Showing papers on "Finite difference published in 1976"
Book•
01 Jan 1976
TL;DR: On Engineering By J T Oden J N Reddy ONLINE SHOPPING for NUMBER InTRODUCTION
Abstract: On Engineering By J T Oden J N Reddy ONLINE SHOPPING FOR NUMBER INTRODUCTION AN INTRODUCTION. MATHEMATICAL LEARNING THEORY R C ATKINSON. AN INTRODUCTION TO THE MATHEMATICAL THEORY OF INVERSE. AN INTRODUCTION TO THE MATHEMATICAL THEORY OF FINITE. AN INTRODUCTION TO THE MATHEMATICAL THEORY OF WAVES. AN INTRODUCTION TO THE MATHEMATICAL THEORY OF INVERSE. AN INTRODUCTION TO THE MATHEMATICAL THEORY OF THE NAVIER. AN INTRODUCTION TO THE MATHEMATICAL THEORY OF VIBRATIONS. AN INTRODUCTION TO THE MATHEMATICAL THEORY OF INVERSE. INTRODUCTION MATHEMATICAL THEORY FINITE ELEMENTS ABEBOOKS. AN INTRODUCTION TO THE MATHEMATICAL THEORY OF WAVES. AN INTRODUCTION TO THE
686 citations
TL;DR: The integrated finite difference method (IFDM) as mentioned in this paper is a powerful numerical technique for solving problems of groundwater flow in porous media, which combines the advantages of an integral formulation with the simplicity of finite difference gradients and is convenient for handling multidimensional heterogeneous systems composed of isotropic materials.
Abstract: The theoretical basis for the integrated finite difference method (IFDM) is presented to describe a powerful numerical technique for solving problems of groundwater flow in porous media. The method combines the advantages of an integral formulation with the simplicity of finite difference gradients and is very convenient for handling multidimensional heterogeneous systems composed of isotropic materials. Three illustrative problems are solved to demonstrate that two- and three-dimensional problems are handled with equal ease. Comparison of IFDM with the well-known finite element method (FEM) indicates that both are conceptually similar and differ mainly in the procedure adopted for measuring spatial gradients. The IFDM includes a simple criterion for local stability and an efficient explicit-implicit iterative scheme for marching in the time domain. If such a scheme can be incorporated in a new version of FEM, it should be possible to develop an improved numerical technique that combines the inherent advantages of both methods.
642 citations
TL;DR: In this article, the displacement discontinuity method (DSM) is proposed for solving complex boundary value problems in plane elastostatics, which is similar to integral equation or influence function techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field.
Abstract: This paper is concerned with the development of a numerical procedure for solving complex boundary value problems in plane elastostatics. This procedure—the displacement discontinuity method—consists simply of placing N displacement discontinuities of unknown magnitude along the boundaries of the region to be analyzed, then setting up and solving a system of algebraic equations to find the discontinuity values that produce prescribed boundary tractions or displacements. The displacement discontinuity method is in some respects similar to integral equation or ‘influence function’ techniques, and contrasts with finite difference and finite element procedures in that approximations are made only on the boundary contours, and not in the field. The method is illustrated by comparing computed results with the analytical solutions of two boundary value problems: a circular disc subjected to diametral compression, and a circular hole in an infinite plate under a uniaxial stress field. In both cases the numerical results are in excellent agreement with the exact solutions.
505 citations
TL;DR: In this article, the authors consider the application of explicit Isecond-order, one-sided or "upwind," difference schemes for the numerical solution of hyperbolic systems in conservation-law form.
