Showing papers on "Numerical analysis published in 1982"

Electromagnetic scattering by surfaces of arbitrary shape

Abstract: The electric field integral equation (EFIE) is used with the moment method to develop a simple and efficient numerical procedure for treating problems of scattering by arbitrarily shaped objects. For numerical purposes, the objects are modeled using planar triangular surfaces patches. Because the EFIE formulation is used, the procedure is applicable to both open and closed surfaces. Crucial to the numerical formulation is the development of a set of special subdomain-type basis functions which are defined on pairs of adjacent triangular patches and yield a current representation free of line or point charges at subdomain boundaries. The method is applied to the scattering problems of a plane wave illuminated flat square plate, bent square plate, circular disk, and sphere. Excellent correspondence between the surface current computed via the present method and that obtained via earlier approaches or exact formulations is demonstrated in each case.

The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors

Abstract: 1. Introduction and Simple Properties.- 1.1. Introduction.- 1.2. Chaotic Ordinary Differential Equations.- 1.3. Our Approach to the Lorenz Equations.- 1.4. Simple Properties of the Lorenz Equations.- 2. Homoclinic Explosions: The First Homoclinic Explosion.- 2.1. Existence of a Homoclinic Orbit.- 2.2. The Bifurcation Associated with a Homoclinic Orbit.- 2.3. Summary and Some General Definitions.- 3. Preturbulence, Strange Attractors and Geometric Models.- 3.1. Periodic Orbits for the Hopf Bifurcation.- 3.2. Preturbulence and Return Maps.- 3.3. Strange Attractor and Homoclinic Explosions.- 3.4. Geometric Models of the Lorenz Equations.- 3.5. Summary.- 4. Period Doubling and Stable Orbits.- 4.1. Three Bifurcations Involving Periodic Orbits.- 4.2. 99.524 100.795. The x2y Period Doubling Window.- 4.3. 145 166. The x2y2 Period Doubling Window.- 4.4. Intermittent Chaos.- 4.5. 214.364 ?. The Final xy Period Doubling Window.- 4.6. Noisy Periodicity.- 4.7. Summary.- 5. From Strange Attractor to Period Doubling.- 5.1. Hooked Return Maps.- 5.2. Numerical Experiments.- 5.3. Development of Return Maps as r Increases: Homoclinic Explosions and Period Doubling.- 5.4. Numerical Experiments on Periodic Orbits.- 5.5. Period Doubling and One-Dimensional Maps.- 5.6. Global Approach and Some Conjectures.- 5.7. Summary.- 6. Symbolic Description of Orbits: The Stable Manifolds of C1 and C2.- 6.1. The Maxima-in-z Method.- 6.2. Symbolic Descriptions from the Stable Manifolds of C1 and C2.- 6.3. Summary.- 7. Large r.- 7.1. The Averaged Equations.- 7.2. Analysis and Interpretation of the Averaged Equations.- 7.3. Anomalous Periodic Orbits for Small b and Large r.- 7.4. Summary.- 8. Small b.- 8.1. Twisting Around the z-Axis.- 8.2. Homoclinic Explosions with Extra Twists.- 8.3. Periodic Orbits Without Extra Twisting Around the z-Axis.- 8.4. Heteroclinic Orbits Between C1 and C2.- 8.5. Heteroclinic Bifurcations.- 8.6. General Behaviour When b = 0.25.- 8.7. Summary.- 9. Other Approaches, Other Systems, Summary and Afterword.- 9.1. Summary of Predicted Bifurcations for Varying Parameters ?, b and r.- 9.2. Other Approaches.- 9.3. Extensions of the Lorenz System.- 9.4. Afterword - A Personal View.- Appendix A. Definitions.- Appendix B. Derivation of the Lorenz Equations from the Motion of a Laboratory Water Wheel.- Appendix C. Boundedness of the Lorenz Equations.- Appendix D. Homoclinic Explosions.- Appendix E. Numerical Methods for Studying Return Maps and for Locating Periodic Orbits.- Appendix F. Computational Difficulties Involved in Calculating Trajectories which Pass Close to the Origin.- Appendix G. Geometric Models of the Lorenz Equations.- Appendix H. One-Dimensional Maps from Successive Local Maxima in z.- Appendix I. Numerically Computed Values of k(r) for ? = 10 and b = 8/3.- Appendix J. Sequences of Homoclinic Explosions.- Appendix K. Large r the Formulae.

