Generalized finite element enrichment functions for discontinuous gradient fields

International Journal for Numerical Methods in Engineering

Volumn 82, Issue 2, 2010, Pages 242-268

  Aragón, Alejandro M ^a   Armando Duarte, C ^a   Geubelle, Philippe H ^b  

^a University of Illinois at Urbana Champaign   (United States)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CORRECTION FUNCTION; DISCONTINUITY LINES; DISCRETE LINE; EMBEDDED NETWORK; FINITE ELEMENT; FINITE ELEMENT SOLUTION; GRADIENT FIELDS; HETEROGENEOUS MATERIALS; HOMOGENEOUS MEDIA; IN-LINE; INTERFACE ENRICHMENT; LINEAR APPROXIMATIONS; MICROVASCULAR; NUMBER OF DEGREES OF FREEDOM; OPTIMAL CONVERGENCE; QUADRATIC APPROXIMATION; TRIPLE JUNCTION; TWO-DIMENSIONAL PROBLEM;

NONLINEAR CONTROL SYSTEMS; POLYCRYSTALLINE MATERIALS; STRUCTURAL ANALYSIS;

PROBLEM SOLVING;

EID: 77949993380     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2772     Document Type: Article

References (47)

1. Shipton LA. Thermal management applications for microvascular systems. Master's Thesis, University of Illinois at Urbana-Champaign, 2007.
2. Sukumar N, Srolovitz DJ, Baker TJ, Prévost JH. Brittle fracture in polycrystalline microstructures with the extended finite element method. International Journal for Numerical Methods in Engineering 2003; 56(14):2015-2037. Available from: http://dx.doi.org/10.1002/nme.653.
3. Duarte CA. The hp cloud method. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, U.S.A., December 1996.
4. Duarte CA, Oden JT. An h-p adaptive method using clouds. Computer Methods in Applied Mechanics and Engineering 1996; 139(1-4):237-262. Available from: http://dx.doi.org/10.1016/S0045-7825(96)01085-7.
5. Duarte CA, Oden JT. H-p clouds-an h-p meshless method. Numerical Methods for Partial Differential Equations 1996; 12(6):673-705. Available from: http://dx.doi.org/10.1002/(SICI)1098-2426(199611)12:6<673::AID-NUM3>3.0.CO;2-P.
6. Oden JT, Duarte CAM, Zienkiewicz OC. A new cloud-based hp finite element method. Computer Methods in Applied Mechanics and Engineering 1998; 153(1-2):117-126. Available from: http://dx.doi.org/10.1016/S0045-7825(97)00039-X.
7. Melenk JM, Babuška I. The partition of unity finite element method: Basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996; 139(1-4):289-314. Available from: http://dx.doi.org/10.1016/S0045-7825(96)01087-0.
8. Babuška I, Melenk JM. The partition of unity method. International Journal for Numerical Methods in Engineering 1997; 40(4):727-758. Available from: http://dx.doi.org/10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N.
9. Belytschko T, Gracie R, Ventura G. A review of extended/generalized finite element methods for material modeling. Modelling and Simulation in Materials Science and Engineering 2009; 17(4):043001 (24pp). Available from: http://stacks.iop.org/0965-0393/17/043001.
10. Duarte CA, Babuška I, Oden JT. Generalized finite element methods for three-dimensional structural mechanics problems. Computers and Structures 2000; 77(2):215-232. Available from: http://dx.doi.org/10.1016/S0045-7949(99)00211-4.
11. Simone A, Duarte CA, Van der Giessen E. A generalized finite element method for polycrystals with discontinuous grain boundaries. International Journal for Numerical Methods in Engineering 2006; 67(8):1122-1145. Available from: http://dx.doi.org/10.1002/nme.1658.
12. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 1999; 45(5):601-620. Available from: http://dx.doi.org/10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S.
13. Möes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46(1):131-150. Available from: http://dx.doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J.
14. Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T. Arbitrary branched and intersecting cracks with the extended finite element method. International Journal for Numerical Methods in Engineering 2000; 48(12):1741-1760. Available from: http://dx.doi.org/10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L.
