Showing papers on "Discretization published in 1985"

Computational Methods for Transient Analysis

Abstract: Preface. 1 . An Overview of Semidiscretization and Time Integration Procedures (T. Belytschko). 2 . Analysis of Transient Algorithms with Particular Reference to Stability Behavior (T.J.R. Hughes). 3 . Partitioned Analysis of Coupled Systems (K.C. Park and C.A. Felippa). 4 . Boundary-Element Methods for Transient Response Analysis (T.L. Geers). 5 . Dynamic Relaxation (P. Underwood). 6 . Dispersion of Semidiscretized and Fully Discretized Systems (H.L. Schreyer). 7 . Silent Boundary Methods for Transient Analysis (M. Cohen and P.C. Jennings). 8 . Explicit Lagrangian Finite-Difference Methods (W. Hermann and L.D. Bertholf). 9 . Implicit Finite Element Methods (M. Geradin, M. Hogge and S. Idelsohn). 10 . Arbitrary Lagrangian-Eulerian Finite Element Methods (J. Donea). Indices.

Block Preconditioning for the Conjugate Gradient Method

Abstract: Block preconditionings for the conjugate gradient method are investigated for solving positive definite block tridiagonal systems of linear equations arising from discretization of boundary value problems for elliptic partial differential equations. The preconditionings rest on the use of sparse approximate matrix inverses to generate incomplete block Cholesky factorizations. Carrying out of the factorizations can be guaranteed under suitable conditions. Numerical experiments on test problems for two dimensions indicate that a particularly attractive preconditioning, which uses special properties of tridiagonal matrix inverses, can be computationally more efficient for the same computer storage than other preconditionings, including the popular point incomplete Cholesky factorization.

Icosahedral Discretization of the Two-Sphere

Abstract: We present an almost uniform triangulation of the two-sphere, derived from the icosahedron, and describe a procedure for discretization of a partial differential equation using this triangular grid. The accuracy of our procedure is described by a strong theoretical estimate, and verified by large-scale numerical experiments. We also describe a data structure for this spherical discretization that allows fast computation on either a vector computer or an asynchronous parallel computer.

Time discretization of parabolic problems by the discontinuous Galerkin method

Abstract: Analyse dans un contexte general de la methode de Galerkin discontinue pour la discretisation temporelle de problemes de type parabolique. Etablissement d'estimations d'erreur aux points nodaux pour des donnees initiales regulieres ou non

