Showing papers on "Finite difference published in 1973"

Analysis of Embedded Shock Waves Calculated by Relaxation Methods

Abstract: The requirements for uniqueness of the calculated jump conditions across embedded shock waves are investigated for type-dependent difference systems used in transonic flow studies. A mathematical analysis shows that sufficient conditions are (1) the equations should be differenced in conservative form and (2) a special difference operator should be used when switching from a hyperbolic to an elliptic operator. The latter results in a consistency condition on the integral equations, rather than the differential, at these points. Calculated jump conditions for several embedded and detached shock waves are analyzed in the physical and hodograph planes. Comparisons are made with previous results, a time-dependent calculation, and data.

Transient shell response by numerical time integration

Abstract: In using the finite element method to compute a transient response, two choices must be made. First, some form of mass matrix must be decided upon. Either the consistent mass matrix prescribed by the finite element method can be employed or some form of diagonal mass matrix may be introduced. Secondly, some particular time integration procedure must be adopted. The procedures available divide themselves into two classes: the conditionally stable explicit schemes and the unconditionally or conditionally stable implicit schemes. The choices should be guided by both economy and accuracy. Using exact discrete solutions compared to the exact solutions of the differential equations, the results of these choices are displayed. Concrete examples of well-matched methods, as well as ill-matched methods, are identified and demonstrated. In particular, the diagonal mass matrix and the explicit central difference time integration method are shown to be a good combination in terms of accuracy and economy.

Uni-moment method of solving antenna and scattering problems

Abstract: It has been shown by this investigator and numerous others [6], [7], [8] that exterior boundary value problems involving localized inhomogeneous media are most conveniently solved using finite difference or finite element techniques together with integral equations or harmonic expansions, which satisfy the radiation conditions. The methods result in large matrices that are partly full and partly sparse; and methods to solve them, such as iteration or banded matrix methods are not very satisfactory. The unimoment method alleviates the difficulties by decoupling exterior problems from the interior boundary value problems. This is done by solving the interior problem many times so that N linearly independent solutions are generated. The continuity conditions are then enforced by a linear combination of the N independent solutions, which may be done by solving much smaller matrices. Methods of generating solutions of the interior problems are discussed.

Multidimensional Stefan Problems

Abstract: An implicit finite difference method for the multidimensional Stefan problem is discussed. The classical problem with discontinuous enthalpy is replaced by an approximate Stefan problem with continuous piecewise linear enthalpy. An implicit time approximation reduces this formulation to a sequence of monotone elliptic problems which are solved by finite difference techniques. It is shown that the resulting nonlinear algebraic equations are solvable with a Gauss-Seidel method and that the discretized solution converges to the unique weak solution of the Stefan problem as the time and space mesh size approaches zero.

Comparison of matrix methods applied to the radial Schrödinger eigenvalue equation: The Morse potential

Abstract: This paper demonstrates the application of matrix methods based on finite differences and on variational (Rayleigh‐Ritz‐Galerkin) procedures to solution of the radial Schrodinger equation for bound states of the Morse potential. It demonstrates sources of numerical inaccuracy: truncation, termination, tolerance, and quadrature. Cubic splines, harmonic oscillators, floating Gaussians, and sines are used as basis functions.

On the dynamics of a system of rigid bodies

Abstract: The existent treatises on rational mechanics do not pay attention to the dynamics of a system of rigid bodies, confining their interest to the general theorems of mechanics, the theory of rotation of a rigid body about a fixed point, some problems of analytical mechanics and the theory of stability. At the same time, the problems whose formulation is reduced precisely to the dynamics of a system of rigid bodies are quite common nowadays, at least during the last 30 years. An attempt to solve such problems on the basis of Lagrange’s second method leads, as a rule, to cumbersome differential equations with a large number of terms; the mechanical interpretation of those is not at all easy. Even computers are then of little help. A solution of the problem of motion of a system of rigid bodies in terms of a linear approximation is a superposition of harmonic motions, and is, therefore, sufficiently transparent. However, in order to get the important qualitative characteristics of the motion, it is necessary to take into account the non-linear terms of initial differential equations and to integrate these equations over a time interval considerably longer than the period of partial oscillations of the associated linear system. However, such integration leads to a rapid accumulation of errors because the actual calculations by the method of finite differences involve a large number of steps.

