Showing papers on "Finite difference method published in 1972"

Numerical solutions of the Navier-Stokes equations for the supersonic laminar flow over a two-dimensional compression corner

Abstract: Numerical solutions have been obtained for the supersonic, laminar flow over a two-dimensional compression corner. These solutions were obtained as steady-state solutions to the unsteady Navier-Stokes equations using the finite difference method of Brailovskaya, which has second-order accuracy in the spatial coordinates. Good agreement was obtained between the computed results and wall pressure distributions measured experimentally for Mach numbers of 4 and 6.06, and respective Reynolds numbers, based on free-stream conditions and the distance from the leading edge to the corner. In those calculations, as well as in others, sufficient resolution was obtained to show the streamline pattern in the separation bubble. Upstream boundary conditions to the compression corner flow were provided by numerically solving the unsteady Navier-Stokes equations for the flat plate flow field, beginning at the leading edge. The compression corner flow field was enclosed by a computational boundary with the unknown boundary conditions supplied by extrapolation from internally computed points.

Computer Analysis of the Fundamental and Higher Order Modes in Single and Coupled Microstrip

Abstract: By means of finite difference methods, dispersion curves are obtained for the fundamental and higher order hybrid modes in both single and coupled microstrip. Structures of realistic proportions are investigated by the use of a graded finite difference mesh. Variational methods are used in deriving the finite difference equations. The higher order modes are found to be similar to LSM slab line modes. A spurious nonphysical class of solutions is found to exist in this and similar formulations, the characteristics of which are described.

On the use of Fast Methods for Separable Finite Difference Equations for the Solution of General Elliptic Problems

Abstract: A well-known source of sparse matrix problems is the systems of linear algebraic equations which arise when we solve elliptic boundary value problems by finite difference methods, A great deal of effort has been devoted to the design and study of iterative methods for the solution of such linear systems (Varga (1962A), Wachspress (1966A) and Young (1971B)). Among direct methods, i.e. methods which give an exact solution of the finite difference equations in absence of round-off errors, Gaussian elimination and its variants are undoubtedly the best known. In this paper we will concentrate on recently developed direct methods other than Gaussian elimination. The best known of these are due to Hockney and Buneman (Buzbee, Golub and Nielson (1970A), Dorr (1970A), Golub (1971A) and Hockney (1965A), (1970A)). Originally the methods of Buneman and Hockney were used only for Poisson’s equation on rectangular domains. The two methods are of comparable speed and very fast. According to Hockney (1970A) his method produces an accurate solution of the standard five-point difference approximation of Poisson’s equation on a 128 × 128 mesh in a time corresponding to that of 3 steps of a successive over-relaxation method for a problem of the same size. The amount of storage needed is about the same as that required for an iterative method and these direct methods therefore compare very favorably with standard band or block Gaussian elimination methods even in that respect.

Simulation for Design Purposes of Magnetic Fields in Turbogenerators with Symmetrical and Asymmetrical Rotors Part I-Model Development and Solution Technique

Abstract: A model the magnetic fields in turbo- generators with either symmetrical or asymmetrical rotor has been developed, and mathematically described and solutions obtained using the digital computer. The model was developed specifically for use in the design stage. It is applied to the entire cross-section of the generator and is applicable to a wide range of machine geometries.

Temporal Operators for Nonlinear Structural Dynamics Problems

Abstract: Three of the most often used numerical integration schemes for the geometrically nonlinear response analysis of structural components are evaluated based on the ease of problem formulation, machine strage required, and speed and accuracy of solution. The particular integration methods considered are the implicit Houbolt and constant-average-acceleration Newmark methods and the explicit central finite difference scheme. The methods are evaluated by a series of numerical experiments with one and multiple degree-of-freedom system, both with and without damping, with emphasis on the characteristics of these temporal operators as regards stability and artificial attenuation or viscosity.

Strang-Type Difference Schemes for Multidimensional Problems

Abstract: Using Strang’s idea, explicit difference schemes of second order accuracy and of optimal stability are obtained for solving partial differential equations in d dimensions.

