International Journal of Fracture

, Volume 204, Issue 1, pp 55–78 | Cite as

Linear elastic fracture simulation directly from CAD: 2D NURBS-based implementation and role of tip enrichment

  • X. Peng
  • E. AtroshchenkoEmail author
  • P. Kerfriden
  • S. P. A. Bordas
Original Paper


We propose a method for simulating linear elastic crack growth through an isogeometric boundary element method directly from a CAD model and without any mesh generation. To capture the stress singularity around the crack tip, two methods are compared: (1) a graded knot insertion near crack tip; (2) partition of unity enrichment. A well-established CAD algorithm is adopted to generate smooth crack surfaces as the crack grows. The M integral and \(J_k\) integral methods are used for the extraction of stress intensity factors (SIFs). The obtained SIFs and crack paths are compared with other numerical methods.


Isogeometric analysis NURBS Linear elastic fracture Boundary element method Crack growth 



The first and last authors would like to acknowledge the financial support of the Framework Programme 7 Initial Training Network Funding under grant number 289361 ‘Integrating Numerical Simulation and Geometric Design Technology’. S. P. A. Bordas also thanks partial funding for his time provided by the UK Engineering and Physical Science Research Council (EPSRC) under grant EP/G069352/1 Advanced discretization strategies for ‘atomistic’ nano CMOS simulation; the EPSRC under grant EP/G042705/1 ‘Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method’ and the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) entitled ‘Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery’. E. Atroshchenko was partially supported by Fondecyt grant number 11130259 entitled ‘Boundary element modeling of crack propagation in micropolar materials’.


  1. Akkerman I, Bazilevs Y, Calo VM, Hughes TJR, Hulshoff S (2008) The role of continuity in residual-based variational multiscale modeling of turbulence. Comput Mech 41(3):371–378CrossRefGoogle Scholar
  2. Aour B, Rahmani O, Nait-Abdelaziz M (2007) A coupled FEM/BEM approach and its accuracy for solving crack problems in fracture mechanics. Int J Solids Struct 44(7–8):2523–2539CrossRefGoogle Scholar
  3. Arnold D, Saranen J (1984) On the asymptotic convergence of spline collocation methods for partial differential equations. SIAM J Numer Anal 21(3):459–472CrossRefGoogle Scholar
  4. Auricchio F, da Veiga LB, Lovadina C, Reali A (2010a) The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations. Comput Methods Appl Mech Eng 199(5–8):314–323CrossRefGoogle Scholar
  5. Auricchio F, Veiga LBD, Hughes TJR, Reali A, Sangalli G (2010b) Isogeometric collocation methods. Math Models Methods Appl Sci 20(11):2075–2107CrossRefGoogle Scholar
  6. Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput Mech 38(4–5):310–322CrossRefGoogle Scholar
  7. Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2010) Isogeometric analysis using T-splines. Comput Methods Appl Mech Eng 199(5–8):229–263CrossRefGoogle Scholar
  8. Becker A (1992) The boundary element methods in engineering. McGraw-Hill Book Company, New YorkGoogle Scholar
  9. Beer G, Marussig B, Zechner J (2015) A simple approach to the numerical simulation with trimmed CAD surfaces. Comput Methods Appl Mech Eng 285:776–790CrossRefGoogle Scholar
  10. Benson DJ, Bazilevs Y, Hsu MC, Hughes TJR (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput Methods Appl Mech Eng 199(5–8):276–289CrossRefGoogle Scholar
  11. Blandford GE, Ingraffea AR, Liggett JA (1981) Two-dimensional stress intensity factor computations using the boundary element method. Int J Numer Methods Eng 17(3):387–404CrossRefGoogle Scholar
  12. Bonnet M, Maier G, Polizzoto C (1998) Symmetric Galerkin boundary element method. Appl Mech Rev 51:669–704CrossRefGoogle Scholar
  13. Bordas S, Nguyen PV, Dunant C, Guidoum A, Nguyen-Dang H (2007) An extended finite element library. Int J Numer Methods Eng 71(6):703–732CrossRefGoogle Scholar
