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Gaussian Quadrature (gaussian + quadrature)
Selected AbstractsA hypersingular time-domain BEM for 2D dynamic crack analysis in anisotropic solidsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2009M. Wünsche Abstract A hypersingular time-domain boundary element method (BEM) for transient elastodynamic crack analysis in two-dimensional (2D), homogeneous, anisotropic, and linear elastic solids is presented in this paper. Stationary cracks in both infinite and finite anisotropic solids under impact loading are investigated. On the external boundary of the cracked solid the classical displacement boundary integral equations (BIEs) are used, while the hypersingular traction BIEs are applied to the crack-faces. The temporal discretization is performed by a collocation method, while a Galerkin method is implemented for the spatial discretization. Both temporal and spatial integrations are carried out analytically. Special analytical techniques are developed to directly compute strongly singular and hypersingular integrals. Only the line integrals over an unit circle arising in the elastodynamic fundamental solutions need to be computed numerically by standard Gaussian quadrature. An explicit time-stepping scheme is obtained to compute the unknown boundary data including the crack-opening-displacements (CODs). Special crack-tip elements are adopted to ensure a direct and an accurate computation of the elastodynamic stress intensity factors from the CODs. Several numerical examples are given to show the accuracy and the efficiency of the present hypersingular time-domain BEM. Copyright © 2008 John Wiley & Sons, Ltd. [source] Optimal stress sampling points of plane triangular elements for patch recovery of nodal stressesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2003S. Rajendran Abstract The existence of optimal stress sampling points in finite elements was first observed by Barlow. Knowledge of optimal stress sampling points is important in stress-recovery methods such as the superconvergent patch recovery (SPR). Recently, MacNeal observed that Barlow points and Gaussian quadrature points are the same for the linear and quadratic bar elements, and different for the cubic bar element. Prathap proposed the best-fit approach to predict the optimal sampling points, and showed that the best-fit points coincide with Gaussian quadrature points not only for the linear and quadratic bar elements but also for the cubic bar element. In this paper, the best-fit approach for predicting the optimal sampling points is extended to the linear and quadratic plane triangular elements, and the effectiveness of Barlow points, Gaussian points and best-fit points as candidates of sampling points for the patch recovery of nodal stresses with these triangular elements is investigated for typical problems. The numerical results suggest that Barlow points do not exist for all strain/stress components, and Gaussian quadrature points which are the same as or close to the best-fit points are better candidates for patch recovery. Copyright © 2003 John Wiley & Sons, Ltd. [source] Adaptive charting schemes based on double sequential probability ratio testsQUALITY AND RELIABILITY ENGINEERING INTERNATIONAL, Issue 1 2009Yan Li Abstract Sequential probability ratio test (SPRT) control charts are shown to be able to detect most shifts in the mean or proportion substantially faster than conventional charts such as CUSUM charts. However, they are limited in applications because of the absence of the upper bound on the sample size and possibly large sample numbers during implementation. The double SPRT (2-SPRT) control chart, which applies a 2-SPRT at each sampling point, is proposed in this paper to solve some of the limitations of SPRT charts. Approximate performance measures of the 2-SPRT control chart are obtained by the backward method with the Gaussian quadrature in a computer program. On the basis of two industrial examples and simulation comparisons, we conclude that the 2-SPRT chart is competitive in that it is more sensitive and economical for small shifts and has advantages in administration because of fixed sampling points and a proper upper bound on the sample size. Copyright © 2008 John Wiley & Sons, Ltd. [source] Valuing credit derivatives using Gaussian quadrature: A stochastic volatility frameworkTHE JOURNAL OF FUTURES MARKETS, Issue 1 2004Nabil Tahani This article proposes semi-closed-form solutions to value derivatives on mean reverting assets. A very general mean reverting process for the state variable and two stochastic volatility processes, the square-root process and the Ornstein-Uhlenbeck process, are considered. For both models, semi-closed-form solutions for characteristic functions are derived and then inverted using the Gauss-Laguerre quadrature rule to recover the cumulative probabilities. As benchmarks, European call options are valued within the following frameworks: Black and Scholes (1973) (represents constant volatility and no mean reversion), Longstaff and Schwartz (1995) (represents constant volatility and mean reversion), and Heston (1993) and Zhu (2000) (represent stochastic volatility and no mean reversion). These comparisons show that numerical prices converge rapidly to the exact price. When applied to the general models proposed (represent stochastic volatility and mean reversion), the Gauss-Laguerre rule proves very efficient and very accurate. As applications, pricing formulas for credit spread options, caps, floors, and swaps are derived. It also is shown that even weak mean reversion can have a major impact on option prices. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:3,35, 2004 [source] High-order ENO and WENO schemes for unstructured gridsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2007W. R. Wolf Abstract This work describes the implementation and analysis of high-order accurate schemes applied to high-speed flows on unstructured grids. The class of essentially non-oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third- and fourth-order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2-D Euler equations in a cell centred finite volume context. High-order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge,Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high-order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high-speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd. [source] Gaussian approximation of exponential type orbitals based on B functionsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 2 2009Didier Pinchon Abstract This work gives new, highly accurate optimized gaussian series expansions for the B functions used in molecular quantum mechanics. These functions are generally chosen because of their compact Fourier transform, following Shavitt. The inverse Laplace transform in the square root of the variable is used for Gauss quadrature in this work. Two procedures for obtaining accurate gaussian expansions have been compared for the required extended precision arithmetic. The first is based on Gaussian quadratures and the second on direct optimization. Both use the Maple computer algebra system. Numerical results are tabulated and compared with previous work. Special cases are found to agree before pushing the optimization technique further. The optimal gaussian expansions of B functions obtained in this work are available for reference. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source] | ||||||||||