Solving the radial Schrödinger equation by using cubic‐spline basis functions

Abstract: This paper demonstrates the use of piecewise continuous (class C2) polynomial basis functions (B splines or hill functions) in solving the l=0 radial Schrodinger equation, with examples of scattering from Eckart, exponential, and static hydrogen potentials, and eigenvalues for Coulomb, harmonic oscillator, and Morse potentials. Simple nonlinear placement of spline centroids can improve accuracy by orders of magnitude. Comparisons demonstrate the greater accuracy of the Galerkin method, compared with collocation, simple finite difference, and Numerov methods.

Complexity Bounds for Regular Finite Difference and Finite Element Grids

Abstract: Sharp lower bounds are obtained for multiplications and storage in the sparse system arising from the application of finite difference or finite element techniques to linear boundary value problems on plane regions yielding regular $n \times n$ grids. Graph-theoretic techniques are used to take advantage of the simplicity of the underlying combinatorial structure of the problem.

Solutions of plane crack problems by mapping technique

Abstract: The analysis of crack problems in plane elasticity has intrigued mathematicians for nearly sixty years. Inglis [1] found the solution for a single crack in an infinite sheet with the use of elliptic coordinates. Since then, many mathematical approaches with wide ranges of sophistication have been applied to a variety of crack configurations and loading conditions. It is easy to appreciate the mathematical interest in a problem area in which solution techniques span such diverse topics as analytic function theory, integral equations, transform methods, conformal mapping, boundary collocation, finite differences, finite elements, asymptotic methods, etc.

Estimating the Eigenvalues of Sturm–Liouville Problems by Approximating the Differential Equation

Abstract: This paper is concerned with computing accurate approximations to the eigenvalues and eigenfunctions of regular Sturm–Liouville differential equations The method consists of replacing the coefficient functions of the given problem by piecewise polynomial functions and then solving the resulting simplified problem Error estimates in terms of the approximate solutions are established and numerical results are displayed Since the asymptotic properties for Sturm–Liouville systems are preserved by the approximation, the relative error in the higher eigenvalues is much more uniform than is the case for finite difference or Rayleigh–Ritz methods

Incorporation of a flux model for radiation into a finite-difference procedure for furnace calculations

Abstract: Calculations are described of the flow, heat-transfer, and chemical-reaction processes occurring within a gaseous-fired furnace. A particular feature of the computations is the incorporation of a four-flux model of radiative heat transfer. The calculation procedure is a two-dimensional one, in which the main hydrodynamic variables are the vorticity and stream function. The turbulent transport properties are obtained from a two-equation turbulence model, in which turbulence energy and energy-dissipation rate are the dependent variables. Mass-transfer and chemical reactions are calculated from a model which assumes a single-step chemical reaction, and physical control: the effects of these processes thus embodied in a single differential equation for the mixture fraction. Heat transfer is determined by solution of differential equations for the specific enthalpy, and for the sums of the radiative fluxes for each of the coordinate directions. Calculations are performed for the conditions of the “M1 Trials” carried out on the furnaceof the International Flame Foundation, IJmuiden. Encouraging agreement is obtained between the predicted and measured distributions of temperature and radiant heat transfer along the furnace walls. However, the agreement within the flow is unsatisfactory, due, it is believed, to neglect in the computations of the effects of the “unmixedness” phenomenon.

Computation of Space Shuttle Flowfields Using Noncentered Finite-Difference Schemes

Abstract: Second- and third-order, noncentered finite-difference schemes are described for the numerical solution of the hyperbolic equations of fluid dynamics. The advantages of noncentered methods over the more conventional centered schemes are: simpler programming logic, nonhomogeneous terms are easily included, and generalization to multidimensional problems is direct. Second- and third-order methods are compared with regard to dissipative and dispersive errors and shock-capturing ability. These schemes are then used in a shock-capturing technique to determine the inviscid, supersonic flow field surrounding space shuttle vehicles (SSV). Resulting flow fields about typical pointed and blunted, delta-winged SSVs at angle of attack are presented and compared with experiment.

Technique for implicit dynamic routing in rivers with tributaries

Abstract: The prediction of transient flow in a river having a major tributary poses a challenging problem for the streamflow forecaster. The interaction of storage and dynamic effects between the two rivers can be simulated efficiently by a mathematical model consisting of the two unsteady flow differential equations and of known stage time, discharge time, or stage discharge relationships at the extremities of the rivers. Numerical solutions of discharge and water surface elevation are obtained from the differential equations at specified time intervals by an implicit finite difference technique. This produces successive systems of nonlinear equations that are efficiently solved by the Newton-Raphson iterative method in combination with an extrapolation procedure and a specialized direct method for solving a system of linear equations. The length of the specified time interval is not limited by computational stability; however, accuracy constraints may limit its size. Some numerical results are presented to illustrate the interaction between a river and a tributary when they are subjected to a flood wave of long duration.