  14. Bordas SPA, Rabczuk T, Zi G (2008) Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. Eng Fract Mech 75(5):943–960CrossRefGoogle Scholar
  15. Bordas SPA, Duflot M (2007) Derivative recovery and a posteriori error estimate for extended finite elements. Comput Methods Appl Mech Eng 196(35–36):3381–3399CrossRefGoogle Scholar
  16. Bordas S, Moran B (2006) Enriched finite elements and level sets for damage tolerance assessment of complex structures. Eng Fract Mech 73(9):1176–1201CrossRefGoogle Scholar
  17. Borden MJ, Verhoosel CV, Scott MA, Hughes TJR, Landis CM (2012) A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Eng 217–220:77–95CrossRefGoogle Scholar
  18. Bouchard PO, Bay F, Chastel Y, Tovena I (2000) Crack propagation modelling using an advanced remeshing technique. Comput Methods Appl Mech Eng 189(3):723–742CrossRefGoogle Scholar
  19. Chang JH, Wu DJ (2007) Stress intensity factor computation along a non-planar curved crack in three dimensions. Int J Solids Struct 44(2):371–386CrossRefGoogle Scholar
  20. Cisilino AP, Aliabadi MH (2004) Dual boundary element assessment of three-dimensional fatigue crack growth. Eng Anal Boundary Elem 28(9):1157–1173CrossRefGoogle Scholar
  21. Cotterell B, Rice JR (1980) Slightly curved or kinked cracks. Int J Fract 16(2):155–169CrossRefGoogle Scholar
  22. Crouch SL (1976) Solution of plane elasticity problems by the displacement discontinuity method. I. Infinite body solution. Int J Numer Methods Eng 10(2):301–343CrossRefGoogle Scholar
  23. Davis BR, Wawrzynek PA, Carter BJ, Ingraffea AR (2016) 3-D simulation of arbitrary crack growth using an energy-based formulation—part II: non-planar growth. Eng Fract Mech 154:111–127CrossRefGoogle Scholar
  24. De Luycker E, Benson DJ, Belytschko T, Bazilevs Y, Hsu MC (2011) X-FEM in isogeometric analysis for linear fracture mechanics. Int J Numer Methods Eng 87(6):541–565CrossRefGoogle Scholar
  25. Deng J, Chen F, Li X, Hu C, Tong W, Yang Z, Feng Y (2008) Polynomial splines over hierarchical T-meshes. Graph Models 70(4):76–86CrossRefGoogle Scholar
  26. Dokken T, Lyche T, Pettersen KF (2013) Polynomial splines over locally refined box-partitions. Comput Aided Geom Des 30(3):331–356CrossRefGoogle Scholar
  27. Dominguez J, Ariza MP (2000) A direct traction BIE approach for three-dimensional crack problems. Eng Anal Bound Elem 24(10):727–738CrossRefGoogle Scholar
  28. Dong L, Atluri SN (2013) Fracture & fatigue analyses: SGBEM-FEM or XFEM? part 2: 3D solids. Comput Model Eng Sci 90(5):3379–3413Google Scholar
  29. Duflot M, Bordas SPA (2008) A posteriori error estimation for extended finite elements by an extended global recovery. Int J Numer Methods Eng 76(8):1123–1138CrossRefGoogle Scholar
  30. Eischen JW (1987) An improved method for computing the J2 integral. Eng Fract Mech 26(5):691–700CrossRefGoogle Scholar
  31. Erdogan F, Sih G (1963) On the crackextension in plates under plane loading and transverse shear. J Basic Eng 85:519–527CrossRefGoogle Scholar
  32. Feischl M, Gantner G, Praetorius D (2015) Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations. Comput Methods Appl Mech Eng 290:362–386CrossRefGoogle Scholar
  33. Frangi A (2002) Fracture propagation in 3D by the symmetric Galerkin boundary element method. Int J Fract 116(4):313–330CrossRefGoogle Scholar
  34. Ghorashi S, Valizadeh N, Mohammadi S (2012) Extended isogeometric analysis for simulation of stationary and propagating cracks. Int J Numer Methods Eng 89(9):1069–1101CrossRefGoogle Scholar
  35. González-Estrada OA, Ródenas JJ, Nadal E, Bordas SPA, Kerfriden P (2011) Equilibrated patch recovery for accurate evaluation of upper error bounds in quantities of interest. In: Proceedings of V adaptive modeling and simulation (ADMOS)Google Scholar
  36. González-Estrada OA, Nadal E, Ródenas JJ, Kerfriden P, Bordas SPA, Fuenmayor FJ (2014) Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Comput Mech 53(5):957–976CrossRefGoogle Scholar
