Showing papers on "Discretization published in 1978"
TL;DR: In this article, two general approaches used in the soluation of coupled structures and fluid systems in which effects of large scale flow are excluded are presented. But the numerical problems posed, discretization methods used and possible simplifications are discussed.
Abstract: The paper presents an introduction to two general approaches used in the soluation of coupled structures and fluid systems in which effects of large scale flow are excluded. In the first approach, the Lagrangian, the fluid is simply treated as a ‘solid’ with a negligible shear modulus. In the second method, Eulerian, a single pressure variable is used in the fluid.
The numerical problems posed, discretization methods used and possible simplifications are discussed.
299 citations
TL;DR: This article exposes the common structural principle of all these techniques for error estimation and iterative improvement in discretization algorithms and exhibits the principal modes of its implementation in a discretized context.
Abstract: Recently, a number of closely related techniques for error estimation and iterative improvement in discretization algorithms have been proposed. In this article, we expose the common structural principle of all these techniques and exhibit the principal modes of its implementation in a discretization context.
278 citations
TL;DR: In this article, a variational principle in terms of displacements in the fluid and the structure with a penalty for irrotationality of displacement is developed for the analysis of harmonic vibrations of ideal compressible fluid and elastic structure systems.
Abstract: A variational principle in terms of displacements in the fluid and the structure with a penalty for irrotationality of displacement in the fluid is developed for the analysis of harmonic vibrations of ideal compressible fluid and elastic structure systems. Its Discretization by the finite element method leads to an algebraic eigenvalue problem with a positive definite symmetric banded matrix. Numerical examples obtained for pure acoustic cases and coupled cases show the efficiency of the method.
234 citations
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.
105 citations
TL;DR: In this article, a variable-mesh model is used with several grid configurations, each having uniform high resolution over a specified area of interest and lower resolution elsewhere to produce short-term forecasts over this area without the necessity of high resolution everywhere.
Abstract: A barotropic primitive-equation model using the finite-element method of space discretization is generalized to allow variable resolution. The overhead incurred in going from a uniform mesh to a variable mesh having the same number of degrees of freedom is found to be approximately 20% overall. The variable-mesh model is used with several grid configurations, each having uniform high resolution over a specified area of interest and lower resolution elsewhere to produce short-term forecasts over this area without the necessity of high resolution everywhere. It is found that the forecast produced on a uniform high-resolution mesh can be essentially reproduced for a limited time over the limited area by a variable-mesh model having only a fraction of the number of degrees of freedom and requiring significantly less computer time. As expected, the period of validity of forecasts on variable meshes can be lengthened by refining the mesh in the outer region. It is concluded that from the point of view ...
91 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.
75 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)
73 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.
60 citations
TL;DR: In this paper, various discretizations of the sine-Gordon equation are studied and the associated soliton solutions, B\"acklund transformations, conservation laws, and inverse scattering equations are obtained.
Abstract: Various discretizations of the sine-Gordon equation are studied. Hirota's discretization scheme is extended and two alternative discretization schemes are constructed. The associated soliton solutions, B\"acklund transformations, conservation laws, and the inverse scattering equations are obtained.
49 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: 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.
TL;DR: In this article, a method of coupling between finite elements and integral representation is described, where the numerical scheme is obtained by means of a finite element discretization of a continuous variational problem.
TL;DR: In this article, a quasi-projection of the Galerkin solution was proposed to derive superconvergence results for linear parabolic and hyperbolic problems in a single space variable.
