Showing papers on "Finite difference published in 1969"
TL;DR: In this paper, 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.
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.
554 citations
TL;DR: In this paper, ten different finite-difference schemes for numerical integration of the primitive equations for the free-surface model are tested for stability and accuracy, and the integrations show that the quadratic conservative and the total energy conservative schemes are more stable than the usual second-order conservative scheme.
Abstract: Ten different finite-difference schemes for the numerical integration of the primitive equations for the free-surface model are tested for stability and accuracy. The integrations show that the quadratic conservative and the total energy conservative schemes are more stable than the usual second-order conservative scheme. But the most stable schemes are those in which the finite-difference approximations to the advection terms are calculated over nine grid points in space and therefore contain a form of smoothing, and the generalized Arakawa scheme, which for nondivergent flow conserve mean vorticity, mean kinetic energy, and mean square vorticity. If the integrations are performed for more than 3 days, it is shown that more than 15 grid points per wavelength are probably needed to describe with accuracy the movement and development of the shortest wave that initially is carrying a significant part of the energy. This is true even if a fourth-order scheme in space is used. Long-term integrations ...
158 citations
TL;DR: In this article, a review of variational methods for the solution of electromagnetic field problems is presented, including the Rayleigh-Ritz approach for determining the minimizing sequence, and a brief description of the finite element method.
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.
119 citations
TL;DR: In this article, a finite difference solution technique is used to solve Maxwell's equations directly in the treatment of electromagnetic pulse scattering in a time-varying inhomogeneous medium, where axial symmetry is obtained.
Abstract: A finite difference solution technique is used to solve Maxwell's equations directly in the treatment of electromagnetic pulse scattering in a time-varying inhomogeneous medium. In particular the scattering from a cylindrical rod inside a cylindrical waveguide is considered where axial symmetry is obtained.
113 citations
01 Jan 1969
TL;DR: In this article, the Adams-Bashforth formulae were used for numerical integration of stiff systems of ODEs, where the problem is of the type y1 = Py + Q(x) where P is a constant and Q(X) a polynomial of degree q. The method is exact if the problem was of the kind y1 + Q (x) = Py+Q(x).
Abstract: This paper gives new finite difference formulae which are suitable for the numerical integration of stiff systems of ordinary differential equations. The method is exact if the problem is of the type y1 = Py + Q(x) where P is a constant and Q(x) a polynomial of degree q. When P = 0 the method is identical with the Adams-Bashforth formulae.
86 citations
TL;DR: In this article, the equations for elastic wave propagation were solved by a finite difference scheme for the case of an elastic quarter plane, where a point source emitting a compressional pulse was located along the diagonal inside the quarter plane.
Abstract: The equations for elastic wave propagation are solved by a finite difference scheme for the case of an elastic quarter plane. A point-source emitting a compressional pulse is located along the diagonal inside the quarter plane. Free-surface conditions are assumed on the boundary lines, so that the problem is nonseparable.
Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The effect of different finite difference formulations for the boundary conditions and the effect of different mesh sizes are studied.
Various reflected volume and surface waves are identified, corner-generated surface waves are clearly seen in the seismograms and their particle motion is studied. The amplitude of the pulse observed at the corner is three times the amplitude of the initial pulse.
82 citations
TL;DR: Galerkin's method, using a local basis, provides unconditionally stable, implicit generalized finite-difference schemes for a large class of linear and nonlinear problems.
Abstract: Finite-difference schemes for initial boundary-value problems for partial differential equations lead to systems of equations which must be solved at each time step. Other methods also lead to systems of equations. We call a method a generalized finite-difference scheme if the matrix of coefficients of the system is sparse. Galerkin's method, using a local basis, provides unconditionally stable, implicit generalized finite-difference schemes for a large class of linear and nonlinear problems. The equations can be generated by computer program. The schemes will, in general, be not more efficient than standard finite-difference schemes when such standard stable schemes exist. We exhibit a generalized finite-difference scheme for Burgers' equation and solve it with a step function for initial data. U
76 citations
TL;DR: In this paper, a general computer program has been developed which solves the finite difference analogue of the conservation equations in boundary layer form for laminar film condensation, and closed form analytical solutions based on the Nusselt assumptions have been extended to include the effect of a nonisothermal condenser wall.
