Abstract
In a recent paper with Gillette and Sukumar an upper bound was derived for the gradients of Wachspress barycentric coordinates in simple convex polyhedra. This bound provides a shape-regularity condition that guarantees the convergence of the associated polyhedral finite element method for second order elliptic problems. In this paper we prove the optimality of the bound using a family of hexahedra that deform a cube into a tetrahedron.
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References
Floater, M.S., Gillette, A., Sukumar, N.: Gradient bounds for wachspress coordinates on polytopes. SIAM J. Numer. Anal. 52, 515–532 (2014)
Floater, M.S., Kos, G., Reimers, M.: Mean value coordinates in 3D. Comput. Aided Geom. Des. 22, 623–631 (2005)
Gillette, A., Rand, A., Bajaj, C.: Error estimates for generalized barycentric interpolation. Adv. Comput. Math. 37, 417–439 (2012)
Ju, T., Schaefer, S., Warren, J.: Mean value coordinates for closed triangular meshes. ACM Trans. on Graph. 24, 561–566 (2005)
Ju, T., Schaefer, S., Warren, J., Desbrun, M.: A geometric construction of coordinates for convex polyhedra using polar duals, symposium on geometry processing. Eurographics Assoc. 2005, 181–186 (2005)
Meyer, M., Barr, A., Lee, H., Desbrun, M.: Generalized barycentric coordinates for irregular polygons. J. Graph. Tools 7, 13–22 (2002)
Rand, A., Gillette, A., Bajaj, C.: Interpolation error estimates for mean value coordinates over convex polygons. Adv. Comput. Math. 39, 327–347 (2013)
Sukumar, N., Tabarraei, A.: Conforming polygonal finite elements. Int. J. Numer. Methods Eng. 61, 2045–2066 (2004)
Talischi, C., Paulino, G.H., Le, C.H.: Honeycomb wachspress finite elements for structural topology optimization. Struct. Multidisc. Optim. 37, 569–583 (2009)
Wachspress, E.L.: A Rational Finite Element Basis. Academic Press, New York (1975)
Warren, J.: Barycentric coordinates for convex polytopes. Adv. Comput. Math. 6, 97–108 (1996)
Warren, J., Schaefer, S., Hirani, A.N., Desbrun, M.: Barycentric coordinates for convex sets. Adv. Comput. Math. 27, 319–338 (2007)
Wicke, M., Botsch, M., Gross, M.: A finite element method on convex polyhedra. In: Proceedings of Eurographics 2007, 255–364 (2007)
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Floater, M.S. (2015). Optimality of a Gradient Bound for Polyhedral Wachspress Coordinates. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2014. Lecture Notes in Computer Science(), vol 9213. Springer, Cham. https://doi.org/10.1007/978-3-319-22804-4_16
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DOI: https://doi.org/10.1007/978-3-319-22804-4_16
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