Showing papers on "Finite difference published in 1980"

The Finite Difference Method in Partial Differential Equations

Abstract: Keywords: elements : finis ; equations : differentielles ; methodes de : calcul ; mathematiques Reference Record created on 2005-11-18, modified on 2016-08-08

Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow

Abstract: An implicit finite-difference procedure for unsteady three-dimensional flow capable of handling arbitrary geometry through the use of general coordinate transformations is described. Viscous effects are optionally incorporated with a "thin-layer" approximation of the Navier-Stokes equations. An implicit approximate factorization technique is employed so that the small grid sizes required for spatial accuracy and viscous resolution do not impose stringent stability limitations. Results obtained from the program include transonic inviscid or viscous solutions about simple body configurations. Comparisons with existing theories and experiments are made. Numerical accuracy and the effect of three-dimensional coordinate singularities are also discussed.

Application of the Finite-Difference Time-Domain Method to Sinusoidal Steady-State Electromagnetic-Penetration Problems

Abstract: A numerical method for predicting the sinusoidal steady-state electromagnetic fields penetrating an arbitrary dielectric or conducting body is described here. The method employs the finite-difference time-domain (FD-TD) solution of Maxwell's curl equations implemented on a cubic-unit-cell space lattice. Small air-dielectric loss factors are introduced to improve the lattice truncation conditions and to accelerate convergence of cavity interior fields to the sinusoidal steady state. This method is evaluated with comparison to classical theory, method-of-moment frequency-domain numerical theory, and experimental results via application to a dielectric sphere and acylindrical metal cavity with an aperture. Results are also given for a missile-like cavity with two different types of apertures illuminated by an axial-incidence plane wave.

A finite-element/finite-difference simulation of the injection-molding filling process

Abstract: A detailed formulation is presented for simulating the injection-molding filling of thin cavities of arbitrary planar geometry. The modelling is in terms of generalized Hele-Shaw flow for an inelastic, non-Newtonian fluid under non-isothermal conditions. A hybrid numerical scheme is employed in which the planar coordinates are described in terms of finite elements and the gapwise and time derivatives are expressed in terms of finite differences. The simulation is applied to the filling of a two-gated plate mold having an intentionally unbalanced runner system. Good agreement is obtained with experimental results in terms of short-shot sequences, weldline formation and pressure traces at prescribed points in the cavity.

Sur le spectre des opérateurs aux différences finies aléatoires

Abstract: We study a class of random finite difference operators, a typical example of which is the finite difference Schrodinger operator with a random potential which arises in solid state physics in the tight binding approximation. We obtain with probability one, in various situations, the exact location of the spectrum, and criterions for a given part in the spectrum to be pure point or purely continuous, or for the static electric conductivity to vanish. A general formalism is developped which transforms the study of these random operators into that of the asymptotics of a multiple integral constructed from a given recipe. Finally we apply our criterions and formalism to prove that, with probability one, the one-dimensional finite difference Schrodinger operator with a random potential has pure point spectrum and developps no static conductivity.

On Implementing a Numeric Huygen's Source Scheme in a Finite Difference Program to Illuminate Scattering Bodies

Abstract: A numerical procedure is described that will simplify the analysis of the EMP response of structures with dielectric or poorly conducting segments.

Stable and entropy satisfying approximations for transonic flow calculations

Abstract: Finite difference approximations for the small disturbance equation of transonic flow are developed and analyzed. New schemes of the Cole-Murman type are presented fpr which nonlinear stability is proved. The Cole-Murman scheme may have entropy violating expansion shocks as solutions. In the new schemes the switch between the subsonic and supersonic domains is designed such that these nonphysical shocks are guaranteed not to occur. Results from numercial calculations are given which illustrate these conclusions

A computer program to generate two-dimensional grids about airfoils and other shapes by the use of Poisson's equation

Abstract: A method for generating two dimensional finite difference grids about airfoils and other shapes by the use of the Poisson differential equation is developed. The inhomogeneous terms are automatically chosen such that two important effects are imposed on the grid at both the inner and outer boundaries. The first effect is control of the spacing between mesh points along mesh lines intersecting the boundaries. The second effect is control of the angles with which mesh lines intersect the boundaries. A FORTRAN computer program has been written to use this method. A description of the program, a discussion of the control parameters, and a set of sample cases are included.

Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows

Abstract: A spectral method for numerical computation of 3-D time-dependent incompressible flows between two plane parallel plates is presented. Fourier expansions in the coordinates parallel to the walls and expansions in Chebyshev polynomials in the normal coordinate are used. The time coordinate is discretized with second order finite differences, treating the viscous terms implicitly. An efficient direct solution procedure for the implicit equations is developed which reduces the 3-D problem to a set of essentially tridiagonal linear equations in one space coordinate. Boundary and continuity conditions are satisfied exactly, apart from round-off errors.

