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Showing papers on "Discretization published in 1974"


Journal ArticleDOI
TL;DR: In this article, the authors extended the direct method of cyclic reduction to linear systems which result from discretization of separable elliptic equations with Dirichlet, Neumann, or periodic boundary conditions.
Abstract: This paper extends the direct method of cyclic reduction to linear systems which result from the discretization of separable elliptic equations with Dirichlet, Neumann, or periodic boundary conditions. For an $m \times n$ net, the operation count is proportional to $mn\log _2 n$ and $mn$ storage locations are required.

272 citations


Journal ArticleDOI
TL;DR: An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented and is based on a discretization studied earlier by H. B. Keller.
Abstract: An adaptive finite difference method for first order nonlinear systems of ordinary differential equations subject to multipoint nonlinear boundary conditions is presented. The method is based on a discretization studied earlier by H. B. Keller. Variable order is provided through deferred corrections, while a built-in natural asymptotic estimator is used to automatically refine the mesh in order to achieve a required tolerance. Extensive numerical experimentation and a FORTRAN program

87 citations


Journal ArticleDOI
TL;DR: The relationship between the continuous and the discrete parameters yields a simple method of maximum likelihood estimation of the continuous parameters from a discretely sampled data.
Abstract: Discretization of a continuous autoregressive moving average process at an equispaced sampling interval results in a discrete autoregressive moving average process The relationship between the continuous and the discrete parameters yields a simple method of maximum likelihood estimation of the continuous parameters from a discretely sampled data A technique is described for modeling of continuous processes from discrete observations and is illustrated with analysis of the yearly Wolfer's sunspot numbers data

84 citations


Journal ArticleDOI
TL;DR: An algorithm is derived for solving a large class of Ito random integral equations and gives a sample pathwise solution and is readily implementable on a digital computer.
Abstract: An algorithm is derived for solving a large class of Ito random integral equations. The derivation of the algorithm involves approximate discretization of the given Ito equation. The Ito integrals arising out of discretization are expressed as functions of normal random variables. The algorithm gives a sample pathwise solution and is readily implementable on a digital computer.

79 citations


Journal ArticleDOI
Isaac Fried1
TL;DR: Incompressibility is gradually introduced into the finite elements with the mesh refinement in such a way as to balance it with the residual discretization energy in order to ensure fastest convergence to the incompressible solution with best conditioned stiffness matrix to minimize round-off errors in the computations.

69 citations


Journal ArticleDOI
TL;DR: In this article, a mixed finite-difference scheme is presented for the free vibration analysis of simply supported laminated orthotropic circular cylinders, based on the linear three-dimensional theory of orthotropic elasticity, and the governing equations are reduced to six firstorder ordinary differential equations in the thickness coordinate.

52 citations


Journal ArticleDOI
TL;DR: In this article, a computational procedure for the large deformation dynamic response of solids is presented and the underlying mechanics, the constitutive theories of interest, the spatial discretization, and the time integration scheme are each discussed.

49 citations


Book ChapterDOI
TL;DR: The initial-boundary value problem for a linear parabolic equation with the Dirichlet boundary condition is solved approximately by applying the finite element discretized in the space dimension and three types of finite-difference discretizations in time: the backward, the Crank-Nicolson and the Calahan discretization.
Abstract: The initial-boundary value problem for a linear parabolic equation with the Dirichlet boundary condition is solved approximately by applying the finite element discreti- zation in the space dimension and three types of finite-difference discretizations in time: the backward, the Crank-Nicolson and the Calahan discretization. New error bounds are derived. 1. Introduction. A number of years ago, engineers applied the finite element method to the solution of the heat conduction problem. We mention the papers by Visser (7) and by Wilson and Nickell (8). Their idea is that in the space dimension a finite element discretization is used whereas in time a finite-difference method is applied. Recently, there appeared papers in mathematical journals where these methods were analyzed as well as new methods proposed, some of them of higher order of accuracy, and where error bounds of a different kind were derived. We mention the papers by Douglas and Dupont (3), Hlavacek (5) and Bramble and

46 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of utilizing experimental data to characterize the stress constitutive function for a nonlinear elastic solid is formulated as an inverse boundary value problem and the use of finite element discretization is extended by introducing a technique of material parameterization.

