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Showing papers on "Numerical analysis published in 1977"


Journal ArticleDOI
TL;DR: The main purpose of this paper is to provide an algorithm with a restart procedure that takes account of the objective function automatically and to study a multiplying factor that occurs in the definition of the search direction of each iteration.
Abstract: The conjugate gradient method is particularly useful for minimizing functions of very many variables because it does not require the storage of any matrices. However the rate of convergence of the algorithm is only linear unless the iterative procedure is "restarted" occasionally. At present it is usual to restart everyn or (n + 1) iterations, wheren is the number of variables, but it is known that the frequency of restarts should depend on the objective function. Therefore the main purpose of this paper is to provide an algorithm with a restart procedure that takes account of the objective function automatically. Another purpose is to study a multiplying factor that occurs in the definition of the search direction of each iteration. Various expressions for this factor have been proposed and often it does not matter which one is used. However now some reasons are given in favour of one of these expressions. Several numerical examples are reported in support of the conclusions of this paper.

1,588 citations


Book
01 Jan 1977
TL;DR: In this article, a mathematical analysis of spectral methods for mixed initial-boundary value problems is given, and the development of a mathematical theory that explains why spectral methods work and how well they work.
Abstract: : This monograph gives a mathematical analysis of spectral methods for mixed initial-boundary value problems. Spectral methods have become increasingly popular in recent years, especially since the development of fast transform methods, with applications in numerical weather prediction, numerical simulations of turbulent flows, and other problems where high accuracy is desired for complicated solutions. The development of a mathematical theory is given that explains why spectral methods work and how well they work.

925 citations




Journal ArticleDOI
TL;DR: In this paper, the log derivative and renormalized numerov were used to calculate bound-state solutions of the one-dimensional Schroedinger equation, and a useful interpolation formula for calculating eigenfunctions at nongrid points was derived.
Abstract: Two new numerical methods, the log derivative and the renormalized Numerov, are developed and applied to the calculation of bound‐state solutions of the one‐dimensional Schroedinger equation. They are efficient and stable; no convergence difficulties are encountered with double minimum potentials. A useful interpolation formula for calculating eigenfunctions at nongrid points is also derived. Results of example calculations are presented and discussed.

522 citations


Journal ArticleDOI
TL;DR: In this article, a three-dimensional finite-difference model of air flow over an irregular lower surface is described, and the model is nonhydrostatic and the anelastic approximation has been used to filter sound waves.

409 citations


Journal ArticleDOI
TL;DR: In this paper, an iterative method for solving nonsymmetric linear systems based on the Tchebychev polynomials in the complex plane is discussed, and the iteration is shown to converge whenever the eigenvalues of the linear system lie in the open right half complex plane.
Abstract: In this paper an iterative method for solving nonsymmetric linear systems based on the Tchebychev polynomials in the complex plane is discussed. The iteration is shown to converge whenever the eigenvalues of the linear system lie in the open right half complex plane. An algorithm is developed for finding optimal iteration parameters as a function of the convex hull of the spectrum.

292 citations


Journal ArticleDOI
TL;DR: In this paper, two efficient numerical methods for dealing with the stability of linear periodic systems are presented, which combine the use of multivariable Floquet-Liapunov theory with an efficient numerical scheme for computing the transition matrix at the end of one period.
Abstract: Two efficient numerical methods for dealing with the stability of linear periodic systems are presented. Both methods combine the use of multivariable Floquet-Liapunov theory with an efficient numerical scheme for computing the transition matrix at the end of one period. The numerical properties of these methods are illustrated by applying them to the simple parametric excitation problem of a fixed end column. The practical value of these methods is shown by applying them to some helicopter rotor blade aeroelastic and structural dynamics problems. It is concluded that these methods are numerically efficient, general and practical for dealing with the stability of large periodic systems.

269 citations


Book
01 Jan 1977

259 citations


Journal ArticleDOI
TL;DR: In this paper, the first bi-harmonic problem on general two-dimensional domains was solved using a mixed finite element method, where the continuous problem has been approximated by an appropriate mixed-finite element method.
Abstract: We describe in this report various methods, iterative and "almost direct," for solving the first biharmonic problem on general two-dimensional domains once the continuous problem has been approximated by an appropriate mixed finite element method. Using the approach described in this report we recover some well known methods for solving the first biharmonic equation as a system of coupled harmonic equations, but some of the methods discussed here are completely new, including a conjugate gradient type algorithm. In the last part of this report we discuss the extension of the above methods to the numerical solution of the two dimensional Stokes problem in p- connected domains (p $\geq$ 1) through the stream function-vorticity formulation.

