Two families of two‐dimensional finite element interpolation functions of general degree with general continuity

• Published in: International journal for numerical methods in engineering Vol. 124; no. 10; pp. 2171 - 2195
• Main Author: Moitinho de Almeida, J. P.
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 30.05.2023; Wiley Subscription Services, Inc
• Subjects: Apexes; Approximation; C1$$ {C}^1 $$ continuity; C2$$ {C}^2 $$ continuity; Continuity (mathematics); Cr$$ {C}^r $$ continuity; finite elements; Interpolation; interpolation functions; Mathematical analysis; Quadrilaterals
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.7205

This work presents a generalization of the functions used in Fraeijs de Veubeke's quadrilateral (FVQ) element and in Hsieh, Clough and Tocher's (HCT) triangle, allowing for an arbitrary degree of the approximation, d$$ d $$, and for arbitrary order of continuity, r$$ r $$. These functions can be used to define finite elements that lead by direct assembly to Cr$$ {C}^r $$ approximations. Their determination for each element is procedural, generalizing the reasoning used by Zienkiewicz to explain the FVQ element. The completeness of each basis is confirmed by comparing its dimension with that of a broken basis, defined primitive element wise, upon which the continuity constraints are successively imposed. The order of continuity at the vertices that is required to enable the independent definition of the normal derivatives at the sides is confirmed numerically. The corresponding interpolation functions are symbolically derived using Mathematica, and simple matlab codes using the data thus obtained are made available, to illustrate their application.