Showing papers on "Numerical analysis published in 1969"
TL;DR: In this paper, Cook et al. gave an algorithm which computes the coefficients of the product of two square matrices A and B of order n with less than 4. 7 n l°g 7 arithmetical operations (all logarithms in this paper are for base 2).
Abstract: t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmetical operations (all logarithms in this paper are for base 2, thus tog 7 ~ 2.8; the usual method requires approximately 2n 3 arithmetical operations). The algorithm induces algorithms for invert ing a matr ix of order n, solving a system of n linear equations in n unknowns, comput ing a determinant of order n etc. all requiring less than const n l°g 7 arithmetical operations. This fact should be compared with the result of KLYUYEV and KOKOVKINSHCHERBAK [1 ] tha t Gaussian elimination for solving a system of l inearequations is optimal if one restricts oneself to operations upon rows and columns as a whole. We also note tha t WlNOGRAD [21 modifies the usual algorithms for matr ix multiplication and inversion and for solving systems of linear equations, trading roughly half of the multiplications for additions and subtractions. I t is a pleasure to thank D. BRILLINGER for inspiring discussions about the present subject and ST. COOK and B. PARLETT for encouraging me to write this paper. 2. We define algorithms e~, ~ which mult iply matrices of order m2 ~, by induction on k: ~ , 0 is the usual algorithm, for matr ix multiplication (requiring m a multiplications and m 2 ( m t) additions), e~,k already being known, define ~ , ~ +t as follows: If A, B are matrices of order m 2 k ~ to be multiplied, write
2,581 citations
TL;DR: In this paper, a numerical method for the dynamic analysis of infinite continuous systems is developed, applicable to systems for which all exciting forces and geometrical irregularities are confined to a limited region and is applicable to both transient and steady state problems.
Abstract: A numerical method for the dynamic analysis of infinite continuous systems is developed. The method is applicable to systems for which all exciting forces and geometrical irregularities are confined to a limited region and is applicable to both transient and steady state problems. The infinite system is replaced by a system consisting of a finite region subjected to a boundary condition which simulates an energy absorbing boundary. The resulting systems may be analyzed by the finite element method. Examples applying the method to foundation vibration problems are presented. Good agreement with existing solutions is found and new results for embedded footings are presented.
2,172 citations
TL;DR: In this paper, a numerical method for time dependent compressible Navier-Stokes equations applied to axisymmetric flow field produced by hypervelocity impact, examining viscous effects is presented.
Abstract: Numerical method for time dependent compressible Navier-Stokes equations applied to axisymmetric flow field produced by hypervelocity impact, examining viscous effects
1,156 citations
TL;DR: In this paper, a numerical solution capability is developed for the solution of problems in three dimensional elastostatics, which utilizes singular integral equations which can be solved numerically for the unknown surface tractions and displacements for the fully mixed boundary value problem.
521 citations
204 citations
TL;DR: In this paper, a numerical iterative method of computing the actual charge distribution around the conductor in the corona is presented, which is applicable to any general configuration for which the space-charge-free field can be calculated.
Abstract: Theoretical calculation of corona losses for practical unipolar dc transmission line configurations presents considerable difficulty because of the nonlinear nature of the equations describing the space-charge fields. The application of numerical methods to obtain solutions of practical interest is discussed. One of the difficulties in the analysis of space-charge fields is the determination of the actual charge distribution around the conductor in the corona. A numerical iterative method of computing this charge distribution is presented. The method is applicable to any general configuration for which the space-charge-free field can be calculated. The line- to-plane configuration is considered. A method of including the effect of conductor surface irregularities in the theoretical calculation of corona losses is outlined, and it is suggested that, by the same method, Popkov's formula may also be modified to make it applicable to lines with practical transmission-line conductors. Calculations by the method of analysis developed as well as by the modified Popkov equation are compared with experimental results.
192 citations
TL;DR: In this paper, an optimal feedback control for a linear time-varying system with time delay is presented, where the performance criterion is quadratic with a fixed, finite upper limit and results in a set of differential equations with boundary conditions.
