Showing papers on "Numerical analysis published in 1976"

Numerical methods in finite element analysis

Abstract: Numerical methods in finite element analysis , Numerical methods in finite element analysis , مرکز فناوری اطلاعات و اطلاع رسانی کشاورزی

Dynamics of an expanding circular fault

Abstract: We study a plane circular model of a frictional fault using numerical methods. The model is dynamic since we specify the effective stress at the fault. In one model we assume that the fault appears instantaneously in the medium; in another, that the rupture nucleates at the center and that rupture proceeds at constant subsonic velocity until it suddenly stops. The total source slip is larger at the center and the rise time is also longer at the center of the fault. The dynamic slip overshoots the static slip by 15 to 35 per cent. As a consequence, the stress drop is larger than the effective stress and the apparent stress is less than one half the effective stress. The far-field radiation is discussed in detail. We distinguish three spectral regions. First, the usual constant low-frequency level. Second, an intermediate region controlled by the fault size and, finally, the high-frequency asymptote. The central region includes the corner frequency and is quite complicated. The corner frequency is shown to be inversely proportional to the width of the far-field displacement pulse which, in turn, is related to the time lag between the stopping phases. The average corner frequency of S waves v s is related to the final source radius, a , by v s = 0.21 β/α . The corner frequency of P waves is larger than v s by an average factor of 1.5.

Numerical methods used in atmospheric models

Abstract: Methods used for the solution of hydrodynamic governing equations in numerical models of the atmosphere are discussed. In particular grid point finite difference methods and problems and methods used for time and horizontal space differencing are covered. Specific problems relating to the numerical solution of the advection and gravity wave equations are discussed. Volume 1

SLIC (Simple Line Interface Calculation)

Abstract: SLIC is an alternating-direction method for the geometric approximation of fluid interfaces. It may be used in one, two, or three space dimensions, and it is characterized by the following features: (1) Fluid surfaces are represented locally for each mixed- fluid zone, and these surfaces are defined as a composition of one space dimensional components, one for each coordinate direction. (2) These onedimensional components are composed entirely of straight lines, either perpendicular to or parallel to that coordinate direction. (3) The one-dimensional surface approximations for a mixed fluid cell are completely determined by testing whether or not the various fluids in the mixed cell are present or absent in the zone just to the left and to the right in the coordinate direction under consideration. (4) Because of the completely one-dimensional nature of the SLIC interface description, it is relatively easy to advance the fluid surfaces correctly in time. With the SLIC fluid-surface definitions, it should be possible to incorporate any one space dimensional method for advancing contact discontinuities. This makes SLIC very practical for the numerical solution of fluid dynamical problems.

Rupture propagation with finite stress in antiplane strain

Abstract: Rupture propagation in antiplane strain is investigated by using both analytic and numerical methods. Under the assumption that a solid will absorb energy irreversibly when it is strained at a sufficiently large shear stress it is found that energy must be absorbed at the rupture front in addition to the work done against the sliding friction stress. The energy absorbed increases with propagation distance, so it is not negligible at any length scale and is much larger than the ideal surface energy of molecular cohesion. The concept of a critical crack length carries over to the case of a finite stress-slip law on a fault plane but does not carry over to a homogeneous inelastic medium. In a dynamic slip event, while a typical value of particle velocity is proportional to stress drop, the peak value near the fault is proportional to material strength.

Dissipative two-four methods for time-dependent problems

Abstract: A generalization of the Lax-Wendroff method is presented. This generalization bears the same relationship to the two-step Richtmyer method as the KreissOliger scheme does to the leapfrog method. Variants based on the MacCormack method are considered as well as extensions to parabolic problems. Extensions to two dimensions are analyzed, and a proof is presented for the stability of a Thommentype algorithm. Numerical results show that the phase error is considerably reduced from that of second-order methods and is similar to that of the Kreiss-Oliger method. Furthermore, the (2, 4) dissipative scheme can handle shocks without the necessity for an artificial viscosity.

Fitting surfaces to scattered data

Abstract: A variety of numerical methods for fitting a function to data given at a set of points scattered throughout a domain in the plane are surveyed. Four classes of methods are discussed: (1) global interpolation; (2) local interpolation; (3) global approximation; and (4) local approximation. Also, two-stage methods and contouring are discussed. The surfaces constructed include polynomials, spline functions, and rational functions, among others.

Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations

Abstract: The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

On the failure of ellipticity of the equations for finite elastostatic plane strain

Abstract: In this paper we establish necessary and sufficient conditions, in terms of the local principal stretches, for ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. The material under consideration is assumed to be homogeneous and isotropic, but its strain-energy density is otherwise unrestricted. We also determine the directions of the characteristic curves appropriate to plane elastostatic deformations that are accompanied by a failure of ellipticity.

The use of second-order information in the approximation of discreate-time linear systems

Abstract: It is common practice to partially characterize a filter with a finite portion of its impulse response, with the objective of generating a recursive approximation. This paper discusses the use of mixed first and second information, in the form of a finite portion of the impulse response and autocorrelation sequences. The discussion encompasses a number of techniques and algorithms for this purpose. Two approximation problems are studied: an interpolation problem and a least squares problem. These are shown to be closely related. The linear systems which form the solutions to these problems are shown to be stable. An efficient algorithm for obtaining solutions is given and shown to be closely related to a well-known algorithm of Levinson and the Jury stability test. The close connection between these algorithms suggests that they are fundamental in the numerical analysis of stable discrete-time linear systems.

The mechanics of locating earthquakes

Abstract: A complete reexamination of Geiger9s method in the light of modern numerical analysis indicates that numerical stability can be insured by use of the QR algorithm and the convergence domain considerably enlarged by the introduction of step-length damping. In order to make the maximum use of all data, the method is developed assuming a priori estimates of the statistics of the random errors at each station. Numerical experiments indicate that the bulk of the joint probability density of the location parameters is in the linear region allowing simple estimates of the standard errors of the parameters. The location parameters are found to be distributed as one minus chi squared with m degrees of freedom, where m is the number of parameters, allowing the simple construction of confidence levels. The use of the chi-squared test with n-m degrees of freedom, where n is the number of data, is introduced as a means of qualitatively evaluating the correctness of the earth model.

A generalized conjugate gradient method for nonsymmetric systems of linear equations

Abstract: We consider a generalized conjugate gradient method for solving systems of linear equations having nonsymmetric coefficient matrices with positive-definite symmetric part. The method is based on splitting the matrix into its symmetric and skew-symmetric parts, and then accelerating the associated iteration using conjugate gradients, which simplifies in this case, as only one of the two usual parameters is required. The method is most effective for cases in which the symmetric part of the matrix corresponds to an easily solvable system of equations. Convergence properties are discussed, as well as an application to the numerical solution of elliptic partial differential equations.

Numerical solution of singular control problems using multiple shooting techniques

Abstract: Algorithms for calculating the junction points between optimal nonsingular and singular subarcs of singular control problems are developed. The algorithms consist in formulating appropriate initialvalue and boundary-value problems; the boundary-value problems are solved with the method of multiple shooting. Two examples are detailed to illustrate the proposed numerical methods.

Gradient Optimization and Nonlinear Control

Abstract: The book represents an introduction to computation in control by an iterative, gradient, numerical method, where linearity is not assumed. The general language and approach used are those of elementary functional analysis. The particular gradient method that is emphasized and used is conjugate gradient descent, a well known method exhibiting quadratic convergence while requiring very little more computation than simple steepest descent. Constraints are not dealt with directly, but rather the approach is to introduce them as penalty terms in the criterion. General conjugate gradient descent methods are developed and applied to problems in control.

New Method for Computing the Resonant Frequencies of Dielectric Resonators

Abstract: A new method is developed for accurately predicting resonant frequencies of dielectric resonators used in microwave circuits. By introducing an appropriate approximation in the field distribution outside the resonator, an analytical formulation becomes possible. Two coupled eigenvalue equations thus derived are subsequently solved by a numerical method. The accuracy of the results computed by the present method is demonstrated by comparison with previously published data.

Squeezing flows of Newtonian liquid films - An analysis including fluid inertia

Abstract: A theoretical study is made of the flow behavior of thin Newtonian liquid films being “squeezed” between two flat plates. Solutions to the problem are obtained by using a numerical method, which is found to be stable for all Reynolds numbers, aspect ratios, and grid sizes tested. Particular emphasis is placed on including in the analysis the inertial terms in the Navier-Stokes equations. Comparison of results from the numerical calculation with those from Ishizawa's perturbation solution is made. For the conditions considered here, it is found that the perturbation series is divergent, and that in general one must use a numerical technique to solve this problem.

