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Showing papers on "Discretization published in 1979"


Book
Roger Temam1
01 Jan 1979
TL;DR: This paper presents thediscretization of the Navier-Stokes Equations: General Stability and Convergence Theorems, and describes the development of the Curl Operator and its application to the Steady-State Naviers' Equations.
Abstract: I. The Steady-State Stokes Equations . 1. Some Function Spaces. 2. Existence and Uniqueness for the Stokes Equations. 3. Discretization of the Stokes Equations (I). 4. Discretization of the Stokes Equations (II). 5. Numerical Algorithms. 6. The Penalty Method. II. The Steady-State Navier-Stokes Equations . 1. Existence and Uniqueness Theorems. 2. Discrete Inequalities and Compactness Theorems. 3. Approximation of the Stationary Navier-Stokes Equations. 4. Bifurcation Theory and Non-Uniqueness Results. III. The Evolution Navier-Stokes Equations . 1. The Linear Case. 2. Compactness Theorems. 3. Existence and Uniqueness Theorems. (n < 4). 4. Alternate Proof of Existence by Semi-Discretization. 5. Discretization of the Navier-Stokes Equations: General Stability and Convergence Theorems. 6. Discretization of the Navier-Stokes Equations: Application of the General Results. 7. Approximation of the Navier-Stokes Equations by the Projection Method. 8. Approximation of the Navier-Stokes Equations by the Artificial Compressibility Method. Appendix I: Properties of the Curl Operator and Application to the Steady-State Navier-Stokes Equations. Appendix II. (by F. Thomasset): Implementation of Non-Conforming Linear Finite Elements. Comments.

4,486 citations


Journal ArticleDOI
TL;DR: In this paper, a numerical technique was developed to solve the three-dimensional (3-D) potential distribution about a point source of current located in or on the surface of a half-space containing an arbitrary 3-D conductivity distribution.
Abstract: A numerical technique has been developed to solve the three‐dimensional (3-D) potential distribution about a point source of current located in or on the surface of a half‐space containing an arbitrary 3-D conductivity distribution. Self‐adjoint difference equations are obtained for Poisson’s equation using finite‐difference approximations in conjunction with an elemental volume discretization of the lower half‐space. Potential distribution at all points in the set defining the subsurface are simultaneously solved for multiple point sources of current. Accurate and stable solutions are obtained using full, banded, Cholesky decomposition of the capacitance matrix as well as the recently developed incomplete Cholesky‐conjugate gradient iterative method. A comparison of the 2-D and 3-D simple block‐shaped models, for the collinear dipole‐dipole array, indicates substantially lower anomaly indices for inhomogeneities of finite strike‐extent. In general, the strike‐extents of inhomogeneities have to be approxi...

391 citations


Journal ArticleDOI
TL;DR: PDECOL, new computer software package for numerically solving coupled systems of nonlinear partial differential equations (PDE's) in one space and one time dimension, is discussed.
Abstract: PDECOL, new computer software package for numerically solving coupled systems of nonlinear partial differential equations (PDE's) in one space and one time dimension, is discussed. The package implements finite element collocation methods based on piecewise polynomials for the spatial discretization techniques. The time integration process is then accomplished by widely acceptable procedures that are generalizations of the usual methods for treating time-dependent partial differental equations. PDECOL is unique because of its flexibiility both in the class of problems it addresses and in the variety of methods it provides for use in the solution process. High-order methods (as well as low-order ones) are readily available for use in both the spatial and time discretization procedures. The time integration methods used feature automatic time step size and integration formula order selection so as to solve efficiently the problem at hand and yet achieve a user-specific time integration error level. PDECOL consists of a collection of 19 subroutines written in reasonably standard Fortran, and therefore is quite portable. No special hardware features are required. PDECOL is designed to solve broad classes of difficult systems of partial differential equations that descrobe physical processes. 4 figures, 1 table. (RWR)

294 citations


Journal ArticleDOI
TL;DR: In this paper, a generalization to three dimensions of the discrete wave number representation method was presented to study the near field of a 3D seismic source embedded in a layered medium, where elastic wave fields are represented by a superposition of plane waves propagating in discrete directions.
Abstract: We present the generalization to three dimensions of the discrete wave number representation method of Bouchon and Aki (1977). The method is developed to study the near field of a three-dimensional seismic source embedded in a layered medium. The elastic wave fields are represented by a superposition of plane waves propagating in discrete directions. The discretization is exact and results from a periodic two-dimensional arrangement of sources. The accuracy of the method is checked, in the case of a rectangular dislocation source radiating in an infinite medium, by comparing the results obtained with Madariaga's (1978) exact solution. Examples of the calculation of strong ground motion produced by a thrust fault and a strike slip fault are presented.