  37. Gosz M, Moran B (2002) An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions. Eng Fract Mech 69(3):299–319CrossRefGoogle Scholar
  38. Goury O (2015) Computational time savings in multiscale fracture mechanics using model order reduction. PhD thesis, Cardiff UniversityGoogle Scholar
  39. Gravouil A, Moës N, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets-part II: level set update. Int J Numer Methods Eng 53(11):2569–2586CrossRefGoogle Scholar
  40. Gu J, Zhang J, Li G (2012) Isogeometric analysis in BIE for 3-D potential problem. Eng Anal Bound Elem 36(5):858–865CrossRefGoogle Scholar
  41. Guiggiani M, Krishnasamy G, Rudolphi TJ, Rizzo FJ (1992) A general algorithm for the numerical solution of hypersingular boundary integral equations. J Appl Mech 59(3):604–614CrossRefGoogle Scholar
  42. Heltai L, Arroyo M, DeSimone A (2012) Nonsingular isogeometric boundary element method for Stokes flows in 3D. Comput Methods Appl Mech Eng. 268:514–539. doi: 10.1016/j.cma.2013.09.017
  43. Henshell RD, Shaw KG (1975) Crack tip finite elements are unnecessary. Int J Numer Methods Eng 9(3):495–507CrossRefGoogle Scholar
  44. Hong H, Chen J (1988) Derivations of integral equations of elasticity. J Eng Mech 114(6):1028–1044CrossRefGoogle Scholar
  45. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39–41):4135–4195CrossRefGoogle Scholar
  46. Ingraffea AR, Grigoriu M (1990) Probabilistic fracture mechanics: a validation of predictive capability. Department of Structure Engineering, Cornell University, Report, pp 90–98Google Scholar
  47. Juhl P (1998) A note on the convergence of the direct collocation boundary element method. J Sound Vib 212(4):703–719CrossRefGoogle Scholar
  48. Kaczmarczyk U, Nezhad MM, Pearce C (2014) Three-dimensional brittle fracture: configurational-force-driven crack propagation. Int J Numer Methods Eng 97(7):531–550CrossRefGoogle Scholar
  49. Karihaloo BL, Xiao QZ (2001) Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity. Eng Fract Mech 68(15):1609–1630CrossRefGoogle Scholar
  50. LaGreca R, Daniel M, Bac A (2005) Local deformation of NURBS curves. Math Methods Curves Surf Tromso 2004:243–252Google Scholar
  51. Lan M, Waisman H, Harari I (2013) A direct analytical method to extract mixed-mode components of strain energy release rates from Irwin’s integral using extended finite element method. Int J Numer Methods Eng 95(12):1033–1052CrossRefGoogle Scholar
  52. Li S, Mear ME, Xiao L (1998) Symmetric weak-form integral equation method for three-dimensional fracture analysis. Comput Methods Appl Mech Eng 151(3–4):435–459CrossRefGoogle Scholar
  53. Liew KM, Cheng Y, Kitipornchai S (2007) Analyzing the 2D fracture problems via the enriched boundary element-free method. Int J Solids Struct 44(11–12):4220–4233CrossRefGoogle Scholar
  54. Li K, Qian X (2011) Isogeometric analysis and shape optimization via boundary integral. Comput Aided Des 43(11):1427–1437CrossRefGoogle Scholar
  55. Liu Y, Rudolphi TJ (1991) Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations. Eng Anal Bound Elem 8(6):301–311CrossRefGoogle Scholar
  56. Lutz E, Ingraffea AR, Gray LJ (1992) Use of ‘simple solutions’ for boundary integral methods in elasticity and fracture analysis. Int J Numer Methods Eng 35(9):1737–1751CrossRefGoogle Scholar
  57. Martinez J, Dominguez J (1984) On the use of quarter-point boundary elements for stress intensity factor computations. Int J Numer Methods Eng 20(10):1941–1950CrossRefGoogle Scholar
  58. Martin PA, Rizzo FJ (1996) Hypersingular integrals: how smooth must the density be? Int J Numer Methods Eng 39(4):687–704CrossRefGoogle Scholar
  59. Marussig B, Zechner J, Beer G, Fries TP (2015) Fast isogeometric boundary element method based on independent field approximation. Comput Methods Appl Mech Eng 284:458–488CrossRefGoogle Scholar
  60. Melenk JM, Babuška I (1996) The partition of unity finite element method: Basic theory and applications. Comput Methods Appl Mech Eng 139(1–4):289–314CrossRefGoogle Scholar