44 citations
TL;DR: The evolution of a model of the flow in a layer of fluid suddenly heated from below at a Rayleigh number sufficient for a laboratory flow to ultimately become turbulent is investigated with numerical experiments as discussed by the authors.
Abstract: The evolution of a model of the flow in a layer of fluid suddenly heated from below at a Rayleigh number sufficient for a laboratory flow to ultimately become turbulent is investigated with numerical experiments The numerical study simulates the flow by means of the mean field equations along the lines of Herring's (1963, 1964) pioneering study but uses a different finite difference technique and concentrates attention on the flow development rather than on the final statistically steady state These solutions are compared with previous and some new simulations in two dimensions The solutions confirm Herring's work and in addition show that the mean field equations, and in particular the weak-coupling approximations, describe the gross features of the model sufficiently well for the mean field equations to be used with reasonable confidence in evolutionary studies
34 citations
33 citations
TL;DR: In this paper, a finite-difference method has been developed for the solutions to the governing partial differential equations for the constant-property turbulent boundary layer Prandtl's mixinglength concept was used to express the apparent turbulent shearing stress according to a hypothesized mixing-length distribution through the boundary layer.
Abstract: A finite-differe nce method has been developed for the solutions to the governing partial differential equations for the constant-property turbulent boundary layer Prandtl's mixinglength concept was used to express the apparent turbulent shearing stress according to a hypothesized mixing-length distribution through the boundary layer The model for the mixing-length distribution is in general agreement with experimental data for a variety of flow conditions The method, an always stable explicit finite-difference formulation that requires no iterative procedures, is numerically more direct than other methods recently proposed and is unique in its evaluation of the apparent shearing stress by using an assumed mixing-length distribution as the only empirical input The predicted skin-friction coefficients and velocity profiles agree well with experimental data for the several comparisons made, which included flows in both favorable and adverse pressure gradients as well as flat plate cases with and without blowing The numerical method presented is not restricted to use with the particular mixing-length model tentatively proposed in this work, but can be used to compare various models and help establish their properties and range of applicability, thereby serving to further understanding of the most fundamental aspects of turbulent flow a c Cf gc I
TL;DR: In this paper, a finite-difference approximation of the mass fluxes from one hexagonal cell to the next through their common boundary is proposed, which conserves the total mass, the total momentum, and the total kinetic energy of the fluid as well as the total squared vorticity of a non-veto flow.
Abstract: The hexagonal grid based on a partition of the icosahedron has distinct geometrical qualities for the mapping of a sphere and also presents some indexing difficulties. The applicability of this grid to the primitive equations of fluid dynamics is demonstrated, and a finite-difference approximation of these equations is proposed. The basic variables are the mass fluxes from one hexagonal cell to the next through their common boundary. This scheme conserves the total mass, the total momentum, and the total kinetic energy of the fluid as well as the total squared vorticity of a nondivergent flow. A computational test was performed using a hexagonal grid to describe space periodic waves on a nonrotating plane. The systematic variation of total kinetic and potential energy is less than 10−5 after 1,000 time steps.
TL;DR: In this paper, a theoretical analysis is presented for the problem of flat rectangular plates undergoing large deflections due to the action of a uniform lateral pressure or combinations of uniform lateral pressures and compressive edge loading.
Abstract: A theoretical analysis is presented for the problem of flat rectangular plates undergoing large deflections due to the action of a uniform lateral pressure or combinations of uniform lateral pressure and compressive edge loading. The governing partial differential equations are replaced by their finite difference equivalents and the resulting difference equations solved by over-relaxation using an I.C.T. 1905 computer.The results for plates subjected to lateral loads only are in good agreement with the few existing solutions. For plates subjected to lateral loads and combinations of lateral and edge loading, the results show good agreement with the results of an experimental programme carried out at the University of Strathclyde.
