Conforming polygonal finite elements

International Journal for Numerical Methods in Engineering

Volumn 61, Issue 12, 2004, Pages 2045-2066

  Sukumar, N ^a   Tabarraei, A ^a  

^a University of California   (United States)

Author keywords

Laplace interpolant; Mean value co ordinates; Meshfree methods; Natural element method; Natural neighbour interpolants; Wachspress basis functions

Indexed keywords

APPROXIMATION THEORY; BOUNDARY VALUE PROBLEMS; COMPUTATIONAL GEOMETRY; COMPUTER GRAPHICS; FUNCTIONS; MECHANICS; TOPOLOGY;

CONVEX POLYGONS; POLYGON MESHES; SOLID MECHANICS;

FINITE ELEMENT METHOD;

FINITE ELEMENT METHOD; MATHEMATICAL METHOD;

EID: 10244239270     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.1141     Document Type: Article

References (49)

1. Hughes TJR. The Finite Element Method. Prentice-Hall: Englewood Cliffs, NJ, 1987.
2. Bathe KJ. Finite Element Procedures. Prentice-Hall: Englewood Cliffs, NJ, 1996.
3. Wachspress EL. A Rational Finite Element Basis. Academic Press: New York, NY, 1975.
4. Gout JL. Rational Wachspress-type finite elements on regular hexagons. IMA Journal of Numerical Analysis 1985; 5:59-77.
5. Dahmen W, Dikshit HP, Ojha A. On Wachspress quadrilateral patches. Computer Aided Geometric Design 2000; 17:879-890.
6. Malsch EA, Dasgupta G. Shape functions for concave quadrilaterals. In Proceedings of the First MIT Conference on Fluid and Solid Mechanics, vol. 2, Bathe KJ (ed.). Amsterdam. Elsevier Press: The Netherlands, 2001; 1617-1622.
7. Dasgupta G. Interpolants within convex polygons: Wachspress' shape functions. Journal of Aerospace Engineering 2003; 16(1 ):1-8.
8. Dasgupta G. Integration within polygonal finite elements. Journal of Aerospace Engineering 2003; 16(1): 9-18.
9. Martin RB, Burr DB, Sharkey AN. Skeletal Tissue Mechanics. Springer-Verlag: New York, NY, U.S.A., 1998.
10. Cueto E, Calvo B, Doblaré M. Modelling three-dimensional piece-wise homogeneous domains using the α-shape-based natural element method. International Journal for Numerical Methods in Engineering 2002; 54(6): 871-897.
11. Peters JF, Heymsfield E. Application of the 2-D constant strain assumption to FEM elements consisting of an arbitrary number of nodes. International Journal for Numerical Methods in Engineering 2003; 40: 143-159.
12. Dohrmann CR, Key SW, Heinstein MW. A method for connecting dissimilar finite element meshes in two dimensions, International Journal for Numerical Methods in Engineering 2000; 48:655-678.
13. Diaz AR, Bénard A. Designing materials with prescribed elastic properties using polygonal cells, International Journal for Numerical Methods in Engineering 2003; 57(3):301-314.
14. Ghosh S, Moorthy S. Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite-element method. Computer Methods in Applied Mechanics and Engineering 1995; 121 1-4):373-409.
15. Okabe A, Boots B, Sugihara K. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons: Chichester, England, 1992.
16. Christie I, Hall C. The maximum principle for bilinear elements, International Journal for Numerical Methods in Engineering 1984; 20:549-553.
17. Strang G, Fix G. An Analysis of the Finite Element Method. Prentice-Hall: Englewood Cliffs, NJ, 1973.
18. Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P. Meshless methods: An overview and recent developments. Computer Methods in Applied Mechanics and Engineering 1996; 139:3-47.
19. Li S, Liu WK. Meshfree and particle methods and their applications. Applied Mechanics Review 2002; 55(1):1-34.
20. Lancaster P, Salkauskas K. Surfaces generated by moving least squares methods. Mathematics of Computation 1981; 37:141-158.
21. Babuška I, Banerjee U, Osborn JE. Survey of meshless and generalized finite element methods: A unified approach. Acta Numerica 2003; 12:1-125.
22. Braun J, Sambridge M. A numerical method for solving partial differential equations on highly irregular evolving grids. Nature 1995; 376:655-660.
23. Sukumar N. The Natural Element Method in Solid Mechanics. Ph.D. Thesis, Theoretical and Applied Mechanics, Northwestern University, Evanston, IL, U.S.A., June 1998. URL: http://dilbert.engr.ucdavis.edu/~suku/nem/thesis.html.