15. Duarte CA, Hamzeh ON, Liszka TJ, Tworzydlo WW. A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Computer Methods in Applied Mechanics and Engineering 2001; 190(15-17):2227-2262. Available from: http://dx.doi.org/10.1016/S0045-7825(00)00233-4.
16. Chahine E, Laborde P, Renard Y. A quasi-optimal convergence result for fracture mechanics with XFEM. Comptes Rendus Mathematique 2006; 342(7):527-532. Available from: http://dx.doi.org/10.1016/j.crma.2006.02.002.
17. Merle R, Dolbow J. Solving thermal and phase change problems with the extended finite element method. Computational Mechanics 2002; 28(5):339-350. Available from: http://dx.doi.org/10.1007/s00466-002-0298-y.
18. Dolbow J, Fried E, Ji H. Chemically induced swelling of hydrogels. Journal of the Mechanics and Physics of Solids 2004; 52(1):51-84. Available from: http://dx.doi.org/10.1016/S0022-5096(03)00091-7.
19. Dolbow J, Fried E, Ji H. A numerical strategy for investigating the kinetic response of stimulus-responsive hydrogels. Computer Methods in Applied Mechanics and Engineering 2005; 194(42-44):4447-4480. Available from: http://dx.doi.org/10.1016/j.cma.2004.12.004.
20. Ji H, Mourad H, Fried E, Dolbow J. Kinetics of thermally induced swelling of hydrogels. International Journal of Solids and Structures 2006; 43(7-8):1878-1907. Available from: http://dx.doi.org/10.1016/j.ijsolstr.2005.03.031.
21. Strouboulis T, Copps K, Babuška I. The generalized finite element method. Computer Methods in Applied Mechanics and Engineering 2001; 190(32-33):4081-4193. Available from: http://dx.doi.org/10.1016/S0045-7825(01)00188-8.
22. Sukumar N, Chopp DL, Moës N, Belytschko T. Modeling holes and inclusions by level sets in the extended finite-element method. Computer Methods in Applied Mechanics and Engineering 2001; 190(46-47):6183-6200. Available from: http://dx.doi.org/10.1016/S0045-7825(01)00215-8.
23. Moës N, Cloirec M, Cartraud P, Remacle JF. A computational approach to handle complex microstructure geometries. Computer Methods in Applied Mechanics and Engineering 2003; 192(28-30):3163-3177. Available from: http://dx.doi.org/10.1016/S0045-7825(03)00346-3.
24. Babuška I, Osborn JE. Generalized finite element methods: their performance and their relation to mixed methods. SIAM Journal on Numerical Analysis 1983; 20(3):510-536. Available from: http://www.jstor.org/stable/2157269.
25. Srinivasan KR, Matouš K, Geubelle PH. Generalized finite element method for modeling nearly incompressible bimaterial hyperelastic solids. Computer Methods in Applied Mechanics and Engineering 2008; 197:4882-4893. Available from: http://dx.doi.org/10.1016/j.cma.2008.07.014.
26. Belytschko T, Möes N, Usui S, Parimi C. Arbitrary discontinuities in finite elements. International Journal for Numerical Methods in Engineering 2001; 50(4):993-1013. Available from: http://dx.doi.org/10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M.
27. Robbins J, Voth TE. An extended finite element formulation for modeling the response of polycrystalline materials to shock loading. 15th APS Topical Conference on Shock Compression of Condensed Matter, Kohala Coast, HI, 24-29 June 2007.
28. Fries TP. A corrected xfem approximation without problems in blending elements. International Journal for Numerical Methods in Engineering 2008; 75(5):503-532. Available from: http://dx.doi.org/10.1002/nme.2259.
29. Möes N, Béchet E, Tourbier M. Imposing dirichlet boundary conditions in the extended finite element method. International Journal for Numerical Methods in Engineering 2006; 67(12):1641-1669. Available from: http://dx.doi.org/10.1002/nme.1675.
30. Babuška I, Banerjee U, JE O. Survey of meshless and generalized finite element methods: A unified approach. Acta Numerica 2003; 12:1-125.
