Direct determination of shape functions for isoparametric elements with arbitrary node configuration

Abstract Shape functions have been derived to describe different forms of elements, notably triangles and rectangles in 2-D, and tetrahedrons, cuboids, and triangular prisms in 3-D. There are generalised solutions for some regular node configurations, and hierarchical correction algorithms help with more difficult node distributions. But to this point there is no single formula or set of formulae that allows the direct determination of shape functions for any node configuration without restrictions. This paper shows how a general set of formulae can be derived which is applicable to any isoparametric element type with arbitrary node configuration. This formulation is in such a form that it is clear and concise. The approach is based on the Lagrange polynomial considering up to three Cartesian and four volume coordinates. Additionally, the correction procedure that is inherent in the formulation to guarantee an appropriate evaluation of the generalised shape functions and to fulfil all four isoparametric shape function criteria is discussed. The proof of validity illustrates the correctness of the method.