Showing papers on "Finite difference published in 1988"

Fourth-order finite-difference P-SV seismograms

Abstract: I describe the properties of a fourth-order accurate space, second-order accurate time two-dimensional P-Sk’ finite-difference scheme based on the MadariagaVirieux staggered-grid formulation. The numerical scheme is developed from the first-order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses. The Madariaga-Virieux staggered-grid scheme has the desirable quality that it can correctly model any variation in material properties, including both large and small Poisson’s ratio materials, with minimal numerical dispersion and numerical anisotropy. Dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints/wavelength. The scheme can be used to accurately simulate wave propagation in mixed acoustic-elastic media, making it ideal for modeling marine problems. Explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free-surface or within a layer and to satisfy free-surface boundary conditions. Benchmark comparisons of finite-difference and analytical solutions to Lamb’s problem are almost identical, as are comparisons of finite-difference and reflectivity solutions for elastic-elastic and acoustic-elastic layered models.

Generation of finite difference formulas on arbitrarily spaced grids

Abstract: On etablit des recurrences simples pour le calcul des poids dans les formules aux differences finies compactes pour les derivees de tous ordres avec une precision d'ordre arbitraire sur des grilles a une dimension d'espacement arbitraire

Handbook of Numerical Heat Transfer

Abstract: A comprehensive presentation is given of virtually all numerical methods that are suitable for the analysis of the various heat transverse and fluid flow problems that occur in research, practice, and university instruction. After reviewing basic methodologies, the following topics are covered: finite difference and finite element methods for parabolic, elliptic, and hyperbolic systems; a comparative appraisal of finite difference versus finite element methods; integral and integrodifferential systems; perturbation methods; Monte Carlo methods; finite analytic methods; moving boundary problems; inverse problems; graphical display methods; grid generation methods; and programing methods for supercomputers.

Simplified second-order Godunov-type methods

Abstract: Harten, Lax, and van Leer showed how to construct a simple approximate Riemann solution which contains only one intermediate step. This construction assumes that you have a priori bounds on the smallest and largest signal velocities in the exact Riemann solution.Here we propose a number of algorithms for obtaining these bounds. Heuristic arguments are presented to support our choice of bounds. We derive first-order schemes which use these approximate Riemann solutions and show their relationship to known finite difference schemes.Next, we use the approach of van Leer et al. [1] to construct second-order schemes based on these approximate Riemann solutions. Of particular interest is a central difference scheme requiring no upwind switches. This scheme is only slightly more complex than standard predictor–corrector finite difference schemes. Preliminary numerical results are presented which show that these schemes are nonoscillatory, have good shock resolution and produce results which are competitive with those produced by more complex second order Godunov-type schemes.

Time-domain finite difference approach to the calculation of the frequency-dependent characteristics of microstrip discontinuities

Abstract: The frequency-dependent characteristics of the microstrip discontinuities have previously been analyzed using full-wave approaches. The time-domain finite-difference (TD-FD) method presented here is an independent approach and is relatively new in its application for obtaining the frequency-domain results for microwave components. The validity of the TD-FD method in modeling circuit components for MMIC CAD applications is established. >

On convergence of block-centered finite differences for elliptic-problems

Abstract: We consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains. We demonstrate that with sufficiently smooth data, the discrete $L^2 $-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids. Extensions to nonselfadjoint and parabolic problems are discussed.

The Random Walk Method in Pollutant Transport Simulation

Abstract: Standard finite difference and finite element solution methods of the pollutant transport equation require restrictive spatial discretization in order to avoid numerical dispersion. The random walk method offers a robust alternative if for reasons of calculational effort discretization requirements cannot be met. The method is discussed for the case of an ideal tracer starting out from the Ito-Fokker-Planck-equation. Features such as chemical reactions and adsorption can be incorporated. Besides being an alternative to other solution methods for the classical transport equation the random walk deserves attention due to its generalizability allowing the incorporation of non-Fickian dispersion. A shortcoming of the method results from the general roughness of simulated distributions in space and time due to statistical fluctuations and resolution problems. The method is applied to a field case of groundwater pollution by chlorohydrocarbons.

Fluid Flow and Mixed Convection Transport From a Moving Plate in Rolling and Extrusion Processes

Abstract: The heat transfer arising due to the movement of a continuous heated plate in processes such as hot rolling and hot extrusion has been studied. Of particular interest were the resulting temperature distribution in the solid and the proper imposition of the boundary conditions at the location where the material emerges from a furnace or die. These considerations are important in the simulation and design of practical systems. A numerical study of the thermal transport process has been carried out, assuming a two-dimensional steady circumstance. The boundary layer equations, as well as full governing equations including buoyancey effects, are solved employing finite difference techniques. The effect of various physical parameters, which determine the temperature and flow fields, is studied in detail. The significance of these results in actual manufacturing processes is discussed.

Ocean Acoustic Propagation by Finite Difference Methods

Abstract: Foreword. Preface. Introduction. Ocean acoustic wave propagation problems. Finite difference schemes. Initial and boundary conditions. Range step size analysis. Wide-angle capability. Applicable solution methods other than the implicit finite difference scheme. Representative test examples. Listing of computer codes. Reference citation index. Subject index.

