Gradient Bounds for Wachspress Coordinates on Polytopes

Abstract

Abstract. 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 hypercubes and simplices. Additionally, we provide an implementation of the Wachspress coordinates on convex polyhedra using Matlab and employ them in a 3D finite element solution of the Poisson equation on a non-trivial polyhedral mesh. As expected from the upper bound derivation, the H 1 -norm of the error in the method converges at a linear rate with respect to the size of the mesh elements.

Serendipity face and edge VEM spaces

“…However it should be clear from the very beginning that VEMs allow much more general geometries. For these more general geometries the comparison should actually be done between VEMs and other methods designed for polytopes, as for instance [11], [15], [19], [21], [23], [26], [27], [28], [31], [33], [34], [37]. The natural comparison, within Finite Elements, of our V f k,k−1,k−1 elements are clearly the BDM spaces as described in (2.20) for triangles (see Figure 1).…”

The finite cell method for polygonal meshes: poly-FCM

“…The two-dimensional formulation has been extended to three dimensions (convex polyhedra) by Warren [79,80]. A Matlab code that is based on yet another definition of Wachspress coordinates is provided by Floater et al [81]. Their definition is based on the unit normal outward vector n i corresponding to edge x i x i+1 and the perpendicular distance h i of x to that edge…”

“…In the following, we briefly discuss an extension to complex three-dimensional problems and show that such an extension is straightforward. To this end, three-dimensional generalized barycentric coordinates are deployed as demonstrated in various articles, such as [15,79,81,84,87,101,102]. The spacetree-based adaptive integration technique could be reduced to the already known case encountered in the tetrahedral version of the FCM [105,107].…”

PolyTop++: an efficient alternative for serial and parallel topology optimization on CPUs & GPUs

“…The Cantilever Beam problem was solved with our GPU EbEPCG solution using two meshes: one with polyhedron elements and other one with hexahedral. For the polyhedral case, we used Wachspress coordinates as defined in Floater et al (2014) and analyzed in Warren et al (2007). These coordinates are valid for elements that are not only convex but also simple.…”