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Algebraic Construction of Smooth Interpolants on Polygonal Domains

# Concave Polygons with Linearity on the Sides

The perim[a,b,c] function is replaced with perimLin[d,a,b,c,e]. Similarly the varPerim[x,y] function is modified so that four vertices define the behavior of the minimum along a line. The constructed interpolants on the quadrilateral and pentagon are linear on the sides.

The interpolants for a quadrilateral can be constructed as follows.

Figure 6. Interpolants that are smooth and linear on the sides of a concave polygon.

The formulation for a concave element applies to any shape. For linearity on a side the area function must be modified.

Otherwise, the side node would cause singular behavior along the boundary.

Figure 7. Linearity on the sides.

This works even with multiple side nodes.

Figure 8. Side node interpolants that are smooth and linear on the sides.