Finite element analysis of slow non-newtonian channel flow

Abstract: A finite element method is applied to isothermal slow channel flow of power-law fluids. The fully developed flow is normal to the channel cross section. The method and results are compared with a finite difference method for rectangular channels and with exact solutions for the Newtonian case. An advantage of finite element methods is the flexibility of the mesh of elements approximating the continuum, chosen to suit the particular problem. Arbitrary boundary shapes can be handled as illustrated by a rectangular channel with rounded corners.

Review of methods for numerical solution of the hollow-waveguide problem

Abstract: A review is given of current methods for numerically solving the hollow-waveguide problem. Attention is drawn to the relative advantages of the various methods, particularly with regard to their performance with different waveguide shapes. Any requirement for a completely automatic algorithm, or for the computation of fields as well as cutoff values, is also found to put some methods to particular advantage. An attempt is made to compare and contrast the various formulations of finite-difference and finite-element methods (and other similar approaches) which result in a standard form of matrix-eigenvalue problem. Other methods studied are those of point-matching, integral equations and conformal transformations.

Mathematical and Experimental Modeling of the Circulation Patterns in Glass Melts

Abstract: Natural convection currents in a rectangular two-dimensional enclosure representative of the longitudinal section of an industrial glass-melting furnace have been established by both model experiments and numerical calculation. For the latter a finite-difference method has been employed to solve the time-dependent coupled flow and energy equations. The highly generalized mathematical model makes allowance for buoyancy, temperature-dependent viscosity, and diffusive radiation. Generalized boundary conditions are employed to permit specification of any combination of temperature, flux, or mixed thermal boundary conditions. Representative temperature and flow contour maps obtained from the calculations are shown to agree well with experimental results obtained with a 1/20 scale model in which glycerine was employed as the modeling fluid.

Calculation of three-dimensional laminar boundary-layer flows.

Abstract: Accuracy tests have been made of the approximation, proposed recently by Wang (1971), which reduces the computation of three-dimensional, laminar, compressible, boundary-layer equations to the problem of solving two-dimensional type boundary-layer equations. The tests were made by a comparison of the results achieved using Wang's method with a fully three-dimensional boundary-layer calculation. Results of the comparison are tabulated and are found to be in good agreement.

Numerical methods for Orr–Sommerfeld problems

Abstract: Five previously employed numerical methods for the solution of Orr–Sommerfeld problems have been compared to each other and to a new method, the differential method of near-orthonormalized integration. Brief summaries of each method are included. The comparison, based on seven factors, reflects the results of an implementation of a computer program for each method for the classic Orr–Sommerfeld problem of plane Poiseuille flow. This comparison shows that the new method and the algebraic finite difference method are currently the best available numerical solution methods for the problems in this class, with the new method being less problem dependent.

Numerical computation of multishocked, three-dimensional supersonic flow fields with real gas effects.

Abstract: A computational procedure is presented which is capable of determining the supersonic flow field surrounding three-dimensional wing-body configurations such as a delta-wing space shuttle. The governing equations in conservation-law form are solved by a finite difference method using a second-order noncentered algorithm between the body and the outermost shock wave, which is treated as a sharp discontinuity. Secondary shocks which form between these boundaries are captured automatically, and the intersection of these shocks with the bow shock posed no difficulty. Resulting flow fields about typical blunt nose shuttle-like configurations at angle of attack are presented. The differences between perfect and real gas effects for high Mach number flows are shown.

Collapse Analysis for Shells of General Shape; Volume 1. Analysis

Abstract: : F33615-69-C-1523, AF-1467146703, AFFDLTR-71-8-Vol-1(*shells(structural forms), failure(mechanics)), structural properties, numerical analysis, buckling, plastic properties, tensor analysis, computer programming newton- raphson method, collapse, plates(structural members), finite difference theory, stags computer program, structural analysis. The report presents a theory for nonlinear collapse analysis of shells with general shape. The theory combines energy principals and finite difference methods to obtain a system of nonlinear equations; these are solved by a modified Newton-Raphson technique. For greater economy and flexibility in the analysis a capability is provided for use of variable spacing finite difference grids. Inelastic material behavior, as predicted by the White-Besseling Theory, is incorporated into the analysis. A computer code, STAGS, based on the theory has been written and used to solve a number of sample problems. Results for these problems are presented.