234 citations



ReportDOI
15 Aug 1977
TL;DR: In this paper, the authors have developed methods for solving integral equations which work well in spite of the presence of singularities, in which the new approximation methods which were developed do work well for singularities.
Abstract: : One encounters difficulties in the solution of integral equations, namely, the occurrence of singularities in the kernel, and the occurence of unknown-type singularities in the solution of the integral equation. Standard methods of approximation based on exactness for polynomials up to a certain degree are very poor, and frequently fail for functions having such singularities. The purpose of this contract was to develop methods for solving integral equations which work well in spite of the presence of singularities. This goal has been accomplished, in that the new approximation methods which were developed do work well in the presence of singularities. (Author)

Journal ArticleDOI
01 Jan 1977
TL;DR: In this article, the authors present a method that numerically solves the full two-dimensional Navier-Stokes equations with species transport, mixing, and chemical reaction between species using the Implicit Continuous-fluid Eulerian (ICE) technique.
Abstract: In this paper we present a method that numerically solves the full two-dimensional, time-dependent Navier-Stokes equations with species transport, mixing, and chemical reaction between species. The generality of the formulation permits the solution of flows in which deflagrations, detonations, or transitions from deflagration to detonation are found. The solution procedure is embodied in the RICE computer program. RICE is an Eulerian finite difference computer code that uses the Implicit Continuous-fluid Eulerian (ICE) technique to solve the governing equations. We first present the differential equations of motion and the solution procedure of the RICE program. Next, a method is described for artificially thickening the combustion zone to dimensions resolvable by the computational mesh. This is done in such a way that the physical flame speed and jump conditions across the flame front are preserved. Finally, the results of two example calculations are presented. In the first, the artificial thickening technique is used to solve a one-dimensional laminar flame problem. In the second, the results of a full two-dimensional calculation of unsteady combustion in two connected chambers are detailed.

BookDOI
01 Jan 1977
TL;DR: In this article, Jacobi et al. introduce the Euler-Maclaurin Formula and the Laplace Summation Formula, which is used by Laplace, Legendre, and Gauss in the Sixteenth and Early Seventeenth Centuries.
Abstract: 1. The Sixteenth and Early Seventeenth Centuries.- 1.1. Introduction.- 1.2. Napier and Logarithms.- 1.3. Briggs and His Logarithms.- 1.4. Burgi and His Antilogarithms.- 1.5. Interpolation.- 1.6. Vieta and Briggs.- 1.7. Kepler.- 2. The Age of Newton.- 2.1. Introduction.- 2.2. Logarithms and Finite Differences.- 2.3. Trigonometric Tables.- 2.4. The Newton-Raphson and Other Iterative Methods.- 2.5. Finite Differences and Interpolation.- 2.6. Maclaurin on the Euler-Maclaurin Formula.- 2.7. Stirling.- 2.8. Leibniz.- 3. Euler and Lagrange.- 3.1. Introduction.- 3.2. Summation of Series.- 3.3. Euler on the Euler-Maclaurin Formula.- 3.4. Applications of the Summation Formula.- 3.5. Euler on Interpolation.- 3.6. Lunar Theory.- 3.7. Lagrange on Difference Equations.- 3.8. Lagrange on Functional Equations.- 3.9. Lagrange on Fourier Series.- 3.10. Lagrange on Partial Difference Equations.- 3.11. Lagrange on Finite Differences and Interpolation.- 3.12. Lagrange on Hidden Periodicities.- 3.13. Lagrange on Trigonometric Interpolation.- 4. Laplace, Legendre, and Gauss.- 4.1. Introduction.- 4.2. Laplace on Interpolation.- 4.3. Laplace on Finite Differences.- 4.4. Laplace Summation Formula.- 4.5. Laplace on Functional Equations.- 4.6. Laplace on Finite Sums and Integrals.- 4.7. Laplace on Difference Equations.- 4.8. Laplace Transforms.- 4.9. Method of Least Squares.- 4.10. Gauss on Least Squares.- 4.11. Gauss on Numerical Integration.- 4.12. Gauss on Interpolation.- 4.13. Gauss on Rounding Errors.- 5. Other Nineteenth Century Figures.- 5.1. Introduction.- 5.2. Jacobi on Numerical Integration.- 5.3. Jacobi on the Euler-Maclaurin Formula.- 5.4. Jacobi on Linear Equations.- 5.5. Cauchy on Interpolation.- 5.6. Cauchy on the Newton-Raphson Method.- 5.7. Cauchy on Operational Methods.- 5.8. Other Nineteenth Century Results.- 5.9. Integration of Differential Equations.- 5.10. Successive Approximation Methods.- 5.11. Hermite.- 5.12. Sums.

