Showing papers on "Numerical analysis published in 1968"

An accurate numerical steady-state one-dimensional solution of the P-N junction

Abstract: A numerical method of solution of the fundamental semiconductor steady-state one-dimensional transport equations, already available in the literature, is improved and extended, and is applied to a single-junction device. A reduced set of ‘exact’ relations is derived directly from the fundamental set with none of the conventional assumptions or approximations, and is solved numerically by a simple iterative procedure. Freedom is available in the choice of the doping profile, recombination law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. In spite of the generality of the original method, its analytical formulation is shown to be unsuitable for generating a sound numerical algorithm sufficiently acccurate and valid for high reverse-bias conditions. Difficulties and limitations are exposed, and overcomeby an improved formulation extended to any bias condition. Emphasis is on the selection of a numerical algorithm sufficiently sound and efficient to cope with the several fundamental difficulties present in the numerical analysis, and on achieving a high degree of accuracy in the final results (the most delicate problem). As a simple application of the improved formulation, ‘exact’ and first-order theory results for an idealized structure are presented and compared. The poorness of some of the basic assumptions of the conventional first-order theory is exposed, in spite of a satisfactory agreement between the exact and first-order results of the terminal properties for particular bias conditions. The computation time for the achievement of one set of very accurate solutions for a specified applied voltage amounts to approx 1 min on an IBM 7094/7040 shared-file system.

Digital calculation of response spectra from strong-motion earthquake records

Abstract: This report presents a numerical method for computing response spectra from strong-motion earthquake records. The method is based on the exact solution to the governing differential equation and gives a three to four-fold saving in computing time compared to a third order Runge-Kutta method of comparable accuracy. An analysis was made of the errors introduced at various stages in the calculation of spectra so that allowable errors could 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. Included as appendices to the report are computer programs in Fortran IV, with instructions for their use, for computing spectra, for correction of the baseline of the digitized record, and for the computation of ground velocity and displacement.

TEM wave properties of microstrip transmission lines

Abstract: Wave-propagation properties of microstrip transmission lines can be determined accurately if an exact electrostatic-field solution can be found for a pair of charged conductors separated by a dielectric sheet. The latter problem is framed as an integral equation for whose solution simple numerical methods are available. To determine the kernel function of this integral equation, the classical method of images is generalised to include, multiple partial images; the kernel function is then given by well convergent infinite series. Wave impedances calculated using this theory yield very good agreement with experiment. Detailed results are given for the propagation velocity in microstrip lines with very thin strip conductors, and the method used to solve thick-strip problems is described.

Use of Analyticity in the Calculation of Nonrelativistic Scattering Amplitudes

Abstract: A new method of calculating nonrelativistic scattering amplitudes is presented. The scattering amplitude is first calculated as a function of the complex energy below the scattering threshold, and the numerical results are then analytically continued to the physical region. The method is used to calculate two-body and two-channel scattering amplitudes. The numerical analytic continuation is accomplished by a rational-fraction representation similar to the Pad\'e method. Several techniques of numerical analytic continuation by rational fractions are described, and some examples are discussed.

Representation of the Magnetization Characteristic of DC Machines for Computer Use

Abstract: The calculation of flux densities in dc machines using digital computers requires a representation of the magnetization curve of the materials either as H = f1(B) or B = f2(H). A figure of merit of the approximation is defined. The representation can be achieved by the use of simple algebraic or transcendental functions. For satisfactory fit, however, the B-H characteristic has to be subdivided into several parts. As a better alternative a numerical method is proposed in which the magnetization curve is subdivided into a large number of sections. A method of linear interpolation is then employed. A computer program to accomplish this task is described.

Numerical methods of higher-order accuracy for diffusion-convection equations

Abstract: A numerical formulation of high-order accuracy, based on variational methods, is proposed for the solution of multi-dimensional diffusion-convection type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that very accurate solutions of a one-domensional problem can be obtained in the neighborhood of a sharp front without doing a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in 2 dimensions. (14 refs.)