Ocean Tides. Part 1. Global Ocean Tidal Equations

Abstract: A detailed derivation of improved ocean tidal equations in continuous (COTEs) and discrete (DOTEs) forms is presented. These equations feature the Boussinesq linear eddy dissipation law with a novel eddy viscosity that depends on the lateral mesh area, i.e., on mesh size and ocean depth. Analogously, the linear law of bottom friction is used with a new bottom friction coefficient depending on the bottom mesh area. The primary astronomical tide‐generating potential is modified by secondary effects due to the oceanic and terrestrial tides. The fully linearized equations are defined in a single‐layer ocean basin of realistic bathymetry varying from 50 m to 7,000 m. The DOTEs are set up on a 1o by 1o spherically graded grid system, using central finite differences in connection with Richardson's staggered computation scheme. Mixed single‐step finite differences in time are introduced, which enhance decay, dispersion, and stability properties of the DOTEs and facilitate—in Part II of this paper—a unique hydrod...

Convergence of multigrid iterations applied to difference equations

Abstract: Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.

Long-Time Solutions to Heat-Conduction Transients with Time-Dependent Inputs

Abstract: An economical method for obtaining long-time solutions to one, two, or three-dimensional heat-conduction transients with time-dependent forcing functions is presented. The conduction problem is spatially discretized by finite differences or by finite elements to obtain a system of first-order ordinary differential equations. The time-dependent input functions are each approximated by continuous, piecewise-linear functions each having the same uniform time interval. A set of response coefficients is generated by which a long-time solution can be carried out with a considerably lower cost than for conventional methods. The method has applications in problems of air-conditioning calculations for buildings and in the development of load models for solar energy studies.

Analytical and numerical techniques for analyzing an electrically short dipole with a nonlinear load

Abstract: An electrically short dipole with a nonlinear dipole load is analyzed theoretically using both analytical and numerical techniques. The analytical solution is given in terms of the Anger function of imaginary order and imaginary argument and is derived from the nonlinear differential equation for the Thevenin's equivalent circuit of a dipole with a diode. The numerical technique is to solve the nodal equation using a time-stepping finite difference equation method. The nonlinear resistance of the diode is treated using the Newton-Raphson iteration technique. A comparison between the analytical and numerical solutions is given.

Kinematic wave routing incorporating shock fitting

Abstract: An analytical solution to the kinematic wave approximation for unsteady flow routing is presented. The model allows time-dependent lateral inflow with piecewise spatial uniformity and can be applied to complex kinematic cascades. Kinematic shocks are considered as manifestations of higher-order effects such as rnonoclinal flood waves, bores, etc. Within the context of kinematic approximation therefore we retain their dynamic effects by routing the discontinuities as they appear. Certain simplifying assumptions are made which permit closed form solutions and an efficient numerical algorithm, based on the method of characteristics, is employed. The resulting model, called an approximate shock-fitting scheme, preserves the effect of the shocks without the usual computational complications and compares favorably with an implicit finite difference solution. The efficiency and accuracy of the new method are illustrated by computing a variety of unsteady flows, ranging from simple cascades to complex natural watersheds.

A numerical simulation of Newtonian and visco-elastic flow past stationary and rotating cylinders

Abstract: Numerical solutions are presented for the two-dimensional flow past a circular cylinder in an infinite domain. The flow is assumed to be uniform at infinity and the cylinder is allowed to rotate with a constant angular velocity Ω. Ω is chosen to be in the range (0 to 5 W / a ) where a is the radius of the cylinder and W is the mainstream velocity at infinity. To incorporate viscoelastic properties into the flow, an implicit four-constant Oldroyd model is used, and the resulting nonlinear constitutive equations are solved in parallel with the equations of motion as a coupled set of partial differential equations. The method of solution used is a finite difference technique with block over-relaxation. The results are compared with those of other numerical computations as well as with available experimental data. In particular, consideration is given lift experienced by the cylinder and on the streamline patterns and vorticity distribution.

Numerical solution of flow problems using body-fitted coordinate systems

Abstract: The technique of boundary-fitted coordinate systems is based on a method of automatic numerical generation of a general curvilinear coordinate system having a coordinate line coincident with each boundary of a general multi-connected region containing any number of arbitrarily shaped bodies. Once the curvilinear coordinate system is generated, any partial differential system of interest can be solved on this coordinate system by transforming the equations and solving the resulting system in finite difference approximation on the rectangular transformed plane. This method of automatic body-fitted curvilinear coordinate generation is used to construct finite-difference solutions of the full, time dependent Navier-Stokes equations for the unsteady viscous flow about arbitrary two-dimensional airfoils, or any other two-dimensional bodies. Finally, initial results for three-dimensional applications are also presented.

