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Showing papers on "Discretization published in 1977"


Journal ArticleDOI
TL;DR: In this article, a method based on a discrete horizontal wave-number representation of seismic-source wave fields is developed and applied to the study of the near-field of a seismic source embedded in a layered medium.
Abstract: A method based on a discrete horizontal wave-number representation of seismic-source wave fields is developed and applied to the study of the near-field of a seismic source embedded in a layered medium. The discretization results from a periodicity assumption in the description of the source. The problem is basically two-dimensional but its extension to three dimensions is sometimes feasible. The source is quite general and is represented through its body-force equivalents. Tests of the accuracy of the method are made against Garvin's (1956) analytical solution (a buried line source in a half-space) and against Niazy's (1973) results for a propagating fault in an infinite medium. In both cases, a remarkably good agreement is found. The method is applied to the modeling of the San Fernando earthquake, and to the computation of synthetic seismograms at short distance from a complex source in a layered medium. In particular, we show that the high acceleration-high frequency phase of the Pacoima Dam records is due to the Rayleigh wave from the point of ground breakage. Other high-acceleration phases, predicted by our model, are associated with the shear-wave arrival from the hypocenter or result from changes in the fault orientation.

496 citations



Journal ArticleDOI
TL;DR: In this paper, the features of an advanced numerical solution capability for boundary value problems of linear, homogeneous, isotropic, steady-state thermoelasticity theory are outlined.
Abstract: The features of an advanced numerical solution capability for boundary value problems of linear, homogeneous, isotropic, steady-state thermoelasticity theory are outlined. The influence on the stress field of thermal gradient, or comparable mechanical body force, is shown to depend on surface integrals only. Hence discretization for numerical purposes is confined to body surfaces. Several problems are solved, and verification of numerical procedures is obtained by comparison with accepted results from the literature.

341 citations


Journal ArticleDOI
TL;DR: In this article, the familiar quasigeostrophic equations are reconsidered to consistently include forcing and diffusive processes, as well as auxiliary conditions for making the model a fully determined one in both simply and multiply connected domains.

169 citations


Journal ArticleDOI
TL;DR: The numerical solution of ill-posed problems by projection methods is considered Regularization is carried out simply by choosing an optimal discretization parameter It is shown by asymptotic estimates and by numerical examples that this kind of regularization is as efficient as the method of Tikhonov and Phillips.
Abstract: The numerical solution of ill-posed problems by projection methods is considered Regularization is carried out simply by choosing an optimal discretization parameter It is shown by asymptotic estimates and by numerical examples that this kind of regularization is as efficient as the method of Tikhonov and Phillips

127 citations


Journal ArticleDOI
TL;DR: A simulation procedure is presented in this work which can analyze the behavior of any dispersed phase system consisting of particles whose random behavior is specified in terms of probability functions, and is an efficient alternative to the modeling of such systems.
Abstract: A simulation procedure is presented in this work which can analyze the behavior of any dispersed phase system consisting of particles whose random behavior is specified in terms of probability functions. The procedure distinguishes itself from its predecessors in being free from arbitrary discretization of time or any other parameter along which the system evolves, and its ability to predict the behavior of randomly behaving small populations. Where population balance equations, which describe the behavior of particulate systems, cannot be readily solved, the simulation technique presented herein represents an efficient alternative to the modeling of such systems. Tables and a flow chart are given which would enable the use of the method for any system with specified particle behavior.

118 citations


Journal ArticleDOI
TL;DR: In this paper, a general formulation of the incremental equations of motion for structures undergoing large displacement finite strain deformation is presented, based on the Lagrangian frame of reference, in which constitutive models of a variety of types may be introduced.
Abstract: The paper presents the theoretical and computational procedures which have been applied in the design of a general purpose computer code for static and dynamic response analysis of non-linear structures A general formulation of the incremental equations of motion for structures undergoing large displacement finite strain deformation is first presented These equations are based on the Lagrangian frame of reference, in which constitutive models of a variety of types may be introduced The incremental equations are linearized for computational purposes, and the linearized equations are discretized using isoparametric finite element formulation Computational techniques, including step-by-step and iterative procedures, for the solution of non-linear equations are discussed, and an acceleration scheme for improving convergence in constant stiffness iteration is reviewed The equations of motion are integrated using Newmark's generalized operator, and an algorithm with optional iteration is described A solution strategy defined in terms of a number of solution parameters is implemented in the computer program so that several solution schemes can be obtained by assigning appropriate values to the parameters The results of analysis of a few non-linear structures are briefly discussed