280 citations


Journal ArticleDOI
TL;DR: In this article, a direct approach is employed to obtain a general formulation of plate bending problems in terms of a pair of singular integral equations involving displacement, normal slope, bending moment and shear on the plate boundary.

263 citations


Journal ArticleDOI
TL;DR: In this paper, a modified transonic mixed-type equation is proposed to compute transonic flows around cylinders and airfoils with special emphasis on the explicit methods that are suitable for vector processing on the STAR 100 computer.
Abstract: New methods for transonic flow computations based on the full potential equation in conservation form are presented. The idea is to modify slightly the density (due to the artificial viscosity in the supersonic region), and solve the resulting elliptic-like problem iteratively. It is shown that standard discretization techniques (central differencing) as well as some standard iterative procedures (SOR, ADI, and explicit methods) are applicable to the modified transonic mixed-type equation. Calculations of transonic flows around cylinders and airfoils are discussed with special emphasis on the explicit methods that are suitable for vector processing on the STAR 100 computer.

180 citations


Journal ArticleDOI
TL;DR: In this article, the computer code Duvorol, dealing with the computation of three-dimensional rolling contact with dry friction, is described, which is based on the variational principle of Duvaut and Lions for dry friction.
Abstract: In this paper the computer code Duvorol, dealing with the computation of three-dimensional rolling contact with dry friction, is described. It is based on the variational principle of Duvaut and Lions for dry friction, which leads to an incremental theory. The relevant properties of Duvorol are: 1Generality. All half-space steady-state rolling contact problems with Hertzian normal contact can be treated. 2Reliability. The total tangential force is always found with reasonable accuracy by a standard discretization. 3Speed. On an IBM 370/158 the calculation of a case then takes only several seconds.

144 citations


Journal ArticleDOI
TL;DR: In this article, an integral equation method for the numerical solution of the linear problems of elasticity of the homogeneous continuum was proposed. But this method is not suitable for the analysis of plate and axisymmetric cases.
Abstract: This paper concerns an integral equation method for the numerical solution of the linear problems of elasticity of the homogeneous continuum. The mathematical form adopted is sufficiently general to refer to both plane and three dimensional analysis as well as to analysis of plates and specialized axisymmetric cases. Having indirectly deduced the integral equations, the resolving procedure is based on variational principles and uses a convenient discretization of the boundary through the use of finite elements. Particular attention is given to the definition of the asymptotic conditions of the elementary displacement field to theoretically ensure a priori the conditioning of the resolving algebraic equations without any restriction as regards the nature of the boundary and the conditions thereby imposed. Efficiency, praticality and flexibility of use are characteristics particular of this method and have already been widely tested by the author in varying applications. Further information on the above can be found in the bibliography cited.

128 citations


Journal ArticleDOI
TL;DR: In this paper, a study of spatial discretization schemes for the multigroup discrete-ordinates transport equations in slab geometry is described, and the purpose of the study is to determine the most computationally efficient method, defined as the one that produces the minimum error for a given cost.
Abstract: A study of spatial discretization schemes for the multigroup discrete-ordinates transport equations in slab geometry is described. The purpose of the study is to determine the most computationally efficient method, defined as the one that produces the minimum error for a given cost. Cost is defined as the total amount of computer time required to complete one inner iteration, given a limit on storage, and three error norms are used to measure the accuracies of edge fluxes, cell average fluxes, and integral parameters. Three test problems are studied: the first is a model one-group problem examined in detail, while the second and third are more realistic multigroup problems. One conclusion is that a new method, labeled linear characteristic, significantly outperforms all other methods that have been implemented up to the present time. 15 references.

119 citations


Journal ArticleDOI
TL;DR: In this paper, a method for computing the scattering and diffraction of harmonic SH waves by an arbitrarily shaped alluvial valley is presented, where the problem is formulated in terms of a system of Fredholm integral equations of the first kind with the integration paths outside the boundary.
Abstract: A method is presented to compute the scattering and diffraction of harmonic SH waves by an arbitrarily shaped alluvial valley. The problem is formulated in terms of a system of Fredholm integral equations of the first kind with the integration paths outside the boundary. A discretization scheme using line source solutions is employed and the boundary conditions are satisfied in the least-squares sense. Numerical results for amplification spectra for different geometries are presented. Agreement with known analytical solutions is excellent.

