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Quadrature Points (quadrature + point)
Selected AbstractsPositive-definite q -families of continuous subcell Darcy-flux CVD(MPFA) finite-volume schemes and the mixed finite element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Michael G. Edwards Abstract A new family of locally conservative cell-centred flux-continuous schemes is presented for solving the porous media general-tensor pressure equation. A general geometry-permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control-volume distributed subcell flux-continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2:259,290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive-definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical-space schemes are shown to be non-symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non-symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical-space and subcell-space q -families of schemes. M -matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical-space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell-wise constant tensor schemes and that subcell tensor approximation using the control-volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 John Wiley & Sons, Ltd. [source] Convergence study of a family of flux-continuous, finite-volume schemes for the general tensor pressure equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9-10 2006Mayur Pal Abstract In this paper, a numerical convergence study of family of flux-continuous schemes is presented. The family of flux-continuous schemes is characterized in terms of quadrature parameterization, where the local position of continuity defines the quadrature point and hence the family. A convergence study is carried out for the discretization in physical space and the effect of a range of quadrature points on convergence is explored. Structured cell-centred and unstructured cell-vertex schemes are considered. Homogeneous and heterogeneous cases are tested, and convergence is established for a number of examples with discontinuous permeability tensor including a velocity field with singularity. Such cases frequently arise in subsurface flow modelling. A convergence comparison with CVFE is also presented. Copyright © 2006 John Wiley & Sons, Ltd. [source] Computational homogenization of uncoupled consolidation in micro-heterogeneous porous mediaINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 14 2010Fredrik Larsson Abstract Variationally consistent homogenization is exploited for the analysis of transient uncoupled consolidation in micro-heterogeneous porous solids, whereby the classical approach of first-order homogenization for stationary problems is extended to transient problems. Homogenization is then carried out in the spatial domain on representative volume elements (RVE), which are introduced in quadrature points in standard fashion. Along with the classical averages, a higher-order conservation quantity is obtained. An iterative FE2 -algorithm is devised for the case of nonlinear permeability and storage coefficients, and it is applied to pore pressure changes in asphalt-concrete (particle composite). Various parametric studies are carried out, in particular, with respect to the influence of the ,substructure length scale' that is represented by the size of the RVE's. Copyright © 2009 John Wiley & Sons, Ltd. [source] Variationally consistent computational homogenization of transient heat flowINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2010Fredrik Larsson Abstract A framework for variationally consistent homogenization, combined with a generalized macro-homogeneity condition, is exploited for the analysis of non-linear transient heat conduction. Within this framework the classical approach of (model-based) first-order homogenization for stationary problems is extended to transient problems. Homogenization is then carried out in the spatial domain on representative volume elements (RVE), which are (in practice) introduced in quadrature points in standard fashion. Along with the classical averages, a higher order conservation quantity is obtained. An iterative FE2 -algorithm is devised for the case of non-linear diffusion and storage coefficients, and it is applied to transient heat conduction in a strongly heterogeneous particle composite. Parametric studies are carried out, in particular with respect to the influence of the ,internal length' associated with the second-order conservation quantity. Copyright © 2009 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] Convergence study of a family of flux-continuous, finite-volume schemes for the general tensor pressure equationINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 9-10 2006Mayur Pal Abstract In this paper, a numerical convergence study of family of flux-continuous schemes is presented. The family of flux-continuous schemes is characterized in terms of quadrature parameterization, where the local position of continuity defines the quadrature point and hence the family. A convergence study is carried out for the discretization in physical space and the effect of a range of quadrature points on convergence is explored. Structured cell-centred and unstructured cell-vertex schemes are considered. Homogeneous and heterogeneous cases are tested, and convergence is established for a number of examples with discontinuous permeability tensor including a velocity field with singularity. Such cases frequently arise in subsurface flow modelling. A convergence comparison with CVFE is also presented. Copyright © 2006 John Wiley & Sons, Ltd. [source] Distributed Gaussian discrete variable representationINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 1 2005Hasan Karabulut Abstract A discrete variable representation (DVR) made from distributed Gaussians gn(x) = e, (n = ,,, ,, ,) and its infinite grid limit is described. The infinite grid limit of the distributed Gaussian DVR (DGDVR) reduces to the sinc function DVR of Colbert and Miller in the limit c , 0. The numerical performance of both finite and infinite grid DGDVRs and the sinc function DVR is compared. If a small number of quadrature points are taken, the finite grid DGDVR performs much better than both infinite grid DGDVR and sinc function DVR. The infinite grid DVRs lose accuracy due to the truncation error. In contrast, the sinc function DVR is found to be superior to both finite and infinite grid DGDVRs if enough grid points are taken to eliminate the truncation error. In particular, the accuracy of DGDVRs does not get better than some limit when the distance between Gaussians d goes to zero with fixed c, whereas the accuracy of the sinc function DVR improves very quickly as d becomes smaller, and the results are exact in the limit d , 0. An analysis of the performance of distributed basis functions to represent a given function is presented in a recent publication. With this analysis, we explain why the sinc function DVR performs better than the infinite grid DGDVR. The analysis also traces the inability of Gaussians to yield exact results in the limit d , 0 to the incompleteness of this basis in this limit. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [source] | ||||||||||