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Hierarchical high‐order conforming C 1 bases for quadrangular and triangular finite elements
Author(s) -
Ferreira Luan J. F.,
Bittencourt Marco L.
Publication year - 2016
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5308
Subject(s) - mathematics , quadrilateral , hermite polynomials , tensor product , basis (linear algebra) , basis function , projection (relational algebra) , finite element method , tensor (intrinsic definition) , order (exchange) , polynomial , product (mathematics) , mathematical analysis , pure mathematics , geometry , algorithm , thermodynamics , finance , economics , physics
Summary We present three new sets of C 1 hierarchical high‐order tensor‐product bases for conforming finite elements. The first basis is a high‐order extension of the Bogner–Fox–Schmit basis. The edge and face functions are constructed using a combination of cubic Hermite and Jacobi polynomials with C 1 global continuity on the common edges of elements. The second basis uses the tensor product of fifth‐order Hermite polynomials and high‐order functions and achieves global C 1 continuity for meshes of quadrilaterals and C 2 continuity on the element vertices. The third basis for triangles is also constructed using the tensor product of one‐dimensional functions defined in barycentric coordinates. It also has global C 1 continuity on edges and C 2 continuity on vertices. A patch test is applied to the three considered elements. Projection and plate problems with smooth fabricated solutions are solved, and the performance of the h ‐ and p ‐refinements are evaluated by comparing the approximation errors in the L 2 ‐ and energy norms. A plate with singularity is then studied, and h ‐ and p ‐refinements are analysed. Finally, a transient problem with implicit time integration is considered. The results show exponential convergence rates with increasing polynomial order for the triangular and quadrilateral meshes of non‐distorted and distorted elements. Copyright © 2016 John Wiley & Sons, Ltd.