Extended finite element method on polygonal and quadtree meshes

Computer Methods in Applied Mechanics and Engineering

Volumn 197, Issue 5, 2008, Pages 425-438

  Tabarraei, A ^a   Sukumar, N ^a  

^a University of California Davis   (United States)

Author keywords

Crack growth; Discontinuous enrichment; Fracture; Natural neighbors; Nonconvex polygons; Partition of unity; Voronoi tessellation

Indexed keywords

CRACK PROPAGATION; FAILURE (MECHANICAL); FRACTURE; MESH GENERATION; STRESS INTENSITY FACTORS;

DISCONTINUOUS ENRICHMENT; NATURAL NEIGHBORS; NONCONVEX POLYGONS; PARTITION OF UNITY; VORONOI TESSELLATION;

FINITE ELEMENT METHOD;

CRACK PROPAGATION; FAILURE (MECHANICAL); FINITE ELEMENT METHOD; FRACTURE; MESH GENERATION; STRESS INTENSITY FACTORS;

EID: 35448995164     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2007.08.013     Document Type: Article

References (46)

1. Sukumar N., and Tabarraei A. Conforming polygonal finite elements. Int. J. Numer. Methods Engrg. 61 (2004) 2045-2066
2. Idelsohn S.R., Calvo N., and Onate E. Polyhedralization of an arbitrary 3D point set. Comp. Methods Appl. Mech. Engrg. 192 22-24 (2003) 2649-2667
3. Belytschko T., and Black T. Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Engrg. 45 (1999) 601-620
4. Moës N., Dolbow J., and Belytschko T. A finite element method for crack growth without remeshing. Int. J. Numer. Methods Engrg. 46 1 (1999) 131-150
5. Strouboulis T., Copps K., and Babuška I. The generalized finite element method. Comp. Methods Appl. Mech. Engrg. 190 32-33 (2001) 4081-4193
6. Melenk J.M., and Babuška I. The partition of unity finite element method: basic theory and applications. Comp. Methods Appl. Mech. Engrg. 139 (1996) 289-314
7. Babuška I., and Melenk I. Partition of unity method. Int. J. Numer. Methods Engrg. 40 (1997) 727-758
8. Wachspress E.L. A Rational Finite Element Basis (1975), Academic Press, NY
9. Meyer M., Lee H., Barr A.H., and Desbrun M. Generalized barycentric coordinates on irregular polygons. J. Graphics Tools 7 1 (2002) 13-22
10. Floater M.S. Mean value coordinates. Comp. Aided Geometric Des. 20 1 (2003) 19-27
11. Sukumar N. Construction of polygonal interpolants: a maximum entropy approach. Int. J. Numer. Methods Engrg. 61 (2004) 2159-2181
12. Malsch E.A., and Dasgupta G. Shape functions for polygonal domains with interior nodes. Int. J. Numer. Methods Engrg. 61 12 (2004) 1153-1172
13. Malsch E.A., Lin J.J., and Dasgupta G. Smooth two dimensional interpolants: a recipe for all polygons. J. Graphics Tools 10 2 (2005) 11-23
14. Floater M.S., Kós G., and Reimers M. Mean value coordinates in 3D. Comp. Aided Geometric Des. 22 7 (2005) 623-631
15. Hormann K., and Floater M.S. Mean value coordinates for arbitrary planar polygons. ACM Trans. Graphics 25 4 (2006) 1421-1424
16. Sukumar N., and Malsch E.A. Recent advances in the construction of polygonal finite element interpolants. Arch. Comput. Methods Engrg. 13 1 (2006) 129-163
17. Christ N.H., Friedberg R., and Lee T.D. Weights of links and plaquettes in a random lattice. Nuclear Phys. B 210 3 (1982) 337-346
18. Tabarraei A., and Sukumar N. Adaptive computations on conforming quadtree meshes. Finite Elements Anal. Des. 41 (2005) 686-702
19. Tabarraei A., and Sukumar N. Adaptive computations using material forces and residual-based error estimators on quadtree meshes. Comp. Methods Appl. Mech. Engrg. 196 25-28 (2007) 2657-2680
20. Karihaloo B.L., and Xiao Q.Z. Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review. Comp. Struct. 81 3 (2003) 119-129