Digital picture processing : an introduction

Abstract: 1. Introduction.- 1 Fundamentals of the Theory of Digital Signal Processing.- 2. Elements of Signal Theory.- 2.1 Signals as Mathematical Functions.- 2.2 Signal Space.- 2.3 The Most Common Systems of Basis Functions.- 2.3.1 Impulse Basis Functions.- 2.3.2 Harmonic Functions.- 2.3.3 Walsh Functions.- 2.3.4 Haar Functions.- 2.3.5 Sampling Functions.- 2.4 Continuous Representations of Signals.- 2.5 Description of Signal Transformations.- 2.5.1 Linear Transformations.- 2.5.2 Nonlinear Element-by-Element Transforms.- 2.6 Representation of Linear Transformations with Respect to Discrete Bases.- 2.6.1 Representation Using Vector Responses.- 2.6.2 Matrix Representations.- 2.6.3 Representation of Operators by Means of Their Eigenfunctions and Eigenvalues.- 2.7 Representing Operator with Respect to Continuous Bases.- 2.7.1 Operator Kernel.- 2.7.2 Description in Terms of Impulse Responses.- 2.7.3 Description Using Frequency Transfer Functions.- 2.7.4 Description with Input and Output Signals Referred to Different Bases.- 2.7.5 Description Using Eigenfunctions.- 2.8 Examples of Linear Operators.- 2.8.1 Shift-Invariant Filters.- 2.8.2 The Identity Operator.- 2.8.3 Shi ft Operator.- 2.8.4 Sampling Oparator.- 2.8.5 Gating Operator (Multiplier).- 3. Discretization and Quantization of Signals.- 3.1 Generalized Quantization.- 3.2 Concepts of Discretization and Element-by-Element Quantization.- 3.2.1 Discretization.- 3.2.2 Element-by-Element Quantization.- 3.3 The Sampling Theorem.- 3.4 Sampling Theory for Two-Dimensional Signals.- 3.5 Errors of Discretization and Restoration of Signals in Sampling Theory.- 3.6 Other Approaches to Discretization.- 3.7 Optimal Discrete Representation and Dimensionality of Signals.- 3.8 Element-by-Element Quantization.- 3.9 Examples of Optimum Quantization.- 3.9.1 Example: The Threshold Metric.- 3.9.2 Example: Power Criteria for the Absolute Value of the Quantization Error.- 3.9.3 Example: Power Criteria for the Relative Quantization Error.- 3.10 Quantization in the Presence of Noise. Quantization and Representation of Numbers in Digital Processors.- 3.11 Review of Picture-Coding Methods.- 4. Discrete Representations of Linear Transforms.- 4.1 Problem Formulation and General Approach.- 4.2 Discrete Representation of Shift-Invariant Filters for Band-Limited Signals.- 4.3 Digital Filters.- 4.4 Transfer Functions and Impulse Responses of Digital Filters.- 4.5 Boundary Effects in Digital Filtering.- 4.6 The Discrete Fourier Transform (DFT).- 4.7 Shifted, Odd and Even DFTs.- 4.8 Using Discrete Fourier Transforms.- 4.8.1 Calculating Convolutions.- 4.8.2 Signal Interpolation.- 4.9 Walsh and Similar Transforms.- 4.10 The Haar Tansform. Addition Elements of Matrix Calculus.- 4.11 Other Orthogonal Tansforms. General Representations. Review of Applications.- 5. Linear Transform Algorithms.- 5.1 Fast Algorithms of Discrete Orthogonal Transforms.- 5.2 Fast Haar Transform (FHT) Algorithms.- 5.3 Fast Walsh Transform (FWT) Algorithms.- 5.4 Fast Discrete Fourier Transform (FFT) Algorithms.- 5.5 Review of Other Fast Algorithms. Features of Two-Dimensional Transforms.- 5.5.1 Truncated FFT and FWT Algorithms.- 5.5.2 Transition Matrices Between Various Transforms.- 5.5.3 Calculation of Two-Dimensional Transforms.- 5.6 Combined DFT Algorithms.- 5.6.1 Combined DFT Algorithms of Real Sequences.- 5.6.2 Combined SDFT (1/2, 0) Algorithms of Even and Real Even Sequences.- 5.7 Recursive DFT Algorithms.- 5.8 Fast Algorithms for Calculating the DFT and Signal Convolution with Decreased Multiplication.- 6. Digital Statistical Methods.- 6.1 Principles of the Statistical Description of Pictures.- 6.2 Measuring the Grey-Level Distribution.- 6.2.1 Step Smoothing.- 6.2.2 Smoothing by Sliding Summation.- 6.2.3 Smoothing with Orthogonal Transforms.- 6.3 The Estimation of Correlation Functions and Spectra.- 6.3.1 Averaging Local Spectra.- 6.3.2 Masking (Windowing) the Process by Smooth Functions.- 6.3.3 Direct Smoothing of Spectra.- 6.4 Generating Pseudorandom Numbers.- 6.5 Measuring Picture Noise.- 6.5.1 The Prediction Method.- 6.5.2 The Voting Method.- 6.5.3 Measuring the Variance and the Auto-Correlation Function of Additive Wideband Noise.- 6.5.4 Evaluation of the Intensity and Frequency of the Harmonic Components of Periodic Interference and Other Types of Interference with Narrow Spectra.- 6.5.5 Evaluation of the Parameters of Pulse Noise, Quantization Noise and Strip-Like Noise.- 2 Picture Processing.- 7. Correcting Imaging Systems.- 7.1 Problem Formulation.- 7.2 Suppression of Additive Noise by Linear Filtering.- 7.3 Filtering of Pulse Interference.- 7.4 Correction of Linear Distortion.- 7.5 Correction of Amplitude Characteristics.- 8. Picture Enhancement and Preparation.- 8.1 Preparation Problems and Visual Analysis of Pictures.- 8.1.1 Feature Processing.- 8.1.2 Geometric Transformations.- 8.2 Adaptive Quantization of Modes.- 8.3 Preparation by Nonlinear Transformation of the Video Signal Scale.- 8.4 Linear Preparation Methods.- 8.5 Methods of Constructing Graphical Representation: Computer Graphics.- 8.6 Geometric Picture Transformation.- 8.6.1 Bilinear Interpolation.- 8.6.2 Interpolation Using DFT and SDFT.- 9. Measuring the Coordinates of Objects in Pictures.- 9.1 Problem Formulation.- 9.2 Localizing a Precisely Known Object in a Spatially Homogeneous Picture.- 9.3 Uncertainty in the Object and Picture Inhomogeneity. Localization in "Blurred" Pictures.- 9.3.1 "Exhaustive" Estimator.- 9.3.2 Estimator Seeking an Averaged Object.- 9.3.3 Adjustable Estimator with Fragment-by Fraament Optimal Fi1tering.- 9.3.4 Non-Adjustable Estimator.- 9.3.5 Localization in Blurred and Noisy Pictures.- 9.4 Optimal Localization and Picture Contours. Choice of Reference Objects.- 9.5 Algorithm for the Automatic Detection and Extraction of Bench-Marks in Aerial and Space Photographs.- 10-Conclusion.- References.

