Showing papers on "Discretization published in 1978"
TL;DR: In this article, the authors give a functional-analytical treatment of discretization methods such as quadrature formula method for nonlinear integral equations, difference method for nonsmooth boundary value problems, etc.
Abstract: THIS survey paper gives a functional-analytical treatment of discretization methods such as quadrature formula method for nonlinear integral equations, difference method for nonlinear boundary value problems, etc. Two approaches to the convergence problem have been developed. The first of them (Section 3) is applicable to an equation with differentiable operator and rests on a remark that such an operator is locally almost linear. The second, less traditional approach (Section 4) is based on a topological concept, namely the invariance of the fixed point index under suitable approximations of an operator. As regards the approximation concepts, the paper is built on a relatively novel principle of regular convergence of operators (Section 2). In our fixed opinion, this concept is rather appropriate to applications, and we hope that the reader agrees with us familiarizing himself with the proof ideology of Sections 5-7. Another methodological prop of the paper is the concept of discrete convergence (Section 1). In Sections 5-7 the abstract results of Sections l-4 have been applied to the quadrature formula method for nonlinear integral equations and to the collocation, subregion, Galerkin and difference methods for nonlinear boundary value problems. Only ordinary differential equations are considered. For partial differential equations our approaches are still weakly developed : first works (e.g. [l-3]) concern linear equations. Sections l-3 contain more material than is urgently needed for applications, our significant goal. By stars are labelled the sections, propositions etc. that can be omitted if one wishes to get to applications more quickly. The main text contains only few references. For the reference notes, see the end of the paper.
113 citations
TL;DR: In this paper, a method for the calculation of roll-up of the trailing vortex sheet shed from an elliptically loaded wing was proposed, which converges as the discretization is refined.
Abstract: Careful discretization of two-dimensional vortex sheets has led to the discovery of the cause of inconsistency of the multi-vortex representation of such sheets. The method described here is applied to the calculation of roll-up of the trailing vortex sheet shed from an elliptically loaded wing. Contrary to the results obtained with multi-vortex methods, calculations are found to converge as the discretization is refined.
78 citations
TL;DR: In this article, the operator compact implicit spatial discretization method for the second order wave equation when first order terms are present was implemented. And the resulting algorithm is completely analogous to the compact implicit algorithm when lower order terms were not present.
Abstract: : In a previous paper a fourth order compact implicit scheme was presented for the second order wave equation. A very efficient factorization technique was developed when only second order terms were present. In this note we implement the operator compact implicit spatial discretization method for the second order wave equation when first order terms are present. The resulting algorithm is completely analogous to the compact implicit algorithm when lower order terms were not present. For this more general operator compact implicit spatial approximation the same factorization as in our previous paper is developed. (Author)
75 citations
TL;DR: In this article, the numerical solution of the Navier equations discretized by finite elements is studied by various forms of pre-conditioned conjugate gradient methods, and the dependence of the number of iterations is examined as a function of Poisson's ratio.
62 citations
10 Jul 1978
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.
48 citations
TL;DR: In this paper a metal algorithm for the class of IDeC-methods for differential equations is presented and analyzed and conditions are given which guarantee a certain order of accuracy.
Abstract: Iterated Defect Correction (IDeC) is a technique for improving successively an approximate solution of a given problemFy=0. One of the most important fields of application of this principle are differential equations. Here, IDeC can be used as a technique for increasing the order of a discretization method and thus for improving the accuracy. In this paper a metalgorithm for the class of IDeC-methods for differential equations is presented and analyzed. For every component of this metalgorithm conditions are given which guarantee a certain order of accuracy. These conditions are of particular importance for practical applications, as far as the implementation of IDeC-methods is concerned.
46 citations
TL;DR: A CAD method is presented for converting existing continuous-data control systems into digital control systems by means of a digital controller that is synthetized by matching the frequency response of the digital control system to that of the continuous- data system with a minimum weighted mean-square error.
Abstract: A CAD method is presented for converting existing continuous-data control systems into digital control systems by means of a digital controller The digital controller is synthetized by matching the frequency response of the digital control system to that of the continuous-data system with a minimum weighted mean-square error A formula for computing the parameters of the digital controller is obtained as a result The design technique is illustrated with a numerical example and a comparison with previous methods is also presented
45 citations
TL;DR: A high precision unconditionally stable algorithm for computation of linear dynamic structural systems that shares the advantageous property of the amplification matrix preserving a banded form due to discretization in space, which means less computer space and fewer operations are needed.
42 citations
TL;DR: In this article, a method based on a linearized theory and the finite element method is developed to calculate the potential and kinetic energies of a suspension bridge by using finite element technique to decompose the structure into equivalent systems of finite elements.