Abstract: Explicit second-order upwind difference schemes in combination with spatially symmetric schemes can produce larger stability bounds and better numerical resolution than symmetric schemes alone. However, if conservation form is essential, a special operator is required for transition between schemes. An operational approach has been devised for deriving transition operators so that strict conservation and local consistency are maintained. Various aspects of hybrid schemes are studied numerically for model linear and nonlinear equations. To demonstrate the utility of combining two different algorithms, MacCormack's explicit, noncentered, second-order method is combined with a completely upwind version, and numerical solutions of the Euler equations are obtained for two-dimensional, transonic flows with embedded supersonic regions and shock I. Introduction "1T4 this paper we consider the application of explicit Isecond-order, one-sided or "upwind," difference schemes for the numerical solution of hyperbolic systems in conservation-law form = 0
386 citations
TL;DR: A generalization of the Lax-Wendroff method is presented in this article, which bears the same relationship to the two-step Richtmyer method as the KreissOliger scheme does to the leapfrog method.
Abstract: A generalization of the Lax-Wendroff method is presented. This generalization bears the same relationship to the two-step Richtmyer method as the KreissOliger scheme does to the leapfrog method. Variants based on the MacCormack method are considered as well as extensions to parabolic problems. Extensions to two dimensions are analyzed, and a proof is presented for the stability of a Thommentype algorithm. Numerical results show that the phase error is considerably reduced from that of second-order methods and is similar to that of the Kreiss-Oliger method. Furthermore, the (2, 4) dissipative scheme can handle shocks without the necessity for an artificial viscosity.
319 citations
TL;DR: In this article, the upwind differencing method was applied to a two-dimensional model of recirculating flow in a cavity with a sliding top, and it was shown that the false diffusion associated with first order upwind difference approximations can cause the numerical solution to severely misrepresent the physical transport processes.
225 citations
199 citations
TL;DR: In this article, the numerical solution by finite differences of two-dimensional problems in electromagnetic induction is reexamined with a view to generalizing the method to three-dimensional models.
Abstract: Summary
The numerical solution by finite differences of two-dimensional problems in electromagnetic induction is reexamined with a view to generalizing the method to three-dimensional models. Previously published work, in which fictitious values were used to derive the finite difference equations, is discussed and some errors in the theory which appear to have gone undetected so far, are pointed out. It is shown that the previously published B-polarization formulas are incorrect at points where regions of different conductivity meet, and that the E-polarization formulas are inaccurate when the step sizes of the numerical grid around the point are uneven. An appropriately-modified version of the two-dimensional theory is developed on the assumption that the Earth's conductivity is a smoothly-varying function of position, a method which naturally lends itself to three-dimensional generalization. All the required finite-difference formulas are derived in detail, and presented in a form which is suitable for programming. A simple numerical calculation is given to illustrate the application of the method and the results are compared with those obtained from previous work.
149 citations
TL;DR: In this paper, a discrete version of the resistivity problem in two dimensions, which takes into account nonuniform grid spacing, is presented as a system of self-adjoint difference equations.
Abstract: Resistivity surveying is commonly done by using a point‐source dipole. Consequently, a finite‐difference evaluation of apparent resistivity curves implies the use of three‐dimensional simulation models which necessitate prohibitive computer costs. However, if we assume variation of resistivity only in two dimensions and use a line‐source dipole for setting up the finite‐difference model of a given structure, the potential field can be evaluated easily. A discrete version of the resistivity problem in two dimensions, which takes into account nonuniform grid spacing, is presented as a system of self‐adjoint difference equations. Since the iterative solution of such a system does not require grid spacing to be less than a certain critical value, it was successfully used for the development of fast‐convergence finite‐difference models. By examining in detail the characteristics of the matrix associated with the evaluation of the potential field, it is demonstrated that the proposed modeling procedure will rem...
109 citations
TL;DR: In this article, a Fourier series analysis is performed to determine the dissipative and dispersive characteristics of finite difference and finite element methods for solving the convective-dispersive equation.
Abstract: Various finite difference and finite element methods for solving the one-dimensional convective-dispersive equation are investigated. A Fourier series analysis is performed to determine the dissipative and dispersive characteristics of these numerical methods. The analysis indicates that the commonly observed phenomenon of overshoot of a concentration pulse is due to the inability of the numerical schemes to propagate the small wavelengths which are important to the description of the front. Furthermore, the numerical smearing of a sharp front is due to dissipation of these small wavelengths. The finite element method was found to be superior to finite difference methods for solution of the convective-dispersive equation.