Implicit Finite-Difference Simulations of Three-Dimensional Compressible Flow

Abstract: An implicit finite-difference procedure for unsteady three-dimensional flow capable of handling arbitrary geometry through the use of general coordinate transformations is described. Viscous effects are optionally incorporated with a "thin-layer" approximation of the Navier-Stokes equations. An implicit approximate factorization technique is employed so that the small grid sizes required for spatial accuracy and viscous resolution do not impose stringent stability limitations. Results obtained from the program include transonic inviscid or viscous solutions about simple body configurations. Comparisons with existing theories and experiments are made. Numerical accuracy and the effect of three-dimensional coordinate singularities are also discussed.

A conservative implicit finite difference algorithm for the unsteady transonic full potential equation

Abstract: An implicit finite difference procedure is developed to solve the unsteady full potential equation in conservation law form Computational efficiency is maintained by use of approximate factorization techniques The numerical algorithm is first order in time and second order in space A circulation model and difference equations are developed for lifting airfoils in unsteady flow; however, thin airfoil body boundary conditions have been used with stretching functions to simplify the development of the numerical algorithm

Time-dependent difference theory for noise propagation in a two-dimensional duct

Abstract: A time dependent numerical formulation was derived for sound propagation in a two dimensional straight soft-walled duct in the absence of mean flow. The time dependent governing acoustic-difference equations and boundary conditions were developed along with the maximum stable time increment. Example calculations were presented for sound attenuation in hard and soft wall ducts. The time dependent analysis were found to be superior to the conventional steady numerical analysis because of much shorter solution times and the elimination of matrix storage requirements.

Numerical Modeling of Tidal Circulation in Harbors

Abstract: In this study a numerical model has been developed that can be used to predict the two-dimensional tide-induced velocity fields in harbors and estuaries. The model is particularly suited to basins having a narrow entrance where, on the incoming tide, the divergence of the velocity field associated with the jet inlet gives rise to the generation of vorticity. The time-dependent nonlinear equations of motion are formulated to include the effects of bottom roughness, wind action, the earth's rotation, and a simplified version of the turbulent transfer of momentum. These equations are expressed in an alternating-direction implicit finite difference form and are solved by Gaussian elimination. The numerical model has been checked by making comparisons between the computed velocity fields and experimentally measured velocities and path lines for two hydraulic model studies involving various rectangular harbors and a circular reservoir.

A model with non‐reflecting boundaries for use in explicit soil—structure interaction analyses

Abstract: The purpose of this paper is to present an accurate and efficient model for use in explicit soil-structure interaction analyses for seismic excitation. The main body of the paper is concerned with the implementation of non-reflecting boundaries at the base and vertical faces of a two-dimensional finite element or finite difference soil mesh. The paper proposes a scheme for implementing viscous dashpots as energy absorbers at the base of the model where the seismic excitation is applied. To achieve this, the form of the seismic input is modified from the conventional accelerogram to a force time history that can be obtained from a standard deconvolution program with very minor modifications. The proposed scheme is also applicable to frequency-domain solutions. For the lateral boundaries, a superposition non-reflecting boundary formulation is recommended. The paper shows how easily the standard formulation can be modified to accommodate a seismic excitation that is assumed to be vertically propagating from the base of the model. An example is presented to demonstrate the accuracy of the proposed model.

Solving Finite Difference Approximations to Nonlinear Two-Point Boundary Value Problems by a Homotopy Method

Abstract: The Chow–Yorke algorithm is a homotopy method that has been proved globally convergent for Brouwer fixed point problems, classes of zero finding, nonlinear programming, and two-point boundary value problems. The method is numerically stable, and has been successfully applied to several practical nonlinear optimization and fluid dynamics problems. Previous application of the homotopy method to two-point boundary value problems has been based on shooting, which is inappropriate for fluid dynamics problems with sharp boundary layers. Here the Chow–Yorke algorithm is proved globally convergent for a class of finite difference approximations to nonlinear two-point boundary value problems. The numerical implementation of the algorithm is briefly sketched, and computational results are given for two fairly difficult fluid dynamics boundary value problems.

Application of the Finite Element Method to the Nonlinear Inverse Heat Conduction Problem Using Beck’s Second Method

Abstract: The calculation of the surface temperature and surface heat flux from a measured temperature history at an interior point of a body is identified in the literature as the inverse heat conduction problem. This paper presents, to the author’s knowledge, the first application of a solution technique for the inverse problem that utilizes a finite element heat conduction model and Beck’s nonlinear estimation procedure. The technique is applicable to the one-dimensional nonlinear model with temperature-dependent thermophysical properties. The formulation is applied first to a numerical example with a known solution. The example treated is that of a periodic heat flux imposed on the surface of a rod. The computed surface heat flux is compared with the imposed heat flux to evaluate the performance of the technique in solving the inverse problem. Finally, the technique is applied to an experimentally determined temperature transient taken from an interior point of an electrically-heated composite rod. The results are compared with those obtained by applying a finite difference inverse technique to the same data.