Methods of Numerical Mathematics

Abstract: 1 Fundamentals of the Theory of Difference Schemes.- 1.1. Basic Equations and Their Adjoints.- 1.1.1. Norm Estimates of Certain Matrices.- 1.1.2. Computing the Spectral Bounds of a Positive Matrix.- 1.1.3. Eigenvalues and Eigenfunctions of the Laplace Operator.- 1.1.4. Eigenvalues and Eigenvectors of the Finite-Difference Analog of the Laplace Operator.- 1.2. Approximation.- 1.3. Countable Stability.- 1.4. The Convergence Theorem.- 2 Methods of Constructing Difference Schemes for Differential Equations.- 2.1. Variational Methods in Mathematical Physics.- 2.1.1. Some Problems of Variational Calculation.- 2.1.1. The Ritz Method.- 2.1.3. The Galerkin Method.- 2.1.4. The Method of Least Squares.- 2.2. The Method of Integral Identities.- 2.2.1. Method of Constructing Difference Equations for Problems with Discontinuous Coefficients on the Basis of an Integral Identity.- 2.2.2. The Variational Form of an Integral Identity.- 2.3. Difference Schemes for Equations with Discontinuous Coefficients Based on Variational Principles.- 2.3.1. Simple Difference Equations for a Diffusion Based on the Ritz Method.- 2.3.2. Constructions of Simple Difference Schemes Based on the Galerkin (Finite Elements) Method.- 2.4. Principles for the Construction of Subspaces for the Solution of One-Dimensional Problems by Variational Methods.- 2.4.1. A General Approach to the Construction of Subspaces of Piecewise-Polynomial Functions.- 2.4.2. Constructing a Basis Using Trigonometric Functions and Applying It in Variational Methods.- 2.5. Variational-Difference Schemes for Two-Dimensional Equations of Elliptic Type.- 2.5.1. The Ritz Method.- 2.5.2. The Galerkin Method.- 2.5.3. Methods for Constructing Subspaces.- 2.6. Variational Methods for Multi-Dimensional Problems.- 2.6.1. Methods of Choosing the Subspaces.- 2.6.2. Coordinate-by-Coordinate Methods for Multi-Dimensional Problems.- 2.7. The Method of Fictive Domains.- 3 Interpolation of Net Functions.- 3.1. Interpolation of Functions of One Variable.- 3.1.1. Interpolation of Functions of One Variable by Cubic Splines.- 3.1.2. Piecewise-Cubic Interpolation with Smoothing.- 3.1.3. Smooth Construction.- 3.1.4. The Convergence of Spline Functions.- 3.2. Interpolation of Functions of Two or More Variables.- 3.3. An r-Smooth Approximation to a Function of Several Variables.- 3.4. Elements of the General Theory of Splines.- 4 Methods for Solving Stationary Problems of Mathematical Physics.- 4.1. General Concepts of Iteration Theory.- 4.2. Some Iterative Methods and Their Optimization.- 4.2.1. The Simplest Iteration Method.- 4.2.2. Convergence and Optimization of Stationary Iterative Methods.- 4.2.3. The Successive Over-Relaxation Method.- 4.2.4. The Chebyshev Iteration Method.- 4.2.5. Comparison of the Convergence Rates of Various Iteration Methods for a System of Finite-Difference Equations.- 4.3. Nonstationary Iteration Methods.- 4.3.1. Convergence Theorems.- 4.3.2. The Method of Minimizing the Residuals.- 4.3.3. The Conjugate Gradient Method.- 4.4. The Splitting-Up Method.- 4.4.1. The Commutative Case.- 4.4.2. The Noncommutative Case.- 4.4.3. Variational and Chebyshevian Optimization of Splitting-Up Methods.- 4.5. Iteration Methods for Systems with Singular Matrices.- 4.5.1. Consistent Systems.- 4.5.2. Inconsistent Systems.- 4.5.3. The Matrix Analog of the Method of Fictive Regions.- 4.6. Iterative Methods for Inaccurate Input Data.- 4.7. Direct Methods for Solving Finite-Difference Systems.- 4.7.1. The Fast Fourier Transform.- 4.7.2. The Cyclic Reduction Method.- 4.7.3. Factorization of Difference Equations.- 5 Methods for Solving Nonstationary Problems.- 5.1. Second-Order Approximation Difference Schemes with Time-Varying Operators.- 5.2. Nonhomogeneous Equations of the Evolution Type.