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Recent advances in the construction of polygonal finite element interpolants

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Summary

This paper is an overview of recent developments in the construction of finite element interpolants, which areC 0-conforming on polygonal domains. In 1975, Wachspress proposed a general method for constructing finite element shape functions on convex polygons. Only recently has renewed interest in such interpolants surfaced in various disciplines including: geometric modeling, computer graphics, and finite element computations. This survey focuses specifically on polygonal shape functions that satisfy the properties of barycentric coordinates: (a) form a partition of unity, and are non-negative; (b) interpolate nodal data (Kronecker-delta property), (c) are linearly complete or satisfy linear precision, and (d) are smooth within the domain. We compare and contrast the construction and properties of various polygonal interpolants—Wachspress basis functions, mean value coordinates, metric coordinate method, natural neighbor-based coordinates, and maximum entropy shape functions. Numerical integration of the Galerkin weak form on polygonal domains is discussed, and the performance of these polygonal interpolants on the patch test is studied.

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References

  1. N. Agmon, Y. Alhassid and R. D. Levine (1978). An algorithm for determining the Lagrange parameters in the maximal entropy formalism. In M. Tribus and R. D. Levine (Eds.),The Maximum Entropy Formalism, pp. 206–209, Cambridge, MA. MIT Press.

    Google Scholar 

  2. N. Agmon, Y. Alhassid and R. D. Levine (1979). An algorithm for finding the distribution of maximal entropy.Journal of Computational Physics,30, 250–258.

    Article  MATH  Google Scholar 

  3. M. Arroyo and M. Ortiz (2005). Localmaximum-entropy approximation schemes: A seamless bridge between finite elements and meshfree methods.International Journal for Numerical Methods in Engineering, in press.

  4. V.V. Belikov, V.D. Ivanov, V.K. Kontorovich, S.A. Korytnik and A. Yu. Semenov (1997). The non-Sibsonian interpolation: A new method of interpolation of the values of a function on an arbitrary set of points.Computational Mathematics and Mathematical Physics,37(1), 9–15.

    MathSciNet  Google Scholar 

  5. J.-P. Berrut and L.N. Trefethen (2004). Barycentric Lagrange interpolation.SIAM Review,46(3), 501–517.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. D. Buhmann (2000). Radial basis functions.Acta Numerica,9, 1–38.

    Article  MathSciNet  Google Scholar 

  7. N. H. Christ, R. Friedberg and T. D. Lee (1982). Weights of links and plaquettes in a random lattice.Nuclear Physics B,210(3), 337–346.

    Article  MathSciNet  Google Scholar 

  8. H.S.M. Coxeter (1961).Introduction to Geometry. John Wiley and Sons, New York, N.Y.

    MATH  Google Scholar 

  9. E. Cueto, N. Sukumar, B. Calvo, M.A. Martínez, J. Cegonïno and M. Doblaré (2003). Overview and recent advances in natural neighbour Galerkin methods.Archives of Computational Methods in Engineering,10(4), 307–384.

    Article  MathSciNet  MATH  Google Scholar 

  10. W. Dahmen, H.P. Dikshit and A. Ojha (2000). On Wachspress quadrilateral patches.Computer Aided Geometric Design,17, 879–890.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Dasgupta (2003). Integration within polygonal finite elements.Journal of Aerospace Engineering,16(1), 9–18.

    Article  Google Scholar 

  12. G. Dasgupta (2003). Interpolants within convex polygons: Wachspress'shape functions.Journal of Aerospace Engineering,16(1), 1–8.

    Article  Google Scholar 

  13. A.R. Diaz and A. Bénard (2003). Designing materials with prescribed elastic properties using polygonal cells.International Journal for Numerical Methods in Engineering,57(3), 301–314.

    Article  MathSciNet  MATH  Google Scholar 

  14. C.R. Dohrmann, S.W. Key and M.W. Heinstein (2000). A method for connecting dissimilar finite element meshes in two dimensions.International Journal for Numerical Methods in Engineering,48, 655–678.

    Article  MATH  Google Scholar 

  15. M.S. Floater Mean value coordinates.Computer Aided Geometric Design, 20(1): 19–27, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  16. M.S. Floater and K. Hormann (2005). Surface parameterization: a tutorial and survey. In N. A. Dodgson, M. S. Floater, and M. A. Sabin (Eds.),Advances in Multiresolution for Geometric Modelling. Mathematics and Visualization, pp. 157–186. Springer, Berlin, Heidelberg.

