SUMMARYThis paper presents an uniform and unified approach to construct h-and p-shape functions for quadrilaterals, triangles, hexahedral and tetrahedral based on the tensorial product of one-dimensional Lagrange and Jacobi polynomials. The approach uses indices to denote the one-dimensional polynomials in each tensorization direction. The appropriate manipulation of the indices allows to obtain hierarchical or nonhierarchical and inter-element C 0 continuous or non-continuous bases. For the one-dimensional elements, quadrilaterals, triangles and hexahedral, the optimal weights of the Jacobi polynomials are determined, the sparsity profiles of the local mass and stiffness matrices plotted and the condition numbers calculated. A brief discussion of the use of sum factorization and computational implementation is considered.
SUMMARYIn this work, we choose the points and weights of the Gauss-Jacobi, Gauss-Radau-Jacobi and GaussLobatto-Jacobi quadrature rules that optimize the number of operations for the mass and stiffness matrices of the high-order finite element method. The procedure is particularly applied to the mass and stiffness matrices using the tensor-based nodal and modal shape functions given in (Int. J. Numer. Meth. Engng 2007; 71(5):529-563). For square and hexahedron elements, we show that it is possible to use tensor product of the 1D mass and stiffness matrices for the Poisson and elasticity problem. For the triangular and tetrahedron elements, an analogous analysis given in (Int. J. Numer. Meth. Engng 2005; 63(2):1530-1558) was considered for the selection of the optimal points and weights for the stiffness matrix coefficients for triangles and mass and stiffness matrices for tetrahedra.
Relation between two sequences of orthogonal polynomials, where the associated measures are related to each other by a first degree polynomial multiplication (or division), is well known. We use this relation to study the monotonicity properties of the zeros of generalized orthogonal polynomials. As examples, the Jacobi, Laguerre and Charlier polynomials are considered. 2005 Elsevier Inc. All rights reserved.
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