118 citations


Journal ArticleDOI
G. Chavent1
TL;DR: A synthetic presentation of the identification methods proposed in the litterature and the minimization of the output least square seems to be the most popular, though its implementation may vary considerably from one author to the other.

Journal ArticleDOI
TL;DR: Collocation at Gauss points is shown to be a high order accurate discretization of certain unconstrained optimal control problems and best possible convergence rates are established along with superconvergence results.
Abstract: Collocation at Gauss points is shown to be a high order accurate discretization of certain unconstrained optimal control problems. Best possible convergence rates are established along with superconvergence results.

Journal ArticleDOI
TL;DR: In this article, the error to the discretization in time of a parabolic evolution equation by a single-step method or by a multistep method when the initial condition is not regular was studied.
Abstract: We study the error to the discretization in time of a parabolic evolution equation by a single-step method or by a multistep method when the initial condition is not regular. Introduction. The problem we are considering is the parabolic evolution equation 5 u'(t)+Au(t)=O, O 3 is documented in [8] and [2]. It is shown in [8] that for p > 3, rp is in fact strongly A(0p)-stable for some 0 < Op < ir/2. For small p, Op is close to ir/2 and in the special cases p = 3, 4, rp is A-stable. Examples of rational approximations to eZ which are strongly A(O)-stable with r(oo) = 0 are provided by the family r,,(z) developed in [2]. In the second part, we investigate error estimates when the discretization in time is carried out by means of a multistep method. Zlamal gives an error bound under the assumption that the operator A is selfadjoint and the method strongly A(O)-stable. Here, error estimates are obtained if the operator A is maximal sectorial and the method strongly A(0)-stable (O < 0 < ir/2). I. Semidiscretization in Time by a Single-Step Method.

Journal ArticleDOI
TL;DR: It is shown that the definition without limiting procedure by a formal series expansion has the same ambiguities as the definition through discretization, these last ones being related to ordering problems.
Abstract: We study functional integrals in phase space and show that the definition without limiting procedure by a formal series expansion has the same ambiguities as the definition through discretization, these last ones being related to ordering problems. We exhibit the way to obtain the prescription that makes the series expansion unambiguous and study the mechanism that makes the expansion independent of the chosen discretization.

Journal ArticleDOI
TL;DR: In this article, a finite element procedure for the dynamic analysis of fluid-structure systems is presented and evaluated, where the fluid is assumed to be inviscid and compressible and is described using an updated Lagrangian formulation.

Journal ArticleDOI
TL;DR: In this paper, the Onsager-Machlup Lagrangian defined by the functional-integral representation of general continuous Markov processes is derived explicitly and concisely based on continuous and differentiable paths connecting fixed end points in the convariant propagator, and the application of a recently developed Fourier-series analysis of these trajectories in locally flat spaces.
Abstract: The Onsager-Machlup Lagrangian defined by the functional-integral representation of general continuous Markov processes is derived explicitly and concisely. Attention is given both to the mathematics and its physical interpretation. The method is based on (i) the consideration of continuous and differentiable paths connecting fixed end points in the convariant propagator, (ii) the application of a recently developed Fourier-series analysis of these trajectories in locally flat spaces, and (iii) the use of the proper transformations between locally Euclidean and globally Riemannian geometries. The spectral analysis allows for arbitrary paths rather than an a priori straight line even in the short-time propagator and avoids any ad hoc discretization rule. Specialized to globally flat spaces the result agrees with formulas given by Stratonovich, Horsthemke and Bach, Graham, and Dekker. It is demonstrated that a unique covariant path integral is equivalent to a whole class of stochastically equivalent lattice expressions.

Journal ArticleDOI
TL;DR: In this article, an accurate theory of fluctuations and noise in computer simulation of plasma is developed and the analytic method describes the space and time discretization exactly and the results reduce simply and correctly to the standard results of plasma kinetic theory.
Abstract: An accurate theory of fluctuations and noise in computer simulation of plasma is developed The analytic method describes the space and time discretization exactly and the results reduce simply and correctly to the standard results of plasma kinetic theory in the limit of small space and time step If the particles are imagined to be a Monte‐Carlo sampling of the phase space, then the fluctuations in this sampling, as modified by collective effects, are a concern of this paper This theory is of interest in theoretical and empirical studies to understand the character of representation by simulation methods of plasma processes such as transport

