Showing papers on "Discretization published in 1986"

The construction of preconditioners for elliptic problems by substructuring. I

Abstract: In earlier parts of this series of papers, we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems. The resulting algorithms are well suited for implementation on computers with parallel architecture. In this paper, we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners. The iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns. These preconditioners involve an incomplete Chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work. Theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods.

Random field finite elements

Abstract: The probabilistic finite element method (PFEM) is formulated for linear and non-linear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in non-linear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem and a two-dimensional plane-stress beam bending problem. The moments calculated compare favourably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.

Absorbing boundary conditions for difference approximations to the multi-dimensional wave equation

Abstract: Construction des conditions aux limites absorbantes pour l'equation des ondes multidimensionnelle a l'aide d'approximations aux differences. Formulation sous forme canonique des conditions aux limites absorbantes analytiques

Split-step methods for the solution of the nonlinear Schro¨dinger equation

Abstract: A split-step method is used to discretize the time variable for the numerical solution of the nonlinear Schrodinger equation. The space variable is discretized by means of a finite difference and a Fourier method. These methods are analyzed with respect to various physical properties represented in the NLS. In particular it is shown how a conservation law, dispersion and instability are reflected in the numerical schemes. Analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived. These results are demonstrated by numerical examples.

Computer modeling of heat flow in welds

Abstract: This paper summarizes progress in the development of methods, models, and software for analyzing or simulating the flow of heat in welds as realistically and accurately as possible. First the fundamental equations for heat transfer are presented and then a formulation for a nonlinear transient finite element analysis (FEA) to solve them is described. Next the magnetohydrodynamics of the arc and the fluid mechanics of the weld pool are approximated by a flux or power density distribution selected to predict the temperature field as accurately as possible. To assess the accuracy of a model, the computed and experimentally determined fusion zone boundaries are compared. For arc welds, accurate results are obtained with a power density distribution in which surfaces of constant power density are ellipsoids and on radial lines the power density obeys a Gaussian distribution. Three dimensional, in-plane and cross-sectional kinematic models for heat flow are defined. Guidelines for spatial and time discretization are discussed. The FEA computed and experimentally measured temperature field,T(x, y, z, t), for several welding situations is used to demonstrate the effect of temperature dependent thermal properties, radiation, convection, and the distribution of energy in the arc.

Generalised meshes for quantum mechanical problems

Abstract: A new method to discretise Schrodinger equations on a mesh is described. This method is based on an accurate approximation of a variational calculation. The regularly spaced mesh and meshes based on the zeros of orthogonal polynomials are studied in detail. It is shown that with each type of mesh is associated a particular kinetic energy operator and an optimal formula for its discretized form. The applications to some simple potential problems show that the method is very accurate as well as very simple. Applications to many-body problems indicate that the accuracy of the results is improved by an order of magnitude with respect to conventional mesh calculations.

Estimation of the coefficients of a diffusion from discrete observations

Abstract: Estimating the parameters of a diffusio,. we give a measure of the loss of information due to discretization, when the sampling interval is. We emphasize differences that depend on whether σ2 is known or unknown.

On the multi-level splitting of finite element spaces

Abstract: In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log κ)2) where κ is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like $$O\left( {\left( {\log \frac{1}{h}} \right)^2 } \right)$$ instead of $$O\left( {\left( {\frac{1}{h}} \right)^2 } \right)$$ for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.

Numerical analysis of parametrized nonlinear equations

Abstract: Some Sample Problems Some Background Material Solution Manifolds and Their Parameterizations Discretization Errors One-Distributions and Augmented Equations A Continuation Method Some Numerical Examples The Computation of Limit Points Differential Equations on Manifolds Error Estimates and Related Topics References Index.

