Showing papers on "Finite difference method published in 1969"

Propagation of elastic waves in layered media by finite difference methods

Abstract: A finite difference equation formulation for the equations of elasticity is presented and applied to the problem of a layered half-space with a buried point source emitting a compressional pulse. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The results for a specific case are compared with those found by a completely different method in order to check the validity of the finite difference methods. The agreement is excellent. The effect of different mesh sizes on the theoretical seismograms is studied next and a suitable grid system selected for the applications that follow. The development of Rayleigh waves on the surface of a half-space and the change of the Rayleigh wave with depth and pulse width are examined. The problem of a layered half-space with a high velocity bottom is considered and the refraction arrivals on the surface and on the interface are studied. The problem of interface waves on the surface separating two semiinfinite media is also examined. Interface waves are found when the physical parameters lie both inside and outside the region determined by the Stoneley equation. Finally, a series of theoretical seismograms for a layered half-space showing the variation of the surface waves as a function of depth and of the density in the lower medium is presented.

Numerical Methods for Partial Differential Equations

Abstract: Introduction. Computer Program Packages. Typical Problems. Classification of Equations. Discrete Methods. Finite Differences and Computational Molecules. Finite Difference Operators. Method of Weighted Residuals. Finite Elements. Method of Lines. Errors. Stability and Convergence. Irregular Boundaries. Choice of Discrete Network. Dimensionless Forms. Parabolic Equations: Introduction. Properties of a Simple Explicit Method. Fourier Stability Method. Implicit Methods. Additional Stability Considerations. Matrix Stability Analysis. Extension of Matrix Stability Analysis. Consistency, Stability, And Convergence. Pure Initial Value Problems. Variable Coefficients. Examples of Equations with Variable Coefficients. General Concepts of Error Reduction. Methods of Lines (MOL) for Parabolic Equations. Weighted Residuals and the Method of Lines. Bubnov-Galerkin Scheme for Parabolic Equations. Finite Elements and Parabolic Equations. Hermite Basis. Finite Elements and Parabolic Equations. General Basis Fucntion. Finite Elements and Parabolic Equations. Special Basis Functions. Explicity Finite Difference Methods for Nonlinear Problems. Further Applications on One Dimentions. Asymmetric Approximations. Elliptic Equations: Introduction. Simple Finite Difference Schemes. Direct Methods. Iterative Methods. Linear Elliptic Equations. Some Poit Iterative Methods. Convergence of Point Iterative Methods. Rates of Convergence. Accelerations. Conjugate Gradient Method. Extensions of SOR. Auliative Examples of Over-Relaxation. Other Point Iterative Methods. Block Iterative Methods. Alternating Direction Methods. Summary of ADI Results. Triangular Elements. Boundary Element Method (BEM). Spectral Methods. Some Nonlinear Examples. Hyperbolic Equations: Introduction. The Quasilinear System. Introductory Examples. Method of Characteristics. Constant States and Simple Waves. Typical Application of Characteristics. Finite Differences for First-Order Equations. Lax-Wendroff Methods and Other Algorithms Dissipation and Dispersion. Explicity Finite Difference Methods. Attenuation. Implicit Methods for Second-Order Equations. Time Quasilinear Examples. Simultaneous First-Order Equations. Explicit Methods. An Implicity Method for First-Order Equations. Hybrid Methods for First-Order Equations. Finite Elements and the Wave Equation. Spectral Methods and Periodic Systems. Gas Dyunamics in One Space Variable. Eulerian Difference Equations. Lagrangian Difference Equations. Hopscotch Methods for Conservation Laws. Explicity-Implicity Schemes for Conservation Laws. Special Topics: Introduction. Singularities. Shocks. Eigenvale Problems. Parabolic Equations in Segeral Space Variables. Additional Comments on Elliptic Equations. Hyperbolic Equations in Higher Dimensions. Mixed Systems. Higher Order Equations in Elasticity and Vibrations. Computational Fluid Mechanics. Stream Function. Vorticity Method for Fluid Mechanics. Primitive Variable Methods for Fluid Mechanics. Vector Potential Methods for Fluid Mechanics. Introduction to Monte Carlo Mehtods. Fast Fourier Transform and Applications. Method of Fractional Steps. Applications of Group Theory in Computation. Computational Ocean Acoustics. Enclosure Methods. Chapter References. Author Index. Subject Index.