An accurate numerical one-dimensional solution of the p-n junction under arbitrary transient conditions

Abstract: A numerical iterative method of solution of the one-dimensional basic two-carrier transport equations describing the behavior of semiconductor junction devices under arbitrary transient conditions is presented. The method is of a very general character: none of the conventional assumptions and restrictions are introduced and freedom is available in the choice of the doping profile, recombination-generation law, mobility dependencies, injection level, and boundary conditions applied solely at the external contacts. For a specified arbitrary input signal of either current or voltage as a function of time, the solution yields terminal properties and all the quantities of interest in the interior of the device (such as mobile carrier and net electric charge densities, electric field, electrostatic potential, particle and displacement currents) as functions of both position and time. Considerable attention is focused on the numerical analysis of the initial-value-boundary-value problem in order to achieve a numerical algorithm sufficiently sound and efficient to cope with the several fundamental difficulties of the problem, such as stability conditions related to the discretization of partial differential equations of the parabolic type, small differences between nearly equal numbers, and the variation of most quantities over extremely wide ranges within short regions. Results for a particular n + - p single-junction structure under typical external excitations are reported. The iterative scheme of solution for a single device is applicable also to ensembles of active and passive circuit elements. As a simple example, resutls for the combination of an n + - p diode and an external resistor, analyzed under switching conditions, are presented. The inductive behavior of the device for high current pulses, and storage and recovery phenomena under forward-to-reverse bias switching, are also illustrated. ‘Exact’ and conventional approximate analytical results are compared and discrepancies are exposed.

Finite-Element Analysis of Thin Shells

Abstract: Equations describing small deformations of thin shells of arbitrary shape are written in terms of the displacements of the middle surface and rotation of the normals to the middle surface. The theory accounts for transverse shear deformations. By then using simple bilinear approximations of the displacement and rotation fields within finite elements of the shell, a consistent discrete model is obtained; the model provides complete interelement compatibility, and applies to the analysis of thin shells of arbitrary shape. A discrete equivalent of the Kirchhoff hypothesis is then introduced, which greatly improves convergence rates, and which forces the finite element model to convergence to the continuous Kirchhoff model. Numerical examples are included.

Molecular Collisions. VIII

Abstract: The set of integral equations describing the molecular scattering process developed in the previous papers of this series is reformulated as a set of differential equations, in which the coupling is associated with a Hermitian matrix. Exact eigenvectors and eigenvalues of this matrix are developed. These eigenvectors may be used as the basis of a numerical solution of the equations. They also lead to an approximate solution which is valid when the difference of the wavenumbers in the entrance and exit channels is small. The approximate solution may be considered as the lowest‐order term in a series development.

Ordinary Differential Equations

Abstract: Introduction First-Order Linear Differential Equations Second-Order Differential Equations Power- Series Descriptions The Wronskian Eigenvalue Problems The Second-Order Linear Nonhomogeneous Equation Expansions in Eigenfunctions The Perturbation Expansion Asymptotic Series Special Functions The Laplace Transform Rudiments of the Variational Calculus Separation of Variables and Product Series Solutions of Partial Differential Equations Nonlinear Differential Equations More on Difference Equations Numerical Methods Singular Perturbation Methods.

On Newton-like methods

Abstract: where x 0 is prechosen and M is some, not necessarily continuous, correspondence between /2 o and L(Y, X). For a practical problem, such as the simultaneous solution of nonlinear equations, NEWTON'S method is usually absurd because of the relative impossibility of doing the calculations necessary to find x,+ 1 exactly. Any number of expediencies suggest themselves, such as the use of a matr ix of difference approximations in place of the Jacobian matrix. Some very ingenious algorithms for the solution of this problem are given by BROYDEN [61 and BROWN and CONTE [7]. All the methods suggested there are of the type listed above. These two papers abound with excellent numerical examples illustrating the imminent practicality of these methods. Many authors have studied methods of this type. Among them [8] and [ t t ] s tudy the method as stated here and the bibliography is composed of references in which some variant of this method is studied. Many more could be listed. In [8], the author, by this approach, obtained a generalization of a theorem due to MYSOVSKIH [t2] on the convergence of NEWTON'S method and a convergence theorem for Newton-like methods. This latter theorem yielded only linear convergence. Here we present sufficient conditions on M(x) for the sequence of error bounds of the iteration to be of order p, 1 <=p<= 2.