Journal ArticleDOI
TL;DR: In this article, a baroclinic primitive equations model is formulated using a variable resolution finite-element discretization in all three space dimensions, and the horizontal domain over which the model is integrated is a rectangle on a polar stereographic projection which approximately covers the Northern Hemisphere.
Abstract: A baroclinic primitive equations model is formulated using a variable resolution finite-element discretization in all three space dimensions. The horizontal domain over which the model is integrated is a rectangle on a polar stereographic projection which approximately covers the Northern Hemisphere. A wall boundary condition is imposed at this rectangular boundary giving rise to a well-posed initial boundary value problem. The mesh is specified to be of Cartesian product form with arbitrary non-uniform spacing. By choosing the mesh to be uniformly high over an area of interest and degrading smoothly away from this area, it is possible to use the model to produce a high-resolution local forecast for a limited time period. This choice of mesh avoids the noise problems of a so-called nested grid. A semi-implicit time discretization is used for efficiency. Some results for forecast periods of 24 and 48 h are also given to demonstrate its viability in an operational context.

Proceedings ArticleDOI
01 Jan 1979
TL;DR: Multi-level adaptive technique (MLAT) as discussed by the authors is a general strategy of solving continuous problems by cycling between coarser and finer levels of discretization, which provides very fast solvers together with adaptive, nearly optimal discretisation schemes to general boundary-value problems in general domains.
Abstract: The multi-level adaptive technique (MLAT) is a general strategy of solving continuous problems by cycling between coarser and finer levels of discretization. It provides very fast solvers together with adaptive, nearly optimal discretization schemes to general boundary-value problems in general domains. Here the state of the art is surveyed, emphasizing steady-state fluid dynamics applications, from slow viscous flows to transonic ones. Various new techniques are briefly discussed, including distributive relaxation schemes, the treatment of evolution problems, the combined use of upstream and central differencing, local truncation extrapolations, and other 'super-solver' techniques.

Journal ArticleDOI
TL;DR: The method of finite elements is found to be superior in improved accuracy, computer time and storage requirements, as well as programming implementation aspects and that results derived from the finite element analysis tend to converge asymptotically to corresponding experimental test data, as the discretization mesh fineness is increased.
Abstract: In this paper, an evaluation of the methods of finite elements and finite differences, as applied to nonlinear magnetic field problems in electrical machines, is presented. The evaluation covers the aspects of effectiveness, numerical accuracy, modeling implementation considerations as well as computer storage and execution time requirements of the two methods. The evaluation includes static as well as sinusoidally time varying fields. The method of finite elements is found to be superior in improved accuracy, computer time and storage requirements, as well as programming implementation aspects. A crucial finding of this investigation is that results derived from the finite element analysis tend to converge asymptotically to corresponding experimental test data, as the discretization mesh fineness is increased. This is not the case for finite differences, where the results strongly indicate that there are lower bounds beyond which inherent numerical error cannot be decreased by an increase in the degree of fineness of the corresponding discretization mesh. Details of the analysis, on which these findings stand, are presented here.

Journal ArticleDOI
TL;DR: In this article, the eigenvalues of a Fokker-Planck equation involving a critical point have been computed by means of a simple discretization technique, which smoothly connects the monostable case above the critical point with the bistable case below it.

Journal ArticleDOI
TL;DR: In this paper, a new numerical method for the reconstitution of the inhomogeneous media conductivity is described, which is based on a space time discretization of an exact integral equation.
Abstract: A new numerical method for the reconstitution of the inhomogeneous media conductivity is described. The conducting medium is nonmagnetic, and its relative permittivity equals one. Its frequency-independent conductivity varies normally to its interface. The medium is normally illuminated by a transverse electromagnetic (TEM) plane wave of causal time dependence. The reconstruction process is a direct approach to the inverse problem. It is based on a space time discretization of an exact integral equation. The conductivity profile is determined step-by-step without approximations other than numerical. Some examples are given to illustrate the most interesting properties of the method; special attention is given to the simulation of experimental errors.

Journal ArticleDOI
TL;DR: In this paper, the authors give a precise definition of the functional integrals involved by means of different discretization prescriptions, and discuss general covariance and derive in a unified way, all the lagrangians proposed in the literature together with their associated discretizations.
Abstract: We give a precise definition of the functional integrals involved by means of different discretization prescriptions. The value of the functional integral is prescription dependent and this leads to a variety of representations with different Lagrangians. We discuss general covariance and we derive in a unified way, all the lagrangians proposed in the literature together with their associated discretizations. We make also a critical examination of previous results.

