Maximum Entropy Coordinates for Arbitrary Polytopes

Maximum Entropy Coordinates for Arbitrary Polytopes

Barycentric coordinates can be used to express any point inside a triangle as a unique convex combination of the triangle’s vertices, and they provide a convenient way to linearly interpolate data that is given at the vertices of a triangle. In recent years, the ideas of barycentric coordinates and barycentric interpolation have been extended to arbitrary polygons in the plane and general polyt...

Maximum-entropy meshfree coordinates in computational mechanics

Barycentric coordinates for convex polytopes

An extension of the standard barycentric coordinate functions for simplices to arbitrary convex polytopes is described. The key to this extension is the construction, for a given convex polytope, of a unique polynomial associated with that polytope. This polynomial, the adjoint of the polytope, generalizes a previous two-dimensional construction described by Wachspress. The barycentric coordina...

Quadratic maximum-entropy serendipity shape functions for arbitrary planar polygons

In this paper, we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons. Drawing on the work of Bompadre et al. [1] and Hormann and Sukumar [2], we adopt a relative entropy measure for signed (positive or negative) shape functions, with nodal prior weight functions that have the appropriate zero-set on the boundary of the polygon. We maximize t...

Gradient Bounds for Wachspress Coordinates on Polytopes

We derive upper and lower bounds on the gradients of Wachspress coordinates defined over any simple convex d-dimensional polytope P . The bounds are in terms of a single geometric quantity h∗, which denotes the minimum distance between a vertex of P and any hyperplane containing a non-incident face. We prove that the upper bound is sharp for d = 2 and analyze the bounds in the special cases of ...