31. Chessa J, Wang H, Belytschko T. On the construction of blending elements for local partition of unity enriched finite elements. International Journal for Numerical Methods in Engineering 2003; 57(7):1015-1038. Available from: http://dx.doi.org/10.1002/nme.777.
32. Ventura G, Gracie R, Belytschko T. Fast integration and weight function blending in the extended finite element method. International Journal for Numerical Methods in Engineering 2009; 77(1):1-29. Available from: http://dx.doi.org/10.1002/nme.2387.
33. Tarancón JE, Vercher A, Giner E, Fuenmayor FJ. Enhanced blending elements for xfem applied to linear elastic fracture mechanics. International Journal for Numerical Methods in Engineering 2009; 77(1):126-148. Available from: http://dx.doi.org/10.1002/nme.2402.
34. Gracie R, Wang H, Belytschko T. Blending in the extended finite element method by discontinuous galerkin and assumed strain methods. International Journal for Numerical Methods in Engineering 2008; 74(11):1645-1669. Available from: http://dx.doi.org/10.1002/nme.2217.
35. Strouboulis T, Babuška I, Copps K. The design and analysis of the generalized finite element method. Computer Methods in Applied Mechanics and Engineering 2000; 181(1-3):43-69. Available from: http://dx.doi.org/10.1016/S0045-7825(99)00072-9.
36. Osher SJ, Fedkiw RP. Level Set Methods and Dynamic Implicit Surfaces (1st edn). Springer: Berlin, 2002.
37. Duarte CA, Reno LG, Simone A. A high-order generalized fem for through-the-thickness branched cracks. International Journal for Numerical Methods in Engineering 2007; 72(3):325-351. Available from: http://dx.doi.org/10.1002/nme.2012.
38. Pereira JP, Duarte CA, Guoy D, Jiao X. Hp-generalized fem and crack surface representation for non-planar 3-d cracks. International Journal for Numerical Methods in Engineering 2009; 77(5):601-633. Available from: http://dx.doi.org/10.1002/nme.2419.
39. Chessa J, Smolinski P, Belytschko T. The extended finite element method (xfem) for solidification problems. International Journal for Numerical Methods in Engineering 2002; 53(8):1959-1977. http://dx.doi.org/10.1002/nme.386.
40. Vaughan B, Smith B, Chopp D. A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources. Communications in Applied Mathematics and Computational Science 2006; 1(1):207-228.
41. Strouboulis T, Copps K, Babuška I. The generalized finite element method: an example of its implementation and illustration of its performance. International Journal for Numerical Methods in Engineering 2000; 47(8):1401-1417. Available from: http://dx.doi.org/10.1002/(SICI)1097-0207(20000320)47:8<1401::AID-NME835>3.0.CO; 2-8.
42. White SR, Sottos NR, Geubelle PH, Moore JS, Kessler MR, Sriram SR, Brown EN, Viswanathan S. Autonomic healing of polymer composites. Nature 2001; 409:794-797.
43. Toohey KS, Sottos NR, Lewis JA, Moore JS, White SR. Self-healing materials with microvascular networks. Nature Materials 2007; 6:581-585.
44. Aragón AM, Hansen CJ, Wu W, Geubelle PH, Lewis JA, White SR. Computational design and optimization of a biomimetic self-healing/cooling material. Proceedings of SPIE 2007; 6526. DOI: 10.1117/12.717064.
45. Aragón AM, Wayer JK, Geubelle PH, Goldberg DE, White SR. Design of microvascular flow networks using multi-objective genetic algorithms. Computer Methods in Applied Mechanics and Engineering 2008; 197 (49-50):4399-4410. Available from: http://dx.doi.org/10.1016/j.cma.2008.05.025.
46. Kays WM, Crawford ME, Weigand B. Convective heat and mass transfer (4th edn). McGraw-Hill: New York, 2004.
47. Murray CD. The physiological principle of minimum work. i. the vascular system and cost of blood volume. Proceedings of the National Academy of Sciences of the United States of America 1926; 12(3):207-214. Available from: http://www.pnas.org/content/12/3/207.short.