Showing papers on "Discretization published in 1981"
TL;DR: In this paper, it was shown that the Green9s functions for an elastic layered medium can be expressed as a double integral over frequency and horizontal wavenumber, which can be exactly represented by a discrete summation.
Abstract: Green9s functions for an elastic layered medium can be expressed as a double integral over frequency and horizontal wavenumber. We show that, for any time window, the wavenumber integral can be exactly represented by a discrete summation. This discretization is achieved by adding to the particular point source an infinite set of specified circular sources centered around the point source and distributed at equal radial interval. Choice of this interval is dependent on the length of time desired for the point source response and determines the discretized set of horizontal wavenumbers which contribute to the solution. Comparisons of the results obtained with those derived using the two-dimensional discretization method (Bouchon, 1979) are presented. They show the great accuracy of the two methods.
928 citations
261 citations
TL;DR: The Hough transform was improved by O'Gorman and Clowes by taking into account the gradient direction and the resulting scatter of the peaks can be reduced by using a weighting function in the transform.
196 citations
IBM1
TL;DR: In this paper, numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data, and two separate topics are studied: the first deals with the solution of the acoustic wave equation and the second topic treats wave phenomena whose direction of propagation is restricted within ±90 degrees from a given axis.
Abstract: Numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data. Two separate topics are studied. The first deals with the solution of the acoustic wave equation. The second topic treats wave phenomena whose direction of propagation is restricted within ±90 degrees from a given axis. In the numerical methods developed here, the wave field is advanced in time by using standard time differencing schemes. On the other hand, expressions including space derivative terms are computed by Fourier transform methods. This approach to computing derivatives minimizes truncation errors. Another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves, e.g., to avoid multiple reflections. Constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain. The error analysis of the algorithms shows that truncation errors are due mainly to time discretization. Such er...
188 citations
TL;DR: In this article, the functional coefficients imbedded in a two-dimensional partial differential equation which governs unsteady groundwater flow are optimally identified using a modified Gauss-Newton algorithm.
Abstract: This paper presents a systematic procedure whereby the functional coefficients imbedded in a two-dimensional partial differential equation which governs unsteady groundwater flow are optimally identified. The coefficients to be identified are transmissivities which vary spatially. Finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region. A modified Gauss-Newton algorithm is used for parameter optimization. Covariance analysis is used to estimate the reliability of the estimated parameters. As the dimension of the unknown parameter increases, the modeling error represented by a least squares criterion will generally decrease, but errors in data would be propagated to a greater degree into the estimated parameters, thus reducing the reliability of estimation. The reliability of the estimated parameters is characterized by a norm of the covariance matrix. This information is used for the determination of the optimum dimension in parameterization.
177 citations
TL;DR: In this paper, the discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy.
Abstract: The discretization by finite elements of a model variational problem for a clamped loaded beam is studied with emphasis on the effect of the beam thickness, which appears as a parameter in the problem, on the accuracy. It is shown that the approximation achieved by a standard finite element method degenerates for thin beams. In contrast a large family of mixed finite element methods are shown to yield quasioptimal approximation independent of the thickness parameter. The most useful of these methods may be realized by replacing the integrals appearing in the stiffness matrix of the standard method by Gauss quadratures.
158 citations
30 Nov 1981
TL;DR: In this article, an outline of the derivation of the differential equation governing linear subsonic and supersonic potential flow is given, and the use of Green's Theorem to obtain an integral equation over the boundary surface is discussed.
Abstract: An outline of the derivation of the differential equation governing linear subsonic and supersonic potential flow is given. The use of Green's Theorem to obtain an integral equation over the boundary surface is discussed. The engineering techniques incorporated in the PAN AIR (Panel Aerodynamics) program (a discretization method which solves the integral equation for arbitrary first order boundary conditions) are then discussed in detail. Items discussed include the construction of the compressibility transformations, splining techniques, imposition of the boundary conditions, influence coefficient computation (including the concept of the finite part of an integral), computation of pressure coefficients, and computation of forces and moments.
138 citations
TL;DR: Weak conditions are presented for approximating dynamic programming models and results about convergence preserving properties of supremum value functions and integrals are extended to so-called upper-semi-continuous convergence sequences and are related to discretization procedures by using projections.