Finite Deformations of Shallow Shells

Abstract: A method is described for the numerical solution of nonlinear shell equations. By application of the Rayleigh-Ritz procedure the differential shell equations are represented by a set of nonlinear algebraic equations which are solved by the Newton-Raphson iteration procedure; the occurrence of double roots is avoided by the use, when necessary, of a suitable displacement parameter as the independent variable. The numerical method of analysis is applied to a limited range of doubly-curved, shallow shells, which are rectangular in planform and loaded with a uniform pressure. The complex solution paths of symmetrical configuration states are traced completely from the unloaded to inverted configurations. Bifurcations from these primary paths are detected and some are traced. They are found to be the solution curves of unstable unsymmetrical configurations.

Computer Simulation of Unsteady Flows in Waterways

Abstract: THE SET OF NONLINEAR, PARTIAL DIFF. EQUATIONS, DESCRIBING ONE-DIMENSIONAL TRANSLATORY WAVE MOTION PROVIDES THE MATHEMATICAL MODEL USED IN THE DEVELOPMENT OF THREE DISTINCTLY DIFFERENT TECHNIQUES FOR THE DIGITAL SIMULATION OF UNSTEADY FLOWS IN RIVERS AND ESTUARIES. THE FIRST TECHNIQUE IS BASED UPON POWER SERIES METHODS AND USES A MACLAURIN SERIES EXPANSION OF THE PARTIAL DIFFERENTIAL EQUATIONS. THE SECOND TECHNIQUE IS PREMISED UPON THE METHOD OF CHARACTERISTICS AND USES A NUMERICAL EVALUATION PROCESS AT SUCCESSIVE SPECIFIED TIME INTERVALS. THE THIRD SIMULATION TECHNIQUE RELIES UPON AN IMPLICIT METHOD OF FLOW SIMULATION WHEREIN THE PARTIAL DIFFERENTIAL EQUATIONS ARE TRANSFORMED TO FINITE DIFFERENCE EQUATIONS. THE EFFECTS OF FLUID FRICTION, VARIABLE CHANNEL GEOMETRY, WIND, LATERAL INFLOW OR OUTFLOW, THE CORIOLIS ACCELERATION, AS WELL AS OVERBANK STORAGE, ARE INCLUDED. FLOWS ARE CONSIDERED TO BE OF HOMOGENOUS DENSITY. EACH OF THE METHODS IS PROGRAMMED FOR HIGH SPEED DIGITAL COMPUTER. COMPARISONS OF THE SIMULATED FLOWS OBTAINED USING EACH OF THE SIMULATION TECHNIQUES WITH THE APPROPRIATE FIELD MEASURED TRANSIENT FLOWS INDICATE GENERALLY GOOD AGREEMENT /ASCE/

Plate Stress Analysis by a General Integral Method

Abstract: An integral method that can directly deal with the three basic problems in simply- or multiply-connected domains is described. The method consists in considering the body as immersed in the indefinite plane and trying to reproduce the actual field inside the corresponding area by means of auxiliary forces distributed on lines externally surrounding the boundary. The parameters defining this distribution are obtained by solving a system of equations, and from these parameters stresses and displacements can be determined at any point of the body. The determination of particular integrals of the field equations when arbitrary body forces are present is also reduced to the determination of the effects of forces distributed on lines in the indefinite plane although the method is more general than other integral methods, it proves to give more accurate results than the finite-element method.

On two arbitrarily located identical parallel antennas

Abstract: The conventional problem of two arbitrarily located parallel antennas is solved by using an integral equation technique. The two simultaneous integral equations for the two antennas are first decoupled into two independent integral equations and then solved by an approximate method with currents represented by five trigonometric functions, three for the symmetric and two for the antisymmetric parts. Typical current distributions and input admittances are obtained for half-wave and full-wave antennas in nonstaggered, in 45\deg echelon, and in collinear arrangements. For the nonstaggered case, the results agree with experimental data. For the other two arrangements, no experimental data are yet available. However, the current distribution is also obtained by a numerical method. The two theoretical results agree favorably for all three cases. The five-term method can be extended to a general array of N -parallel elements. This is reserved for a further report.