The fast adaptive composite grid (FAC) method for elliptic equation

Abstract: The fast adaptive composite grid (FAG) method is a systematic process for solving differential boundary value problems. FAC uses global and local uniform grids both to define the composite grid problem and to interact for its fast solution. It can with little added cost substantially improve accuracy of the coarse grid solution and is very suitable for vector and parallel computation. This paper develops both the theoretical and practical aspects of FAC as it applies to elliptic problems, 1. Introduction. The need for local resolution in physical models occurs frequently in practice. Special local features of the forcing function, operator coefficients, boundary, and boundary conditions can demand resolution in restricted regions of the domain that is much finer than the required global resolution. It is important that the discretization and solution processes account for this locally, that is, that the local phenomena do not precipitate a dramatic increase in the overall computation. Unfortunately, this objective of efficiently adapting to local features is often in conflict with the solution process: equation solvers can degrade or even fail to apply in the presence of varying discretization scales; data structures that account for irregular grids can be cumbersome; the computer architecture may not be able to effectively account for grid irregularity (e.g., " vectorizability" may be inhibited); etc. In fact, even the discretization process itself may find difficulty with this objective: for finite differences, it is problematic to develop accurate difference formulae for irregular grids; for finite elements, this objective is reflected in the substantial overhead costs needed to automate the discretization. The fast adaptive composite grid method (FAC (11)) is a discretization and solution method designed to achieve efficient local resolution by constructing the

The Finite-Difference-Time-Domain Method and its Application to Eigenvalue Problems

Abstract: This paper describes the application of the finite-difference method in the time domain to the solution of three-dimensional (3-D) eigenvalue problems. Maxwell's equations are discretized in space and time, and steady-state solutions are then obtained via Fourier transform. While achieving the same accuracy and versatility as the TLM method, the finite-difference-time-domain (FD-TD) method requires less than half the CPU time and memory under identical simulation conditions. Other advantages over the TLM method include the absence of dielectric boundary errors in the treatment of 3-D inhomogeneous planar structures, such as microstrip. Some numerical results, including dispersion curves of a microstrip on anisotropic substrate, are presented.

A three‐dimensional finite‐element model for simulating water flow in variably saturated porous media

Abstract: A three-dimensional finite-element model for simulating water flow in variably saturated porous media is presented. The model formulation is general and capable of accommodating complex boundary conditions associated with seepage faces and infiltration or evaporation on the soil surface. Included in this formulation is an improved Picard algorithm designed to cope with severely nonlinear soil moisture relations. The algorithm is formulated for both rectangular and triangular prism elements. The element matrices are evaluated using an “influence coefficient” technique that avoids costly numerical integration. Spatial discretization of a three-dimensional region is performed using a vertical slicing approach designed to accommodate complex geometry with irregular boundaries, layering, and/or lateral discontinuities. Matrix solution is achieved using a slice successive overrelaxation scheme that permits a fairly large number of nodal unknowns (on the order of several thousand) to be handled efficiently on small minicomputers. Six examples are presented to verify and demonstrate the utility of the proposed finite-element model. The first four examples concern one- and two-dimensional flow problems used as sample problems to benchmark the code. The remaining examples concern three-dimensional problems. These problems are used to illustrate the performance of the proposed algorithm in three-dimensional situations involving seepage faces and anisotropic soil media.

Very High Order Accurate TVD Schemes

Abstract: A systematic procedure for constructing semi-discrete families of 2m - 1 order accurate, 2m order dissipa-tive, variation diminishing, 2m + 1 point band width, conservation form approximations to scalar conservation laws is presented. Here m is an integer between 2 and 8. Simple first order forward time discretization, used together with any of these approximations to the space derivatives, also results in a fully discrete, variation diminishing algorithm. These schemes all use simple flux limiters, without which each of these fully discrete algorithms is even linearly unstable. Extensions to systems, using a nonlinear field-by-field decomposition are presented, and shown to have many of the same properties as in the scalar case. For linear systems, these nonlinear approximations are variation diminishing, and hence convergent. A new and general criterion for approximations to be variation diminishing is also given. Finally, numerical experiments using some of these algorithms are presented.

Time discretization of an integro-differential equation of parabolic type

Abstract: The stability and convergence properties of two time discretizations of an integro-differential equation of parabolic type are studied. The methods reduce to the backward-Euler and Crank–Nicolson methods if the integral term is absent. The integral term is approximated in each case by a quadrature rule with relatively high-order truncation error, so that a relatively large time step can be used for the quadrature, in order to reduce the memory and computational requirements of the method.