Abstract: Weak conditions are presented for approximating dynamic programming models For a sequence of these models, continuous convergence of the sequence of associated optimal value functions is obtained under the condition that state and action space converge in the sense of Kuratowski, and that the mappings of admissible actions as well as the transition law, the discount factors and the reward functions converge continuously Further a relation for the associated sets of optimal actions is given The analysis is based on results about convergence preserving properties of supremum value functions and integrals The approximation results are extended to so-called upper-semi-continuous convergent sequences and are related to discretization procedures by using projections
121 citations
TL;DR: The finite-difference method is a direct, versatile, and reasonably efficient means of solving the two-dimensional cochlear model.
Abstract: A current, linear, two-dimensional mathematical model of the mechanics of the cochlea is solved numerically by using a finite difference approximation of the model equations. The finite-difference method is used to discretize Laplace's equation over a rectangular region with specified boundary conditions. The resulting matrix equation for fluid pressure is solved by using a Gaussian block-elimination technique. Numerical solutions are obtained for fluid pressure and basilar membrane displacement as a function of distance from the stapes. The finite difference method is a direct, versatile, and reasonably efficient means of solving the two-dimensional cochlear model.
118 citations
TL;DR: In this paper, the authors proposed a numerical approach to probabilistic calculations with great simplicity, power, and elegance, and an application to seismic risk assessment is outlined, where the philosophy and technique of this approach are described, some pitfalls are pointed out, and some application examples are outlined.
Abstract: If the point of view is adopted that in calculations of real-world phenomena we almost invariably have significant uncertainty in the numerical values of our parameters, then, in these calculations, numerical quantities should be replaced by probability distributions and mathematical operations between these quantities should be replaced by analogous operations between probability distributions. Also, practical calculations one way or another always require discretization or truncation. Combining these two thoughts leads to a numerical approach to probabilistic calculations having great simplicity, power, and elegance. The philosophy and technique of this approach is described, some pitfalls are pointed out, and an application to seismic risk assessment is outlined.
116 citations
TL;DR: In this paper, a standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed.
Abstract: A standard Galerkin method for a quasilinear equation of Sobolev type using continuous, piecewise-polynomial spaces is presented and analyzed. Optimal order error estimates are established in various norms, and nodal superconvergence is demonstrated. Discretization in time by explicit single-step methods is discussed.
TL;DR: In this paper, the authors examined the discrete eigenstructure of a well-known Lagrange multiplier formulation for linear elasticity or Stokes-flow and showed how the weak incompressibility constraint is reflected in the elementary divisor structure of the eigenproblem whose solution determines the finite element approximation to the natural modes.
TL;DR: The treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrids algorithm but only for fast convergence.
Abstract: The treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence. For the linear equations which arise from the discretization of the Poisson equation, a convergence factor 0,5 is established independent of the shape of the domain and of the regularity of the solution.
TL;DR: In this article, a numerical method is presented to compute one unknown constitutive parameter of an inhomogeoeous lossy dielectric slab from the reflected field in the time domain.
Abstract: A numerical method is presented to compute one unknown constitutive parameter of an inhomogeoeous lossy dielectric slab from the reflected field in the time domain. The method is based upon a space-time discretization of the integral equation for the reflected field. In the inversion, especially those space-time points where the numerical computation of the electric-field strength in the slab is most accurate are taken into account. This is achieved by computing the unknown parameter iteratively. Alternately solving equations for an approximate direct-scattering problem and an approximate inverse-scattering problem yields successive approximations for the electric field in the slab and the unknown constitutive coefficient. Both problems lead to an infinite system of linear equations from which a finite subsystem is selected. General criteria for this selection are presented. Various profiles have been reconstructed numerically from the reflected field due to a sine-squared incident pulse.
TL;DR: In this paper, the quantum inverse scattering method was applied to a discretization of the generalized Toda field theory and a family of solutions of the factorization equations were obtained.
TL;DR: In this article, an entirely new formulation of the classical steady-state skin effect problem is presented, where the magnetic vector potential is obtained directly from the projective solution of a single integrodifferential equation.