Abstract: A method is developed based on a linearized theory and the finite-element method. The method involves two distinct steps: (1)Specification of the potential and kinetic energies of the bridge, (2)use of finite element technique to: (a)discretize the structure into equivalent systems of finite elements; (b)select the displacement model most closely approximating the real case; (c)derive the element and assemblage stiffness and inertia properties; and finally (d)form the matrix equations of motion and the resulting eigenproblems. A numerical example is presented to illustrate the applicability of the analysis and to investigate the dynamic characteristics of laterally vibrating suspension bridges. This method eliminates the need to solve transcendental frequency equations, simplifies the determination of the energy stored in different members of the bridge, and represents a simple, fast, and accurate tool for calculating the natural frequencies and modes of lateral vibration by means of a digital computer.
40 citations
12 Oct 1978
TL;DR: The multilevel (multigrid) adaptive technique, a general strategy of solving continuous problems by cycling between coarser and finer levels of discretization is described, which provides very fast general solvers, together with adaptive, nearly optimal discretized schemes.
Abstract: The multilevel (multigrid) adaptive technique, a general strategy of solving continuous problems by cycling between coarser and finer levels of discretization is described. It provides very fast general solvers, together with adaptive, nearly optimal discretization schemes. In the process, boundary layers are automatically either resolved or skipped, depending on a control function which expresses the computational goal. The global error decreases exponentially as a function of the overall computational work, in a uniform rate independent of the magnitude of the singular-perturbation terms. The key is high-order uniformly stable difference equations, and uniformly smoothing relaxation schemes.
38 citations
TL;DR: In this paper, a numerical approach based on a rigorous integral formulation is described to the problem of an inhomogeneous dispersive slab illuminated by incident TEM plane wave with arbitrary time dependence, where the slab is assumed to be nonmagnetic and its complex permittivity only varies normally to its interfaces.
Abstract: A numerical approach is described, based on a rigorous integral formulation, to the problem of an inhomogeneous dispersive slab illuminated by incident TEM plane wave with arbitrary time dependence. The slab is assumed to be nonmagnetic and its complex permittivity only varies normally to its interfaces. The numerical process consists of a space-time discretization. The solution can be determined step by step from simple recurrence formulas. Some examples are given for normal incidence in order to illustrate the most interesting features of the method and its possible field of applications.
TL;DR: In this paper, an efficient free-vibration analysis procedure of two-dimensional structures was developed by employing a discretization technique based on a recently developed concept of finite dynamic elements, involving higher order dynamic correction terms in the associated stiffness and inertia matrices.
Abstract: The paper develops an efficient free-vibration analysis procedure of two-dimensional structures. This is achieved by employing a discretization technique based on a recently developed concept of finite dynamic elements, involving higher order dynamic correction terms in the associated stiffness and inertia matrices. A plane rectangular dynamic element is developed in detail. Numerical solution results of free-vibration analysis presented herein clearly indicate that these dynamic elements combined with a suitable quadratic matrix eigenproblem solution technique effect a most economical and efficient solution for such an analysis when compared with the usual finite element method.
TL;DR: The first part consists of a theoretical development of a phenomenon that often occurs in practice, namely, that the number of iterations for Newton's method to converge to within a fixed tolerance and for a fixed starting vector is essentially independent of the mesh size.
Abstract: The object of this paper is the numerical solution of nonlinear two-point boundary value problems by Newton's method applied to the discretized problem on successively refined grids. The first part consists of a theoretical development of a phenomenon that often occurs in practice, namely, that the number of iterations for Newton's method to converge to within a fixed tolerance and for a fixed starting vector is essentially independent of the mesh size. The second part develops a process based on these results for determining an efficient mesh refinement strategy. Numerical results are also provided.
TL;DR: In this article, the first kind associated with strictly monotone Volterra integral operators are solved by projecting the exact solution of such an equation into the spaceS m (?1) (Z N ) of piecewise polynomials of degreem?0, possessing jump discontinuities on the setZ N of knots.
Abstract: In the present paper integral equations of the first kind associated with strictly monotone Volterra integral operators are solved by projecting the exact solution of such an equation into the spaceS m (?1) (Z N ) of piecewise polynomials of degreem?0, possessing jump discontinuities on the setZ N of knots. Since the majority of "direct" one-step methods (including the higher-order block methods) result from particular discretizations of the moment integrals occuring in the above projection method we obtain a unified convergence analysis for these methods; in addition, the above approach yields the tools to deal with the question of the connection between the location of the collocation points used to determine the projection inS m (?1) (Z N ) and the order of convergence of the method.