Abstract: A method is presented whereby an optimal control may be obtained for a linear time-varying system with time delay. The performance criterion is quadratic with a fixed, finite upper limit, and results in a set of differential equations with boundary conditions whose solution yields an optimal feedback control. A numerical technique is developed for the solution of the differential equations, and two examples are worked.
176 citations
TL;DR: In this paper, a numerical method for computing response spectra from strong-motion earthquake records is developed, based on the exact solution to the governing differential equation, which gives a three to fourfold saving in computing time compared to a third order Runge-Kutta method of comparable accuracy.
Abstract: A numerical method for computing response spectra from strong-motion earthquake records is developed, based on the exact solution to the governing differential equation. The method gives a three to four-fold saving in computing time compared to a third order Runge-Kutta method of comparable accuracy. An analysis also is made of the errors introduced at various stages in the calculation of spectra so that allowable errors can be prescribed for the numerical integration. Using the proposed method of computing or other methods of comparable accuracy, example calculations show that the errors introduced by the numerical procedures are much less than the errors inherent in the digitization of the acceleration record.
174 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 non-iterative integral solution for collision problems utilizing an integral equation formalism is presented. The approach is noniterative and applies equally well to purely local interactions or problems with combined local and non-local interactions.
Abstract: A new method (homogeneous integral solution) for solving collision problems utilizing an integral equation formalism is presented. The approach is noniterative and applies equally well to purely local interactions or problems with combined local and nonlocal interactions. The method involves transforming the integral equation of scattering into a Volterra integral equation of the second kind. It follows that even for nonlocal interactions, the wavefunction may be determined noniteratively from knowledge of its value at a single point (the origin). A simple numerical procedure is proposed for solution of the Volterra equation which allows one to avoid matrix inversions completely. At the end of the calculation, the scattering T or R matrix is obtained directly from the solutions. The method is illustrated by a calculation of the singlet and triplet s‐wave Hartree–Fock phase shifts for electron–H‐atom scattering.
TL;DR: In this paper, an integral equation on the area of the crack for the relative displacement across the crack is presented. But the kernel of this integral equation is non-integrable and a method for discretizing it and a numerical method of solution is carried out.
Abstract: An infinite elastic medium is initially at rest in a prestressed state of plane- or anti-plane strain. At time t = 0 a plane crack comes into existence which occupies a strip parallel to the y axis and whose width varies in time. Assuming that the components of the traction are known on the crack surface it is possible to set up an integral equation on the area of the crack for the relative displacement across the crack. Although the kernel of this integral equation is non-integrable a method is found for discretizing it and a numerical method of solution is carried out. The results, which in some cases are the solutions of diffraction problems, are presented graphically.
TL;DR: In this paper, a numerical method is developed for solving the governing nonlinear differential equations of this problem, and two types of loading conditions are included-an impulsive load and an instantaneous ly applied step load with infinite duration.
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.
TL;DR: In this article, a constructive method is presented for the calculation of the Taylor vortices as branching solutions of the nonlinear Navier-Stokes boundary value problem, based upon an abstract version of the theory of Schmidt-Lyapunov.
Abstract: : A constructive method is presented for the calculation of the Taylor vortices as branching solutions of the nonlinear Navier-Stokes boundary-value problem. The method is based upon an abstract version of the theory of Schmidt-Lyapunov. The efficiency of the procedure is tested by comparison of numerical results with various experimental data for the mean torque and the higher harmonics. (Author)
TL;DR: In this article, a modified version of the finite element method is used to solve a series of real world problems associated with GEOLOGICAL STRUCTURES, such as the formation of mullions.
Abstract: A NUMERICAL APPROACH TO THE STUDY OF FINITE, QUASI-STATIC PLANE DEFORMATIONS OF VISCOUS SOLIDS IS INTRODUCED. THE PROBLEM IS FORMULATED IN TERMS OF A SERIES OF INCREMENTAL PROBLEMS. A MODIFIED VERSION OF THE FINITE ELEMENT METHOD IS EMPLOYED IN THE SOLUTION OF THIS SERIES OF PROBLEMS. IT IS HOPED THAT THIS APPROACH MAY PROVE CONVENIENT FOR THE STUDY OF CERTAIN PROBLEMS ASSOCIATED WITH GEOLOGICAL STRUCTURES. THE EXAMPLE OF THE FINITE COMPRESSION OF A BODY WITH AN INITIALLY IMPERFECT FREE SURFACE MIGHT SHED SOME LIGHT ON THE FORMATION OF MULLIONS. /AUTHOR/
TL;DR: A method of computation that uses discrete space techniques depending on concepts of invariance to solve realistic radiative transfer problems with arbitrary internal, external source distributions and scattering diagrams.