- 5.3. Splitting-Up Methods for Nonstationary Problems.- 5.3.1. The Stabilization Method.- 5.3.2. The Predictor-Corrector Method.- 5.3.3. The Component-by-Component Splitting-Up Method.- 5.3.4. Some General Remarks.- 5.4. Multi-Component Splitting.- 5.4.1. The Stabilization Method.- 5.4.2. The Predictor-Corrector Method.- 5.4.3. The Component-by-Component Splitting-Up Method Based on the Elementary Schemes.- 5.4.4. Splitting-Up of Quasi-Linear Problems.- 5.5. General Approach to Component-by-Component Splitting.- 5.6. Methods of Solving Equations of the Hyperbolic Type.- 5.6.1. The Stabilization Method.- 5.6.2. Reduction of the Wave Equation to an Evolution Problem.- 6 Richardson's Method for Increasing the Accuracy of Approximate Solutions.- 6.1 Ordinary First-Order Differential Equations.- 6.2. General Results.- 6.2.1. The Decomposition Theorem.- 6.2.2. Acceleration of Convergence.- 6.3. Simple Integral Equations.- 6.3.1. The Fredholm Equation of the Second Kind.- 6.3.2. The Volterra Equation of the First Kind.- 6.4. The One-Dimensional Diffusion Equation.- 6.4.1. The Difference Method.- 6.4.2. The Galerkin Method.- 6.5. Nonstationary Problems.- 6.5.1. The Heat Equation.- 6.5.2. The Splitting-Up Method for the Evolutionary Equation.- 6.6. Richardson's Extrapolation for Multi-Dimensional Problems.- 7 Numerical Methods for Some Inverse Problems.- 7.1. Fundamental Definitions and Examples.- 7.2. Solution of the Inverse Evolution Problem with a Constant Operator.- 7.2.1. The Fourier Method.- 7.2.2. Reduction to the Solution of a Direct Equation.- 7.3. Inverse Evolution Problems with Time-Varying Operators.- 7.4. Methods of Perturbation Theory for Inverse Problems.- 7.4.1. Some Problems of the Linear Theory of Measurements.- 7.4.2. Conjugate Functions and the Notion of Value.- 7.4.3. Perturbation Theory for Linear Functionals.- 7.4.4. Numerical Methods for Inverse Problems and Design of Experiment.- 7.5. Perturbation Theory for Complex Nonlinear Models.- 7.5.1. Fundamental and Adjoint Equations.- 7.5.2. The Adjoint Equation in Perturbation Theory.- 7.5.3. Perturbation Theory for Nonstationary Problems.- 7.5.4. Spectral Methods in Perturbation Theory.- 8 Methods of Optimization.- 8.1. Convex Programming.- 8.2. Linear Programming.- 8.3. Quadratic Programming.- 8.4. Numerical Methods in Convex Programming Problems.- 8.5. Dynamic Programming.- 8.6. Pontrjagin's Maximum Principle.- 8.7. Extremal Problems with Constraints and Variational Inequalities.- 8.7.1. Elements of the General Theory.- 8.7.2. Examples of Extremal Problems.- 8.7.3. Numerical Methods in Extremal Problems.- 9 Some Problems of Mathematical Physics.- 9.1. The Poisson Equation.- 9.1.1. The Dirichlet Problem for the One-Dimensional Poisson Equation.- 9.1.2. The One-Dimensional von Neumann Problem.- 9.1.3. The Two-Dimensional Poisson Equation.- 9.1.4. A Problem of Boundary Conditions.- 9.2. The Heat Equation.- 9.2.1. The One-Dimensional Problem of Heat Conduction.- 9.2.2. The Two-Dimensional Problem of Heat Conduction.- 9.3. The Wave Equation.- 9.4. The Equation of Motion.- 9.4.1. The Simplest Equations of Motion.- 9.4.2. The Two-Dimensional Equation of Motion with Variable Coefficients.- 9.4.3. The Multi-Dimensional Equation of Motion.- 9.5. The Neutron Transport Equation.- 9.5.1. The Nonstationary Equation.- 9.5.2. The Transport Equation in Self-Adjoint Form.- 10 A Review of the Methods of Numerical Mathematics.- 10.1. The Theory of Approximation, Stability, and Convergence of Difference Schemes.- 10.2. Numerical Methods for Problems of Mathematical Physics.- 10.3. Conditionally Well-Posed Problems.- 10.4. Numerical Methods in Linear Algebra.- 10.5. Optimization Problems in Numerical Methods.- 10.6. Optimization Methods.- 10.7. Some Trends in Numerical Mathematics.- References.- Index of Notation.