Capacitance Microphone Dynamic Membrane Deflections

Abstract: In order to describe the operation of capacitance microphones accurately, it is necessary to have solutions for both the static and dynamic deflections of the moving electrode The authors have previously presented numerical solution approaches to the static deflections caused by the electrical biasing of the microphone The present paper describes an application of finite difference methods to the differential equations describing the dynamic membrane motion due to the acoustic excitation The solutions include the effects of the static deflections and the motion of the thin air film between the microphone's electrodes The numerical results are compared with published results for simplified stationary electrode geometries and with experimental data on more complicated geometries at different tension levels The predicted mode shapes and resonant frequencies change greatly depending upon the magnitudes of the membrane tensions and the back chamber volume

Fast finite-difference solution of biharmonic problems

Abstract: Setting the Reynolds number equal to zero, in a method for solving the Navier-Stokes equations numerically, results in a fast numerical method for biharmonic problems. The equation is treated as a system of two second order equations and a simple smoothing process is essential for convergence. An application is made to a crack-type problem.

Convergent finite difference schemes for nonlinear parabolic equations

Abstract: In this paper, initial-boundary value problems for a general class of nonlinear parabolic equations are studied. We show the solutions of certain associated finite difference equations converge to the solution of the initial-boundary value problem with $O(h^2 )$ rate of convergence..

Effects of Nonlinear Refraction at 10.6 μm on the Far-Field Irradiance Distribution*

Abstract: Theoretical evaluation of the effects of absorption at 10.6 μm on the diffracted field has been investigated. The physical model of the nonlinear interaction process includes the effects of absorption by CO2 and H2O, transverse flow and vibrational relaxation effects associated with atmospheric absorption by CO2. The analysis is based upon the scalar wave equation in the Fresnel limit and the governing equation is integrated numerically using a stable, finite difference method. Diffraction patterns in the far field for an unfocused beam and at the focal point for a focused beam are presented. The atmosphere acts as a nonlinear lens with aberrations and the irradiance distributions exhibit a complicated filamentary pattern at the focal point.

Numerical investigation of the atmospheric dispersion of stack effluents

Abstract: This report describes a numerical method based on the best gradient-transfer theory currently available for computing pollutant concentration distributions downwind from a stack. The vertical inhomogeneity of the atmosphere and ground roughness are included in the model. Vertical wind and temperature profiles are calculated numerically from given values of ground roughness and wind speed and relative temperature at stack height. An equation governing the plume from the stack is solved by a finite difference method. The numerical results, compared with several experiments, suggest that ground roughness is an important parameter and that is agreement between different sets of experimental data may be due to different values of this parameter. The effect of wind is found to be small under neutral conditions. The effective mean wind decreases to a minimum value a short distance from the stack and then increases downwind.

Numerical method for determining the electromagnetic field in saturated steel plates

Abstract: An explicit finite-difference method based on that of Dufort-Frankel is described that is suitable for the solution of the nonlinear diffusion equation that arises in the study of flux and current penetration into saturated steel. The method is applied to study the process by which energy is dissipated in the material.

Large deflection of elastic plates under patch loading

Abstract: Two methods based on: (1) the direct finite difference approach; and (2) the dynamic relaxation method are presented for the treatment of the elastic large deflection behavior of plates under transverse loading. The merits of the methods in respect of formulation, accuracy, storage and computing time requirements are pointed out together with the main aspects of the finite element method for the analysis of the same problem. The large deflection behavior of square plates under a central patch loading (a concentrated load distributed over a finite area) is investigated. Numerical solutions are offered in a general and condensed form for plates with simply supported and clamped boundary conditions, and for a range of patch sizes. The solutions are shown to be of particular value in the assessment of stresses under a patch loading in practical plate problems. The solutions can serve as a useful guide line for the advanced design of the plate components of plated structures.

Dynamics of Nuclear Systems

Abstract: T. J. Thompson in his paper on the interaction of reactors dynamics with nuclear safety, emphasizes the importance of the safety and the operational characteristics of the plants which should on no account be compromised in the interest of dynamic response characteristics. He also brings out the nature and effects of slow and fast acting transients on the effective performance of the system and stresses the necessity of correct solutions in real problems.