Time‐Dependent Viscous Flow over a Circular Cylinder

Abstract: A finite difference method is extended to high Reynolds number flow about circular cylinders with particular emphasis given to the quantitative description of fine flow features. The method is of the explicit type and includes a directional difference scheme for the nonlinear terms which enhances calculational stability at high Reynolds numbers. A cell structure is chosen which provides local cell dimensions consistent with the structure of solutions expected. Solutions are presented for a range of Reynolds numbers from 1 to 3 × 105 in which the flow is started impulsively from rest, and the development is studied up to the approach of the steady‐state or the limit cycle condition, whichever is appropriate to the particular Reynolds number.

Computation of Electromagnetic Fields

Abstract: This paper reviews some of the more useful, current and newly developing methods for the solution of electromagnetic fields. It begins with an introduction to numerical methods in general, including specific references to the mathematical tools required for field analysis, e.g., solution of systems of simultaneous linear equations by direct and iterative means, the matrix eigenvalue problem, finite difference differentiation and integration, error estimates, and common types of boundary conditions. This is followed by a description of finite difference solution of boundary and initial value problems. The paper reviews the mathematical principles behind variational methods, from the Hilbert space point of view, for both eigenvalue and deterministic problems. The significance of natural boundary conditions is pointed out. The Rayleigh-Ritz approach for determining the minimizing sequence is explained, followed by a brief description of the finite element method. The paper concludes with an introduction to the techniques and importance of hybrid computation.

Axisymmetric dynamic snap-through of elastic clamped shallow spherical shells.

Abstract: The behavior of axisymmetric dynamic snap-through of elastic, clamped shallow spherical shells under a uniform pressure has been investigated by.several authors. In most of the previous work, approximate methods were used, and as yet no positive conclusion has been made on the critical load for snap-through. In this paper, a numerical method is developed for solving the governing nonlinear differential equations of this problem. Two types of loading conditions are included-—an impulsive load and an instantaneous ly applied step load with infinite duration. In the case of impulsive loading, it is found from the quasi-static problem under zero load that the only equilibrium, position of the shell is its undeformed configuration. Hence, if we define the dynamic snap-through based on the finite jump behavior of deflection then there is no possibility for dynamic snap-through under impulsive loading. In the case of step loading, the snap-through loads are evaluated for a wide range of the geometrical parameter of the shell. Comparison of the present calculated critical loads with the previous results from approximate methods is made. It is found that the present critical load checks closely with an experimental value of the critical load for one value of the geometrical parameter of the shell.

Calculation of nonequilibrium hypersonic turbulent boundary layers and comparisons with experimental data

Abstract: Compressible hypersonic turbulent boundary layers solution by finite difference method, relating mixing length to velocity profile shape factor

Theory of Sand Filtration

Abstract: A set of nonlinear partial differential equations with dimensionless variables, for filtration of dilute unisized suspension in water through homogeneous sand beds under constant rates of vertically downward flow, is presented. The equations describe the space-time variation of suspension concentration and specific deposit in a filter medium. An expression for dimensionless filter coefficient relating all physical variables of filtration is obtained by dimensional analysis. Solutions of the equations were obtained with the aid of a digital computer using the finite difference method. The analytically determined suspension concentrations at different time intervals and depths of bed agreed adequately with the experimental values. An equation relating head loss per unit depth and specific deposit within the filter bed is derived and the validity of the proposed equation is verified with the experimental head loss data.

An iterative finite-difference method for hyperbolic systems

Abstract: Abstract. An iterative finite-difference scheme for initial value problems is presented. It is applied to the quasi-linear hyperbolic system representing the one-dimensional time dependent flow of a compressible polytropic gas. The emphasis in this research was on the handling of discontinuities, such as shock waves, and overcoming the post-shock oscillations resulting from nonlinear instabilities. The linear stability is investigated as well. The success of the method is indicated by the monotonie profiles which were obtained for almost all the cases tested.

Analysis of Coaxial Line Discontinuities by Boundary Relaxation

Abstract: Discontinuities in coaxial lines may in general be represented in equivalent networks by lumped capacitances. The calculation of discontinuity capacitance is possible by means of mode-matching techniques for very simple discontinuities; for more complex cases, direct numerical methods are preferable. A new numerical technique is presented for solving the field problem in a region bounded on two sides by infinitely extending coaxial lines. The approach used is to define operators by means of which the potentials at a given cross-sectional plane of the coaxial line are related to potentials at another plane. The problem of a discontinuity region between two infinitely long lines is thereby converted into a finite problem with prescribed boundary operators in place of boundary values. Standard methods may be used to solve the problem in a finite region. Subsequent reformulation of the discontinuity capacitance in terms of stored energy permits calculation of this capacitance from the potential values in only a minimal region. The resulting computer programs are at least an order of magnitude faster than previously published ones.