39 citations


01 Jan 1974
TL;DR: In this article, a method for the solution of the electromagnetic field inside resonant cavities and waveguides of arbitrary shape, whether homogeneously or inhomogeneously filled, is described.
Abstract: AZrstract-A method is described for the solution of the electromagnetic field inside resonant cavities and waveguides of arbitrary shape, whether homogeneously or inhomogeneously filled. The method, suitably programmed for use with a digital computer, is based on the direct discxetization of the Maxwell equations in integral form. Since the method works with the components of the electromagnetic field, the numerical solution directly gives the distributions of the field in the structure, in addition to the resonant frequencies of cavities or the propagation constants of wavegnides. Some numerical applications of the method are given.

37 citations


Journal ArticleDOI
TL;DR: In this paper, an application of isoparametric elements for the elastic-plastic dynamic analysis of shells of revolution is presented, which can analyze axisymmetric structures subjected to axially symmetric loading as well as plane stress problems.

Journal ArticleDOI
TL;DR: In this paper, a method is described for the solution of the electromagnetic field inside resonant cavities and waveguides of arbitrary shape, whether homogeneously or inhomogeneously filled.
Abstract: A method is described for the solution of the electromagnetic field inside resonant cavities and waveguides of arbitrary shape, whether homogeneously or inhomogeneously filled The method, suitably programmed for use with a digital computer, is based on the direct discretization of the Maxwell equations in integral form Since the method works with the components of the electromagnetic field, the numerical solution directly gives the distributions of the field in the structure, in addition to the resonant frequencies of cavities or the propagation constants of waveguides Some numerical applications of the method are given

Journal ArticleDOI
TL;DR: In this paper, a polynomial expansion of the displacement model is proposed to improve the stiffness of the model, e.g. by direct approximation of the strain field as opposed to its derivation from the displacement field.

Book ChapterDOI
01 Jan 1974
TL;DR: In this article, the relative efficiency of three types of spatial differencing used in setting up differential-difference equations for the 1-periodic parabolic problem ut = uxx was evaluated.
Abstract: 0. Summary. Three subjects are considered in this paper. First, the notion of the resolving power of approximation methods, i.e. the number of intervals (or function values) per wavelength necessary to attain a preassigned error when approximating a given frequency, permits the evaluation of the relative efficiency of three types of spatial differencing used in setting up differential-difference equations for the 1-periodic parabolic problem ut = uxx. The three types of high-order spatial schemes considered are (explicit) centered differencing, the super-convergent smooth spline-Galerkin schemes discovered by Thomee, and the very high-order implicit schemes (Mehrstellenverfahren) which generalize Numerov's method. Seven time discretizations are introduced, namely Euler, backward-Euler, duFort-Frankel, 2nd order explicit, trapezoidal, Calahan-Zlamal, and 4th order Pade. The computational work necessary to solve each full discretization is minimized, for given error requirements, by balancing the number of intervals per wavelength against the number of time intervals per eth-life. The resulting data is used to compare the relative efficiency of these finite element and finite difference methods. Secondly, some corresponding results for the hyperbolic problem ut = ux are briefly reviewed. Finally, as Numerov-trapezoidal differencing turns out to be almost optimal for the heat equation, a tridiagonal implicit difference formula which extends Numerov differencing to the general 2nd order linear differential operator in one space dimension is presented. The technique used in deriving this scheme inspires certain difference analogs of some finite element schemes. It also leads to a curious modification of the diamond difference scheme for ut = ux. This “alternating kite” scheme has O(Δ2 + Δx3) truncation error with no more computational work than the diamond scheme itself.

Journal ArticleDOI
TL;DR: In this paper, the transient heat conduction problem can be solved by application of Galerkin's method to space as well as time discretization, which corresponds to the procedure known as finite elements in time and space.