101 citations


Journal ArticleDOI
TL;DR: This paper construct and analyze classes of single step methods of arbitrary order for homogeneous linear initial boundary value problems for parabolic equations with time-independent coefficients with Galerkin-type methods.
Abstract: In this paper we construct and analyze classes of single step methods of arbitrary order for homogeneous linear initial boundary value problems for parabolic equations with time-independent coefficients. The spatial discretization is done by means of general Galerkin-type methods.

75 citations


Journal ArticleDOI
TL;DR: In this article, a new finite element scheme is presented for overcoming the problem of numerical oscillation associated with the steady state convection-diffusion equation when its convective terms are important.

64 citations


Journal ArticleDOI
TL;DR: In this paper, a barotropic primitive equation model using the finite-element method of space discretization is presented and compared with those from both finite-difference and spectral models.
Abstract: A barotropic primitive equation model using the finite-element method of space discretization is presented. The semi-implicit method of time discretization is implemented by taking a suitable form of the governing equations. Finite-element forecasts are then compared with those from both finite-difference and spectral models. From the point of view of both efficiency and accuracy it is concluded that the finite-element method of space discretization appears to he viable for the practical problem of numerical weather prediction.

63 citations


Journal ArticleDOI
TL;DR: In this paper, an improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields.
Abstract: An improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields. The technique provides higher-order approximation of the unknown function over a bounding surface described by two-parameter, non-planar elements. The integral equation is discretized through the Rayleigh–Ritz procedure. Convergence to the solution for operators having a positive-definite component is guaranteed. Kernel singularities are treated by removing them from the relevant integrals and dealing with them analytically. A successive element iterative process, which produces the solution of the large dense matrix of the complete structure, is described. The discretization and equation solution take place one element at a time resulting in storage and computational savings. Results obtained for classical test models, involving scalar electrostatic potential and vector elastostatic displacement fields, demonstrate the technique for the solution of the Fredholm integral equation of the first kind. Solution of the Fredholm equation of the second kind is to be reported subsequently.

Journal ArticleDOI
TL;DR: In this paper, a finite-element formulation for the vertical structure of primitive equation models has been developed, which is a variant of the Galerkin procedure in which the dependent variables are expanded in a finite, set of basis functions and then the truncation error is orthogonalized to each of the basis functions.
Abstract: A finite-element formulation for the vertical structure of primitive equation models has been developed. The finite-element method is a variant of the Galerkin procedure in which the dependent variables are expanded in a finite, set of basis functions and then the truncation error is orthogonalized to each of the basis functions. In the present case, the basis functions are Châpeau functions in sigma, the vertical coordinate. The procedure has been designed for use with a semi-implicit time discretization algorithm. Although this vertical representation has been developed for ultimate implementation in a three-dimensional finite-element model, it has been first tested in a spherical harmonic, baroclinic, primitive equations model. Short-range forecasts made with this model are very encouraging.

Journal ArticleDOI
TL;DR: In this article, the authors examined the close-approach problem associated with flow calculation methods based on vortex-lattice theory using two-dimensional discretized vortex sheets.
Abstract: The close-approach problem associated with flow calculation methods based on vortex-lattice theory was examined numerically using two-dimensional discretized vortex sheets. The analysis first yields a near-field radius of approximately the distance apart of the vortices in the lattice; only within this distance from the sheet are the errors arising from the discretization significant. Various modifications to the discrete vortices are then considered with the objective of reducing the errors. This leads to a near-field model in which a vortex splits into an increasing number of subvortices as it is approached. The subvortices, whose strengths vary linearly from the vortex position, are evenly distributed along an interpolated curve passing through the basic vortices. This subvortex technique can be extended to the three-dimensional case and is efficient because the number of vortices is effectively increased, but only where and when needed.