Linear stability of plane couette flow of an upper convected maxwell fluid

Abstract: We investigate the linear stability of plane Couette flow of an upper convected Maxwell fluid using a spectral method to compute the eigenvalues. No instabilities are found. This is in agreement with the results of Ho and Denn [1] and Lee and Finlayson [2], but contradicts “proofs” of instability by Gorodtsov and Leonov [3] and Akbay and Frischmann [4,5]. The errors in those arguments are pointed out. We also find that the numerical discretization can generate artificial instabilities (see also [1,6]). The qualitative behavior of the eigenvalue spectrum as well as the artificial instabilities is discussed.

Multi‐dimensional discretization scheme for the hydrodynamic model of semiconductor devices

Abstract: A discretization technique is proposed for the multi‐dimensional, steady‐state hydrodynamic model of semiconductor devices, and a derivation of the model's appropriate boundary conditions is given. The model includes the complete balance equations for charge, momentum and energy, coupled with Poisson's equation, thus accounting for both diffusion and convection phenomena. The technique, like the Scharfetter—Gummel scheme for the simpler drift‐diffusion model, provides an efficient method for solving the hydrodynamic equations, allowing for a more detailed investigation of carrier dynamics in semiconductor devices.

A mesh-independence principle for operator equations and their discretizations

Abstract: The mesh-independence principle asserts that, when Newton’s method is applied to a nonlinear equation between some Banach spaces as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration and, as a consequence, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved only for certain classes of boundary value problems. In this paper a proof is presented for a general class of operator equations and discretizations. It covers the earlier results and extends them well beyond the cases that have been considered before.

Weakly Singular Discrete Gronwall Inequalities

Abstract: Generalizations of the classical Gronwall inequality when the kernel of the associated integral equation is weakly singular are presented. The continuous and discrete versions are both given; the former is included since it suggests the latter by analogy. This work is motivated by convergence studies of discretization methods for Volterra integral and integro-differential equations. The results are all given in a form designed to be of most use to numerical analysts. Es werden Verallgemeinerungen der klassischen Gronwallschen Ungleichung fur den Fall angegeben, das der Kern der zugeordneten Integralgleichung schwach singular ist. Es wird sowohl die stetige als auch die diskrete Version angegeben. Die stetige wurde mit einbezogen, weil sie eine analoge Behandlung der diskreten anregt. Die Arbeit wurde durch Konvergenzuntersuchungen von Diskretisierungsmethoden fur Volterrasche Integral- und Integro-Differentialgleichungen motiviert. Samtliche Ergebnisse werden in einer Form gegeben, die sehr brauchbar fur Spezialisten in Numerischer Analysis ist.

Numerical study of a relaxed variational problem from optimal design

Abstract: We revisit a well-known problem of optimal design, the placement of two elastic materials in the cross-section of a rod for maximum torsional rigidity. Another interpretation is the arrangement of two viscous fluids in a pipe for maximum flux under Poiseuille flow. The existence theory allows mixing on a microscopic scale, producing composite materials, and solving a relaxed version of the original design problem. This paper demonstrates that relaxation is as important for calculation as it is for existence. We minimize a discretized version of the relaxed problem using Newton's method; each quadratic approximation is solved by a multigrid method. This allows for greater resolution than previously published calculations, which were based on gradient flow.

A stability analysis of incomplete LU factorizations

Abstract: : The combination of iterative methods with preconditionings based on incomplete LU factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations. In this paper, we show that there are some settings in which the incomplete LU preconditioners are not effective, and we demonstrate that their poor performance is due to numerical instability. Our analysis consists of an analytic and numerical study of a sample two-dimensional non-self-adjoint elliptic problem discretized by several finite difference schemes. (Author)

A unified definition for the discrete-time, discrete-frequency, and discrete-time/Frequency Wigner distributions

Abstract: Various discrete definitions of the Wigner distribution (WD) for discrete-time signals have been proposed in previous works. The formulation developed in this paper leads to natural and unified definitions of discrete versions of the WD. They are directly related to the continuous and preserve most of its properties. The discretization is first considered in the time domain (DTWD), in the frequency domain (DFWD), and then in both domains simultaneously (DTFWD). In each case, the aliasing problem is studied and generalized interpolation formulas allowing the reconstruction of the continuous WD are derived. The DTFWD is particulary relevant for computer implementation of the WD.

