Coons’ interpolation as a vehicle to derive large isoparametric elements

Abstract

This chapter explains that Coons interpolation, which chronologically is the first formula for the mathematical representation of surface patches in Computational Geometry, can be used to derive the closed-form analytical expressions of shape functions that appear in classical isoparametric finite elements of the Serendipity family. Moreover, it is shown that not only Lagrange polynomials but also reduced natural cardinal cubic B-splines as well as most of other interpolations discussed in Chap. 2 can be also used as trial functions along the four edges of a Coons patch macroelement (“C-element”). The latter is generally a large isoparametric finite element with nodal points along the boundary of the patch. In the case of using Curry-Schoenberg (de Boor) B-splines or NURBS along each edge in the Coons patch, the nodal points are merely replaced by the control points. The performance of all these elements is thoroughly investigated through ten examples in two-dimensional and axisymmetric potential and elasticity problems.