Journal ArticleDOI
TL;DR: In this paper, the Navier-Stokes equations in the form of a single, fourth order equation for stremfunction are recast into arestricted variational principles, which form finite element discretization.

Journal ArticleDOI
TL;DR: In this paper, a unified treatment of modal control of spinning flexible spacecraft is presented, where the equations of motion of the spacecraft are hybrid, i.e., they consist of ordinary differential equations for the rotational motion and partial differential equation for the elastic motion.
Abstract: A unified treatment of modal control of spinning flexible spacecraft is presented. The equations of motion of the spacecraft are hybrid, i.e., they consist of ordinary differential equations for the rotational motion and partial differential equations for the elastic motion. Problems involving control of distributed-p arameter systems are generally discretized and the actual control is implemented on the discrete systems. An efficient method for control of linear gyroscopic systems is via model synthesis. There remains, however, the question as to how modal control of discrete systems is related to the control of the actual distributed-parameter systems. It is the object of this paper to provide the information concerning the spatial distribution of the sensors and actuators and to relate this information to the discrete modal control.

Journal ArticleDOI
TL;DR: In this paper, a third order accurate method is proposed for the numerical solution of the one-dimensional Stefan problem, which is specially appropriate for the computation of solutions which admit singularities at the initial time or on the boundary.

Journal ArticleDOI
TL;DR: This paper is concerned with finding numerical solutions of MPBVP's by converting these to equivalent initial value problems by converting the discretization of certain BVP's for partial differential equations over irregular domains with the method of lines.

Journal ArticleDOI
TL;DR: In this paper, a special time-optimal parabolic boundary-value control problem describing a one-dimensional heat-diffusion process is solved numerically using a bang-bang principle recently proved by Lempio.
Abstract: A special time-optimal parabolic boundary-value control problem describing a one-dimensional heat-diffusion process is solved numerically. Using a bang-bang principle recently proved by Lempio, this problem can be transformed in such a way that the variables are jumps of bang-bang controls. A discretization is performed in two steps, and the convergence of the approximate solutions is proved. Finally, an algorithm to solve the discrete problem is developed and some numerical results are discussed.

Journal ArticleDOI
TL;DR: A class of preconditioned conjugate gradient methods for the numerical solution of the homogeneous Dirichlet problem for the biharmonic operator, which shows a low computational complexity in both number of computer operations and demand of storage.
Abstract: The homogeneous Dirichlet problem for the biharmonic operator is solved as the variational formulation of two coupled second-order equations. The discretization by a mixed finite element model results in a set of linear equations whose coefficient matrix is sparse, symmetric but indefinite. We describe a class of preconditioned conjugate gradient methods for the numerical solution of this linear system. The precondition matrices correspond to incomplete factorizations of the coefficient matrix. The numerical results show a low computational complexity in both number of computer operations and demand of storage.

Journal ArticleDOI
TL;DR: The Graph-Theoretical Field Model (GTM) as mentioned in this paper provides a unifying approach for developing numerical models of field and continuum problems by deriving discrete statements of the physical laws which govern the field behaviour.
Abstract: The Graph-Theoretical Field Model provides a unifying approach for developing numerical models of field and continuum problems. The methodology examines the field problem from the first stages of conceptualization without recourse to the governing differential equations of the field problem; this is accomplished by deriving discrete statements of the physical laws which govern the field behaviour. There are generally three laws, and these are modelled by the “cutset equations”, the “circuit equations”, and the “terminal equations”. In order to establish these three sets of equations it is expedient first to spatially discretize the field in a manner similar to the finite difference method and then to associate a linear graph (denoted as the field graph) with the spatial discretization. The concept of “through” and “across” variables, which underlies the cutset and circuit equations respectively, enables one to define the graph in an unambiguous manner such that each “edge” of the graph identifies a pair of complementary variables. From a knowledge of the constitutive properties and the boundary conditions of the field it is possible to associate terminal equations with sets of edges. Since the resulting sets of equations represent the field equations, these equations provide the basis for a complete (but approximate) solution to the field or continuum problem. In fact, this system approach uses a two part model: one for the components and another for the interconnection pattern of the components which renders the formulation procedures totally independent of the solution procedure. This paper presents the theoretical basis of the model and several graph-theoretic formulations for steady-state problems. Examples from heat conduction and small- deformation elasticity are included.