  61. Mi Y, Aliabadi MH (1992) Dual boundary element method for three-dimensional fracture mechanics analysis. Eng Anal Bound Elem 10(2):161–171CrossRefGoogle Scholar
  62. Mi Y, Aliabadi MH (1994) Discontinuous crack-tip elements: application to 3D boundary element method. Int J Fract 67(3):R67–R71CrossRefGoogle Scholar
  63. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150CrossRefGoogle Scholar
  64. Moës N, Gravouil A, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets-part I: mechanical model. Int J Numer Methods Eng 53(11):2549–2568CrossRefGoogle Scholar
  65. Mukherjee YX, Shah K, Mukherjee S (1999) Thermoelastic fracture mechanics with regularized hypersingular boundary integral equations. Eng Anal Bound Elem 23(1):89–96CrossRefGoogle Scholar
  66. Muthu N, Falzon BG, Maiti SK, Khoddam S (2014) Modified crack closure integral technique for extraction of {SIFs} in meshfree methods. Finite Elem Anal Des 78:25–39CrossRefGoogle Scholar
  67. Natarajan S, Song C (2013) Representation of singular fields without asymptotic enrichment in the extended finite element method. Int J Numer Methods Eng 96(13):813–841CrossRefGoogle Scholar
  68. Nguyen VP, Kerfriden P, Bordas SPA (2014) Two- and three-dimensional isogeometric cohesive elements for composite delamination analysis. Compos Part B Eng 60:193–212CrossRefGoogle Scholar
  69. Nikishkov GP, Park JH, Atluri SN (2001) SGBEM-FEM alternating method for analyzing 3D non-planar cracks and their growth in structural components. Comput Model Eng Sci 2(3):401–422Google Scholar
  70. Paluszny A, Zimmerman RW (2013) Numerical fracture growth modeling using smooth surface geometric deformation. Eng Fract Mech 108:19–36CrossRefGoogle Scholar
  71. Partheymüller P, Haas M, Kuhn G (2000) Comparison of the basic and the discontinuity formulation of the 3D-dual boundary element method. Eng Anal Bound Elem 24(10):777–788CrossRefGoogle Scholar
  72. Peake MJ, Trevelyan J, Coates G (2013) Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems. Comput Methods Appl Mech Eng 259:93–102CrossRefGoogle Scholar
  73. Piegl L, Tiller W (1995) The NURBS book. Springer, BerlinCrossRefGoogle Scholar
  74. Politis C, Ginnis AI, Kaklis PD, Belibassakis K, Feurer C (2009) An isogeometric BEM for exterior potential-flow problems in the plane. In: 2009 SIAM/ACM joint conference on geometric and physical modeling, SPM ’09, ACM, New York, NY, USA, pp 349–354Google Scholar
  75. Portela A (2011) Dual boundary-element method: simple error estimator and adaptivity. Int J Numer Methods Eng 86(12):1457–1480CrossRefGoogle Scholar
  76. Portela A, Aliabadi MH, Rooke DP (1992a) The dual boundary element method: effective implementation for crack problems. Int J Numer Methods Eng 33(6):1269–1287CrossRefGoogle Scholar
  77. Portela A, Aliabadi MH, Rooke DP (1992b) Dual boundary element analysis of cracked plates: singularity subtraction technique. Int J Fract 55(1):17–28CrossRefGoogle Scholar
  78. Rabczuk T, Bordas S, Zi G (2010) On three-dimensional modelling of crack growth using partition of unity methods. Comput Struct 88(23–24):1391–1411CrossRefGoogle Scholar
  79. Rahimabadi AA (2014) Error controlled adaptive multiscale method for fracture in polycrystalline materials. PhD thesis, Cardiff UniversityGoogle Scholar
  80. Rigby RH, Aliabadi MH (1998) Decomposition of the mixed-mode J-integral-revisited. Int J Solids Struct 35(17):2073–2099CrossRefGoogle Scholar
  81. Ródenas JJ, González-Estrada OA, Tarancón JE, Fuenmayor FJ (2008) A recovery-type error estimator for the extended finite element method based on singular+smooth stress field splitting. Int J Numer Methods Eng 76(4):545–571CrossRefGoogle Scholar
  82. Rudolphi TJ (1991) The use of simple solutions in the regularization of hypersingular boundary integral equations. Math Comput Model. 15(3–5):269–278CrossRefGoogle Scholar
  83. Scott MA, Li X, Sederberg TW, Hughes TJR (2012) Local refinement of analysis-suitable T-splines. Comput Methods Appl Mech Eng 213–216:206–222CrossRefGoogle Scholar
