementing the DIF3D export package on several large scale computers. Optimized iteration methods for the solution of large-scale fast-reactor finite-difference diffusion theory calculations are presented, along with their theoretical basis. The computational and data management considerations that went into their formulation are discussed. The methods utilized include a variant of the Chebyshev acceleration technique applied to the outer fission source iterations and an optimized block successive over-relaxation method for the within-group iterations. A nodal solution option intended for analysis of LMFBR designs in two- and three-dimensional hexagonal geometries is incorporated in the DIF3D package and is documented in a companion report, ANL-83-1.

Diffusion of electromagnetic fields into a two-dimensional earth; a finite-difference approach

Abstract: We describe a numerical method for time‐stepping Maxwell’s equations in the two‐dimensional (2-D) TE‐mode, which in a conductive earth reduces to the diffusion equation. The method is based on the classical DuFort‐Frankel finite‐difference scheme, which is both explicit and stable for any size of the time step. With this method, small time steps can be used at early times to track the rapid variations of the field, and large steps can be used at late times, when the field becomes smooth and its rates of diffusion and decay slow down. The boundary condition at the earth‐air interface is handled explicitly by calculating the field in the air from its values at the earth’s surface with an upward continuation based on Laplace’s equation. Boundary conditions in the earth are imposed by using a large, graded grid and setting the values at the sides and bottom to those for a haft‐space. We use the 2-D model to simulate transient electromagnetic (TE) surveys over a thin vertical conductor embedded in a half‐space...

The stability of explicit Euler time‐integration for certain finite difference approximations of the multi‐dimensional advection–diffusion equation

Abstract: A comprehensive study is presented regarding the numerical stability of the simple and common forward Euler explicit integration technique combined with some common finite difference spatial discretizations applied to the advection-diffusion equation. One-dimensional results are obtained using both the matrix method (for several boundary conditions) and the classical von Neumann method of stability analysis and arguments presented showing that the latter is generally to be preferred, regardless of the type of boundary conditions. The less-well-known Godunov-Ryabenkii theory is also applied for a particular (Robin) boundary condition. After verifying portions of the one-dimensional theory with some numerical results, the stabilities of the two- and three-dimensional equations are addressed using the von Neumann method and results presented in the form of a new stability theorem. Extension of a useful scheme from one dimension, where the pure advection limit is known variously as Leith's method or a Lax-Wendroff method, to many dimensions via finite elements is also addressed and some stability results presented.

Methods for the numerical solution of the nonlinear Schroedinger equation

Abstract: Optimal L2 rates of convergence are established for several fully-discrete schemes for the numerical solution of the nonlinear Schroedinger equation. Both finite differences and finite elements are considered for the discretization in space, while the integration in time is treated either by the leap-frog technique or by a modified Crank-Nicolson procedure, which generalizes the one suggested by Delfour, Fortin and Payne and possesses two useful conserved quantities.

TVD finite difference schemes and artificial viscosity

Abstract: The total variation diminishing (TVD) finite difference scheme can be interpreted as a Lax-Wendroff scheme plus an upwind weighted artificial dissipation term. If a particular flux limiter is chosen and the requirement for upwind weighting is removed, an artificial dissipation term which is based on the theory of TVD schemes is obtained which does not contain any problem dependent parameters and which can be added to existing MacCormack method codes. Numerical experiments to examine the performance of this new method are discussed.

Dispersion Simulation in Two‐dimensional Tidal Flow

Abstract: An accurate numerical method for the mathematical modeling of contaminant dispersion in two‐dimensional tidal currents is developed and applied. The method avoids the excessive numerical damping or oscillations associated with most finite difference and finite element schemes for advection by using a characteristics approach with high order Hermite bicubic interpolation. The split‐operator algorithm, incorporating a Crank‐Nicolson operator for diffusion, provides a relatively simple and economic method for accurate simulation of pollutant dispersion on a fixed, Eulerian mesh. A special procedure for Lagrangian calculation of the dispersion of concentration fields which are small compared to the mesh size simulates the early stages of growth of point source plumes. These various procedures are described in detail, and their performance is demonstrated by application to schematic test cases and to the Bay of Saint‐Brieuc, France.