On the transport-diffusion algorithm and its applications to the Navier-Stokes equations

Abstract: This paper deals with an algorithm for the solution of diffusion and/or convection equations where we mixed the method of characteristics and the finite element method. Globally it looks like one does one step of transport plus one step of diffusion (or projection) but the mathematics show that it is also an implicit time discretization of thePDE in Lagrangian form. We give an error bound (h+Δt+h×h/Δt in the interesting case) that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation).

A Numerical Method for Solving the Equations of Compressible Viscous Flow

Abstract: Although much progress has already been made In solving problems in aerodynamic design, many new developments are still needed before the equations for unsteady compressible viscous flow can be solved routinely. This paper describes one such development. A new method for solving these equations has been devised that 1) is second-order accurate in space and time, 2) is unconditionally stable, 3) preserves conservation form, 4) requires no block or scalar tridiagonal inversions, 5) is simple and straightforward to program (estimated 10% modification for the update of many existing programs), 6) is more efficient than present methods, and 7) should easily adapt to current and future computer architectures. Computational results for laminar and turbulent flows at Reynolds numbers from 3 x 10(exp 5) to 3 x 10(exp 7) and at CFL numbers as high as 10(exp 3) are compared with theory and experiment.

The value of the stochastic solution in stochastic linear programs with fixed recourse

Abstract: Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.

On the energy release rate and the J -integral for 3-D crack configurations

Abstract: In this paper an analytical expression for the energy release rate has been derived and put in a form suitable for a numerical analysis of an arbitrary 3-D crack configuration. The virtual crack extension method can most conveniently be used for such a derivation. This method was originally developed from finite element considerations and the resulting expressions were, therefore, based on the finite element matrix formulation [1–5]. In this paper the derivation of the energy release rate leads to an expression which is independent of any specific numerical procedure. The formulation is valid for general fracture behavior including nonplanar fracture and shear lips and applies to elastic materials as well as materials following the deformation theory of plasticity. The body force effect is also included. For 3-D fracture problems it is of advantage to use both an average and a local form of the energy release rate and definitions for both forms are suggested. For certain restrictions on the crack geometry it is shown that the energy release rate reduces to the 3-D form of the J-integral.

Time-domain skin-effect model for transient analysis of lossy transmission lines

Abstract: A skin-effect equivalent circuit consisting of resistors and inductors is derived from the skin-effect differential equations for simulating the loss of a transmission line. A numerical method is used to analyze the transmission-line differential equations and the skin-effect equivalent circuit, yielding a model which relates the new values of node voltages and line currents to their values at the previous time step. Based on this model, a very simple program was written on a desk-top computer for the transient analysis of lossy trammission lines. Two examples are presented. The first example is an analysis of the step and pulse responses of a 600-m RG-8/U coaxial cable. The computed results show excellent agreement with measured data. The second example studies the current at the end of a 12-in 7-Ω strip line under different loading conditions. Very good agreement has been obtained between the calculated steady-state solution and that obtained by the frequency-domain method.

The effective index method and its application to semiconductor lasers

Abstract: By the effective index method a two-dimensional field problem is transformed to a problem for a one-dimensional effective waveguide. This method is applied to semiconductor lasers having a gradual lateral variation in the complex permittivity. For the special case of a parabolic variation, analytical formulas for the required gain in the center and the half width of the intensity distribution are derived. The results are compared with a numerical method and very good agreement is found except in some cases where convergence problems occur for the numerical method. This agreement is taken as evidence for the validity of results obtained using the effective index method for analysis of semiconductor laser structures.

A Review of Neutron Transport Approximations

Abstract: Numerical methods for solving the integrodifferential, integral, and surface-integral forms of the neutron transport equation are reviewed. The solution methods are shown to evolve from only a few ...

A sparsity-exploiting variant of the bartels-golub decomposition for linear programming bases

Abstract: We describe a sparsity-exploiting variant of the Bartels—Golub decomposition for linear programming bases. It includes interchanges that, whenever this is possible, avoid the use of any eliminations (with consequent fill-ins) when revising the factorization at an iteration. Test results on some medium scale problems are presented and comparisons made with the algorithm of Forrest and Tomlin.