  84. Scott MA, Simpson RN, Evans JA, Lipton S, Bordas SPA, Hughes TJR, Sederberg TW (2013) Isogeometric boundary element analysis using unstructured T-splines. Comput Methods Appl Mech Eng 254:197–221CrossRefGoogle Scholar
  85. Shivakumar KN, Raju IS (1992) An equivalent domain integral method for three-dimensional mixed-mode fracture problems. Eng Fract Mech 42(6):935–959CrossRefGoogle Scholar
  86. Simpson R, Trevelyan J (2011) A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics. Comput Methods Appl Mech Eng 200(1–4):1–10CrossRefGoogle Scholar
  87. Simpson RN, Bordas SPA, Trevelyan J, Rabczuk T (2012) A two-dimensional isogeometric boundary element method for elastostatic analysis. Comput Methods Appl Mech Eng 209–212:87–100CrossRefGoogle Scholar
  88. Simpson RN, Scott MA, Taus M, Thomas DC, Lian H (2013) Acoustic isogeometric boundary element analysis. Comput Methods Appl Mech Eng 269:265–290. doi: 10.1016/j.cma.2013.10.026
  89. Smith DJ, Ayatollahi MR, Pavier MJ (2001) The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading. Fatigue Fract Eng Mater Struct 24(2):137–150CrossRefGoogle Scholar
  90. Snyder MD, Cruse TA (1975) Boundary-integral equation analysis of cracked anisotropic plates. Int J Fract 11(2):315–328CrossRefGoogle Scholar
  91. Sukumar N, Chopp DL, Béchet E, Moës N (2008) Three-dimensional non-planar crack growth by a coupled extended finite element and fast marching method. Int J Numer Methods Eng 76(5):727–748CrossRefGoogle Scholar
  92. Sutradhar A, Paulino GH (2004) Symmetric Galerkin boundary element computation of T-stress and stress intensity factors for mixed-mode cracks by the interaction integral method. Eng Anal Bound Elem 28(11):1335–1350CrossRefGoogle Scholar
  93. Sutula D, Bordas SPA (2013) Global energy minimization for all crack increment directions in the framework of XFEM. Internal report, University of LuxembourgGoogle Scholar
  94. Tanaka M, Sladek V, Sladeck J (1994) Regularization techniques applied to boundary element methods. Appl Mech Rev 47(10):457–499CrossRefGoogle Scholar
  95. Taus M, Rodin GJ, Hughes TJR (2015) Isogeometric analysis of boundary integral equations. ICES report 15–12Google Scholar
  96. Telles JCF (1987) A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals. Int J Numer Methods Eng 24(5):959–973CrossRefGoogle Scholar
  97. Ventura G, Xu JX, Belytschko T (2002) A vector level set method and new discontinuity approximations for crack growth by EFG. Int J Numer Methods Eng 54(6):923–944CrossRefGoogle Scholar
  98. Verhoosel CV, Scott MA, de Borst R, Hughes TJR (2011) An isogeometric approach to cohesive zone modeling. Int J Numer Methods Eng 87(1–5):336–360CrossRefGoogle Scholar
  99. Wang Y, Benson DJ, Nagy AP (2015) A multi-patch nonsingular isogeometric boundary element method using trimmed elements. Comput Mech 56(1):173–191CrossRefGoogle Scholar
  100. Westergaard HM (1939) Bearing pressures and cracks. J Appl Mech 6:A49–A53Google Scholar
  101. Wyart E, Duflot M, Coulon D, Martiny P, Pardoen T, Remacle J-F, Lani F (2008) Substructuring FE-XFE approaches applied to three-dimensional crack propagation. J Comput Appl Math 215(2):626–638CrossRefGoogle Scholar
  102. Yau JF, Wang SS, Corten HT (1980) A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J Appl Mech 47:333–341CrossRefGoogle Scholar
  103. Zamani NG, Sun W (1993) A direct method for calculating the stress intensity factor in BEM. Eng Anal Bound Elem 11(4):285–292CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • X. Peng
    • 1
  • E. Atroshchenko
    • 2
    Email author
  • P. Kerfriden
    • 1
  • S. P. A. Bordas
    • 1
    • 3
    • 4
  1. 1.Institute of Mechanics Materials and Advanced ManufacturingCardiff UniversityCardiffUK
  2. 2.Department of Mechanical EngineeringUniversity of ChileSantiagoChile
  3. 3.Faculté des Sciences, de la Technologie et de la CommunicationUniversité du LuxembourgLuxembourgLuxembourg
  4. 4.University of Western AustraliaCrawleyAustralia

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