Journal ArticleDOI
TL;DR: Practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed and procedures for circumventing such difficulties are suggested.
Abstract: The penalty function approach has been recently formalized as a general technique for adjoining constraint conditions to algebraic equation systems resulting from variational discretization of boundary value problems by finite difference or finite element techniques. This paper studies the numerical behaviour of the penalty function method for the special case of individual equation constraints imposed on a symmetric system of linear algebraic equations. Constraint representation and computational roundoff error components are distinguished and asymptotically characterized in terms of the penalty function weight coefficients. On the basis of this study, practical rules for the automatic assignment of values to those coefficients within the linear equation solver are proposed. Numerical problems encountered in the case of more general constraints are briefly discussed, and procedures for circumventing such difficulties are suggested.

Journal ArticleDOI
TL;DR: In this paper, the coupled integro-differential Hartree-Fock equations are solved directly in coordinate space for a simple effective interaction, and expressions to correct the total energy to second order in the mesh spacing are derived.

Journal ArticleDOI
TL;DR: In this article, the numerical solution of the multigroup neutron diffusion equations by computer codes was investigated. But the results were not analyzed in every detail and the reasons for the discrepancies are very small and more of theoretical than practical interest.
Abstract: This investigation concentrates on the numerical solution of the multigroup neutron diffusion equations by computer codes. For a realistic model liquid-metal fast breeder reactor, several benchmark problems in two and three space dimensions were derived and calculations were performed by eight different computer programs. The effect on k/sub eff/ and the neutron fluxes of the refinement of the discretization mesh is studied. Very good agreement (approximately 0.05%) of the results was found in those cases where the computer programs use the same discretization scheme of mesh-edged discretization formulas, although the codes employ different methods of solution. On the other hand, minor discrepancies remain between results obtained by codes using mesh-edged and mesh-centered discretization formulas, even for fine-mesh grids. The reasons are not understood in every detail. Fortunately, these discrepancies are very small and more of theoretical than practical interest. The effect of a simple group condensation scheme of k/sub eff/ was also investigated by considering several different energy group structures. Spatial mesh refinements and resolution of the energy range were found to be well decoupled. As the main result, one may take the fact that spatial and energetic mesh refinements may influence the results rather strongly, unless the mesh stepmore » is comparable to the minimum diffusion length and unless enough energy groups are used.« less

Journal ArticleDOI
TL;DR: It is demonstrated that methods can readily be applied to solve problems involving nonlinear Neumann boundary conditions in the problem of designing an optimum distributed parameter system.

Journal ArticleDOI
TL;DR: In this paper, a new way of estimating local discretization errors (based on an idea due to P. E. Zadunaisky) is introduced, which leads to an extension of the domain of applicability, when compared with the variants used by Fox and Pereyra.
Abstract: A new way of estimating local discretization errors (based on an idea due to P. E. Zadunaisky) is introduced. If error estimates obtained by this method are used in connection with the general class of iterated deferred correction algorithms, they lead to an extension of the domain of applicability, when compared with the variants used by Fox and Pereyra.

Journal ArticleDOI
TL;DR: In this paper, it was shown that various (discrete) methods for the approximate solution of Volterra integral equations of the first kind correspond to some discrete version of the method of (recursive) collocation in the space of (continuous) piecewise polynomials.
Abstract: We show that various (discrete) methods for the approximate solution of Volterra (and Abel) integral equations of the first kind correspond to some discrete version of the method of (recursive) collocation in the space of (continuous) piecewise polynomials. In a collocation method no distinction has to be made between equations with regular or weakly singular kernels; the regularity or nonregularity of the given integral operator becomes only relevant when selecting a discretization procedure for the moment integrals resulting from collocation. Similar results hold for equations of the second kind.

Book ChapterDOI
01 Jan 1977
TL;DR: In this paper, the authors describe numerical modeling by the TLM method for transmission-line modeling, where the continuous properties of the field space have been concentrated into discontinuous packets, and the next step is to solve the lumped network in the time domain.
Abstract: Publisher Summary This chapter describes numerical modeling by the TLM method. A lumped electrical network provides a means of discretizing the space derivatives in partial differential equations. The idea is not new, and certainly the lumped network models for diffusion and wave phenomena are well known. They provide the intermediate step for transmission-line modeling. The lumped network may also be regarded as the exact electrical realization of a set of first-order ordinary differential equations obtained from the space discretization process. The network can, therefore, provide useful information about the properties of these ordinary differential equations. Simple differencing techniques lead to the approximation of partial differential equations by lumped networks. The continuous properties of the field space have been concentrated into discontinuous packets. The next step is to solve the lumped network in the time domain, and this is achieved by transmission-line modeling.