103 citations
TL;DR: In this article, a method for the prediction of dynamic characteristics of rectangular plates with cutouts is described, based on the use of variational principles in conjunction with finite difference technique.
TL;DR: In this paper, a finite difference formulation for computing the frequency domain electromagnetic fields due to a point source in the presence of two-dimensional conductivity structures was developed, where computing costs were minimized by reducing the full three-dimensional problem to a series of 2D problems.
Abstract: A finite difference formulation is developed for computing the frequency domain electromagnetic fields due to a point source in the presence of two‐dimensional conductivity structures. Computing costs are minimized by reducing the full three‐dimensional problem to a series of two‐dimensional problems. This is accomplished by Fourier transforming the problem into the x-wavenumber (kx) domain; here the x-direction is parallel to the structural strike. In the kx domain, two coupled partial differential equations for H⁁x(kx,y,z) and E⁁x(kx,y,z) are obtained. These equations resemble those of two coupled transmission sheets. For a requisite number of kx values these equations are solved by the finite difference method on a rectangular grid on the y-z plane. Application of the inverse Fourier transform to the solutions thus obtained gives the electric and magnetic fields in the space domain. The formulation is general; complex two‐dimensional structures containing either magnetic or electric dipole sources can ...
01 Feb 1976
TL;DR: In this article, the Galerkin finite element method can produce highly accurate results provided that the consistent mass formulation is used, and the reasons for the superiority of these results to those using mass lumping are presented.
Abstract: By obtaining the approximate solution to the advection-diffusion equation in both one- and two-dimensions, for both purely advective (hyperbolic) and advection-dominated flows, and by comparison to finite difference results, one demonstrates, contrary to the admonition put forth by Strang and Fix (1973), that the Galerkin finite element method can produce highly accurate results provided that the consistent mass formulation is used. The reasons for the superiority of these results to those using mass lumping are presented. Finally, it is shown that a recommended mass lumping procedure for parabolic isoparametric elements generates completely spurious results for advection-diffusion.
TL;DR: In this article, a method for modeling and inverting the class of multiple reflections involving the near-perfect reflector at the free surface is presented. But it is based on a 1D subsurface model.
Abstract: Starting with a 1-D subsurface model, a method is developed for modeling and inverting the class of multiple reflections involving the near‐perfect reflector at the free surface. A solution to the practical problem of estimating the source waveform is discussed, and application of the 1-D algorithm to field data illustrates the successful elimination of seafloor and peg‐leg multiples. Extending the analysis to waves in two dimensions, we make the approximation that the subsurface behaves as an acoustic medium. Based on several numerical and theoretical considerations, the scalar wave equation is split into two separate partial differential equations: one governing propagation of upcoming waves and a second describing downgoing waves. The result is a pair of propagation equations which are coupled where reflectors exist. Finite difference approximations to the initial boundary value problem are developed to integrate numerically the surface reflection seismogram. Use of the 2‐D algorithm for modeling free‐...
TL;DR: In this paper, the authors justify the diffusion model in the framework of a statistical theory of turbulence and show that the difference between the two diffusion equations that have been proposed is related to an ambiguity in the definition of a Lagrangian hydrodynamical calculation by finite differences in the presence of turbulence.
Abstract: Several recent investigations of post--main-sequence stellar evolution have used a diffusion model of turbulent mixing, and each author has used a different variation. We justify the diffusion model in the framework of a statistical theory of turbulence and show that the difference between the two diffusion equations that have been proposed is related to an ambiguity in the definition of a Lagrangian hydrodynamical calculation by finite differences in the presence of turbulence. We present difference equations for solving the diffusion equation which theory and experience indicate are free of numerical difficulties. (AIP)
TL;DR: In this article, a general method of characteristics for solving the multigroup transport equations is developed, which is combined with an adaptive difference scheme, called the modified diamond scheme, and is then applied to the finite difference form of the equation.