Abstract: An entirely new formulation of the classical steady-state skin effect problem is presented. The magnetic vector potential is obtained directly from the projective solution of a single integrodifferential equation. In this new form of the steady-state diffusion equation, the known measurable total current in conductors replaces the usual unknown source current density vector. The validity and correctness of the new formulation are demonstrated by a simple, easily verifiable example based on finite element discretization.
TL;DR: In this article, the usefulness of quasi-Newton methods for the solution of nonlinear systems of equations is demonstrated, after a review of the Newton iterative method, several quasiNewton updates are presented and tested, and a complete discussion of different methods used, of the convergence rates and of the associated computer costs.
TL;DR: In this article, the estimation of wind-wave generation using a new discrete spectral model is compared to Hasselmann et al.'s (1976) parametric model and to models driven primarily by direct transfer of energy from the atmosphere into the surface waves.
Abstract: The estimation of wind-wave generation using a new discrete spectral model is compared to Hasselmann et al.'s (1976) parametric model and to models driven primarily by direct transfer of energy from the atmosphere into the surface waves. The main source term in this new model is a new parameterization of the net energy transfer due to nonlinear wave-wave interactions. After calibration of the wave-wave interaction source term to resemble the form of the solution to the complete Boltzmann integrals, the discrete spectral model is able to reproduce the fetch-limited results of Hasselmann et al. and fits well within the envelope of duration-limited growth curves from recent investigations. Since this model is discretized into frequency and direction components, a finite-difference scheme is used to model propagation effects. The formulation of this model allows simulations in oceanic conditions to consider both wave growth under local winds and swell decay of waves passing through a region simultane...
TL;DR: In this paper, an efficient numerical method for computing electromagnetic scattering of arbitrary 3-D local inhomogeneities buried in a uniform or two-layered earth is presented, where the inhomogeneity is enclosed by a volume whose conductivity is discretized by a finite-element mesh and whose boundary is only a slight distance away from the inhomeogeneity.
Abstract: We present an efficient numerical method for computing electromagnetic (EM) scattering of arbitrary three‐dimensional (3-D) local inhomogeneities buried in a uniform or two‐layered earth. In this scheme the inhomogeneity is enclosed by a volume whose conductivity is discretized by a finite‐element mesh and whose boundary is only a slight distance away from the inhomogeneity. The scheme uses two sets of independent equations. The first is a set of finite‐element equations derived from a variational integral, and the second is a mathematical expression for the fields at the boundany in terms of electric fields inside the boundary. The Green’s function is used to derive the second set of equations. An iterative algorithm has been developed to solve these two sets of equations. The solutions are the electric fields at nodes inside the finite‐element mesh. The scattered fields anywhere may then be obtained by performing volume integrations over the inhomogeneous region. The scheme is used for modeling 3-D inho...
TL;DR: In this paper, a methodology is presented for numerical solution of transient two-dimensional diffusion-type problems (e.g., heat conduction) in which one of the boundaries of the solution domain moves with time.
TL;DR: A new direct Newton-like method for computing turning points ofL is described that requires per step only the evaluation of one Jacobian and 5 function values ofH to solve a linear system of dimensionn+1 with 4 different right hand sides.
Abstract: Let the space curveL be defined implicitly by the (n, n+1) nonlinear systemH(u)=0. A new direct Newton-like method for computing turning points ofL is described that requires per step only the evaluation of one Jacobian and 5 function values ofH. Moreover, a linear system of dimensionn+1 with 4 different right hand sides has to be solved per step. Under suitable conditions the method is shown to converge locally withQ-order two if a certain discretization stepsize is appropriately chosen. Two numerical examples confirm the theoretical results.
TL;DR: Two one-parameter families of fourth order HODIE discretizations of the Helmholtz equation are derived and a discretization optimal with respect to a certain norm of the truncation error is identified.
Abstract: In this paper we construct and analyze high order finite difference discretizations of a class of elliptic partial differential equations. In particular, two one-parameter families of fourth order HODIE discretizations of the Helmholtz equation are derived and a discretization optimal with respect to a certain norm of the truncation error is identified. The use of compact nine-point formulas of positive type admits both fast direct methods and standard iterative methods for the solution of the resulting systems of linear equations. Extensions yielding sixth order accuracy for the Helmholtz equation and fourth order accuracy for a more general operator are given. Finally, numerical results demonstrating the effectiveness of the discretizations for a wide range of problems are presented.