Book
01 Jul 1977
TL;DR: In this article, a method is presented for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries in a general multi-connected two-dimensional region containing any number of arbitrarily shaped bodies.
Abstract: A method is presented for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected two-dimensional region containing any number of arbitrarily shaped bodies. No restrictions are placed on the shape of the boundaries, which may even be time-dependent, and the approach is not restricted in principle to two dimensions. With this procedure the numerical solution of a partial differential system may be done on a fixed rectangular field with a square mesh with no interpolation required regardless of the shape of the physical boundaries, regardless of the spacing of the curvilinear coordinate lines in the physical field, and regardless of the movement of the coordinate system in the physical plane. A number of examples of coordinate systems and application thereof to the solution of partial differential equations are given. The FORTRAN computer program and instructions for use are included.

Journal ArticleDOI
TL;DR: In this paper, a linear regression by the method of least squares is made on the geometric variables that occur in the equation for elliptical contact deformation, and the ellipticity and the complete elliptic integrals of the first and second kind are expressed as a function of the x,y-plane principal radii.
Abstract: A linear regression by the method of least squares is made on the geometric variables that occur in the equation for elliptical-contact deformation. The ellipticity and the complete elliptic integrals of the first and second kind are expressed as a function of the x,y-plane principal radii. The ellipticity was varied from 1 (circular contact) to 10 (a configuration approaching line contact). The procedure for solving these variables without the use of charts or a high-speed computer would be quite tedious. These simplified equations enable one to calculate easily the elliptical-contact deformation to within 3 percent accuracy without resorting to charts or numerical methods.

Book
01 Jul 1977
TL;DR: In this paper, boundary value problems for semilinear elliptic equations are considered, where the nonlinear terms do not grow too fast as a function of the gradient of the dependent variable and ordered upper and lower solutions are known.
Abstract: : Boundary value problems for semilinear elliptic equations are considered. If the nonlinear terms do not grow too fast as a function of the gradient of the dependent variable and ordered upper and lower solutions are known, then maximal and minimal solutions can be obtained by an iteration procedure. Other results concerning the existence of additional solutions follow from topological principles.

Journal ArticleDOI
TL;DR: In this paper, an interative approach is proposed for the numerical analysis of elastic-plastic continua, which gives after convergence an implicit scheme of integration of the evolution problem, and is concerned with elastic-perfectly plastic materials and with hardening standard materials.
Abstract: An interative approach is proposed for the numerical analysis of elastic–plastic continua. This approach gives after convergence an implicit scheme of integration of the evolution problem, and is concerned with elastic-perfectly plastic materials and with hardening standard materials. Under a generalized assumption of positive hardening, the proof of convergence of the iterative solutions is given. Some numerical examples by the finite element method are also discussed.


Journal ArticleDOI
TL;DR: In this article, a simple numerical solution for the coupled-power equation in optical fibers was obtained by using the finite-difference method of numerical analysis, which yields all the quantities of interest in the interior of the fiber: power distribution, attenuation, and far-field radiation pattern as functions of length.
Abstract: By use of the finite-difference method of numerical analysis, a simple numerical solution is obtained for the coupled-power equation in optical fibers. For a specified arbitrary coupling coefficient and launching condition, the solution yields all the quantities of interest in the interior of the fiber: power distribution, attenuation, and far-field radiation pattern as functions of length. Results for buffered and cabled Corning fibers are reported. Attention is mainly focused on the influence of the microbends on the optical losses.

Journal ArticleDOI
TL;DR: In this paper, the problem of onedimensional heat transfer in finite slabs of freezing food materials cooled from both sides is considered, and a simple formula, based on an approximate analytical solution, is at least as accurate as a complex finite difference formulation.
Abstract: The problem of onedimensional heat transfer in finite slabs of freezing food materials cooled from both sides is considered. The authors’ modification to Plank's equation is compared with a numerical scheme. The paper shows that in the important slab freezing problem a simple formula, based on an approximate analytical solution, is at least as accurate as a complex finite difference formulation, and has some substantial practical advantages.

Journal ArticleDOI
TL;DR: In this article, a numerical method is given to compute all solutions of systemsT ofn polynomial equations inn unknowns on the only premises that the sets of solutions of these systems are finite.
Abstract: In this paper a numerical method is given to compute all solutions of systemsT ofn polynomial equations inn unknowns on the only premises that the sets of solutions of these systems are finite. The method employed is that of "embedding", i.e. the systemT is embedded in a set of systems which are successively solved, starting with one having solutions easily to compute and proceding toT in a finite series of steps. An estimation of the number of steps necessary is given. The practicability of the method is proved for all systemsT. Numerical examples and results are contained.