Journal ArticleDOI
TL;DR: In this article, a general procedure is presented which permits the units to be described by both distributed and lumped parameter models, and the resulting large set of time-dependent ordinary differential equations is solved simultaneously using a Gear-type integrator.

Journal ArticleDOI
TL;DR: In this paper, transient dynamic finite element solutions are undertaken for both double cantilever beam (DCB) and pipeline problems with propagation of the crack being permitted, with an explicit (central difference) scheme being employed for time integration.
Abstract: Recent developments in numerical techniques for dynamic transient stress analysis have ensured that realistic models can now be employed in crack propagation studies. In this paper transient dynamic finite element solutions are undertaken for both double cantilever beam (DCB) and pipeline problems with propagation of the crack being permitted. Standard parabolic isoparametric elements are employed for spatial discretization with an explicit (central difference) scheme being employed for time integration. Both critical stress and energy balance crack propagation criteria are considered.

Journal ArticleDOI
D. Trigiante1
TL;DR: By imposing some conditions on the discrete dynamical systems generated by one step discretization methods, some non linear one step methods of the first and second order are constructed, which turn out to be A-stable or L-stable according to the current definitions.
Abstract: By imposing some conditions on the discrete dynamical systems generated by one step discretization methods, we construct some non linear one step methods of the first and second order, which turn out to beA-stable orL-stable according to the current definitions. Numerical experiments are carried out on some common stiff test problems, confirming the validity of the methods.

Journal ArticleDOI
TL;DR: In this paper, a direct numerical integration of the equations of motion of a rotational shell of revolution under conservative loading is presented. But, the method is only suitable for static problems, and it is only approximate for dynamic response.

Journal ArticleDOI
TL;DR: Inhomogeneous but time-homogeneous linear hyperbolic initial boundary value problems are solved using Galerkin procedures for the space discretization and Runge-Kutta methods for the time discretizations.
Abstract: Inhomogeneous but time-homogeneous linear hyperbolic initial boundary value problems are solved using Galerkin procedures for the space discretization and Runge-Kutta methods for the time discretization. The space discretized system is not transformed a-priori in a linear system of first order. For the difference of the Ritz projection of the exact solution and the numerical approximation error estimates are derived under the assumption that the applied Runge-Kutta methods have a non-empty interval of absolute stability. It is shown that this class of schemes is not empty in the present case of second order systems, too.

Journal ArticleDOI
W. L. Wood1
TL;DR: In this paper, it was shown that the forcing function can be turned on or off in such a way as to make the noise produced by it considerably reduced by making it proportional to 1 −e −βt.
Abstract: The stiff equations resulting from the finite element discretization of the heat conduction equation are very susceptible to Crank-Nicolson noise. This paper shows that the forcing function can be turned on (or the surface temperature raised to a steady value) in such a way as to make the noise produced by it considerably reduced by making it proportional to 1–e −βt.

Journal ArticleDOI
TL;DR: In this article, a generalized complementary energy principle is used for the derivation of the element matrices for the calculation of natural frequencies, where degrees-of-freedom are not defined on nodal points but in an abstract way.
Abstract: Based on a generalized complementary energy principle the derivation of the element matrices is presented for calculation of natural frequencies. The degrees-of-freedom are not defined on nodal points but in an abstract way. No restrictions about the number of interpolation functions in the interior and at the boundaries of the element have been introduced. The exact solution of the discretized element equations leads to the dynamic stiffness matrix while the approximate solution results in a linear eigenvalue problem. Plate bending problems are used to study the convergence of frequencies depending on the degrees of interpolation functions within the element and on its boundaries and on the number of elements.

Journal ArticleDOI
TL;DR: In this paper, the existence, uniqueness, and convergence of approximate solutions of transport equations by methods of the Galerkin type (where trial and weighting functions are the same) are discussed.

Journal ArticleDOI
TL;DR: The proposed method simplifies the computational complexities and gives reasonably accurate results, and the numeric results closely agree with a published result where the same problem is treated by another method.
Abstract: The purpose of this correspondence is to outline a method to study time optimal control of linear diffusion systems by spatial discretization procedure. The proposed method simplifies the computational complexities and gives reasonably accurate results. The numeric results closely agree with a published result where the same problem is treated by another method.