The Finite Difference Solution of Microwave Circuit Problems

Abstract: Using finite difference methods this paper shows how solutions may be obtained with the aid of a digital machine to a wide range of microwave circuit problems These problems include the parameters of TEM-mode transrnission lines, the equivalent circuits of obstacles in these lines, the cutoff frequencies of the fundamental mode in a waveguide of very general cross section, and the equivalent circuits of obstacles in rectangular waveguide Methods for deriving the appropriate finite difference equations are presented and optimum methods for their solution set out; singularities are also included in the treatment The paper ends with a resume of some typical results to problems of practical interest which have been obtained by these methods

Accuracy of Difference Formulation of Navier‐Stokes Equations

Abstract: An error analysis is formulated under the assumption that Lax's equivalence theorem is valid for quasilinear equations. Analysis is carried out for a class of difference approximations of the Burgers' equation. Explicit estimates of the errors due to various sources are obtained and verified by numerical integration. The boundary errors do not decay rapidly away from the boundary and often dominate the truncation errors. The boundary errors led to significant differences between apparently smooth and physically reasonable results of carefully executed computations of the same problem. The results based on difference formulations that are more nearly consistent with the differential formulation are likely to be more accurate. A conservative difference formulation is proposed to meet this consistency requirement as far as is practically possible. Calculations of multidimensional flow examples guided by these inferences are given. The results tend to support such inferences.

Behavior of stiffened curved plate model

Abstract: The solution of the orthotropic plate equation, in polar coordinates, is obtained by the finite difference technique. The boundary conditions are imposed such that simple supports along the radial edges and free supports along the angular edges are prescribed. Fifteen various mesh pattern equations are developed; six of these patterns are described. The solution of these equations, representing a particular subdivided plate sector, is solved by a computer program. The internal forces throughout the plate are also evaluated by the finite difference method, utilizing the deflection data and a computer program. To evaluate the validity of the technique, a stiffened curved steel plate model was tested. Separate stiffness model tests were also conducted to evaluate experimentally the angular and radial stiffnesses and torsional stiffness, which were compared to the calculated stiffnesses. Proper evaluation of the plate stiffnesses resulted in analytical deflections and strains, which correlated very well with the experimental data.

A Discrete Method for Parameter Identification in Linear Systems with Transport Lags

Abstract: Linear dynamical systems with transport lags are characterized by linear differential-difference equations. The task of identifying unknown parameters in such systems from the input-output data is difficult due to mathematical complications associated with differentialdifference equations. This paper presents a method which solves the identification problem. The method is digitally oriented and shows how a continuous-time system can be identified by discrete techniques. The solution is based on Kalman's least square method. The identification procedure essentially involves two steps: 1) discretizing the continuous system via finite difference approximation, and 2) estimating the parameters through the identification of the resulting discrete model. Experimental results have verified the validity of the proposed method.

Axisymmetric Finite Deflection of Circular Plates

Abstract: The axisymmetric finite deflection of circular disks and annular plates is examined. The investigation has two objectives. One is to develop a numerical technique which can be used to solve some nonlinear boundary value problems. The other objective is to obtain information on the deflection and stress distribution of circular plates which have large deflection. The nonlinear equations of equilibrium and boundary conditions are derived by the variational method. These equations are expanded into finite difference equations, and solved by a modified iteration technique. In this modified iteration technique, the loading is increased by small increments. The solution of each loading step is predicted by extrapolating the solutions of previous loading steps. The calculation is manipulated in such a fashion that the solution converges very rapidly. Numerical results of various loading conditions and boundary conditions are obtained. In all these calculations, the proposed iteration method was found to be very effective. The solution for circular plates fixed at the boundary, and subjected to uniform pressure, is in excellent agreement with the existing solution.

Paper 35: Stability, Convergence, and Accuracy of Two-Dimensional Streamline Curvature Methods Using Quasi-Orthogonals

Abstract: The streamline curvature technique for the numerical solution of the equations of motion for two-dimensional flow has been analysed. By considering a simple flow case an optimum damping factor for the quasi-orthogonal method was derived as a function of grid aspect ratio, Mach number, and differentiation formula, and the number of iterations required for a given accuracy was found. The stability and accuracy properties of numerical methods for finding the second derivative of a curve were also investigated, and it was found that, in this context, finite difference methods using polynomials were the best and spline methods the worst.

The Dynamic Response of Cylindrical and Conical Panels

Abstract: The dynamic response of cylindrical and conical panels subjected to arbitrary time-varying load distributions is studied and appropriate equations are presented. Convenient trigonometric series are used, in conjunction with finite-difference methods, to reduce the governing equations to sets of matrix equations. The numerical solution procedure involves time integration, using an unconditionally stable implicit (Houbolt) scheme, together with a Gaussian elimination technique particularly suited to the banded matrices involved. Calculated results treat the effects of conicity and various support conditions on the structural response. Individual cases show quantitatively how cylindrical panel response is more sensitive than that of conical panels to changes in edge restraint.