Journal ArticleDOI
TL;DR: In this paper, a self-correcting approach based on load and displacement incrementation is presented for pre-and post-buckling analysis of finite element systems. But, the postbuckling problem has been less actively pursued probably because of the inherent numerical difficulties encountered.
Abstract: The prediction of nonlinear structural behavior by the finite element method wherein buckling does not occur has received considerable attention and, with it, reasonable success has been achieved. However, the post-buckling problem has been less actively pursued probably because of the inherent numerical difficulties encountered. This note reviews very briefly the numerical methods currently being used for pre- and post-buckling analysis and presents a self-correcting approach based on load and displacement incrementation which is shown to be efficient, reliable, and easy to program. Numerical solutions are presented which demonstrate the effectiveness of the method.

Journal ArticleDOI
TL;DR: In this article, 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, and two coupled eigenvalue equations are subsequently solved by a numerical method.
Abstract: A new method is develop 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

Journal ArticleDOI
TL;DR: In this paper, the Boussinesq approximation of single-mode equations describing thermal convection is constructed by expanding the fluctuating velocity and temperature fields in a complete set of functions (or planforms) of the horizontal coordinates and retaining just one term.
Abstract: In the Boussinesq approximation, single-mode equations describing thermal convection are constructed by expanding the fluctuating velocity and temperature fields in a complete set of functions (or planforms) of the horizontal coordinates and retaining just one term. Numerical solutions of the single-mode equations are investigated, chief consideration being given to hexagonal planforms. Extensive surveys of steady solutions are presented for various Rayleigh numbers, Prandtl numbers, and horizontal wavenumbers. The dependences on Rayleigh number and Prandtl number at very large Rayleigh number are in satisfactory agreement with the results of asymptotic expansions.

Journal ArticleDOI
TL;DR: In this article, a new numerical technique to solve dynamic wave propagation problems in layered systems is presented, which is based on the closed-form analytical solution in the direction parallel to the layering, and arbitrary displacement expansions in a direction perpendicular to it.
Abstract: The paper presents a new numerical technique to solve dynamic wave propagation problems in layered systems. The method is based on the closed-form analytical solution in the direction parallel to the layering, and arbitrary displacement expansions in the direction perpendicular to it. Detailed information is given for the use of the theory for the particular case of a linear expansion. The main advantage of the theory is that a great reduction in the number of degrees-of-freedom necessary to model a system, as compared to conventional finite element models, can be attained. The advantage is achieved at the expense of having to solve the system in the frequency domain, and having to compute the stiffness matrix for each individual frequency. The method is, therefore, limited to linear systems. The relatively modest size of the problem allows a solution in fast memory, without having to resort to peripheral storage allocation.