Second- and Third-Order Noncentered Difference Schemes for Nonlinear Hyperbolic Equations

Abstract: Second- and third-order, explicit finite-difference schemes are described for the numerical solution of the hyperbolic equations of fluid dynamics. The schemes are uncentered in the sense that spatial derivatives are generally approximated by forward or backward difference quotients. The advantages of noncentered methods over the more conventional centered schemes are: programing logic is simpler, nonhomogeneous terms are easily included, and generalization to multidimensional problems is direct. The von Neumann stability analysis for the proposed methods is reviewed and second- and third-order methods are compared with regard to dissipative and dispersive errors and shock-capturing ability.

Finite Difference Analysis of Rayleigh Wave Scattering at Vertical Discontinuities

Abstract: The finite difference iterative method yields the full wave solution to problems without exact solution that involve the scattering of elastic surface waves at vertical discontinuities in homogeneous media. The technique successfully predicts the results for a problem for which an analytical solution does exist, that of a Rayleigh wavelet propagating on a homogeneous, isotropic, semi-infinite half space. For a Rayleigh wave of unit amplitude incident normally at a 90° corner in a homogeneous medium of Poisson's ratio σ = 0.245, the amplitude coefficients for transmission and reflection were found to be 0.64 ± 0.02 and 0.36 ± 0.02, respectively, whereas the corresponding phase shifts were −79 ± 5° and 38 ± 5°. About 45% of the incident energy is converted into body waves at the corner, and more than 90% of this energy is radiated back into a sector of the plane included between lines making angles of 15° with the two free surfaces. All these results, which are independent of wavelength, agree well with other published data. In the related problem of Rayleigh wave scattering at a downward step discontinuity, the dependence on step height of the transmission and reflection coefficients and of the phase shifts for a given wavelength component has also been investigated. Results show good agreement with both experimental curves and earlier theoretical work. This type of numerical simulation may be applied to other two-dimensional geometries, including layered media problems.

Secondary flow in a curved tube

Abstract: The work of Dean and that of McConalogue & Srivastava on the steady motion of an incompressible fluid through a curved tube of circular cross-section is extended through the entire range of Reynolds numbers for which the flow is laminar. The coupled nonlinear system of partial differential equations which defines the motion is solved numerically by finite differences. Computer calculations are described and physical implications are discussed.

High Order Rectangular Shallow Shell Finite Element

Abstract: The stiffness matrix for a high order shallow shell finite element is presented explicitly. The element is of rectangular plan and possesses three constant radii of curvature: two principal ones and a twist one. Each of the three displacement functions is assumed as the product of one-dimensional, first-order Hermite interpolation formulas. An eigenvalue analysis performed on the element stiffness matrix shows that the six rigid-body displacements are included. Convergence studies are carried out for a cylindrical shell, a translational shell with two constant principal radii of curvature, and a hyperbolic paraboloidal shell with a constant twist radius of curvature. Excellent agreements are found when comparing the present results with the alternative series and finite difference solutions. A review of the previously developed shell finite elements shows that the present element is highly efficient in terms of convergence rate or computational effort.

Numerical marching techniques for fluid flows with heat transfer

Abstract: The finite difference formulation and method of solution is presented for a wide variety of fluid flow problems with associated heat transfer. Only a few direct results from these formulations are given as examples, since the book is intended primarily to serve a discussion of the techniques and as a starting point for further investigations; however, the formulations are sufficiently complete that a workable computer program may be written from them. In the appendixes a number of topics are discussed which are of interest with respect to the finite difference equations presented. These include a very rapid method for solving certain sets of linear algebraic equations, a discussion of numerical stability, the inherent error in flow rate for confined flow problems, and a method for obtaining high accuracy with a relatively small number of mesh points.

High order finite difference solution of differential equations.

Abstract: These seminar notes give a detailed treatment of finite difference approximations to smooth nonlinear two-point boundary value problems for second order differential equations. Consistency, stability, convergence, and asymptotic expansions are discussed. Most results are stated in such a way as to indicate extensions to more general problems. Successive extrapolations and deferred corrections are described and their implementations are explored thoroughly. A very general deferred correction generator is developed and it is employed in the implementation of a variable order, variable (uniform) step method. Complete FORTRAN programs and extensive numerical experiments and comparisons are included together with a set of 48 references.