    Chapter  Google Scholar 

  17. M.S. Floater, K. Hormann, and G. Kós (2005). A general construction of barycentric coordinates over convex polygons.Advances in Computational Mathematics. in press.

  18. S. Ghosh and S. Moorthy (1995). Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite-element method.Computer Methods in Applied Mechanics and Engineering,121(1–4), 373–409.

    Article  MATH  Google Scholar 

  19. J.L. Gout (1985). Rational Wachspress-type finite elements on regular hexagons.IMA Journal of Numerical Analysis,5(1), 59–77.

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Grünbaum (1967).Convex polytopes. John Wiley and Sons, New York.

    MATH  Google Scholar 

  21. H. Hiyoshi and K. Sugihara (1999). Two generalizations of an interpolant based on Voronoi diagrams.International Journal of Shape Modeling,5(2), 219–231.

    Article  Google Scholar 

  22. K. Hormann (2004). Barycentric coordinates for arbitrary polygons in the plane. Technical Report, Clausthal University of Technology, September.

  23. T.J.R. Hughes (1987).The Finite Element Method. Prentice-Hall, Englewood Cliffs, N.J.

    MATH  Google Scholar 

  24. E.T. Jaynes (1957). Information theory and statistical mechanics.Physical Review,106(4), 620–630.

    Article  MathSciNet  Google Scholar 

  25. E.T. Jaynes (1989).Concentration of Distributions at Entropy Maxima, pp. 317–336. In R. D. Rosenkrantz Kluwer Academic Publishers, Dordrecht, The Netherlands.

    Google Scholar 

  26. E.T. Jaynes (2003).Probability Theory: The Logic of Science. Cambridge University Press, Cambridge, UK, 1st. Edition.

    MATH  Google Scholar 

  27. J.N. Kapur (1993).Maximum-Entropy Models in Science and Engineering. John Wiley & Sons, Inc., New Delhi, India, 1st. (revised)Edition.

    Google Scholar 

  28. A. Khinchin (1957).Mathematical Foundations of Information Theory. Dover, New York, N.Y.

    MATH  Google Scholar 

  29. E.A. Malsch (2003).Test functions for elliptic operators satisfying essential edge conditions on both convex and concave polygonal domains. PhD thesis, Columbia University.

  30. E. A. Malsch and G. Dasgupta (2001). Shape functions for concave quadrilaterals. In K. J. Bathe, editor,Proceedings of the first MIT Conference on Fluid and Solid Mechanics, Volume 2, pp. 1617–1622, Amsterdam, The Netherlands, Elsevier Press.

    Google Scholar 

  31. E.A. Malsch and G. Dasgupta (2004). Interpolation constraints and thermal distributions: a method for all non-concave polygons.International Journal of Solids and Structures,41(8), 2165–2188.

    Article  MathSciNet  MATH  Google Scholar 

  32. E.A. Malsch and G. Dasgupta (2004). Shape functions for polygonal domains with interior nodes.International Journal for Numerical Methods in Engineering,61(12), 1153–1172.

    Article  MATH  Google Scholar 

  33. E.A. Malsch and G. Dasgupta (2005). Algebraic construction of smooth interpolants on polygonal domains.Mathematica Journal,9(3).

  34. E.A. Malsch, J.J. Lin, and G. Dasgupta (2005). Smooth two dimensional interpolants: a recipe for all polygons.Journal of Graphics Tools,10(2).

  35. M. Meyer, H. Lee, A. Barr and M. Desbrun (2002). Generalized barycentric coordinates on irregular polygons.Journal of Graphics Tools,7(1), 13–22.

    MATH  Google Scholar 

  36. J. Nocedal and S.J. Wright (1999).Numerical Optimization. Springer-Verlag, New York. 27.

    MATH  Google Scholar 

  37. A. Okabe, B. Boots and K. Sugihara (1992).Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, Chichester, England.

    MATH  Google Scholar 

  38. U. Pinkall and K. Polthier (1993). Computing discrete minimal surfaces and their conjugates.Experimental Mathematics,2(1), 15–36.

    MathSciNet  MATH  Google Scholar 

  39. W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling (1992).Numerical Recipes in Fortran: The Art of Scientific Computing. Cambridge University Press, New York, NY, 2nd. Edition.

    Google Scholar 

  40. A. Rényi (1961). On measures of entropy and information. InProceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, vol. 1, pages 547–561, Berkeley, CA, University of California Press.