21. Bordas S., Nguyen P.V., Dunant C., Guidoum A., and Nguyen-Dang H. An extended finite element library. Int. J. Numer. Methods Engrg. 71 6 (2007) 703-732
22. Sibson R. A vector identity for the Dirichlet tesselation. Math. Proc. Camb. Philos. Soc. 87 (1980) 151-155
23. Sukumar N., Moran B., Semenov A.Y., and Belikov V.V. Natural neighbor Galerkin methods. Int. J. Numer. Methods Engrg. 50 1 (2001) 1-27
24. Du Q., Faber V., and Gunzburger M. Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev. 41 (1999) 637-676
25. J. Burkardt, CCVT_BOX: Centroidal Voronoi Tessellation Constrained to a Box. , 2004.
26. Samet H. Application of Spatial Data Structure (1990), Addison-Wesley, New York, NY
27. Kagan P., Fischer A., and Bar-Yoseph P.Z. Mechanically based models: adaptive refinement for B-spline finite element. Int. J. Numer. Methods Engrg. 57 8 (2003) 1145-1175
28. Krysl P., Trivedi A., and Zhu B. Object-oriented hierarchical mesh refinement with CHARMS. Int. J. Numer. Methods Engrg. 60 8 (2004) 1401-1424
29. Krysl P., Grinspun P., and Schröder E. Natural hierarchical refinement for finite element methods. Int. J. Numer. Methods Engrg. 56 8 (2003) 1109-1124
30. Laguardia J.J., Cueto E., and Doblaré M. A natural neighbour Galerkin method with quadtree structure. Int. J. Numer. Methods Engrg. 63 (2005) 789-812
31. Sukumar N., and Prévost J.-H. Modeling quasi-static crack growth with the extended finite element method. Part I: computer implementation. Int. J. Solids Struct. 40 26 (2003) 7513-7537
32. Béchet E., Minnebo H., Moës N., and Burgardt B. Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int. J. Numer. Methods Engrg. 64 8 (2005) 1033-1056
33. Laborde P., Pommier J., Renard Y., and Salaün M. High-order extended finite element method for cracked domains. Int. J. Numer. Methods Engrg. 64 3 (2005) 354-381
34. P. Pommier, Y. Renard, Getfem++. An Open Source Generic C++ Library for Finite Element Methods. , 2005.
35. Vaughan Jr. B.L., Smith B.G., and Chopp D.L. A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources. Commun. Appl. Math. Comput. Sci. 1 1 (2006) 207-228
36. Gamma E., Helm R., Johnson R., and Vlissides J. Design Patterns: Elements of Reusable Object-Oriented Software (1995), Addison-Wesley Longman Inc., Reading, MA
37. Haines E. Point in polygon strategies. In: Heckbert P. (Ed). Graphics Gems vol. IV (1994), Academic Press 24-46
38. J.W. Ratcliff, Efficient Polygon Triangulation. , 2000.
39. J.W. Demmel, J.R. Gilbert, X.S. Li, SuperLU User's Guide, Technical Report LBNL-44289, Lawrence Berkeley National Laboratory, Berkeley, CA. 2003.
40. Yau J., Wang S., and Corten H. A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J. Appl. Mech. 47 (1980) 335-341
41. Moran B., and Shih C.F. Crack tip and associated domain integrals from momentum and energy balance. Engrg. Fract. Mech. 27 (1987) 615-642
42. Tada H., Paris P.C., and Irwin G.R. The Stress Analysis of Cracks Handbook (2000), ASME Press, New York, NY
43. Aliabadi M.H., Rooke D.P., and Cartwright D.J. Mixed-mode Bueckner weight functions using boundary element analysis. Int. J. Fract. 34 (1987) 131-147
44. Bellec J., and Dolbow J.E. A note on enrichment functions for modelling crack nucleation. Commun. Numer. Methods Engrg. 19 12 (2003) 921-932
45. Bocca P., Carpinteri A., and Valente S. Size effects in the mixed mode crack propagation: softening and snap-back analysis. Engrg. Fract. Mech. 35 1-3 (1990) 159-170
46. Bouchard P.O., Bay F., Chastel Y., and Tovena I. Crack propagation modelling using an advanced remeshing technique. Comp. Methods Appl. Mech. Engrg. 189 3 (2000) 723-742