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Quadratic serendipity finite elements over convex polyhedra
Author(s) -
Sinu A.,
Natarajan Sundararajan,
Shankar K.
Publication year - 2017
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.5605
Subject(s) - mathematics , barycentric coordinate system , polygon (computer graphics) , polyhedron , quadratic equation , rate of convergence , geometry , computer science , telecommunications , computer network , channel (broadcasting) , frame (networking)
Summary The conventional approach to construct quadratic elements for an n ‐sided polygon will yield n ( n +1)/2 shape functions, which increases the computational effort. It is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion. Floater and Lai proposed a systematic way to construct higher‐order approximations over arbitrary polygons using the generalized barycentric and triangular coordinates. This approach ensures 2 n shape functions with nodes only on the boundary of the polygon. In this paper, we extend the polygonal splines approach to 3 dimensions and construct serendipity shape functions over hexahedra and convex polyhedra. This is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates. The quadratic shape functions possess Kronecker delta property and satisfy constant, linear, and quadratic precision. The accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests. The numerical results show that the quadratic serendipity elements pass the patch test, yield optimal convergence rate, and can tolerate extreme mesh distortion.