Showing papers on "Numerical analysis published in 1974"
TL;DR: In this paper, a numerical method for the computation of electrostatic fields is described, based on the use of fictitious line charges as particular solutions of Laplace's and Poisson's equations.
Abstract: A numerical method for the computation of electrostatic fields is described. The basis of the method is the use of fictitious line charges as particular solutions of Laplace's and Poisson's equations. Details are given of a digital computer program developed for field calculations by means of this method, and its application is illustrated by practical examples involving two-and three-dimensional geometries.
652 citations
TL;DR: In this paper, the modified (underrelaxed, damped) Newton method is extended in such a way as to apply to the solution of ill-conditioned systems of nonlinear equations, i.e. systems having a "nearly singular" Jacobian at some iterate.
Abstract: In this paper the well-known modified (underrelaxed, damped) Newton method is extended in such a way as to apply to the solution of ill-conditioned systems of nonlinear equations, i.e. systems having a "nearly singular" Jacobian at some iterate. A special technique also derived herein may be useful, if only bad initial guesses of the solution point are available. Difficulties that arose previously in the numerical solution of nonlinear two-point boundary value problems by multiple shooting techniques can be removed by means of the results presented below.
453 citations
TL;DR: The methods are intimately based on the recurrence of matrix factorizations and are linked to earlier work on quasi-Newton methods and quadratic programming.
Abstract: This paper describes two numerically stable methods for unconstrained optimization and their generalization when linear inequality constraints are added. The difference between the two methods is simply that one requires the Hessian matrix explicitly and the other does not. The methods are intimately based on the recurrence of matrix factorizations and are linked to earlier work on quasi-Newton methods and quadratic programming.
338 citations
01 Jan 1974
TL;DR: The basic principles of GRG are discussed, the logic of a computer program implementing this algorithm is presented, and a specific GRG algorithm is constructed by means of flow charts and discussion.
Abstract: : Generalized Reduced Gradient (GRG) methods are algorithms for solving nonlinear programs of general structure. This paper discusses the basic principles of GRG, and constructs a specific GRG algorithm. The logic of a computer program implementing this algorithm is presented by means of flow charts and discussion. A numerical example is given to illustrate the functioning of this program.
304 citations
TL;DR: In this paper, a numerical method for predicting three-dimensional, steady viscous flow in ducts is described, which utilizes approximate governing equations which are applicable to flows having strong convection in one primary flow direction.
173 citations
TL;DR: This paper extends techniques that have been used in piecewise polynomial approximation which permit the construction of equidistributing meshes to meshes on which the local truncation error of the method is approximately constant in some norm.
Abstract: In order to use finite difference approximations with non-uniform meshes in boundary value problems, it is necessary to develop procedures for mesh selection. In this paper we extend techniques that have been used in piecewise polynomial approximation which permit the construction of equidistributing meshes. By this term we mean meshes on which the local truncation error of the method is approximately constant in some norm. Improved error estimates for methods which use equidistributing meshes are obtained.
136 citations
120 citations
TL;DR: In this paper, the coupled integro-differential equations are reduced to linear algebraic equations and solved using a technique which has good numerical stability and which gives good linear independence in the solution vectors.
Abstract: For pt.I, see abstr. A7075 of 1973. The coupled integro-differential equations, described in Paper I, are reduced to linear algebraic equations. The finite difference formulae used are described in detail. The linear algebraic equations are solved using a technique which has good numerical stability and which gives good linear independence in the solution vectors. The method is economic in the use of computer time and storage.
108 citations
TL;DR: The finite element has been used to develop two numerical methods of calculating the flow characteristics of rigid networks of planar fractures as discussed by the authors, one method uses triangular elements to investigate details of laminar flow in fractures of irregular cross section combined with that of a permeable rock matrix.
Abstract: The finite element has been used to develop two numerical methods of calculating the flow characteristics of rigid networks of planar fractures. One method uses triangular elements to investigate details of laminar flow in fractures of irregular cross section combined with that of a permeable rock matrix. The other method uses line elements and is designed only for flow in networks of planar fractures in an impermeable matrix. As an example of the application of the finite element approach the line element method was used to develop a series of dimensionless graphs that characterize seepage in idealized fracture systems beneath dams. These methods treat two-dimensional flow in the laminar regime for networks of fractures of arbitrary orientation and aperture distribution.
TL;DR: In this article, a new numerical method for use in the solution of classical equations of motion is described, accurate to third order in the coordinates and second-order in the velocities, which has the unique property of preserving the energy and total linear and angular momenta at their initial values in the computation.
TL;DR: In this paper, a numerical perturbation method is proposed for the determination of the nonlinear forced response of structural elements when modal interactions take place due to the complicated nature of the response.