Abstract: The classical methods that have been devised to analyze theoretically the transfer of radiation in plane parallel atmospheres may produce an analytic solution provided that the medium is assumed to be homogeneous. Even then, when the results have been expressed in terms of tabulated functions, practical computations are difficult and tedious. It is therefore essential to employ numerical methods for solving realistic radiative transfer problems. We briefly describe a method of computation that uses discrete space techniques depending on concepts of invariance. The solution algorithms compute internal and external light fields for inhomogeneous plane parallel atmospheres with arbitrary internal, external source distributions and scattering diagrams. The stability and errors of our algorithms are susceptible to mathematical analysis and make it possible to identify the critical parameters in the calculation with precision. To illustrate our techniques, we briefly discuss the practical problem of ma...
TL;DR: Centrally clamped spinning circular disk free transverse vibration analysis within accuracy of numerical computations as discussed by the authors was used to analyze the transverse vibrations of the transversal vibration.
Abstract: Centrally clamped spinning circular disk free transverse vibration analysis within accuracy of numerical computations
TL;DR: This paper proposes to construct difference schemes for multi-dimensional quasilinear equations of a rather general form, using the lines of intersection of coordinate and characteristic surfaces and a fixed uniform network, and indicates the possibility of applying Scheme I to the calculation of discontinuous solutions.
Abstract: THERE have recently been intensive developments in various numerical methods for solving multi-dimensional problems for partial differential equations [1–9]. In particular, schemes using the characteristic equations of gas dynamics have found wide application [4–8]. Numerical methods, using the characteristics of hyperbolic equations, have definite advantages over ordinary finite-difference methods, in particular, in taking into account the physical nature of the problem, weak variation along the characteristics of some complexes of the required functions, and the possibility of predicting the instant when discontinuities will occur, for instance, suspended jumps. These advantages stand out very sharply in the solution of problems with two independent variables. But when the number of independent variables and the amount of information to be processed is increased a fixed choice of nodes becomes desirable, as in the network method. In this respect the numerical schemes of [6–8] for analyzing supersonic space flows of gas, for which a fixed network and the use of some form of characteristic relations are usual, are most effective. Obviously the name “method of characteristics” must be retained only for numerical schemes in which characteristics curves or surfaces are used in the process of computation. Then, numerical schemes, in which characteristic reations (conditions of compatibility) are used only for the derivation of difference equations at fixed nodes, can be considered as variants of the network method. This is even more advisable because for schemes of such a kind the problem of stability arises even in the case where the Courant-Friedrich-Levy (CFL) condition is satisfied, i.e. the region of dependence for the differential equation lies inside that of the difference equations [10–12]. In multi-dimensional cases there is a wide range in the choice of numerical methods based on characteristic relations, but obviously it is reasonable to have some fairly general and simple method of constructing explicit difference schemes for multi-dimensional problems based on characteristics of the same type as that put forward in [13] for a hyperbolic system in two variables (this method is described in [1, 14]). In the present paper it is proposed to construct such difference schemes for multi-dimensional quasilinear equations of a rather general form, using the lines of intersection of coordinate and characteristic surfaces and a fixed uniform network. To find the required functions at the points of contact of these lines, drawn from the points considered, with the preceding time layer, where the solution is already known, we can use linear or quadratic interpolation. Schemes I and II (Section 1) thus obtained, and also their modifications are investigated by an example of very simple equations with constant coefficients in Section 2. Difference schemes I and II obtained in Section 3 for the equations of gas dynamics are carried out numerically for a problem on the supersonic space flow round bodies (Section 4). The results given there indicate the possibility of applying Scheme I to the calculation of discontinuous solutions.