Abstract: A general method of characteristics for solving the multigroup transport equations is developed. This is combined with an adaptive difference scheme, called the modified diamond scheme, and is then applied to the finite difference form of the equation. This formulation is obtained from the discrete ordinates equation, which in turn derives from the multigroup equation, both on the basis of consistency arguments. In this connection two forms of the multigroup equation are used, and the diffusion and other important limits also have a bearing on the final difference equation. The new approaches resolve a number of theoretical and practical difficulties with S/sub n/-type transport calculations, in particular in curved and multidimensional geometries. They lead to a firmer basis for discrete ordinates quadrature sets and to better control, mesh cell by mesh cell, over flux extrapolation, including methods to smooth out unwanted flux oscillations. The total effect is a more consistent treatment of the transport equation together with improved accuracy, fewer breakdowns, and more speed in the calculations, while keeping close to the physics of the problem and retaining the basic simplicity of the difference approach.
TL;DR: In this paper, a finite difference formulation for time-varying two-phase flow problems is presented. But it does not consider the effects of phase transitions and the release of latent heat or chemical energy.
Abstract: The theoretical study of time-varying two-phase flow problems in several space dimensions introduces such a complicated set of coupled nonlinear partial differential equations that numerical solution procedures for a high-speed computer are required in almost all but the simplest examples. Efficient attainment of realistic solutions for practical problems requires a finite difference formulation that is simultaneously implicit in the treatment of mass convection, equations-of-state, and the momentum coupling between phases. Such a method is described, and the equations on which it is based are discussed. In particular, the capability for calculating physical instabilities and other time-varying dynamics, at the same time avoiding numerical instability are emphasized. The computer code is applicable to problems in reactor safety analysis, the dynamics of fluidized dust beds, raindrops or aerosol transport, and a variety of similar circumstances, including the effects of phase transitions and the release of latent heat or chemical energy.
TL;DR: This paper develops direct and iterative algorithms for the solution of finite difference approximations of the Poisson and Biharmonic equations on a square, using a number of arithmetic units in parallel.
Abstract: In this paper we develop direct and iterative algorithms for the solution of finite difference approximations of the Poisson and Biharmonic equations on a square, using a number of arithmetic units in parallel. Assuming ann×n grid of mesh points, we show that direct algorithms for the Poisson and Biharmonic equations require 0(logn) and 0(n) time steps, respectively. The corresponding speedup over the sequential algorithms are 0(n
2) and 0(n
2logn). We also compare the efficiency of these direct algorithms with parallel SOR and ADI algorithms for the Poisson equation, and a parallel semi-direct method for the Biharmonic equation treated as a coupled pair of Poisson equations.
TL;DR: In this article, the equations of transient and steady-state flow in two-dimensional artesian aquifers are approximated using finite differences, combined with other linear physical and management constraints and a linear objective function, comprise a linear programming (LP) formulation.
Abstract: The equations of transient and steady-state flow in two-dimensional artesian aquifers are approximated using finite differences. The resulting linear difference equations, combined with other linear physical and management constraints and a linear objective function, comprise a linear programming (LP) formulation. Solutions of such LP models are used to determine optimal well distributions and pumping rates to meet given management objectives for a hypothetical transient problem and for a steady-state field problem.
TL;DR: In this article, the problem of non-steady flow of water in a soil-plant system can be described by adding a sink term to the continuity equation for soil water flow.
Abstract: The problem of non-steady flow of water in a soil-plant system can be described by adding a sink term to the continuity equation for soil water flow. In this paper the sink term is defined in two different ways. Firstly it is considered to be dependent on the hydraulic conductivity of the soil, on the difference in pressure head between the soil and the root-soil interface and some root effectiveness function. Secondly the sink is taken to be a prescribed function of the soil water content. The partial differential equation applying to the first problem is solved by both a finite difference (FD 1) and a finite element (FE 1) technique, that applying to the second problem by a finite difference approach (FD 2). The purpose of this paper is to verify the numerical models against field measurements, to compare the results obtained by the three numerical methods and to show how the finite element method can be applied to complex but realistic two-dimensional flow situations. Two examples are given. T...