TL;DR: In this article, the authors show how solutions may be obtained with the aid of a digital machine to a wide range of microwave circuit problems, including the parameters of TEM-mode transrnission lines, the equivalent circuits of obstacles in these lines, cutoff frequencies of the fundamental mode in a waveguide of very general cross section, and the equivalent circuit of obstacles.
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
TL;DR: In this paper, an alternative finite difference formulation of the governing differential equation of the title problem was proposed, which was used over the entire range of symmetric supports (viz. four-corner supports to one single central support).
01 Jan 1969
TL;DR: The solution of the nonlinear differential equation Y(x, Y, Y) with two-point boundary conditions is approximated by a quintic or cubic spline function y(x), which is well suited to nonuniform mesh size and dynamic mesh size allocation.
Abstract: The solution of the nonlinear differential equation Y″ = F(x, Y, Y′) with two-point boundary conditions is approximated by a quintic or cubic spline function y(x). The method is well suited to nonuniform mesh size and dynamic mesh size allocation. For uniform mesh size h, the error in the quintic spline y(x) is O(h4), with typical error one-third that from Numerov's method. Requiring the differential equation to be satisfied at the mesh points results in a set of difference equations, which are block tridiagonal and so are easily solved by relaxation or other standard methods.
TL;DR: In this paper, the orthotropic plate equation is solved by the finite difference technique and the internal forces throughout the plate are also evaluated using the deflection data and a computer program.
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.
TL;DR: This paper presents a method which solves the identification problem of linear dynamical systems with transport lags and is digitally oriented and shows how a continuous-time system can be identified by discrete techniques.
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.
TL;DR: In this article, bounds for the first six eigenvalues of the H-shaped membrane are found, using interpolation of the boundary conditions at a finite number of points, and the effect of rounding on the calculation of these bounds is considered.
Abstract: used to find explicit bounds for the problem of the H-shaped membrane. Bounds for the first six eigenvalues of the H-shaped membrane are found, using interpolation of the boundary conditions at a finite number of points, and the effect of rounding on the calculation of these bounds is considered. Some speculative remarks are added on the difficulties which arise when the boundary conditions are interpolated at a large number of points. We conclude with an estimate of the fundamental eigenvalue of the H-membrane based on finite difference approximations.
01 Jul 1969
TL;DR: In this article, a critical analysis of well-known procedures for the computation of one-dimensional shocked flows is made, in order to show the inconveniences of computing finite differences across a discontinuity and to prove that the use of the equations of motion in conservation form does not make the results any more accurate.
Abstract: : A critical analysis of well-known procedures for the computation of one-dimensional shocked flows is made, in order to show the inconveniences of computing finite differences across a discontinuity and to prove that the use of the equations of motion in conservation form does not make the results any more accurate. A technique is developed to treat one-dimensional inviscid problems and it is applied to the problem of an accelerating piston. Practical and safe ways to predict the formation of a shock and to follow it up in its evolution are given. (Author)
01 Sep 1969
TL;DR: In this article, a finite difference approximation to the heat flow equations of the stator core and windings of a rotating electrical machine has been derived based on a linear linear model.
Abstract: To ensure that the materials used in the construction of a rotating electrical machine are utilized economically it is necessary to be able to predict the temperature distribution within the machine.A resistance network analogue representation of the stator core and windings is derived, based on the finite difference approximation to the heat flow equations. A solution of the analogue is achieved by matrix methods on a digital computer. The advantages of this particular analogue representation are that the effects of axial heat flow in the copper and changes in the air temperature in the radial air ducts are taken into account. The validity and accuracy of the analogue representation are checked by comparing the results with those obtained from detailed temperature measurements on a production machine. It is shown that the present standard methods of measuring winding temperature are inadequate for determining the actual copper temperature.Under certain circumstances high temperatures can be obtained in t...