On Converting Optimal Control Problems into Nonlinear Programming Problems

Abstract: Direct solutions of the optimal control problem are considered. Two discretization schemes are proposed which are based on the parameterization of the control functions and on the parameterization of the control and the state functions, leading to direct shooting and direct collocation algorithms, respectively. The former is advantageous for problems with unspecified final state, the latter for prescribed final state and especially for stiff problems. The sparsity of the Jacobian matrix of the constraints and the Hessian matrix of the Lagrangian must be exploited in the direct collocation method in order to be efficient. The great advantage of the collocation approach lies in the availability of analytical gradients.

Discretization and simulation of stochastic differential equations

Abstract: We discuss both pathwise and mean-square convergence of several approximation schemes to stochastic differential equations. We then estimate the corresponding speeds of convergence, the error being either the mean square error or the error induced by the approximation on the value of the expectation of a functional of the solution. We finally give and comment on a few comparative simulation results.

A survey of preconditioned iterative methods for linear systems of algebraic equations

Abstract: We survey preconditioned iterative methods with the emphasis on solving large sparse systems such as arise by discretization of boundary value problems for partial differential equations. We discuss shortly various acceleration methods but the main emphasis is on efficient preconditioning techniques. Numerical simulations on practical problems have indicated that an efficient preconditioner is the most important part of an iterative algorithm. We report in particular on the state of the art of preconditioning methods for vectorizable and/or parallel computers.

The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes

Abstract: Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation.

Simulating hydrodynamics: A pedestrian model

Abstract: A Hele Shaw cell contains two fluids seperated by an interface. Because the fluids are held in a narrow regions between two plates the cell can be described by a set of two-dimensional hydrodynamic equations, which determine the velocity fields in the fluids as well as the motion of the interface between them. A discretized version of these equations can be implemented in terms of the motion of random walkers. The walkers have the effect of carrying pieces of the fluid from one place to another. They simulate a discrete version of the Laplace equation and obey the appropriate boundary conditions for the fluid. The walker-hydrodynamic connection is explored in the limiting situation in which the viscosity of one of the fluids vanishes. An algorithm is constructed and a few exemplary simulations are shown.

A New Method of Contaminant Plume Analysis

Abstract: This paper develops an analytical expression for contaminant transport from a finite source in a continuous flow regime. The model requires some numerical integration and its degree of accuracy for near-field problems depends on discretization procedures applied to the source boundary. A second model for a continuous source is developed by extending a well-known pulse model. This second model is particularly useful in that it permits the determination of several potential unknowns directly from a concentration distribution. These include the source concentration, source dimensions, the position of the center of mass which is the product of the seepage velocity and the time since the contaminant first entered the ground water, and up to three dispersivities for a three-dimensional problem. As a demonstration of its utility, this second model is applied with reasonable success to a well-defined field condition. A comparison of the two models indicates that, except for minor differences in the very near field, the results from each are virtually identical.

A discrete method of optimal control based upon the cell state space concept

Abstract: A discrete method of optimal control is proposed in this paper. The continuum state space of a system is discretized into a cell state space, and the cost function is discretized in a similar manner. Assuming intervalwise constant controls and using a finite set of admissible control levels (u) and a finite set of admissible time intervals (τ), the motion of the system under all possible interval controls (u, τ) can then be expressed in terms of a family of cell-to-cell mappings. The proposed method extracts the optimal control results from these mappings by a systematic search, culminating in the construction of a discrete optimal control table.

On Deterministic Control Problems: An Approximation Procedure for the Optimal Cost I. The Stationary Problem

Abstract: We study deterministic optimal control problems having stopping time, continuous and impulse controls in each strategy.We obtain the optimal cost, considered as the maximum element of a suitable set of subsolutions of the associated Hamilton–Jacobi equation, using an approximation method. A particular derivative discretization scheme is employed.Convergence of approximate solutions is shown taking advantage of a discrete maximum principle which is also proved.For the numerical solutions of approximate problems we use a method of relaxation type. The algorithm is very simple; it can be run on computers with small central memory.In Part I we study the stationary case, in Part II [SIAM J. Control Optim., 23 (1985), pp. 267–285] we study the nonstationary case.