Petrov-Galerkin formulations with weighting functions dependent upon spatial and temporal discretization: Applications to transient convection-diffusion problems

Abstract: A new Petrov-Galerkin finite element formulation has been proposed for transient convection-diffusion problems. Most Petrov-Galerkin formulations take into account the spatial discretization and the weighting functions so developed give satisfactory solutions for steady state problems. Though these schemes can be used for transient problems, there is scope for improvement. The schemes proposed here, which take into account temporal as well as spatial discretization, provide improved solutions. In view of the generality of the differential equation being solved, these schemes can be implemented for any physical problem which is governed by the transient convection-diffusion equation. It is also expected that these schemes, suitably adapted, will improve the numerical solutions of the compressible Euler and Navier-Stokes equations.

Solution of the nonlinear Poisson equation of semiconductor device theory

Abstract: A new iterative method for solving the discretized nonlinear Poisson equation of semiconductor device theory is presented This method has two main advantages First, it converges for any initial guess (global convergence) Secondly, the values of electric potential are updated at each mesh point by means of explicit formulas (that is, without the solution of simultaneous equations) The first property makes this method quite robust, while the second allows for the implementation of the method on computers with small RAMs Some numerical results obtained by this method are reported

An event-based simulation model of moisture and energy fluxes at a bare soil surface

Abstract: Land surface energy and water balances can be calculated by solving the partial differential equations governing vertical water and heat flow in the soil. Solution methodologies relying on standard discretization procedures are computationally intensive and are therefore poorly suited for long-term-simulations or Monte-Carlo simulations. As an alternative,a certain degree of accuracy may be forfeited in exchange for great reductions in computational effort by using an event-based simulation model. Since it uses closed-form solutions of the governing equations as basic building blocks, the event-based model avoids most of the work associated with discretization. The use of time condensation and simplified soil moisture kinematics allows these closed-form solutions to serve in continuous simulations under randomly varying forcing. A modified force-restore model of soil temperature provides the necessary link between the energy and water balances. In comparison with finite element solutions of a detailed set of partial differential equations governing water and heat transport in soil, the event-based model closely reproduced average energy and water balances and surface temperatures and decreased computational effort by a factor of at least a hundred.

Is SOR Color-Blind?

Abstract: The work of Young in 1950, see Young [1950], [1971], showed that the Red/ Black ordering and the natural rowwise ordering of matrices with Property A, such as those arising from the 5-point discretization of Poisson's equation, lead to SOR iteration matrices with identical eigenvalues. With the advent of parallel computers, multicolor point SOR schemes have been proposed for more complicated stencils on 2-dimensional rectangular grids, see Adams and Ortega [1982] for example, but to our knowledge, no theory has been provided for the rate of convergence of these methods relative to that of the natural rowwise scheme.New results show that certain matrices may be reordered so the resulting multicolor SOR matrix has the same eigenvalues as that for the original ordering. In addition, for a wide range of stencils, we show how to choose multicolor orderings so the multicolor SOR matrices have the same eigenvalues as the natural rowwise SOR matrix. The strategy for obtaining these orderings is based on “data flow” concepts and can be used to reach Young's conclusions above for the 5-point stencil.The importance of these results is threefold. Firstly, a constructive and easy means of finding these multicolorings is a direct consequence of the theory; secondly, multicolor SOR methods can be found that have the same rate of convergence as the natural rowwise SOR method for a wide range of stencils used to discretize partial differential equations; and thirdly, these multicolor SOR methods can be efficiently implemented on a wide class of parallel computers.

An Improved Control Volume Finite-Element Method for Heat and Mass Transfer, and for Fluid Flow Using Equal-Order Velocity-Pressure Interpolation

Abstract: A new control volume-based finite-element method for the solution of heat, mass, and momentum transfer equations is presented. As compared with other similar methods, the new feature of the proposed method lies in the choice of the particular shape function that accounts explicitly for the source terms in the transport equations. The use of such a shape function for interpolating the velocity components while discretizing the continuity equation leads, very naturally, to a consistent equal-order formulation that permits velocity and pressure to be computed at all the grid points in the domain. The method is applied to a number of test problems, and in all cases the results are found to be more accurate than those obtained by its predecessors.