The potential in a charge coupled device with no mobile minority carriers and zero plate separation

Abstract: A two-dimensional analysis of the potential in charge coupled devices is presented. It is assumed that there are no mobile minority carriers, that the plate separation is zero, and that the plate voltage does not vary with time. The depletion layer approximation is used to linearize the equations, which are then solved exactly with the use of Fourier series. Both surface and buried channel devices are analyzed. These solutions can typically be evaluated on a computer in less than a tenth of the time it takes to obtain a solution by the method of finite differences. The solutions obtained here provide an important tool for the designer of charge coupled devices. In addition to describing the method of obtaining the solutions, we evaluate them to show the effects of a number of different design parameters, and compare the cost of these solutions with the cost of obtaining finite difference solutions.

Computational considerations in non-linear diffusion

Abstract: The numerical solution of non-linear diffusion problems is considered from the computational view- point. The advantage of various aspects of the problem solution are considered toward enhancing the computational understanding of the system. First is the analytical solution to the reduced linear case which acts as an error check on the numerical solutions as well as providing a base for a perturbation theory approach. Next are two different numerical solutions which are utilized to generate solution redundancy. These are based upon the method of differential quadrature and the method of finite differences. The application to non-linear transient slab diffusion with a general reservoir boundary condition is shown as an example of the various methods.

Convergence, accuracy and stability of finite element approximations of a class of non‐linear hyperbolic equations

Abstract: (SUMMARY Numerical stability criteria and rates of convergence are derived for finite element approximations of the nonlinear wave equation Un - F(ur) = [(x, t), where F(llr) possesses properties generally encountered in non-linear elasticity. Piecewise linear finite element approximations in x and central difference approximations in t are studied.

Finite-Difference Energy Models versus Finite-Element Models: Two Variational Approaches in One Computer Program

Abstract: Publisher Summary This chapter demonstrates the similarity of the finite-difference energy method and the finite-element method by the application of both methods to problems involving shells of revolution. Curved finite elements are introduced into the BOSOR3 computer program and rates of convergence and computer times are established for stress, buckling, and vibration analyses of an elastic hemisphere. In both the finite-element and finite-difference energy methods, the unknowns of the problem are certain generalized displacement components located at discrete nodes in the domain. Integration can then be performed analytically or numerically. In the finite-difference calculations the “element” properties corresponding to the centroid of each element were provided as input. No interpolation is required because the energy is evaluated at only one point within each element. The eigenvalues and modes are determined by the inverse power iteration method with spectral shifts.

Developing turbulent flow in smooth pipes

Abstract: A numerical and experimental investigation of steady incompressible developing turbulent flow in smooth pipes is presented. Finite difference techniques are used to solve simultaneously the vorticity transport and stream function equations utilising a modified form of the Van Driest effective viscosity model. The numerical solutions are verified experimentally using air as a working fluid at pipe Reynolds 1 × 105, 2 × 105 and 3 × 105.

Analysis of high-rupturing-capacity fuselink prearcing phenomena by a finite-difference method

Abstract: The complex nature of heat flow within a modern current-limiting fuselink precludes direct analysis using classical techniques. This paper describes a method which has been developed for determining the prearcing and steady-state behaviour of such devices. It uses a finite difference technique and is numerical, the calculations being performed by a digital computer. Examples of current and temperature distributions found for some fuse-links are given, and comparisons of calculated and test results for fuselink-clearance times are shown.

Application of the implicit method to surges in open channels

Abstract: A technique is proposed by which continuous numerical computation of flows with surges in open channels can be handled with an implicit formulation. The one-dimensional gradually varied flow equations of continuity and momentum are written in finite difference form for the point (I + ½, J + θ) in the x, t plane, where I is the distance counter, J is the time counter, and θ is a weighting parameter that is varied from 0.5 to 1.0. A computer program was written to solve the problem in which the discharge hydrograph is given at the upstream end of a channel and a stage-discharge relationship is specified at the downstream end. This program was applied to different flow situations in laboratory channels and natural rivers. It was found that θ = 0.5 produced results with a steep wave front but with considerable numerical instability. The use of θ = 1.0 gave results with considerable diffusion of the wave front. The use of θ = 0.60 produced results that had a steep wave front and only a minor degree of numerical instability. This value of θ resulted in good agreement between calculated and measured hydrographs at the downstream end of the channel in most applications. However, the optimum value of θ depends on channel friction and the steepness of the wave front.