Some 2D Results--Concave Quadrilateral Element
Standard literature in finite elements hardly addresses the convexity issue. There is no mention of it in the encyclopedic
references in the two volumes of Zienkiewicz and Taylor . Wachspress addresses the condition from a projective geometry point of view. He develops concavity with curved sides. His
general procedure  for n-gon is not applicable since the curve through the external intersection points intersects the element domain. It is then
prudent to conclude that there has not been any consistent method available to generate the shape functions of a concave polygon.
Recently, Dasgupta utilized the two-dimensional version of the aforementioned code in the form
Contour Plots of the Shape Functions
The following code calls the shape function generator,
is presented here.
The linearity condition along the straight boundary segments of the calculated shape functions is verified:
In the matrix, rows indicate the shape function and columns indicate corresponding sides of the quadrilateral.
The original shape function is displayed with unit value at the concave end and with zero values at the remaining ends.
The following three shape functions are calculated from the linear interpolating condition:
The second shape function below is associated with the bottom-most node.
The third interpolant shown below corresponds to the node facing the concave end.
The following interpolant is associated with the top-most node. Very low values can be observed in the bottom region of the element.
The following analytical forms of the shape functions can be used to generate the element stiffness matrix using the exact integration Mathematica code presented at IAS in 1998, in Murmansk, Russia.
Observe that the Padé form is automatically captured.
Let us consider a quadrilateral with the following nodes.
Let the shape function associated with
A remarkable result is that if
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