Computer Analysis of Dielectric Waveguides: A Finite-Difference Method

Abstract: A method for computing the modes of dielectric guiding structures based on finite differences is described. The numerical computation program is efficient and can be applied to a wide range of problems. We report here solutions for circular and rectangular dielectric waveguides and compare our solutions with those obtained by other methods. Limitations in the commonly used approximate formulas developed by Marcatili are discussed.

Analysis of Two-Dimensional Melting in Rectangular Enclosures in Presence of Convection

Abstract: Two-dimensional melting of a solid phase change material in a rectangular enclosure heated from one side is simulated numerically. The simulations are carried out by dividing the process in a large number of quasi-static steps. In each quasi-static step, steady-state natural convection in the liquid phase is calculated by directly solving the governing equations of motion with a finite difference technique. This is used to predict the shape and motion of the solid-liquid boundary at the beginning of the next step. The predictions are found to be in good agreement with experiment. Influence of some of the governing parameters on the time development of the melting process is studied using the numerical simulation procedure.

A Dissipative Galerkin Scheme for Open‐Channel Flow

Abstract: The finite element method based on the classical Galerkin formulation produces very poor results when applied to discontinuous channel flow. A variance of the Galerkin method is proposed that exhibits highly selective damping characteristics. The dissipation affects only the numerically-generated high-frequency parasitic waves, while maintaining remarkable accuracy in the approximation to the true solution of the problem. In fact, it is shown that the phase error of the finite element simulation is improved by the introduction of dissipation. The resulting model is second-order accurate with respect to the time step and produces a clean, sharp jump structure that agrees favorably with the exact solution of some test problems. The method is based on discontinuous weighting functions that produce “upwind” effects but at the same time maintain the accuracy of a central difference scheme. The dissipation level is selected by analytical investigations, so that the numerical error is minimized. No second-order pseudo viscosity terms are required, which relaxes the inter-element continuity conditions and results in a very simple and inexpensive scheme.

The two-dimensional Gaussian beam synthetic method: Testing and application

Abstract: The Gaussian beam method of Cervený et al. (1982) is an asymptotic method for the computation of wave fields in inhomogeneous media. The method consists of tracing rays and then solving the wave equation in “ray-centered coordinates.” The parabolic approximation is applied to find the asymptotic local solution in the neighborhod of each ray. The approximate global solution for a given source is then constructed by a superposition of Gaussian beams along nearby rays. The Gaussian beam method is tested in a two-dimensional inhomogeneous medium using two approaches. One is the application of the reciprocal theorem for Green's functions in an arbitrarily heterogeneous medium. The discrepancy between synthetic seismograms for reciprocal cases is considered as a measure of the error. The other approach is to apply Gaussian beam synthesis to cases for which solutions are known by other approximate methods. This includes the soft basin problem that has been studied by finite difference, finite element, discrete wavenumber, and glorified optics. We found that the results of these tests were in general satisfactory. We have used the Gaussian beam method for two applications. First, the method is used to study volcanic earthquakes at Mount Saint Helens. The observed large differences in amplitude and arrival time between a station inside the crater and stations on the flanks can be explained by the combined effects of an anomalous velocity structure and a shallow focal depth. The method is also applied to scattering of teleseismic P waves by a lithosphere with randomly fluctuating velocities.

Numerical Experiments in Boundary-Layer Stability

Abstract: Numerical solution of the three-dimensional incompressible Navier-Stokes equations is used to study the instability of a flat-plate boundary layer in a manner analogous to the vibrating-ribbon experiments. Flow field structures are observed which are very similar to those found in the vibrating-ribbon experiment to which computational initial conditions have been matched. Stream wise periodicity is assumed in the simulation so that the evolution occurs in time, but the events that constitute the instability are so similar to the spatially occurring ones of the laboratory that it seems clear the physical processes involved are the same. A spectral and finite difference numerical algorithm is employed in the simulation.

Instability of difference models for hyperbolic initial boundary value problems

Abstract: A th00 eory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary. According to this theory, instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity C ≥ 0. To make this point of view precise, we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries. From these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in l2 norms, which extend earlier results due to Osher and to Gustafsson, Kreiss, and Sundstrom. In particular we investigate l2-instability with respect to both initial and boundary data and show how they are affected by (a) finite versus infinite reflection coefficients and (b) wave radiation with C = 0 versus C > 0.