A simple, complete numerical solution to the problem of diffraction of SH waves by an irregular surface

Abstract: We present a new method to calculate the diffraction of elastic SH waves by irregular surfaces. The technique developed is applicable to boundaries of arbitrary shape and steepness as well as to periodically corrugated surfaces; it is valid at all frequencies. The approach consists of determining the surface forces which cancel the incident stress along the surface. The method relies on the introduction of a periodicity in the surface shape and on a discretization of the boundary at regular spacing. The surface sources which radiate the scattered wave field are obtained by iteration. Examples of calculations and comparisons with other methods are presented.

The curved beam/deep arch/finite ring element revisited

Abstract: An attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformation. It is shown that with elements of finite size (i. e. a practical level of discretization at which reasonably accurate results can be expected), there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem. These lead to errors, described here as of the 'second kind' and a physical phenomenon called 'membrane locking'. It seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam/deep arch/finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints. What is emphasized is that the choice of shape functions, or subsequent operations to determine the discretized functionals, must consistently model the physical requirements the problem imposes on the field variables.

Eigenvalue optimization algorithms for structure/controller design iterations

Abstract: An eigenspace optimization approach is proposed and demonstrated for the design of feedback controllers for the maneuvers/vibration arrests of flexible structures. The algorithm developed is shown to be equally useful in sequential or simultaneous design iterations that modify the structural parameters, sensor/actuator locations, and control feedback gains. The approach is demonstrated using a differential equation model for the "Draper/RPL configuration." This model corresponds to the hardware used for experimental verification of large flexible spacecraft maneuver controls. A number of sensor/actuat or configuration s are studied vis-a-vis the degree of controllabili ty. Linear output feedback gains are determined using a novel optimization strategy. The feasibility of the approach is established, but more research and numerical studies are required to extend these ideas to truly high-dimensioned systems. Parameterization of the Controlled System's Eigenvalues and Eigenvectors C ONSIDER a linear structure (modeled by a finite element or similar discretization scheme) in which the configuration vector jc is governed by the system of differential equations

Synthetic SH seismograms in a laterally varying medium by the discrete wavenumber method

Abstract: Summary. We present a new method to calculate the SH wavefield produced by a seismic source in a half-space with an irregular buried interface. The diffractiog interface is represented by a distribution of body forces. The Green’s functions needed to solve the boundary conditions are evaluated using the discrete wavenumber method. Our approach relies on the introduction of a periodicity in the source-medium configuration and on the discretization of the interface at regular spacing. The technique developed is applicable to boundaries of arbitrary shapes and is valid at all frequencies. Some examples of calculation in simple configurations are presented showing the capabilities of the method.

The numerical solution of the helmholtz equation for wave propagation problems in underwater acoustics

Abstract: The Helmholtz Equation (-delta-K(2)n(2))u=0 with a variable index of refraction, n, and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems. A numerical algorithm was developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range. The equation is discretized using the finite element method, thus allowing for the modeling of complicated geometrices (including interfaces) and complicated boundary conditions. A global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes. The resulting large, non-selfadjoint system of linear equations with indefinite symmetric part is solved using the preconditioned conjugate gradient method applied to the normal equations. A new preconditioner is developed based on the multigrid method. This preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies. A heuristic argument is given that indicates the superior convergence properties of this preconditioner.

Integral equation solution for the transient electromagnetic response of a three-dimensional body in a conductive half-space

Abstract: The time‐domain integral equation for the three‐dimensional vector electric field is formulated as a convolution of the scattering current with the tensor Green’s function. The convolution integral is divided into a sum of integrals over successive time steps, so that a numerical scheme can be formulated with a time stepping approximation of the convolution of past values of the solution with the system impulse response. This, together with spatial discretization, leads to a matrix equation in which previous solution vectors are multiplied by a series of matrices and fed back into the system by adding to the primary field source vector. The spatial discretization, based on a modification of the usual pulse basis formulation in the frequency domain, includes an additional subset of divergence‐free basis functions generated by integrating the Green’s function around concentric closed rectangular paths. The inductive response of the body is more accurately modeled with these additional basis functions, and a...

On the consequence of discretization errors in the numerical calculation of viscoelastic flow

Abstract: Having recalled some theorems pertaining to strong solutions of the viscoelastic flow problem, the authors establish a parallel between these theorems and observations made in the numerical calculation of such flows by means of a mixed method. Numerical errors in the evaluation of the extra-stress tensor have dramatic consequences upon the other field variables for the flow of a Maxwell fluid, and the damage is limited with an Oldroyd-B fluid.