Numerical solutions of high‐Re recirculating flows in vorticity‐velocity form

Abstract: On presente une methode de calcul des ecoulements stationnaires laminaires a grand nombre de Reynolds. Discretisation des equations de Navier-Stokes par differences finies sur une grille en quinconce et utilisation d'une methode de direction alternee implicite

A review of finite difference methods for seismo-acoustics problems at the seafloor

Abstract: Understanding seismic wave propagation in realistic seafloor environments is essential for many problems in marine seismology. Finite difference methods are becoming increasingly popular in solving propagation problems as the limitations of other techniques which apply only at high frequency or for flat-lying media become fully appreciated. The seafloor problem, a high contrast in Poisson's ratio at a rough sharp interface, is particularly challenging, and many published formulations fail to solve it accurately. The purpose of this paper is to summarize the published finite difference formulations for the elastic wave equation and to outline their applicability to seafloor problems.

Seismic energy partitioning and scattering in laterally heterogeneous ocean crust

Abstract: We present finite difference forward models of elastic wave propagation through laterally heterogeneous upper oceanic crust. The finite difference formulation is a 2-D solution to the elastic wave equation for heterogeneous media and implicitly calculates P and SV propagation, compressional to shear conversion, interference effects and interface phenomena. Random velocity perturbations with Gaussian and self-similar autocorrelation functions and different correlation lengths (a) are presented which show different characteristics of secondary scattering. Heterogeneities scatter primary energy into secondary body waves and secondary Stoneley waves along the water-solid interface. The presence of a water-solid interface in the model allows for the existence of secondary Stoneley waves which account for much of the seafloor ‘noise’ seen in the synthetic seismograms for the laterally heterogeneous models.

Sizing design sensitivity analysis of non-linear structural systems. Part II: Numerical method

Abstract: A numerical method is presented to carry out sizing design sensitivity calculations outside established finite element analysis codes, using postprocessing data only. Geometric as well as material non-linearities are treated. To demonstrate the accuracy of the proposed method, numerical results are presented for structural systems with linear elastic material, large displacements, large rotations and small strains. A distributed parameter approach to structural design sensitivity analysis is used to retain the continuum elasticity formulation throughout the derivation of design sensitivity results. Using this approach and an adjoint variable method, design sensitivity computations are carried out. For structural performance functionals stress and displacement are considered. It is shown that computations can be performed with the same computational effort as for sizing design sensitivity analysis of linear structural systems. Accurate design sensitivity results are obtained for both linear and non-linear structural systems without the uncertainty of numerical accuracy and high cost associated with the selection of finite difference perturbations. Also, the method does not require differentiation of element stiffness and mass matrices in conventional finite element models.

A new finite element based Lax-Wendroff/Taylor-Galerkin methodology for computational dynamics

Abstract: The present paper proposes the development of a new and effective methodology of computation for general computational dynamics. Fundamental concepts and characteristic features of the proposed Lax-Wendroff/Taylor-Galerkin algorithm are described and developed in technical detail. The methodology is based on first expressing the finite difference approximations of the transient time-derivative terms in conservation form in terms of a Taylor-series expansion including higher-order time derivatives, which are then evaluated from the governing dynamic equations also expressed in conservation form. The resulting expressions are discretized in space emploting classical Galerkin schemes and quite naturally we advocate the use of finite elements as the principal computational tool for general computational dynamic modeling/analysis. Therein, the concept of average velocity-based formulations is invoked for updating the necessary conservation variables. The stability characteristics and accuracy properties of the proposed formulations are also examined. Comparative sample test cases of numerical model test problems validate the proposed concepts for applicability to general linear/nonlinear computational dynamic problems.

Finite difference schemes

Abstract: This chapter discusses the finite difference schemes. Finite difference methods are a very general purpose numerical scheme whose theory and development have been described clearly. Although the general theory of finite difference methods with regard to consistency, stability, and convergence is not detailed in the chapter, it discusses the theory of the finite difference scheme developed to solve propagation problems. The explicit scheme does not need information on the next advanced level but usually requires a small step size for stability. The implicit scheme is unconditionally stable in most applications. The chapter presents the conventional definition of consistency, which relates the finite difference operator to the true operator.

Numerical simulation of the transport processes in a heat treatment furnace

Abstract: The numerical simulation of the heat transfer and fluid processes in a furnace is carried out using finite difference techniques. A heat treatment furnace for the annealing of steel is considered as an example and a mathematical model of the system is obtained by simplifying the equations that govern the transport in the various components that constitute the system. The resulting set of coupled equations then governs the time-dependent temperature distribution in the furnace. Because of the complexity of the full problem, the transport processes undergone by individual components of the furnace are simulated first, using appropriate boundary conditions that decouple them from the other components. These individual simulations are considered in terms of the underlying physical mechanisms for validation. Finally, these are combined to obtain the simulation of the full, coupled problem. The temperature cycle undergone by the material is computed, along with the time-dependent temperature distributions in other components. A comparison with experimental data indicates close agreement, lending strong support to the numerical model. The relevance of these results to the design of the overall thermal system is also outlined.