TL;DR: In this article, the elastic deuteron scattering was described by using nucleon optical potentials and the Hulthen potential for the np interaction and including approximately the break-up states of the deuterons.
TL;DR: In this paper, a numerical approach of the reconstruction of an inhomogeneous slab is described, the relative permittivity or the index of which are unknown, and a solution of this inverse problem is based upon a space-time discretization of a field integral formulation, in time domain.
Abstract: A numerical approach of the reconstruction of an inhomogeneous slab is described, the relative permittivity or the index of which are unknown. This one-dimensional dielectric medium is assumed to be linear, isotropic and non-magnetic, its conductivity being known, generally equal to zero. Its frequency independent permittivity arbitrarily varies, normally to its interfaces. A TEM plane wave of arbitrary causal time dependence illuminates this slab. A solution of this inverse problem is based upon a space-time discretization of a field integral formulation, in time-domain. A checked iterative process makes it possible to determine the index profile step by step. Some examples are given to illustrate the main features of this reconstruction method; simulation of experimental errors is considered with a special attention.
TL;DR: In this article, a finite-difference resistivity model is proposed, where a given medium is discretized and divided into rectangular blocks by using a very coarse system of vertical and horizontal grid lines, whose distance from the source(s) increases logarithmically.
Abstract: Highly efficient finite‐difference resistivity modeling algorithms which yield accurate results are put forward. The given medium is discretized and divided into rectangular blocks by using a very coarse system of vertical and horizontal grid lines, whose distance from the source(s) increases logarithmically. Expressions are derived to compute the longitudinal conductance and transverse resistance associated with each of these blocks for a parallel‐layer medium followed by a generalized treatment to accommodate arbitrarily shaped structures. Since the values of Dar Zarrouk parameters are derived from the exact resistivity distribution of the given medium, fine features such as a thin but anomalously resistive bed which ordinarily would be missed entirely in coarse discretization can be taken into account. Further reduction in the size of the model is achieved by making use of a symmetry wherever possible. In most cases the computation of the potential field which involves the inversion of a small sparse m...
TL;DR: In this paper, it is shown how different conservative discretizations of the nonlinear term uux govern the discretization error in computational results, especially when the mesh Reynolds number Re Ax is not small.
TL;DR: In this paper, the canonical path integral is recast in a covariant formalism and the quantum action of DeWitt is derived, including the (h 2 6)R curvature term.
1 Jan 1978
TL;DR: In this paper, the variational equations of motion of large structures with rotating substructures are derived by the substructure synthesis approach, whereby a discretization procedure akin to the Rayleigh-Ritz method is used to represent the elastic motion of every substructure by a suitable set of admissible functions.
Abstract: The variational equations of motion of large structures with rotating substructures are derived by the substructure synthesis approach, whereby a discretization procedure akin to the Rayleigh-Ritz method is used to represent the elastic motion of every substructure by a suitable set of admissible functions. Using an inclusion principle for gyroscopic systems, the effects on the predicted system dynamic characteristics of truncating the number of admissible functions used for each substructure can be assessed. Criteria for rational selection of admissible functions are presented.
TL;DR: The Zienkiewicz three-and four-time-level schemes were adapted for the numerical integration of the diffusion equation after finite element discretization in this article, where conditions were found for A0 stability and the methods used by various authors for their representation of the solution of y1=−λy.
Abstract: The Zienkiewicz three− and four−time level schemes as proposed for vibration problems1 and adapted for the numerical integration of the diffusion equation after finite element discretization. Conditions are found for A0 stability and the methods are studied for their representation of the solution of y1=−λy. Methods used by various authors for linear and non−linear parabolic problems are compared. For the three time−level scheme a Liniger exponential fit of e−λ0Δt with a larger value of λ0Δt and the fully implicit scheme give the most promising results. The four time−level scheme is so much less amenable to control as to make it unsuitable for these problems.
TL;DR: This paper suggests the use of only the minimal covering that is sufficient for convergence in the discretized system and suggests the possibility of considerable economy in the cost of obtaining finite element solutions to complex problems, e.g. coupled field problems, three-dimensional problems, stress concentration etc.
TL;DR: The present paper contains a stability concept for discretization methods of a certain, very general classM, which is optimal (in the sense of yielding the best general, two-sided error bounds) without being more restrictive than any of the classical stability definitions.
Abstract: The present paper contains a stability concept for discretization methods of a certain, very general classM, which is optimal (in the sense of yielding the best general, two-sided error bounds) without being more restrictive than any of the classical stability definitions. The optimal stability functional Ψh related to it depends on the linear part of the discretization operator, and has the important property that Ψh [δ] may be of orderq+1, i.e. Ψh [δ] = O(hq+1), even if the local error δ only has orderq, δ = O(hq). This result may be used for the construction of methods with maximum order. Its application to linear cyclic methods, for example, furnishes a new approach to the theory of linearM-cyclick-step methods of maximum order.
