Definition Of A P-interpolating Space Of Hierarchical Bases Of Finite Elements On The Pyramid

This article shows in detail how to construct in a simple and ordered way a set of rational functions defined on the pyramid topology. The set of functions is parameterized by an integer p. It is shown that these functions, defined in a hierarchical way, constitute a basis for a complete polynomial interpolation space of degree p on the pyramid domain. In order to help this definition we use a denumerable sequence of orthogonal polynomials defined on an interval of the real line. A priori, any increasing sequence of polynomials in one variable can be used. The bases are constructed to be used in the class C0 method of finite elements. The rational functions thus defined can be combined to represent any polynomial of degree p. Thus, given an arbitrary number p, one defines a finite element whose geometry is a pyramid that has associated a complete interpolation space of degree p. Moreover, this element is adequate to be used with the p-adaptive technique on heterogeneous meshes of finite elements hierarchical. © 2014 Elsevier Inc.460174204Ayala Bravo, C.M.A., (2000) Um Sistema de Refinamento H-p Adaptativo Utilizando Elementos Finitos Hierárquicos Multidimensionais, , Departamento de Mecânica Computacional da Faculdade de Engenharia Mecânica, UNICAMP Campinas, BrasilBabuska, I., Szabo, B.A., Katz, I.N., The p-version of the finite element method (1981) SIAM J. Numer. Anal., 18 (3), pp. 515-545Campion, S.D., Jarvis, J.L., An investigation of the implementation of the p-version finite element method (1996) Finite Elem. Anal. Des., 23, pp. 1-21Choquet, G., Topology (1966) Pure Appl. Math., 19, p. 503. , ISBN-13: 978-0121734503, Copyright 2010 Elsevier Ltd. All rights reservedDevloo, P.R.B., Ayala Bravo, C.M.A., Rylo, E.C., Systematic and generic construction of shape functions for p-adaptive meshes of multidimensional finite elements (2009) Comput. Methods Appl. Mech. Engrg., 5, pp. 1716-1725Hoffman, R.M., Busby, H.R., Stress-based convergence analysis for p-adaptive hierarchical finite element analysis (2000) Finite elements in analysis and design, 34 (1), pp. 25-35. , DOI 10.1016/S0168-874X(99)00025-6Lax, P.D., Linear Algebra (1997) Pure Appl. Math., , John Wiley & Sons, Inc. 0-471-11111-2Liu, Y., Busby, H.R., P-version hybrid/mixed finite element method (1998) Finite Elements in Analysis and Design, 30 (4), pp. 325-333. , PII S0168874X98000420Owen, S.J., Saigal, S., Formation of pyramid elements for hexahedra to tetrahedra transitions (2001) Computer Methods in Applied Mechanics and Engineering, 190 (34), pp. 4505-4518. , DOI 10.1016/S0045-7825(00)00330-3, PII S0045782500003303Sherwin, S.J., Hierarchical hp finite elements in hybrid domains (1997) Finite Elem. Anal. Des., 27, pp. 109-119Vijayakar, S.M., Busby, H.R., Houser, D.R., Finite element analysis of quasi-prismatic bodies using Chebyshev polynomials (1987) Internat. J. Numer. Methods Engrg., 24, pp. 1461-147