TL;DR: In this paper, a finite difference solution for Prandtl's boundary-layer equations is described in detail for steady, incompressible luminar and turbulent flows, where only boundary sheets will be considered and curvature effects in the direction normal to the wall will be neglected.
TL;DR: In this article, a technique for deriving a finite-difference scheme to solve initial value partial-differential equations is presented, which is based on the variational method and constrains the finite difference scheme to satisfy the conservation law(s).
01 Jan 1976
TL;DR: In this paper, a review of some recently developed numerical methods for the solution of nonlinear equations of mixed type is presented, which use finite difference approximations to the differential equation.
Abstract: A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.
TL;DR: In this paper, the effect of changes in wave amplitude, shape and surface roughness on turbulent boundary-layer flows is investigated and closed via mixing-length hypotheses and at the level of turbulent energy equation and the resulting equations are solved numerically using finite difference approximations.
Abstract: Summary
Flow in a deep turbulent boundary-layer above a rough, rigid, wavy surface is considered. Closure is made via mixing-length hypotheses and at the level of the turbulent energy equation and the resulting equations are solved numerically using finite difference approximations. Results are presented for a typical case representative of flow above gravel waves on the bed of a tidal channel and the effect of changes in wave amplitude, shape and surface roughness are considered. Comparisons are made with recent experimental and theoretical studies. In some computations allowance is made for the effect of streamline curvature on the turbulence structure and the importance of such effects for these flows is assessed.
01 Jan 1976
TL;DR: Two techniques are described which are useful in computational transonic aerodynamics applications, the finite volume method, which simplifies the application of boundary conditions without introducing the constriction associated with small disturbance theory, and configuration design by numerical optimization, which can be used by aircraft designers to develop configurations that satisfy specific geometric performance constraints.
Abstract: Although the development of a finite difference relaxation procedure to solve the steady form of equations of motion gave birth to the study of computational transonic aerodynamics and considerable progress has been made using the small disturbance theory, no general analytical solution method yet exists for transonic flows that include three dimensional unsteady, and viscous effects. Two techniques are described which are useful in computational transonic aerodynamics applications. The finite volume method simplifies the application of boundary conditions without introducing the constriction associated with small disturbance theory. Governing equations are solved in a Cartesian coordinate system using a body-oriented and shock-oriented mesh network. Only the volume and surface normal directions of the volume elements must be known. The other method, configuration design by numerical optimization, can be used by aircraft designers to develop configurations that satisfy specific geometric performance constraints. Two examples of airfoil design by numerical optimization are presented.
TL;DR: The relationship between and relative advantages of finite difference and finite element methods are discussed in this paper, where it is shown how conservation properties are best preserved by the latter methods and, in particular, how the supression of non-linear instabilities in the advection equation is achieved by the Arakawa schemes.
TL;DR: In this article, a numerical model is developed for simulating the development and propagation of landslide generated water waves in reservoirs, based upon a finite difference representation of the depth averaged hydrodynamic equations.
Abstract: A numerical model is developed for simulating the development and propagation of landslide generated water waves in reservoirs. The numerical model is based upon a finite difference representation of the depth averaged hydrodynamic equations. The landslide is formulated as a moving boundary condition, propagating into the reservoir and accelerating the fluid due to physical displacement and viscous drag. Arbitrary reservoir geometry and landslide parameters can be considered. The numerical model results are compared with experimental results obtained on a 1:120 undistorted scale physical model of Libby Dam and Lake Koocanusa in Montana. Landslides were considered reflecting a wide range of landslide volumes and velocities. The wave heights predicted by the numerical model are in good agreement with the wave heights observed in the physical model.