    Google Scholar 

  41. R. D. Rosenkrantz (Ed)(1989).E.T. Jaynes: Paper on Probability, Statistics and Statistical Physics. Kluwer Academic Publishers, Dordrecht, The Netherlands.

    Google Scholar 

  42. V.L. Rvachev, T.I. Sheiko, V. Shapiro and I. Tsukanov (2000). On completeness of RFM solution structures.Computational Mechanics,25(2–3), 305–316.

    Article  MathSciNet  MATH  Google Scholar 

  43. H. Samet (1984). The quadtree and related hierarchical data structure.ACM Computing Surveys,16(2), 187–260.

    Article  MathSciNet  Google Scholar 

  44. C.E. Shannon (1948). A mathematical theory of communication.The Bell Systems Technical Journal,27, 379–423.

    MathSciNet  MATH  Google Scholar 

  45. D. Shepard (1968). A two-dimensional interpolation function for irregularly spaced points. InACM National Conference, pp. 517–524.

  46. R. Sibson (1980). A vector identity for the Dirichlet tesselation.Mathematical Proceedings of the Cambridge Philosophical Society,87, 151–155.

    Article  MathSciNet  MATH  Google Scholar 

  47. A. K. Soh, L. Zhifei and C. Song (2000). Development of a new quadrilateral thin plate element using area coordinates.Computer Methods in Applied Mechanics and Engineering,190(8–10), 979–987.

    Article  MATH  Google Scholar 

  48. G. Strang and G. Fix (1973).An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs, N.J.

    MATH  Google Scholar 

  49. N. Sukumar (2003). Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids.International Journal for Numerical Methods in Engineering,57(1), 1–34.

    Article  MathSciNet  MATH  Google Scholar 

  50. N. Sukumar (2004). Construction of polygonal interpolants: A maximum entropy approach.International Journal for Numerical Methods in Engineering,61(12), 2159–2181.

    Article  MathSciNet  MATH  Google Scholar 

  51. N. Sukumar, B. Moran, A. Yu. Semenov and V.V. Belikov (2001). Natural neighbor Galerkin methods.International Journal for Numerical Methods in Engineering,50(1), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  52. N. Sukumar and A. Tabarraei (2004). Conforming polygonal finite elements.International Journal for Numerical Methods in Engineering 61(12), 2045–2066.

    Article  MathSciNet  MATH  Google Scholar 

  53. A. Tabarraei and N. Sukumar (2005). Adaptive computations on conforming quadtree meshes.Finite Elements in Analysis and Design,41(7–8), 686–702.

    Article  Google Scholar 

  54. C. Tsallis (1988). Possible generalization of Boltzmann-Gibbs statistics.Journal of Statistical Physics,52(1–2), 479–487.

    Article  MathSciNet  MATH  Google Scholar 

  55. E.L. Wachspress (1975).A Rational Finite Element Basis. Academic Press, New York, N.Y.

    MATH  Google Scholar 

  56. S. Wandzura and H. Xiao (2003). Symmetric quadrature rules on a triangle.Computers and Mathematics with Applications,45, 1829–1840.

    Article  MathSciNet  MATH  Google Scholar 

  57. J. Warren (1996). Barycentric coordinates for convex polytopes. Technical report, Department of Computer Science, Rice University.

  58. J. Warren (1996). Barycentric coordinates for convex polytopes.Advances in Computational Mathematics,6(1), 97–108.

    Article  MathSciNet  Google Scholar 

  59. J. Warren (2003). On the uniqueness of barycentric coordinates. InContemporary Mathematics, Proceedings of AGGM02, pp. 93–99.

  60. J. Warren, S. Schaefer, A.N. Hirani and M. Desbrun (2005). Barycentric coordinates for convex sets. preprint.

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Author notes

  1. E. A. Malsch

    Present address: Thorton Tomasetti, 51 Madison Avenue, 10010, New York, NY

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, University of California, One Shields Avenue, 95616, Davis, CA, U.S.A.

    N. Sukumar

  2. Institute of Applied Mechanics, Technische Universität Braunschweig, Spielmannstraße 11, 38023, Braunschweig, Germany

    E. A. Malsch

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Sukumar, N., Malsch, E.A. Recent advances in the construction of polygonal finite element interpolants. Arch Computat Methods Eng 13, 129–163 (2006). https://doi.org/10.1007/BF02905933

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