Concave Shape:

Here node-(2) is the concave node and the opposite concave node is node-(4). The elliptically condition dictates that the maximum and minimum values of all the shape functions must occur at the boundary. The lowest order shape functions are single valued at only one node and zero valued at all others. Along the boundary between the zero and single valued nodes the behavior is linear.

First Attempt: Isoparametric formulation

The first attempt at finding the shape functions along the boundary is applying the isoparametric formulation generally used for trapezoidal domains. Notice that while it is linear along the zero to one boundary and zero on all others the shape function is not contained in the concave domain. Also, the function bifurcates.

C0 formulation: tessellating or meshing the domain

Conformal mapping

Enrichment functions

C1 formulation combining convex sub-parts

The concavity restriction is alleviated by analytically enforcing continuity of the displacement, slope and curvature tensor fields. In particular, for the simplest concave shape, the four-noded element, the shape function associated with the concave node is calculated. The three remaining shape functions are generated by assuring the remaining three resulting functions, along with the first, can reproduce any arbitrary linear field.

  • C1 concave finite element development (pdf )