24. Sukumar N, Moran B, Belytschko T. The natural element method in solid mechanics. International Journal for Numerical Methods in Engineering 1998; 43(5):839-887.
25. Sukumar N, Moran B, Semenov AYu, Belikov VV. Natural neighbor Galerkin methods. International Journal for Numerical Methods in Engineering 2001; 50(1):1-27.
26. Yoo JW, Moran B, Chen JS. Stabilized conforming nodal integration in the natural-element method. International Journal for Numerical Methods in Engineering 2004; 60(5):861-890.
27. Sibson R. A vector identity for the Dirichlet tessellation. Mathematical Proceedings of the Cambridge Philosophical Society 1980; 87:151-155.
28. Christ NH, Friedberg R, Lee TD. Weights of links and plaquettes in a random lattice. Nuclear Physics B 1982; 210(3):337-346.
29. Belikov VV, Ivanov VD, Kontorovich VK, Korytnik SA, Semenov AYu. 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 1997; 37(1):9-15.
30. Hiyoshi H, Sugihara K. Two generalizations of an interpolant based on Voronoi diagrams, International Journal of Shape Modeling 1999; 5(2):219-231.
31. Cueto E, Sukumar N, Calvo B, Martínez MA, Cegon̈ino J, Doblaré M. Overview and recent advances in natural neighbor Galerkin methods. Archives of Computational Methods in Engineering 2003; 10 4):307-384.
32. Sukumar N. Voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids. International Journal for Numerical Methods in Engineering 2003; 57(1):1-34.
33. Sukumar N, Bolander JE. Numerical computation of discrete differential operators on non-uniform grids. CMES: Computer Modeling in Engineering and Sciences 2003; 4(6):691-706.
34. Hiyoshi H, Sugihara K. Improved continuity of Voronoi-based interpolation over Delaunay spheres. Computational Geometry 2002; 22:167-183.
35. Idelsohn SR, Calvo N, Onate E. Polyhedralization of an arbitrary 3D point set. Computer Methods in Applied Mechanics and Engineering 2003; 192(22-24):2649-2667.
36. Idelsohn SR, Onate E, Calvo N, Del Pin F. Meshless finite element method. International Journal for Numerical Methods in Engineering 2003; 58(6):893-912.
37. Coxeter HSM. Introduction to Geometry. Wiley: New York, NY, 1961.
38. Warren J. Barycentric coordinates for convex polytopes. Advances in Computational Mathematics 1996; 6(1):97-108.
39. Rashid MM, Gullett PM. On a finite element method with variable element topology. Computer Methods in Applied Mechanics and Engineering 2000; 190(11-12):1509-1527.
40. Bazeley GR, Cheung YK, Irons BM, Zienkiewicz OC. Triangular elements in plate bending. Conforming and non-conforming solutions. In Proceedings of the 1st Conference on Matrix Methods in Structural Mechanics, Wright-Patterson AFB, 1965.
41. Irons BM, Razzaque A. Experience with the patch test for convergence of finite elements. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Aziz AK (ed.). Academic Press: New York, 1972.
42. Meyer M, Lee H, Barr AH, Desbrun M. Generalized barycentric coordinates for irregular n-gons. Journal of Graphics Tools 2002; 7 1):13-22.
43. Floater MS. Mean value coordinates. Computer Aided Geometric Design 2003; 20(1):19-27.
44. Floater MS. One-to-one piecewise linear mappings over triangulations. Mathematics of Computations 2003; 72(242):685-696.
45. Abramowitz M, Stegun IA. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (10th edn). Applied Mathematics Series • 55. National Bureau of Standards: Washington DC, 1972.
46. Wandzura S, Xiao H. Symmetric quadrature roles on a triangle. Computers and Mathematics with Applications 2003; 45:1829-1840.
47. Xiao H. Private communication. 2003.
48. Sukumar N, Prévost J-H. Modeling quasi-static crack growth with the extended finite element method. Part I: Computer implementation. International Journal of Solids and Structures 2003; 40 26):7513-7537.
49. Tabarraei A, Sukumar N. Adaptive computations on conforming quadtree meshes. Finite Elements in Analysis and Design (Special Issue on Papers Presented at the Robert J. Melosh Finite Element Medal Competition), to appear 2005.