Forbidden Shapes in the Finite Element Method

In two dimensional finite element analysis, increased accuracyis obtained by the addition of a number of nodal points alongthe sides of the elements. Curvilinear coordinates (p, q) wereintroduced to enable the interpolating functions to be obtainedin these complicated situations. Unfortunately as was pointedout by Jordan (1970) and is further outlined here, many usefulelement shapes with side nodes are "forbidden" by this proceduredue to the vanishing of the Jacobian of the transformation fromthe (p, q) system to the fixed (x, y) system. A new technique is introduced in the present paper, based ongeometrical considerations, whereby interpolating functionsare obtained directly in terms of x and y for the triangle andquadrilateral with arbitrarily placed side points. These localfunctions can be used to construct piecewise smooth global interpolatingfunctions with C^° continuity over regions possessing curvedboundaries and composed of elements which are triangles andparallelograms with arbitrarily positioned side points.