TL;DR: In this paper, it is shown that if orthogonal base functions are used with a mixed variational formulation, then consistant diagonal mass matrices and corresponding sets of spatially discretized field equations are obtained.
Abstract: The conventional dynamic variational approach and finite element base functions lead to non-diagonal consistent mass matrics which are inappropriate for use with an explicit time integration scheme. In this work, it is shown that if orthogonal base function are used with a mixed variational formulation, then consistant diagonal mass matrices and corresponding sets of spatially discretized field equations are obtained. Although the approach is quite general, the theory is purposely illustrated by a detailed development for one set of base functions. Central difference time integration is incorporated for applications to one-dimensional wave propagation and to Euler-Bernoulli beams. Numerical examples are provided for elastic and elastic-plastic materials.
TL;DR: The diferential equations governing two-phase flow through a porous medium are derived and the space discretization is carried out by the Galerkin finite element method, and an optimum method is suggested for use as a starting algorithm for more sophisticated time-stepping schemes.
Abstract: The diferential equations governing two-phase flow through a porous medium are derived and the space discretization is carried out by the Galerkin finite element method. An examination of the eigenvalues of the resulting system follows in order to decide on a suitable stability criterion to be satisfied by the time-stepping algorithm. A class of time-stepping schemes is introduced and their behaviour when applied to a specific problem is analyzed. The aim is to achieve an adequate modelling of the water-oil interface and to preserve reasonable values of the capillary pressure. A technique to clustering the mesh around the interface is used to improved the modelling, and a boundary condition based on the relative mobilities of the two fiuids is imposed to enable the program to calculate the water-oil ratio after breakthrough. The effect of lumping the mass matrix is then examined with a view to reducting the surge in the values obtained for the capillary pressure. Finally an optimum method is suggested for use as a starting algorithm for more sophisticated time-stepping schemes.
TL;DR: In this paper, the second partial derivative with respect to the space variable is discretized by means of a second-order central difference, which is proved to be convergent in theory and provided reasonable accuracy.
Abstract: Two accurate and general purpose numerical approaches are presented for the solution of variable‐coefficient parabolic wave equations. (1) An Ordinary Differential Equation Approach: The parabolic equation is treated as a system of ordinary differential equations where the second partial derivative with respect to the space variable is discretized by means of a second‐order central difference. Nonlinear Multistep (NLMS) methods are used as predictor and corrector for solving this system. A variable‐step‐size technique is built in to give the desired accuracy. The theory with regard to consistency, stability, and convergence has been very well developed for NLMS methods thus ensuring the convergence of this procedure. (2) A Finite‐Difference Approach: A finite‐difference technique is derived from the conventional implicit schemes. This technique is proved to be convergent in theory and is found to be general purpose and provides reasonable accuracy. The solution of a range‐dependent problem with nonflat bo...
TL;DR: In this article, a vector-matrix form of the system of differential equations is used to solve the finite-difference equations by the use of the "progonka" process.
TL;DR: The partition technique is designed to remove the restriction on the size of problems that can be efficiently computed by integral equation methods by dividing the domain by partition curves and then applying a standard integral equation method to each of the subdomains.
TL;DR: In this article, the numerical solution of the first kind of Fredholm integral equations was analyzed by means of finite rank and other approximation methods replacing the original problem by T N x=y N,N=1,2,.....
Abstract: This paper analyzes the numerical solution of Fredholm integral equations of the first kindTx=y by means of finite rank and other approximation methods replacingTx=y byT N x=y N ,N=1,2, .... The operatorsT andT N can be viewed as operators from eitherL 2[a, b] toL 2[c,d] or as operators fromL ?[a, b] toL ?[c, d]. A complete analysis of the fully discretized problem as compared with the continuous problemTx=y is also given. The filtered least squares minimum norm solutions (LSMN) to the discrete problem and toT N x=y are compared with the LSMN solution ofTx=y. Rates of convergence are included in all cases and are in terms of the mesh spacing of the quadrature for the fully discretized problem.
TL;DR: In this article, the problem of discretizing the minimal surface over an obstacle with linear finite elements on a regular triangulation of the not necessarily convex domain is considered.
Abstract: Nonlinear locally coercive variational inequalities are considered and especially the minimal surface over an obstacle. Optimal or nearly optimal error estimates are proved for a direct discretization of the problem with linear finite elements on a regular triangulation of the not necessarily convex domain. It is shown that the solution may be computed by a globally convergent relaxation method. Some numerical results are presented.