A comparative study of numerical methods of elastic-plastic analysis.

Abstract: WO general methods have been developed for the elasticplastic analysis of continuous solid bodies. The method of "thermal" or "initial strains"1 is based on the idea of modifying the elastic equations of equilibrium to compensate for the fact that the plastic strains do not cause any change in stress. On the other hand, the tangent modulus method2 is based on the linearity of the incremental laws of plasticity. The load is applied in increments, and at each stage, a new set of coefficients are obtained for the equilibrium equations. Both methods have been used in conjunction with finite element theory. The matrix equations for finite element analysis using the method of initial strains were developed in Refs. 3-5, whereas the equations for the tangent modulus method were developed in Refs. 6-8. Since both methods solve the same problem, there should be a close relation between them, and perhaps a comparison could lead to a better understanding of the original problem. This note addresses itself to such a comparison. II. Elastic-Plastic Stress-Strain Relations The linear relation between the increments of stress and strain developed in Marcal and King8 is taken here as the point of departure. With the definition of the elastic components of the strain increments,

Inelastic Molecular Collisions: Exponential Solution of Coupled Equations for Vibration-Translation Energy Transfer

Abstract: A new numerical method is used to solve the quantum‐mechanical problem of energy transfer between vibrational and translational degrees of freedom in collisions between an oscillator and an atom. The dependence of the probability of vibrational excitation on the energy of the collision and the size of the truncated basis set is examined. The oscillatory dependence of this transition probability on the energy is is shown to be an artifact due to the truncation of the basis set. The new numerical method is the stepwise solution of the coupled first‐order differential equations by exponentiation of the integrated coupling matrices, an application of the general method investigated by Magnus. It is shown to be fast and accurate.

A Synthesis on Wave Run-Up

Abstract: A critical survey of existing theories and experiments on wave run-up is carried out. The theories of run-up of nonbreaking waves are analyzed first. Breaking criteria and the main theories of run-up of breaking waves are presented, including the theories for the run-up of a bore, the nonsaturated breaker theory, and the numerical method for calculating run-up. This set of various theories is compared with experimental investigations. A simple case illustrates how the wave run-up can be enhanced by resonance effects.

Numerical Analysis: Stable numerical methods for obtaining the Chebyshev solution to an overdetermined system of equations

Abstract: An implementation of Stiefel's exchange algorithm for determining a Chebyshev solution to an overdetermined system of linear equations is presented, that uses Gaussian LU decomposition with row interchanges. The implementation is computationally more stable than those usually given in the literature. A generalization of Stiefel's algorithm is developed which permits the occasional exchange of two equations simultaneously.

Upstream Influence in Viscous Interaction Problems

Abstract: A study of hypersonic viscous interaction on a flat plate is presented. It is shown that the numerical solutions develop singular behavior with respect to the initial data employed. Nonsingular solutions are possible only when a parameter in the initial data is fixed at some specific value which is not known a priori. The origin of this singular dependence is then investigated. It is found that the problem is improperly set as an initial value problem because the interaction dynamics contain upstream influence. If the problem is attacked as an initial value problem, then a constraint must be placed on the initial flow profiles if a bounded solution is to result. Finally, an approximate numerical method is presented to develop this nonsingular solution for the original system of coupled nonlinear equations.

Finite-element method for waveguide problems

Abstract: A numerical method is presented from which it is possible to calculate the propagation coefficients of waveguides with arbitrary boundaries and dielectric fillings.

On the quadratic convergence of a generalization of the jacobi method to arbitrary matrices

Abstract: A method of diagonalizing a general matrix is proved to be ultimately quadratically convergent for all normalizable matrices. The method is a slight modification of a method due to P. J. Eberlein, and it brings the general matrix into a normal one by a combination of unitary plane transformations and plane shears (non-unitary). The method is a generalization of the Jacobi Method: in the case of normal matrices it is equivalent to the method given by Goldstine and Horwitz.