Journal ArticleDOI
TL;DR: In this article, a program for discretization in the time dimension of a parabolic time element is described and the coefficients required to form the global system are given, and the efficiency of the process is examined by comparison with the customary difference method.
Abstract: A program is demonstrated which apart from linear finite elements in time also includes elements with shape functions of the second and third degree. The algorithm for discretization in the time dimension is described and, using the example of a parabolic time element, the coefficients required to form the global system are given. By various test examples the efficiency of the process is examined by comparison with the customary difference method. Generally, with finite elements in time, the solution has better stability. Comparing the time required for calculation with the accuracy of the solution it would appear that in examining problems where boundary conditions are constant in time, higher order time elements are no improvement over the linear time element. However, for the purpose of reproducing periodic processes, higher order time elements offer an advantage in that one is not limited to linear variations of the boundary conditions within the element. Thus, for example, the temperature curve for parabolic variation of the surface temperature can be reproduced with close approximation by two time elements per period and a shape function of the third degree.

Journal ArticleDOI
TL;DR: In this paper, the optimal identification of aquifer parameters in a distributed system is formulated as an optimal control problem and the Lagrange multipliers are applied to obtain the set of necessary conditions that is optimal.
Abstract: Optimal identification of aquifer parameters in a distributed system is formulated as an optimal control problem. The dynamics of the head is governed by a second-order nonlinear partial differential equation. The numerical example presented considers that the parameters to be identified are functions of the space variable. Observations on head variations are available at several observation wells distributed within the system. Spatial discretization is first used to transform the distributed system to a lumped system. The least squares criterion function is then established. After introducing the Lagrange multipliers, the maximum principle is applied to obtain the set of necessary conditions that is optimal. These conditions are expressed in terms of a set of canonic equations of two-point boundary value type that is easily solved by the technique of quasi-linearization. Thus aquifer parameters are directly identified on the basis of observational data taken at observation stations. The maximum principle formulation is inherently more accurate and stable, since it minimizes the least squares error over the whole time and space domains. Computationally, it is extremely efficient. The numerical example presented demonstrates simultaneous identification of 11 parameters defined at discretized points along the space variable. Quadratic convergence is also demonstrated by numerical experimentation.

Journal ArticleDOI
TL;DR: In this article, a numerical algorithm for solving stiff boundary value problems with turning points is presented, where the stiff systems are characterized as singularly perturbed differential equations and the numerical method is derived by appropriately discretizing the boundary layer and connection theory for such systems.
Abstract: A numerical algorithm for solving stiff boundary value problems with turning points is presented. The stiff systems are characterized as singularly perturbed differential equations. The numerical method is derived by appropriately discretizing the boundary layer and connection theory for such systems. Numerical results demonstrate the effectiveness of the method. In many cases the calculation proceeds with mesh increments which are orders of magnitude larger than those used by other known methods.

Journal ArticleDOI
TL;DR: In this paper, basic concepts of asymptotic theory for discretization methods have been difined so that convergence is always equivalent to consistency and stanility, and the theory is then applied to a class of discretisation methods for the initial value problems of ordinary differential equations, the weakly nonlinear methods.
Abstract: Basic concepts of asymptotic theory for discretization methods have been difined so that convergence is always equivalent to consistency and stanility. The theory is then applied to a class of discretization methods for the initial value problems of ordinary differential equations, the so-called weakly nonlinear methods.

Journal ArticleDOI
TL;DR: A corrector method applicable to linear and nonlinear problems—the $(2 \times 2)$ interval discretization method—is presented, which is a slight modification of Hansen’s method and enables us to improve previously given results.
Abstract: Two error bounding methods for two-point boundary value problems are presented. Using the simple factorization method and interval analysis we find the numerical solution of two-point boundary value problems of linear type with complete error bounds. The numerical solutionis given as an interval which always contains the exact solution.In the second part of this paper we present a corrector method applicable to linear and nonlinear problems—the $(2 \times 2)$ interval discretization method—which is a slight modification of Hansen’s method [3] and enables us to improve previously given results.

Journal ArticleDOI
TL;DR: DYPLAS is a computer program designed to compute the response history of general three-dimensional structures subjected to transient thermal and mechanical loadings, and considers nonlinearities arising from material behavior as well as from large deformations that may occur in the structure.