The interaction region in the boundary layer of a shock tube

Abstract: Velocity and temperature boundary layers developed on a plane wall by ideal shock-tube flow are considered for weak shock and expansion waves. Analytically, the boundary layer consists of three regions, bounded by (1) expansion-wave head, (2) diaphragm location, (3) contact discontinuity, (4) shock. The flow fields (1, 2) and (3, 4) are, essentially, known. In the interaction region (2, 3), these flow fields merge, the governing equations are ‘singular parabolic’ and admit boundary conditions usually associated with elliptic equations. It is convenient to replace the weak expansion wave in the main flow by a line discontinuity. A consistent linearization scheme can now be devised to obtain the solution in the three regions. In (2, 3), the resulting linear singular parabolic equations for the first-order solutions are solved successfully by an iterative finite difference method, normally applied to elliptic equations.

Refraction of sound by a jet: a numerical study

Abstract: The equations appropriate to the propagation of sound in a realistic jet flow have been solved numerically for the case of a sinusoidal point source on the axis of a subsonic jet. The use of slowly varying quantities (related to phase and amplitude) as independent variables nonlinearized the wave equation but facilitated the application of finite difference methods. The difference equations approximating the partial differential equation were solved by nonlinear block relaxation using a Newton‐like method. The solutions approach the results of ray acoustics in the high‐frequency limit. For the most part, the finite frequency results agree well with experimental directivity patterns for a point source in a jet; they lend further support to the view that the downstream valley in jet noise is due to refraction. The computations have yielded detailed phase and amplitude data throughout the sound field. Unexpected findings are that the distortion of the constant phase surfaces is slight and that the flow beyon...

Creep buckling of a circular cylindrical shell.

Abstract: A theory is presented for the creep behavior and buckling of a circular cylindrical shell under axisymmetrical loads. Elastic deformations and secondary creep are included and the multimembrane model is utilized. The set of differential equations is solved by means of finite difference methods with respect to the axial coordinate and with respect to time. A number of numerical examples are presented which demonstrate the nature of the solution. The deflection pattern due to creep deformations is similar to that given by the elastic solution, and the deflections increase with time in the same way as the elastic deflections do when the load is increased. The deflections approach infinite values within a finite time. The double membrane model was found to yield a very good estimate of the creep rate in comparison with more accurate multimembrane models, and the difference in the critical time did not exceed 5 % in the cases investigated. An approximate buckling criterion was conjectured and used for comparison with available experimental results. A fairly good agreement was noted.

Appendix d: petros 2: a new finite-difference method and program for the calculation of large elastic-plastic dynamically-induced deformations of general thin shells.

Abstract: : The equations of motion, strain-displacement relations, and the constitutive relations for a thin Kirchoff shell of arbitrary shape are derived in tensor notation. Taken into account are arbitrarily-large deformations, and material which exhibits elastic, elastic-plastic, strain hardening, and/or strain-rate sensitive behavior: also the shell may be subject to arbitrary initial velocities and/or transient external forces. The governing differential and algebraic equations then are recast into finite-difference form. These finite-difference equations have been employed to write a FORTRAN IV computer program called PETROS 2. Applications to illustrate various features of this formulation and program are presented. Included herein is the PETROS 2 program together with sample problem input and solution data. Subroutines which will handle shells having the following initial geometries are included: (1) cone, (2) cylinder, and (3) flat plate. The user may readily provide a similar initializing subroutine to permit analyzing a shell of some other initial shape. This report contains appendix D to the main report, AD-708 773.

The Use of Interpolation in Improving Finite Difference Solutions of TEM Mode Structures

Abstract: A finite difference potential solution to a TEM mode transmission line cross section may be used to define a continuous potential function, leading to an upper bound for the capacitance. The accuracy of the capacitance calculation is shown to depend on the potential function fitted. A method is developed for interpolating a suitable potential function; in the cases considered, the use of this potential function gave capacitance solutions with an error approximately one-fifth that obtained using the usual methods.

Numerische Behandlung singulärer Sturm-Liouville-Probleme

Abstract: Generalizing the method of Wendroff [9] and using an estimate for the square integral of a normed eigenfunction outside a compact set, bounds are obtained for the eigenvalues of singular Sturm-Liouville problems from a finite difference method. The number of mesh points necessary to obtain. the accuracy ? behaves like ??½ ln? if? tends to zero. Some numerical examples are given.