Newton's Method

Abstract: Newton's method plays a central role in the development of numerical techniques for optimization In fact, most of the current practical methods for optimization can be viewed as variations on Newton's method It is therefore important to understand Newton's method as an algorithm in its own right and as a key introduction to the most recent ideas in this area One of the aims of this expository paper is to present and analyze two main approaches to Newton's method for unconstrained minimization: the line search approach and the trust region approach The other aim is to present some of the recent developments in the optimization field which are related to Newton's method In particular, we explore several variations on Newton's method which are appropriate for large scale problems, and we also show how quasi-Newton methods can be derived quite naturally from Newton's method

Numerical analysis of transient photoconductivity in insulators

Abstract: The nonlinear partial differential equations which describe transient photoconductivity in insulators are solved numerically. Trapping and recombination are included which allows photoconduction to be studied in the presence of large space charge. Use of methods for ‘‘stiff’’ differential equations insures stable time dependent solutions even for problems which have widely different time constants. The program is applied to a thin film (1000 A) of silicon dioxide to illustrate the salient features of the numerical solutions and the capabilities of the program.

Efficient Sequential Solution of the Nonlinear Inverse Heat Conduction Problem

Abstract: The nonlinear inverse heat conduction problem is the calculation of surface heat fluxes and temperatures by utilizing measured interior temperatures in opaque solids possessing temperature-variable thermal properties. The most widely used numerical method for this problem was developed by Beck. The new sequential procedure presented here reduces the number of computer calculations by a factor of 3 or 4. The general heat conduction model used permits treatment of various one-dimensional geometries (plates, cylinders, and spheres), energy sources, and fin effects. The numerical procedure is illustrated for finite differences, but the basic concepts are also applicable to the finite-element method. Detailed descriptions of the computational algorithms are given and a nonlinear example is provided.

Nonlinear programming via an exact penalty function: Asymptotic analysis

Abstract: In this paper we consider the final stage of a ‘global’ method to solve the nonlinear programming problem. We prove 2-step superlinear convergence. In the process of analyzing this asymptotic behavior, we compare our method (theoretically) to the popular successive quadratic programming approach.

Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons

Abstract: A first order criterion for pseudo-convexity and second order criteria for quasi-convexity and pseudo-convexity are given for twice differentiable functions on open convex sets. The relationships between these second order criteria and other known criteria are also analysed. Finally, the numbers of operations required to verify these criteria are calculated and compared.

A capacitance matrix method for Dirichlet problem on polygon region

Abstract: An efficient algorithm for the solution of linear equations arising in a finite element method for the Dirichlet problem is given. The cost of the algorithm is proportional toN 2log2 N (N=1/h) where the cost of solving the capacitance matrix equations isNlog2 N on regular grids andN 3/2log2 N on irregular ones.

Nonlinear-wave effects on fixed and floating bodies

Abstract: A numerical method for calculating the interaction of steep (nonlinear) ocean waves with large fixed or floating structures of arbitrary shape is described. The interaction is treated as a transient problem with known initial conditions corresponding to still water in the vicinity of the structure and a prescribed incident waveform approaching it. The development of the flow, together with the associated fluid forces and structural motions, are obtained by a time-stepping procedure in which the flow at each time step is calculated by an integral-equation method based on Green's theorem. A few results are presented for two reference situations and these serve to illustrate the effects of nonlinearities in the incident waves.

Analysis of a multilevel iterative method for nonlinear finite element equations

Abstract: The multilevel iterative technique is a powerful technique for solving the systems of equations associated with discretized partial differential equations. We describe how this technique can be combined with a globally convergent approximate Newton method to solve nonlinear partial differential equations. We show that asymptotically only one Newton iteration per level is required; thus the complexity for linear and nonlinear problems is essentially equal.

New Method for Tidal Current Computation

Abstract: The computation of two-dimensional tidal currents using the Alternating Direction Implicit Method (ADI) can be subject to numerical attenuation, parasitic oscillations, and poor reproduction of wave propagation when large time steps are used. The new method described in the paper is designed to overcome these difficulties. It is based on a fractional step method in which momentum advection is calculated using the method of characteristics, horizontal momentum diffusion is calculated using an implicit finite difference scheme, and wave propagation is calculated using an iterative alternating direction implicit algorithm. The resulting method has been incorporated in the CYTHERE-ES1 modelling system, in which tidal flat flooding and drying as well as wind effects and Coriolis acceleration are taken into account. The basic principles of the method, as well as its application to four schematic test cases and two engineering studies, are described.

The finite element method for parabolic equations

Abstract: In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.

Integral equations for fluids of linear molecules

Abstract: A general procedure is described that puts the practice of integral equation theory for molecular fluids on a par with that of simple fluids: any integral equation approximation can be solved for any intermolecular potential with no additional approximations beyond those inherent in numerical analysis. The essential elements are expansions in spherical harmonics and numerical evaluation of the spherical harmonic coefficients of the pair distribution function. An explicit formula is derived giving the Helmholtz free energy from the computed coefficients.