TL;DR: In this paper, the authors derived nonlinear partial differential equations which describe axisymmetric flow in terms of the radial and axial coordinates and obtained the solution by using a Newton-Raphson inner iteration, as well as the usual outer iteration customarily used for a finite difference solution of a partial differential boundary value problem of the elliptic type.
Abstract: Methods are developed for obtaining solutions by finite differences to free streamline axisymmetric, potential fluid flows. The formulation of the boundary value problem considers the velocity potential and Stokes stream function as the independent variables and the radial and axial coordinates as the dependent variables. The nonlinear partial differential equations are derived which describe axisymmetric flow in terms of the radial and axial coordinates. Because of the nonlinear nature of the equations, the solution is obtained by using a Newton-Raphson inner iteration, as well as the usual outer iteration customarily used for a finite difference solution of a partial differential boundary value problem of the elliptic type. Two axisymmetric problems have been investigated. The first is of a jet of inviscid incompressible fluid issuing from a nozzle into the free atmosphere. The second is the cavity flow resulting from a jet flowing past a body of revolution.
01 Apr 1969
TL;DR: In this paper, several algorithms are presented for solving block tridiagonal systems of linear algebraic equations when the matrices on the diagonal are equal to each other and the matrix on the subdiagonals are all equal to another.
Abstract: Several algorithms are presented for solving block tridiagonal systems of linear algebraic equations when the matrices on the diagonal are equal to each other and the matrices on the subdiagonals are all equal to each other. It is shown that these matrices arise from the finite difference approximation to certain elliptic partial differential equations on rectangular regions. Generalizations are derived for higher order equations and non-rectangular regions.
01 Jan 1969
TL;DR: A class of finite difference methods called splitting techniques for the solution of the multigroup diffusion theory reactor kinetics equations in two space dimensions is presented in this paper, and a subset of the above class is shown to be consistent with the differential equations and numerically stable.
Abstract: A class of finite difference methods called splitting techniques are presented for the solution of the multigroup diffusion theory reactor kinetics equations in two space dimensions. A subset of the above class is shown to be consistent with the differential equations and numerically stable. An exponential transformation of the semi-discrete equations is shown to reduce the truncation error of the above methods so that they beoome practical methods for two-dimensional problems. A variety of numerical experiments are presented which illusthate the truncation error, convergence rates, and stability of a particular member of the above class. Thesis Supervisor: Kent F. Hansen Title: Associate Professor of Nuclear Engineering 'AAMakw"A" "AwwwUhAwww" IN M 111mol"t""'.. P 3 TABLE OF CONTENTS Page ABSTRACT 2 LIST OF FIGURES 5 LIST OF TABLES 6 ACKNOWLEDGMENTS 7 BIOGRAPHICAL NOTE 8 Chapter
TL;DR: 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.
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.
TL;DR: In this article, it was shown that completely general finite-difference approximations of linear partial differential operators in space, based on functional (rather than, but including, polynomial) approximation, can be easily proven in a direct method involving only the inversion and multiplication of constant matrices.
Abstract: Methods used in analog computation for the parallel- finite-differences solution of partial differential equations have been almost universally based on the "classical" derivation of finite-difference approximations. That is, the space-dependence of the approximate solution is lo cally assumed to be appropriately represented by a Tay lor (or polynomial) truncated series of low1, or higher7, order. Although known to introduce higher truncation errors than the use of other functional series2,3, this re striction has been mostly accepted for reasons of con venience because no simple method for obtaining more general approximations was available in practice.We show in this paper that completely general finite- difference approximations of linear partial differential operators in space, based on functional (rather than, but including, polynomial) approximations, can be easily ob tained in a direct method involving only the inversion and multiplication of constant matrices.One of the drawbacks of the analog com...