Journal ArticleDOI
TL;DR: In this article, discrete viscous lattices are studied in order to simulate the propagation of general disturbances in semi-infinite, elastic and viscoelastic periodic composites, and exact and approximate solutions for the cases in which viscosity is introduced to the elastic lattice by adding dash pots in both series and parallel with the springs connecting adjacent particles are derived.

Journal ArticleDOI
TL;DR: The conjugate gradient method is adapted to the optimization of linear, distributed-parameter systems and measures for the suboptimality of the numerical solution are established.
Abstract: The conjugate gradient method is adapted to the optimization of linear, distributed-parameter systems. The effects of errors introduced by the discretization process are analyzed. Measures for the suboptimality of the numerical solution are established. In order to demonstrate the concepts involved, four specific examples are considered.


01 Sep 1974
TL;DR: The importance of consistency of coding with physical properties is emphasized in this article, where examples are given of gas dynamical problems treated numerically, where apparently minor changes to the coding eliminate all numerical oscillations and instabilities.
Abstract: : Examples are given of gas dynamical problems treated numerically, where apparently minor changes to the coding eliminate all numerical oscillations and instabilities. In each case, the changes are justified by physical considerations and the importance of consistency of coding with physical properties is emphasized.

Journal ArticleDOI
TL;DR: These canonical forms correspond to the system properties of Controllability, Observability, Reachability, and Constructability and are applicable to the case of continuous systems in which time has been discretized and to the cases of linear sequential machines.
Abstract: Several canonical forms are developed for linear, constant, single-input, single-output systems. These canonical forms correspond to the system properties of Controllability, Observability, Reachability, and Constructability and are applicable to the case of continuous systems in which time has been discretized and to the case of linear sequential machines.

Journal ArticleDOI
TL;DR: Two conceptually important and computationally useful aspects of the integral‐transform (IT) approach have been tested on the He sequence and a judicious blend of mapping and iteration, coupled with a priori discretization and subsequent extrapolation seems to be useful.
Abstract: Two conceptually important and computationally useful aspects of the integral‐transform (IT) approach have been tested on the He sequence. Both generalized scaling and the iteration approach enable one to optimize the shape function; a priori discretization of the IT trial functions by M‐point integration rules is computationally expedient; coupled with extrapolation procedures it can alleviate the integration difficulties inherent in the IT approach. By optimizing a mapping (generalized scaling) the initial choice of a shape function becomes less crucial; for discretized IT functions it results in the reduction of the number of simultaneous parameter optimizations. The iteration approach transforms the nonlinear parameter optimization into a multistage process, with its attendant advantages. It also minimizes the importance of the initial choice of the cubature rule (discretization). A judicious blend of mapping and iteration, coupled with a priori discretization and subsequent extrapolation seems to be ...

Journal ArticleDOI
TL;DR: In this article, the convergence rates of an approximate solution method for many conforming finite element models are determined by theoretical analyses and checked by numerical investigation. But the authors consider only discretization errors; and the solution quantities referred to are for the system energy quantities which correspond to the eigenvalues for eigenvalue problems.

01 Oct 1974
TL;DR: This program documents the theoretical basis for the computer program CEL/NONSAP, a nonlinear structural analysis program that predicts either static or dynamic response of structural systems with material and/or geometric nonlinearities.
Abstract: : This program documents the theoretical basis for the computer program CEL/NONSAP. NONSAP is a nonlinear structural analysis program that predicts either static or dynamic response of structural systems with material and/or geometric nonlinearities. The finite element procedure is used for spatial discretization of the incremental equations of motion, and a step-by-step technique is used to compute the time dependent response. The algorithm used to solve the discrete set of nonlinear equations is a modified Newton iteration. Derivations of the finite element matrices are given for those elements currently operational in the NONSAP program element library.

Book ChapterDOI
01 Jan 1974
TL;DR: Different kinds of approximation, especially chained approximation, are illustrated on linear and nonlinear differential equations, and possibilities of getting exact error bounds are considered.
Abstract: Different kinds of approximation, especially chained approximation, are illustrated on linear and nonlinear differential equations. Particularly possibilities of getting exact error bounds are considered.