TL;DR: It is shown that the Riccati equation can be partitioned and under certain conditions the unstable part deleted.
Abstract: Some methods of handling command inputs via the linear optimal control theory are compared, and the convergence of the associated Riccati matrix differential equation is investigated.
TL;DR: In this article, an integral equation method for computing the conformal mapping of a finite doubly-connected domain onto R <|w|<1, where R is uniquely determined, is presented.
Abstract: This paper describes an integral equation method for computing the conformal mapping of a finite doubly-connected domain ontoR <|w|<1, whereR is uniquely determined. The method is illustrated by numerical examples.
TL;DR: In this article, a numerical method for solving a fundamental nonlinear problem in fluid dynamics is presented, which can be verified easily by direct substitution into the difference equations, and why the method converges is a difficult matter.
Abstract: For many years one of the major criticisms of mathematics was that it provided no methods for solving nonlinear problems. With the development of the high speed computer and related numerical methods, this criticism is now being answered. In this paper we present a numerical method for solving a fundamental nonlinear problem in fluid dynamics. That the numbers presented actually do represent a numerical solution can be verified easily by direct substitution into the difference equations. Why the method converges is a difficult matter and is at present under study.
TL;DR: In this article, a combined analytical-numerical method which involves no linearizations in the hydrodynamical equations and applies to all but surface-tension dominated motions is used to compute a variety of such motions.
Abstract: This paper considers the large-amplitude symmetric and asymmetric irrota-tional motion of an inviscid incompressible fluid with a liquid—vapour interface in an accelerating container of revolution. A combined analytical—numerical method which involves no linearizations in the hydrodynamical equations and applies to all but surface-tension dominated motions is used to compute a variety of such motions. One important aspect of this non-linear method is that it accurately determines the initial development of surface instabilities such as breakers near the wall of the container.
TL;DR: In this paper, a completely numerical method for steady state linear viscoelastic stress analysis is presented by means of the finite element approach, which is developed to obtain steady state solutions to mixed boundary value problems in which the character of the boundary conditions at a point changes with time.
Abstract: A completely numerical method for steady state linear viscoelastic stress analysis is presented by means of the finite element approach. Numerical representations of the measured viscoelastic constitutive relations are used. This method is developed to obtain steady state solutions to mixed boundary value problems in which the character of the boundary conditions at a point changes with time. Such problems cannot be handled by direct application of the correspondence theorem. A numerical example of viscoelastic sheet rolling is presented along with an experimental verification of the solution by photo-viscoelastic observations.
TL;DR: In this paper, a simple approximate method for calculating the early time history of the magnetic field and coil motion is shown to give reasonable agreement with experiment, and experimental and theoretical evidence is presented for the emission of a vapor cloud by the wall into the magnetic volume.
Abstract: Experiments on the production of high magnetic fields in single‐turn coils by means of high‐voltage capacitor banks are described. Fields as high as 3.5 megagauss have been produced. Numerical analysis of the interaction of this field with the metal wall shows that magnetic diffusion and wall compression are the principal interaction phenomena. In addition, experimental and theoretical evidence are presented for the emission of a vapor cloud by the wall into the magnetic‐field volume. Finally, a simple approximate method for calculating the early time history of the magnetic field and coil motion is shown to give reasonable agreement with experiment.
TL;DR: The approach given in this paper leads to numerical methods for Volterra integral equations which avoid the need for special starting procedures as discussed by the authors, which can be seen as a generalization of the approach described in the present paper.
Abstract: The approach given in this paper leads to numerical methods for Volterra integral equations which avoid the need for special starting procedures. Formulae for a typical fourth-order method are derived and some numerical examples presented. A convergence theorem is given for the method described. 1. Introduction. In this paper we consider the numerical solution of the equation
01 Jul 1969
TL;DR: Load distribution calculations for subsonic compressible flow of arbitrary wings and lifting surfaces were performed in this paper for the case of a single wing and a single lifting surface with arbitrary lifting surface.
Abstract: Load distribution calculations for subsonic compressible flow of arbitrary wings and lifting surfaces