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Collocation Points (collocation + point)

Distribution by Scientific Domains

Mathematics and Statistics	33%
Engineering	33%
Chemistry	33%

Selected Abstracts

Boundary element formulation for 3D transversely isotropic cracked bodies

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2004
M. P. Ariza
Abstract The boundary traction integral representation is obtained in elasticity when the classical displacement representation is differentiated and combined according to Hooke's law. The use of both traction and displacement integral representations leads to a mixed (or dual) formulation of the BEM where the discretization effort for crack problems is much smaller than in the classical formulation. A boundary element analysis of three-dimensional fracture mechanics problems of transversely isotropic solids based on the mixed formulation is presented in this paper. The hypersingular and strongly singular kernels appearing in the formulation are regularized by using two terms of the displacement series expansion and one term of the traction expansion, at the collocation point. All the remaining integrals are analytically evaluated or transformed by means of Stokes' theorem into regular or weakly singular integrals, which are numerically computed. The method is general and can be used for elements of any shape including quarter-point crack front elements. No change of co-ordinates is required for the integration. The formulation as presented in this paper is something as clear, general and easy to handle as the classical BE formulation. It is used in combination with three-dimensional quadratic and quarter-point elements to obtain accurate results for several different crack problems. Cracks in boundless and finite transversely isotropic domains are studied. The meshes are simple and include only discretization of the crack and the external boundary. The obtained results are in good agreement with those existing in the literature. Copyright © 2004 John Wiley & Sons, Ltd. [source]

On a quadrature algorithm for the piecewise linear wavelet collocation applied to boundary integral equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2003
Andreas Rathsfeld
Abstract In this paper, we consider a piecewise linear collocation method for the solution of a pseudo-differential equation of order r=0, ,1 over a closed and smooth boundary manifold. The trial space is the space of all continuous and piecewise linear functions defined over a uniform triangular grid and the collocation points are the grid points. For the wavelet basis in the trial space we choose the three-point hierarchical basis together with a slight modification near the boundary points of the global patches of parametrization. We choose linear combinations of Dirac delta functionals as wavelet basis in the space of test functionals. For the corresponding wavelet algorithm, we show that the parametrization can be approximated by low-order piecewise polynomial interpolation and that the integrals in the stiffness matrix can be computed by quadrature, where the quadrature rules are composite rules of simple low-order quadratures. The whole algorithm for the assembling of the matrix requires no more than O(N [logN]3) arithmetic operations, and the error of the collocation approximation, including the compression, the approximative parametrization, and the quadratures, is less than O(N,(2,r)/2). Note that, in contrast to well-known algorithms by Petersdorff, Schwab, and Schneider, only a finite degree of smoothness is required. In contrast to an algorithm of Ehrich and Rathsfeld, no multiplicative splitting of the kernel function is required. Beside the usual mapping properties of the integral operator in low order Sobolev spaces, estimates of Calderón,Zygmund type are the only assumptions on the kernel function. Copyright © 2003 John Wiley & Sons, Ltd. [source]

Analysis of a block red-black preconditioner applied to the Hermite collocation discretization of a model parabolic equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2001
Stephen H. Brill
Abstract We are concerned with the numerical solution of a model parabolic partial differential equation (PDE) in two spatial dimensions, discretized by Hermite collocation. In order to efficiently solve the resulting systems of linear algebraic equations, we choose the Bi-CGSTAB method of van der Vorst (1992) with block Red-Black Gauss-Seidel (RBGS) preconditioner. In this article, we give analytic formulae for the eigenvalues that control the rate at which Bi-CGSTAB/RBGS converges. These formulae, which depend on the location of the collocation points, can be utilized to determine where the collocation points should be placed in order to make the Bi-CGSTAB/RBGS method converge as quickly as possible. Along these lines, we discuss issues of choice of time-step size in the context of rapid convergence. A complete stability analysis is also included. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:584,606, 2001 [source]

State estimation of a solid-state polymerization reactor for PET based on improved SR-UKF

ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING, Issue 2 2010
Ji Liu
Abstract A state estimator for the continuous solid-state polymerization (SSP) reactor of polyethylene terephthalate (PET) is designed in this study. Because of its invalidity in the application to some of the practical examples such as SSP processes, the square-root unscented Kalman filter (SR-UKF) algorithm is improved for the state estimation of arbitrary nonlinear systems with linear measurements. Discussions are given on how to avoid the filter invalidation and accumulating additional error. Orthogonal collocation method has been used to spatially discretize the reactor model described by nonlinear partial differential equations. The reactant concentrations on chosen collocation points are reconstructed from the outlet measurements corrupted with a large noise. Furthermore, the error performance of the developed ISR-UKF is investigated under the influence of various initial parameters, inaccurate measurement noise parameters and model mismatch. Simulation results show that this technique can produce fast convergence and good approximations for the state estimation of SSP reactor. Copyright © 2009 Curtin University of Technology and John Wiley & Sons, Ltd. [source]

Simulation and Optimization of an Adiabatic Multi-Bed Catalytic Reactor for the Oxidation of SO2

CHEMICAL ENGINEERING & TECHNOLOGY (CET), Issue 1 2007
A. Nodehi
Abstract A software package was developed for the simulation and optimization of a multi-bed adiabatic reactor for the catalytic oxidation of SO2, using a heterogeneous plug flow model. The orthogonal collocation (OC) technique with up to eight collocation points was used for the solution of a nonlinear, two-point boundary value differential equation for the catalyst particle, and it was shown that the use of the OC technique with two collocation points can describe the system well. Because of the nonlinear behavior of the effectiveness factor along the bed, optimal catalyst distribution between the beds and corresponding inlet temperatures were determined by two methods, including: the use of (1) intrinsic or (2) actual rate of reaction in the optimization criteria. The results showed that for the second case, the minimum amount of the catalyst can be reached at lower temperatures, the amount of catalyst required is always less, and the number of beds is greater than or equal to that of the first case. [source]

Analytic Element Modeling of Embedded Multiaquifer Domains

GROUND WATER, Issue 1 2006
Mark Bakker
An analytic element approach is presented for the modeling of multiaquifer domains embedded in a single-aquifer model. The inside of each domain may consist of an arbitrary number of aquifers separated by leaky layers. The analytic element solution is obtained through a combination of existing single-aquifer and multiaquifer analytic elements and allows for the analytic computation of head and leakage at any point in the aquifer. Along the boundary of an embedded multiaquifer domain, the normal flux is continuous everywhere; continuity of head across the boundary is met exactly at collocations points and approximately, but very accurately, in between. The analytic element solution compares well with an existing exact solution. A hypothetical example with a river intersecting two embedded domains illustrates the practical application of the proposed approach. [source]