Efficient linear and nonlinear heat conduction with a quadrilateral element

Abstract: A method is presented for performing efficient and stable finite element calculations of heat conduction with quadrilaterals using one-point quadrature. The stability in space is obtained by using a stabilization matrix which is orthogonal to all linear fields and its magnitude is determined by a stabilization parameter. It is shown that the accuracy is almost independent of the value of the stabilization parameter over a wide range of values; in fact, the values 3, 2 and 1 for the normalized stabilization parameter lead to the 5-point finite difference, 9-point finite difference and fully integrated finite element operators, respectively, for rectangular meshes; numerical experiments reported here show that the three have identical rates of convergence in the L2 norm. Eigenvalues of the element matrices, which are needed for stability limits, are also given. Numerical applications are used to show that the method yields accurate solutions with large increases in efficiency, particularly in nonlinear problems.

Analysis of interrupted traffic flow by finite difference methods

Abstract: Three numerical techniques for macroscopic analysis of traffic dynamics at signalized links (isolated or coordinated) are presented. The techniques are based on finite differences in time and space and assist in implementing continuum models at real situations. The general methodology presented allows treatment of any arrival and departure pattern, inclusion of sinks or sources, employment of any desired equilibrium flow-density relationship and arbitrary specified initial conditions. Implementation of the proposed method to a signalized intersection and a coordinated link suggests satisfactory agreement with field data and notable agreement with analytical results, respectively. Comparisons made under progressively realistic assumptions demonstrate substantial improvements in model performance as the complexity of the assumptions increases

A Vertically Averaged Circulation Model Using Boundary-Fitted Coordinates

Abstract: A two-dimensional vertically averaged circulation model using boundary-fitted coordinates has been developed for predicting sea level and currants in estuarine and shelf waters. The basic idea of the approach is to use a set of coupled quasi-linear elliptic transformation equations to map the physical domain to a corresponding transformed plane such that all boundaries are coincident with coordinate lines and the transformed mesh is rectangular. The hydrodynamic equations are then solved by a multi-operation finite difference technique in the rectangular mesh transformed grid. Comparisons of the circulation model predictions for tidally forced flows in a wedge section with both flat and quadratic bottom topography, and in a flat channel with exponential variation in width, were in excellent agreement with corresponding analytic solutions. Simulation of steady-state wind-induced setup in a closed basin formed using elliptic cylindrical coordinates also was in excellent agreement with the analytic ...

Finite Difference Techniques for Partial Differential Equations

Abstract: Publisher Summary Analytical methods of solving partial differential equations are usually restricted to linear cases with simple geometries and boundary conditions. The increasing availability of more and more powerful digital computers has made more common the use of numerical methods for solving such equations, in addition to non-linear equations with more complicated boundaries and boundary conditions. This chapter describes one particular method, the method of numerical finite differences. This method is based on the representation of the continuously defined function τ (x,y,z,t) and its derivatives in terms of values of an approximation τ defined at particular, discrete points called gridpoints. From the appropriate Taylor's series expansions of τ about such gridpoints, forward, backward, and central difference approximations to derivatives of τ can be developed to convert the given partial differential equation and its initial and boundary conditions to a set of linear algebraic equations linking the approximations τ defined at the gridpoints.

On the approximation of infinite optimization problems with an application to optimal control problems

Abstract: This paper is concerned with approximations to infinite optimization problems in Banach spaces. Under the assumption of a first order necessary and a second order sufficient optimality condition we derive convergence results for the optimal solutions and the optimal values of the approximating problems. An application to finite difference approximations of nonlinear optimal control problems with state constraints is given.

A Numerical Heat Transfer Analysis of Strip Rolling

Abstract: The lack of a practical mathematical model to simulate thermal behavior of the metal rolling process has forced mill operators and designers to rely on plant experience and testing, which is time consuming and expensive. An effective finite difference model has been developed to study the temperature profiles of the work roll and the strip. Several finite difference techniques have been successfully employed to cope with the special characteristics of the rolling process, such as very high velocity, high temperature variation in a very thin layer, curved boundary, and bimaterial interface. Typical rolling conditions were analyzed to provide temperature information on the roll and strip. Both cold and hot rollings were considered, and the effect of changing velocities was also studied. Good correspondence is found when present results are compared with either analytical solutions under simplified rolling conditions or measured data.