Abstract: A numerical-perturbation method is proposed for the determination of the nonlinear forced response of structural elements. Purely analytical techniques are capable of determining the response of structural elements having simple geometries and simple variations in thickness and properties, but they are not applicable to elements with complicated structure and boundaries. Numerical techniques are effective in determining the linear response of complicated structures, but they are not optimal for determining the nonlinear response of even simple elements when modal interactions take place due to the complicated nature of the response. Therefore, the optimum is a combined numerical and perturbation technique. The present technique is applied to beams with varying cross sections. ~ 4Y large-amplitude deflection of a beam or a plate which is restrained at its ends or along its edges results in some midplane stretching/One must account for this stretching with nonlinear strain-displacement relationships. The nonlinear equations of motion describing this situation were the basis of a number of earlier investigations and are the basis for the present paper as well. The purpose of the present paper is to present a new scheme for determining the response to a harmonic excitation. Emphasis is placed on the case when the frequency of the excitation is near a natural frequency. A convenient way to attack this nonlinear problem involves representing the deflection curve or surface with an expansion in terms of the linear, free-oscillation modes. The deflection is then determined in two steps. First, the damping, the forcing, and the nonlinear terms are deleted and the linear modes (eigenfunctions) and natural frequencies (eigenvalues) are determined. Second, the time-dependent coefficients in the expansion are obtained from a set of coupled, nonlinear, ordinary, second-order differential equations, the linear modes being used to determine the coefficients in these equations. (The procedure is described in detail in Sec. II.) Generally, one cannot obtain the linear modes analytically for structural elements having complicated boundaries and composition, and one cannot easily determine the character of the timedependent coefficients through numerical integration of the set of nonlinear equations. (The results obtained in the present numerical example are typical of the complicated manner in which the steady-state amplitudes of the various modes making up the response can vary with the amplitude and the frequency of the excitation.) Consequently, an optimal procedure involves a numerical method to determine the linear, free-oscillation modes and an analytical method to determine the time-dependent coefficients. The present procedure combines either a finiteelement or a finite-difference method with the method of multiple scales (see, for example, Ref. 1). The following brief review mentions representative examples of the work that was and is
TL;DR: In this paper, the orthogonal collocation method is used to obtain approximate solutions to the differential equations modeling chemical reactors, which is very often useful in engineering work, where valid approximations are accepted.
Abstract: The orthogonal collocation method is used to obtain approximate solutions to the differential equations modeling chemical reactors. There are two ways to view applications of the orthogonal collocation method. In one view it is a numerical method for which the convergence to the exact answer can be seen as the approximation is refined in successive calculations by using more collocation points, which are similar to grid points in a finite difference method. Another viewpoint considers only the first approximation, which can often be found analytically, and which gives valuable insight to the qualitative behavior of the solution. The answers, however, are of uncertain accuracy, so that the calculation must be refined to obtain useful numbers. However, with experience and appropriate caution, the first approximation is often sufficient and is easy to obtain. Thus it is very often useful in engineering work, where valid approximations are accepted. We present both viewpoints here: we use the first a...
TL;DR: In this article, a variational method is used to determine the propagation characteristics of an optical fiber consisting of a core with an arbitrary refractive-index distribution and a uniform cladding, and the obtained characteristic equations for the simplest case (uniform core case) are compared with analytic solutions to ensure the validity of the analysis.
Abstract: The variational method is used to determine the propagation characteristics of an optical fiber consisting of a core with an arbitrary refractive-index distribution and a uniform cladding. The problem is first translated into a variational problem; the factional is computed upon the TEM-approximation basis. The variational problem is then solved by using the Rayleigh-Ritz method. The computed propagation characteristics are presented for refractive-index distributions of practical interest. The single-mode condition for a quadratic self-focusing fiber is obtained as v < 3.53 where v denotes the conventionally used normalized frequency; this result agrees with the numerical analysis by Dil and Blok. The obtained characteristic equations for the simplest case (uniform core case) are compared with analytic solutions to ensure the validity of the analysis.
Dissertation•
01 Jan 1974
TL;DR: The equivalent source concept of Green's theory affords a generalized manner of formulating static and time-varying electromagnetic problems; material property inhomogeneities are replaced by equivalent source distributions which satisfy a Fredholm integral equation of the second kind as mentioned in this paper.
Abstract: The electromagnetic response of magnetically and electrically inhomogeneous media whose geometries are not amenable to conventional partial differential equation analysis are most readily analysed in an integral equation framework. The equivalent source concept of Green's theory affords a generalized manner of formulating static and time-varying electromagnetic problems; material property inhomogeneities are replaced by equivalent source distributions which satisfy a Fredholm integral equation of the second kind. -- For static field problems, the equivalent source method represents conductivity, permittivity and permeability variations in terms of current source, charge and "magnetic pole" density distributions. In this form, the problems have analogous mathematical forms and the equivalent source satisfies a scalar Fredholm equation. The formalism is readily related to the static field methods used in applied geophysics. -- The time-varying equivalent source formulation represents material property variations in terms of electric and magnetic current densities which satisfy a pair of coupled vector Fredholm equations. Analysis of the integral operators shows that the scattering operator is bimodal for many geophysical problems. This result leads to the analysis of scattering problems in terms of generalized eigenfunctions. The bimodal nature of the scattering operator often leads to highly ill-conditioned matrices when numerical methods are applied to geophysical problems. -- Approximate parametric solution methods of solving the time-varying electric scattering problem are considered. Approximation of the solution by a general functional form and applying minimum criteria reduce the integral equations to matrix equations. The least squares method is applied analysing magnetotelluric responses of 2-dimensional structures and the Galerkin formalism is used to find the eigenfunctions for a thin plate in a whole space. The results are compared with other available numerical and experimental results and assessments of tho methods are given.