Book
01 Jan 1977
TL;DR: In this paper, the origins of the first scheme were discussed and the second and third schemes were executed on the intermediate layer and the final layer, respectively, and the first and second schemes on the third and fourth layers, respectively.
Abstract: 1. General Introduction.- 1.1 Introduction.- 1.2 Boundary Value Problems and Initial Problems.- 1.3 One-Dimensional Unsteady Flow Characteristics.- 1.4 Steady Supersonic Plane or Axi-Symmetric Flow. Equations of Motion in Characteristic Form.- 1.5 Basic Concepts Used in Finite Difference Methods.- References.- 2. The Godunov Schemes.- 2.1 The Origins of Godunov's First Scheme.- 2.2 Godunov's First Scheme. One-Dimensional Eulerian Equations.- 2.3 Godunov's First Scheme in Two and More Dimensions.- 2.4 Godunov's Second Scheme.- 2.5 The Double Sweep Method.- 2.6 Execution of the Second Scheme on the Intermediate Layer.- 2.7 Boundary Conditions on the Intermediate Layer.- 2.8 Procedure on the Final Layer.- 2.9 Applications of the Second Godunov Scheme.- 2.10 Glimm's Method.- 2.11 Outline of Solution for Gas Dynamic Equations.- 2.12 The Glimm Scheme for Simple Acoustic Waves.- 2.13 Random Choice for the Gas Dynamic Equations.- 2.14 Solution of the Riemann Problem.- 2.15 Extension to Unsteady Flow with Cylindrical or Spherical Symmetry.- 2.16 Remarks on Multi-Dimensional Problems.- References.- 3. The BVLR Method.- 3.1 Description of Method for Supersonic Flow.- 3.2 Extensions to Mixed Subsonic-Supersonic Flow. The Blunt Body Problem.- 3.3 The Double Sweep Method for Unsteady Three-Dimensional Flow.- 3.4 Worked Problem. Application to Circular Arc Airfoil.- 3.5 Results and Discussion.- Appendix-Shock Expansion Theory.- References.- 4. The Method of Characteristics for Three-Dimensional Problems in Gas Dynamics.- 4.1 Introduction.- 4.2 Bicharacteristics Method (Butler).- 4.3 Optimal Characteristics Methods (Bruhn and Haack, Schaetz).- 4.4 Near Characteristics Method (Sauer).- References.- 5. The Method of Integral Relations.- 5.1 Introduction.- 5.2 General Formulation. Model Problem.- 5.3 Flow Past Ellipses.- 5.4 The Supersonic Blunt Body Problem.- 5.5 Transonic Flow.- 5.6 Incompressible Laminar Boundary Layer Equations. Basic Formulation.- 5.7 The Method in the Compressible Case.- 5.8 Laminar Boundary Layers with Suction or Injection.- 5.9 Extension to Separated Flows.- 5.10 Application to Supersonic Wakes and Base Flows.- 5.11 Application to Three-Dimensional Laminar Boundary Layers.- 5.12 A Modified Form of the Method of Integral Relations.- 5.13 Application to Viscous Supersonic Conical Flows.- 5.14 Extension to Unsteady Laminar Boundary Layers.- 5.15 Application to Internal Flow Problems.- Model Problem (Chu and Gong).- References.- 6. Telenin's Method and the Method of Lines.- 6.1 Introduction.- 6.2 Solution of Laplace's Equation by Telenin's Method.- 6.3 Solution of a Model Mixed Type Equation by Telenin's Method.- 6.4 Application of Telenin's Method to the Symmetrical Blunt Body Problem.- 6.5 Extension to Unsymmetrical Blunt Body Flows.- 6.6 Application of Telenin's Method to the Supersonic Yawed Cone Problem.- 6.7 The Method of Lines. General Description.- 6.8 Applications of the Method of Lines.- 6.9 Powell's Method Applied to Two Point Boundary Value Problems.- Telenin's Method. Model Problems (Klopfer).- References.

Dissertation
02 Nov 1977
TL;DR: In this article, a discretized description of the kinematics of kinematically indeterminate structures as given in the finite element method is however also a good starting point for the numerical treatment of the analysis of mechanisms.
Abstract: The development of the finite element method for the numerical analysis of the mechanical behaviour of structures has been directed at the calculation of the state of deformation and stress of kinematically determinate structures. The discretized description of the kinematics of kinematically indeterminate structures as given in the finite element method is however also a good starting point for the numerical treatment of the analysis of mechanisms. In the description of the kinematics of mechanisms the relations between deformations and displacements play a central role. For the calculation of the transfer functions of order one and two, being the basic information for the determination of velocity and acceleration, direct methods are presented, applicable to mechanisms consisting of undeformable links. The description is completed with the formulation of dynamics, kinetostatics and vibrations. For mechanisms consisting of deformable links an approximate method is given. The theory is applied to planar mechanisms. Examples demonstrate the use of the theory in kinematic, dynamic and kinetostatic problems.

Journal ArticleDOI
TL;DR: In this article, the authors used linear interpolations along time lines to predict pipeline transients and showed the effects of interpolation, spacing, and grid size on numerical attenuation and dispersion.
Abstract: The method of characteristics with fixed-grid intervals used to predict pipeline transients can be enhanced by projecting characteristics from outside the fundamental grid size. At interior regions linear interpolations are used which satisfy the required stability criterion and at boundaries linear interpolations along time lines complete the numerical solution. An error analysis shows the effects of interpolation, spacing, and grid size on numerical attenuation and dispersion; the results suggest ways to improve the predicted transients by adjusting three finite differencing parameters. Numerical solutions for constant and variable wave speed conditions are presented and show how high-frequency components become less severely suppressed.

Journal ArticleDOI
TL;DR: In this paper, an improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields.
Abstract: An improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields. The technique provides higher-order approximation of the unknown function over a bounding surface described by two-parameter, non-planar elements. The integral equation is discretized through the Rayleigh–Ritz procedure. Convergence to the solution for operators having a positive-definite component is guaranteed. Kernel singularities are treated by removing them from the relevant integrals and dealing with them analytically. A successive element iterative process, which produces the solution of the large dense matrix of the complete structure, is described. The discretization and equation solution take place one element at a time resulting in storage and computational savings. Results obtained for classical test models, involving scalar electrostatic potential and vector elastostatic displacement fields, demonstrate the technique for the solution of the Fredholm integral equation of the first kind. Solution of the Fredholm equation of the second kind is to be reported subsequently.