TL;DR: It is found that in the models considered, small-scale noise can be ascribed to resonance anomalies associated with the method of spatial discretization, and a formal proof is given of the time-stepping stability of a general, discretized form of Laplace's tidal equations.
TL;DR: In this paper, the application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated, and it is shown that certain collocation schemes, in particular those based on Gauss or Lobatto points, do perform very well on such problems, provided that a fine mesh with steps proportional to the layers'' width is used in the layers only, and a coarse mesh, just fine enough to resolve the solution of the reduced problem, was used in between.
Abstract: The application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated. Collocation at Gauss, Radau and Lobatto points is considered, for both initial and boundary value problems for first order systems with constant coefficients. Particular attention is paid to symmetric schemes for boundary value problems; these problems may have boundary layers at both interval ends. .br Our analysis shows that certain collocation schemes, in particular those based on Gauss or Lobatto points, do perform very well on such problems, provided that a fine mesh with steps proportional to the layers'' width is used in the layers only, and a coarse mesh, just fine enough to resolve the solution of the reduced problem, is used in between. Ways to construct appropriate layer meshes are proposed. Of all methods considered, the Lobatto schemes appear to be the most promising class of methods, as they essentially retain their usual superconvergence power for the smooth, reduced solution, whereas Gauss-Legendre schemes do not. .br We also investigate the conditioning of the linear systems of equations arizing in the discretization of the boundary value problem. For a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number, which indicates that these systems can be solved safely by Gaussian elimination with scaled partial pivoting.
TL;DR: In this article, multiple grid methods are applied for the fast solution of the large nonsparse systems of equations that arise from the discretization of Fredholm integral equations of the second kind.
Abstract: In this paper multiple grid methods are applied for the fast solution of the large nonsparse systems of equations that arise from the discretization of Fredholm integral equations of the second kind. Various multiple grid schemes, both with Nystrom and with direct interpolation, are considered. For these iterative methods, the rates of convergence are derived using the collectively compact operator theory by Anselone and Atkinson. Estimates for the asymptotic computational complexity are given, which show that the multiple grid schemes result in e (N2) arithmetic operations.
TL;DR: In this article, the so-called Kupradze functional equations, where the two variables of the kernel belong to two different point sets, are investigated, and the condition number of the corresponding matrices is investigated analytically and numerically.
Abstract: We investigate some integral equations, i. a. the so-called Kupradze functional equations, where the two variables of the kernel belong to two different point sets. An extensive survey of the literature shows the various applications of these equations.
By a discretization of the integral equations they are replaced by systems of linear algebraic equations. The condition number of the corresponding matrices is investigated, analytically and numerically. It is thereby quantitatively found in which way the condition of the matrices deteriorates when the two point sets are moved away from each other.
TL;DR: In this article, a reduced basis technique and a problem-adaptive computational algorithm are presented for the bifurcation and post-buckling analysis of laminated anisotropic plates.
TL;DR: A total Lagrangian formulation for large deformation analysis of shells by the finite element method is presented in this article, where a special discretization in the thickness direction is employed to permit solution of shell problems without numerical difficulties.
TL;DR: In this article, a hybrid finite-element Rayleigh-Ritz technique is used to predict the collapse behavior of shells, and a scalar function is introduced to measure the degree of nonlinearity of the structure for the case of loading applied by means of axial end shortening.
Abstract: A hybrid finite-element Rayleigh-Ritz technique is used to predict the collapse behavior of shells. In this hybrid technique, the modeling versatility of the finite-element method is preserved, and a significant reduction in the number of degrees of freedom is achieved by expressing the nodal displacement vector as a linear combination of a small number of basis vectors. A Rayleigh-Ritz technique is used to approximate the finite-element equations of the discretized shell by a reduced system of nonlinear algebraic equations. A scalar function is introduced to measure the degree of nonlinearity of the structure for the case of loading applied by means of axial end shortening. Also, a quantitative measure for the error of the reduced system of equations is proposed. Some insight is given as to why and when the reduced basis technique works, and the effectiveness of the technique for predicting the collapse behavior of shells is demonstrated by means of a numerical example of elastic collapse of an axially compressed pear-shaped cylinder.