01 Jul 1974
TL;DR: In this paper, a finite difference method for the solution of the transonic flow about a harmonically oscillating wing is presented, where the flow is divided into steady and unsteady perturbation velocity potentials.
Abstract: A finite difference method for the solution of the transonic flow about a harmonically oscillating wing is presented. The partial differential equation for the unsteady transonic flow was linearized by dividing the flow into separate steady and unsteady perturbation velocity potentials and by assuming small amplitudes of harmonic oscillation. The resulting linear differential equation is of mixed type, being elliptic or hyperbolic whereever the steady flow equation is elliptic or hyperbolic. Central differences were used for all derivatives except at supersonic points where backward differencing was used for the streamwise direction. Detailed formulas and procedures are described in sufficient detail for programming on high speed computers. To test the method, the problem of the oscillating flap on a NACA 64A006 airfoil was programmed. The numerical procedure was found to be stable and convergent even in regions of local supersonic flow with shocks.
TL;DR: In this article, two numerical methods are used to evaluate the integrals that express the em fields due to dipole antennas radiating in the presence of a stratified medium, and they can be used for any arbitrary number of layers with general properties.
Abstract: Two numerical methods are used to evaluate the integrals that express the em fields due to dipole antennas radiating in the presence of a stratified medium. The first method is a direct integration by means of Simpson's rule. The second method is indirect and approximates the kernel of the integral by means of the fast Fourier transform. In contrast to previous analytical methods that applied only to two-layer cases the numerical methods can be used for any arbitrary number of layers with general properties.
TL;DR: In this article, the angular momentum reduction of the generalized BHF equations is performed and the iterative schemes and numerical methods employed in the solution of the resulting set of equations are outlined.
TL;DR: In this paper, it is shown that the problem of elasto-plastic finite deformation is governed by a quasi-linear model irrespective of deformation magnitude, which is based on the Galerkin method.
TL;DR: In this paper, an algorithm for solving a linear equation A x = c, where A is an ann×n Hankel or Toeplitz matrix, in O(n 2) arithmetic operations is described.
Abstract: An algorithm is described for solving a linear equationA x=c, whereA is ann×n Hankel or Toeplitz matrix, inO (n 2) arithmetic operations. As contrasted with earlier such algorithms the present one does not require the principal minors ofA to be non-zero except detA.
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.
01 May 1974
TL;DR: A general theory of contract impact problems cast in a variational theorem suitable for implementation with the finite element method is presented in this article, where the numerical scheme is described as is the structural analysis computer code in which it is contained.
Abstract: This report presents a general theory of contract-impact problems cast in a variational theorem suitable for implementation with the finite element method. The numerical scheme is described as is the structural analysis computer code in which it is contained.
TL;DR: In this paper, a numerical method based on the determination of the zero, first and second order moments of the concentration in each cell was shown, through various 2-dimensional numerical experiments, to suppress the spatial truncation error substantially.
Abstract: The advection of substances in a fluid computed with a finite grid resolution is known to smooth out concentration gradients. A numerical method based on the determination of the zero, first and second order moments of the concentration in each cell is shown, through various 2-dimensional numerical experiments, to suppress the spatial truncation error substantially. DOI: 10.1111/j.2153-3490.1974.tb01637.x
TL;DR: In this article, two of the numerical methods most widely used in solving the set of partial differential transport equations for holes, electrons, and electric field in semiconductor devices and the various numerical instability phenomena which can be encountered are described in detail.
Abstract: Two of the numerical methods most widely used in solving the set of partial differential transport equations for holes, electrons, and electric field in semiconductor devices and the various numerical instability phenomena which can be encountered are described in detail. Also presented are approaches, using these methods, to calculate dc static solutions and small-signal solutions, and to simulate devices in voltage-driven, current-driven, and circuit-loaded operation. Sample results are given for each mode of operation for the case of Si avalanche-diode oscillators. The numerical methods and approaches are those developed at our laboratory and sufficient detail is presented to permit the development of similar Fortran codes by others.
TL;DR: In this paper, a fast and numerically stable method for solving the discrete Dirichlet problem for Poisson's equation in case of a rectangle (and mainly, a square) is described.
Abstract: This paper describes a fast and numerically stable method for solving the discrete Dirichlet problem for Poisson's equation in case of a rectangle (and mainly, a square). By using a special calculus for difference operators, the system of linear equations is reduced to a block-triangular system such that the diagonal blocks are heavily diagonally dominant. For a standard version of the algorithm, the number of operations and the computing time are proportional toh ?2 (h=mesh width). The method is one oftotal reduction compared with the method ofblock-cyclic reduction (odd-even reduction) [2], which we describe as a method ofpartial reduction.--Due to the developed calculus, many generalizations are possible.--In a following part II of the paper, the algorithm and numerical results will be described in detail.
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