Abstract: Superconvergence phenomena are demonstrated for Galerkin approxima- tions of solutions of second order parabolic and hyperbolic problems in a single space variable. An asymptotic expansion of the Galerkin solution is used to derive these results and, in addition, to show optimal order error estimates in Sobolev spaces of negative index in multiple dimensions. 1. Introduction. We shall be concerned primarily with the analysis of supercon- vergence phenomena associated with the numerical solution of second order, linear parabolic and hyperbolic equations by Galerkin methods based on piecewise-polyno- mial spaces. Our principal tool will be an asymptotic expansion to high order of the Galerkin solution; this expansion will be obtained by using a sequence of elliptic pro- jections and will be called a quasi-projection. In Sections 4 and 5 we develop the quasi-projection for parabolic Galerkin procedures for problems in one or several space variables for both Neumann and Dirichlet boundary conditions and derive optimal order negative norm estimates for the error in the Galerkin solution. In Section 6 we apply the quasi-projection to de- rive superconvergence results in the case of a single space variable when the Galerkin space consists of piecewise-polynomial functions of degree r. It is well known (4), (6), (7), (9), (10) that, if h is the knot spacing parameter associated with the not necessarily uniform grid, the Galerkin solution for standard parabolic problems con- verges with an error that is at best globally of order 0(hr+ ), as measured in L2 or L°°. Consider a knot at which the smoothness constraint of the Galerkin space re- duces to continuity. We show that the Galerkin solution produces an 0(h2r)-ap- proximation at such a knot. Also, we show that a very simply evaluated weighted quadrature of the Galerkin solution gives an cXft2^-approximation of the flux at the knot; the direct evaluation of the derivative of the Galerkin solution leads to an 0(/1r)-approximation. We summarize briefly in Section 7 results presented in detail elsewhere (3) showing that the superconvergence results above are preserved and that supercon- vergence occurs in the time increment when the Galerkin procedure is discretized in time by a collocation method. In Section 8 we treat continuous-time Galerkin methods for hyperbolic prob- lems and obtain analogous results. Throughout this paper we rely heavily on some earlier results of two of the
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.
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.
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.
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.
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: In this paper, a numerical approximation of a parabolic control problem with a Neumann boundary condition control is considered, based on backward discretization with respect to time and a Galerkin method in the space variables.
Abstract: Numerical approximation of a parabolic control problem with a Neumann boundary condition control is considered. The observation is the final state. The numerical approximation is based on backward discretization with respect to time and a Galerkin method in the space variables. Optimal (except for a logarithmic term) L2 error estimates are derived for the optimal state. Certain error estimates for the optimal control are also given.
TL;DR: In this paper, the idea of a trade-off between the anticipated error and the attainable level of detail in the model estimate is extended to the discrete case, either with even or uneven discretization.
Abstract: summary. Treatments of geophysical inverse problems have tended to polarize into approaches intended to generate models either described by piecewise continuous functions or with some prior discretization. The two approaches are here developed in parallel, and the ideas of a trade-off between the anticipated error and the attainable level of detail in the model estimate are extended to the discrete case, either with even or uneven discretization.
An alternative approach to specifying the potential resolution of a model is to establish upper and lower bounds on parameter values. Linear programming methods are extended to determine bounds which allow for subjective limits on parameter values.
For a non-linear system the possible resolution may be investigated by estimation procedures based on the full set of successful solutions obtained by Monte-Carlo inversion.
TL;DR: In this paper, the authors analyzed least squares and Galerkin's method in a Sobolev space setting geared to integral equations of the first kind with nonsmooth kernels and obtained condition number bounds for the coefficient matrix of the discretized problem.
Abstract: It is well known that first kind integral equations having nonsmooth kernels can sometimes be solved satisfactorily without regularization. Our objective here is to quantify the relationship between the smoothness of the kernel and the amenability of the problem to standard numerical methods. To this end, we analyze least squares and Galerkin’s method in a Sobolev space setting geared to integral equations of the first kind with nonsmooth kernels. Error estimates are obtained, as well as condition number bounds for the coefficient matrix of the discretized problem. The key parameter in these estimates is a smoothing index which gauges the number of derivatives the integral transformation adds to $L^2$ functions to which it is applied.
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.
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.
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 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 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 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: 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.
01 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.