Maximization of a convex quadratic function under linear constraints

Abstract: This paper addresses itself to the maximization of a convex quadratic function subject to linear constraints. We first prove the equivalence of this problem to the associated bilinear program. Next we apply the theory of bilinear programming developed in [9] to compute a local maximum and to generate a cutting plane which eliminates a region containing that local maximum. Then we develop an iterative procedure to improve a given cut by exploiting the symmetric structure of the bilinear program. This procedure either generates a point which is strictly better than the best local maximum found, or generates a cut which is deeper (usually much deeper) than Tui's cut. Finally the results of numerical experiments on small problems are reported.

An efficient numerical method for solving the time-dependent compressible Navier-Stokes equations at high Reynolds number

Abstract: A new numerical method used to drastically reduce the computation time required to solve the Navier-Stokes equations at flight Reynolds numbers is described. The new method makes it possible and practical to calculate many important three-dimensional, high Reynolds number flow fields on computers.

Mathematical programming and the numerical solution of linear equations.

Abstract: : The book is concerned with the use of mathematical programming techniques for solving ill-conditioned systems of linear equations with various kinds of errors in the right hand side vector. The primary motivation for the work was the spectrum unfolding problem of experimental physics, so the treatment also includes the Fredholm integral equation of the first kind, which can be considered to be an infinite dimensional ill-conditioned system. The basic idea of the new techniques which are developed is the use of priori knowledge about the solution in order to greatly reduce the size of the class of solutions which are consistent with the right hand side errors. The methods are designed to give interval estimates for the solution--the sizes of the intervals being determined by the sizes of the errors in the right hand side, and the constraints imposed on the class of acceptable solutions by the a priori information. The basic a priori constraint which is used is that the solution must be non-negative; but it is shown that many other a priori constraints can be reduced to a simple non-negativity constraint by a suitable transformation of variables. When the non-negativity constraint is taken into account, the problem of estimating lower and upper bounds for the solution can be formulated and solved as a mathematical programming problem. The book treats both the case where the right hand side errors are known absolutely to lie in some bounded region and also the case where the errors are normally distributed. (Author)

Discrete modeling and analysis of switching regulators

Abstract: A simplified method for finding and using discrete small-signal models for switching regulators is presented. With introduction of a new "straight-line" approximation, and application of root locus techniques, it is demonstrated that discrete models may be used accurately to predict wide bandwidth closed-loop behavior with methods simple enough to be useful in the initial design phase of a switching regulator. The principal result is a set of converter transfer functions comparable to the set derived by describing function techniques, but not subject to the low frequency restriction of describing function models. Also presented is a set of pulse-width modulator transfer functions which indicates that the potential small-signal transient behavior of a switching regulator is independent of the choice of modulator.

A numerical method for the steady-state probabilities of a g1/g/c queuing system in a general class

Abstract: A numerical method is proposed for solving the balance equations of the steady-state probabilities of a G~/G/c queueing system in a general class The method is based on an iterative calculation of conditional prob­ abilities, instead of absolute probabilities, conditioned by the number of customers in the system By skillfully exploiting a convergence property of the conditional probabilities, it provides a fairly accurate solution of the balance equations with relatively little computational burden In this paper, a numerical method is proposed for solving the balance equa­ tions of the steady-state probabilities of a GI/G/c queueing system in a general class The method is a direct application of the (modified) lumping method introduced in (6) for the stationary distribution of a Markov chain It is based on an iterative calculation of conditional probabilities of the queueing system conditioned by the number of customers in the system By using the conditional probabilities, rather than absolute probabilities, the system of linear equations of the steady-state probabilities is di,'ded into a set of smaller systems of linear equations, and it can be solved with less computational burden by exploiting convergence property of the conditional probabilities Furthermore, errors included in the solution become fairly small The computational time required for solving the balance equations by our method is nearly independent of the value of the utilization factor p Hence, our method is effective even if p is near to l

Centrifugal Instability of a Stokes Layer: Linear Theory

Abstract: The instability of the flow induced by a circular cylinder oscillating in an infinite viscous fluid is investigated. The flow is shown to be unstable to a Taylor vortex mode of instability. A series solution of the partial differential system governing the stability of the flow is obtained. The method used has several advantages over the numerical methods used by different authors for related problems. The instability predicted by the theory leads to a flow with no mean velocity component tangential to the cylinders. The disturbance velocity field decays exponentially at the edge of the Stokes layer. The theoretical results are qualitatively confirmed by an experimental investigation of the problem.