Abstract
A NURBS enhanced extended finite element approach is proposed for the unfitted simulation of structures defined by means of CAD parametric surfaces. In contrast to classical X-FEM that uses levelsets to define the geometry of the computational domain, exact CAD description is considered here. Following the ideas developed in the context of the NURBS-enhanced finite element method, NURBS-enhanced subelements are defined to take into account the exact geometry of the interface inside an element. In addition, a high-order approximation is considered to allow for large elements compared to the size of the geometrical details (without loss of accuracy). Finally, a geometrically implicit/explicit approach is proposed for efficiency purpose in the context of fracture mechanics. In this paper, only 2D examples are considered: It is shown that optimal rates of convergence are obtained without the need to consider shape functions defined in the physical space. Moreover, thanks to the flexibility given by the Partition of Unity, it is possible to recover optimal convergence rates in the case of re-entrant corners, cracks and embedded material interfaces.
Similar content being viewed by others
Notes
Not necessarily
References
Babuška I, Banerjee U (2012) Stable generalized finite element method (SGFEM). Comput Methods Appl Mech Eng 201–204:91–111. doi:10.1016/j.cma.2011.09.012
Bazilevs Y, Bajaj C, Calo V, Hughes T (2010a) Special issue on computational geometry and analysis. Comput. Methods Appl Mech Eng 199(5–8):223, doi:10.1016/j.cma.2009.10.006, http://www.sciencedirect.com/science/article/pii/S0045782509003429
Bazilevs Y, Calo V, Cottrell J, Evans J, Hughes T, Lipton S, Scott M, Sederberg T (2010b) Isogeometric analysis using T-splines. Comput Methods Appl Mech Eng 199(5–8):229–263. doi:10.1016/j.cma.2009.02.036, http://www.sciencedirect.com/science/article/pii/S0045782509000875
Béchet E, Minnebo H, Moës N, Burgardt B (2005) Improved implementation and robustness study of the X-FEM for stress analysis around cracks. Int J Numer Methods Eng 64(8):1033–1056
Béchet E, Moës N, Wohlmuth B (2009) A stable lagrange multiplier space for the stiff interface conditions within the extended finite element method. Int J Numer Methods Eng 78(8):931–954. doi:10.1002/nme.2515
Belytschko T, Parimi C, Moës N, Usui S, Sukumar N (2003) Structured extended finite element methods of solids defined by implicit surfaces. Int J Numer Methods Eng 56:609–635
Benson DJ, Bazilevs Y, De Luycker E, Hsu MC, Scott M, Hughes TJR, Belytschko T (2010) A generalized finite element formulation for arbitrary basis functions: From isogeometric analysis to XFEM. Int J Numer Methods Eng 83(6):765–785. doi:10.1002/nme.2864
Boor CD (1972) On calculation with B-splines. J Approx Theory 6:50–62
Cheng KW, Fries T (2009) Higher-order XFEM for curved strong and weak discontinuities. Int J Numer Methods Eng 82:564–590. doi:10.1002/nme.2768, http://doi.wiley.com/10.1002/nme.2768
Chessa J, Wang H, Belytschko T (2003) On the construction of blending elements for local partition of unity enriched finite elements. Int J Numer Methods Eng 57:1015–1038
Ciarlet P, Raviart PA (1972) Interpolation theory over curved elements, with applications to finite element methods. Comput Methods Appl Mech Eng 1(2):217–249. doi:10.1016/0045-7825(72)90006-0, http://www.sciencedirect.com/science/article/pii/0045782572900060
Cottrell J, Hughes TJ, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FE. Wiley, New York
Cowper G (1973) Gaussian quadrature formulas for triangles. Int J Numer Methods Eng 7:405–408
Cox M (1971) The numerical evaluation of B-splines. Tech. Rep. DNAC 4, National Physical Laboratory, Teddington
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):541–565. doi:10.1002/nme.3121
Dolbow J, Harari I (2009) An efficient finite element method for embedded interface problems. Int J Numer Methods Eng 78(2):229–252. doi:10.1002/nme.2486
Dréau K, Chevaugeon N, Moës N (2010) Studied X-FEM enrichment to handle material interfaces with higher order finite element. Comput Methods Appl Mech Eng 199(29–32):1922–1936. doi:10.1016/j.cma.2010.01.021, http://linkinghub.elsevier.com/retrieve/pii/S0045782510000563
Duster A, Parvizian J, Yang Z, Rank E (2008) The finite cell method for three-dimensional problems of solid mechanics. Comput Methods Appl Mech Eng 197(45–48):3768–3782. doi:10.1016/j.cma.2008.02.036, http://linkinghub.elsevier.com/retrieve/pii/S0045782508001163
Ergatoudis I, Irons B, Zienkiewicz O (1968) Curved, isoparametric, “quadrilateral” elements for finite element analysis. Int J Solids Struct 4(1):31–42. doi:10.1016/0020-7683(68)90031-0, http://www.sciencedirect.com/science/article/pii/0020768368900310
Forsey DR, Bartels RH (1988) Hierarchical B-spline refinement. SIGGRAPH Comput Graph 22(4):205–212. doi:10.1145/378456.378512, http://doi.acm.org.gate6.inist.fr/10.1145/378456.378512
Fries T (2008) A corrected XFEM approximation without problems in blending elements. Int J Numer Methods Eng 75(5):503–532. doi:10.1002/nme.2259
Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304. doi:10.1002/nme.2914
Ghorashi SS, Valizadeh N, Mohammadi S (2012) Extended isogeometric analysis for simulation of stationary and propagating cracks. Int J Numer Methods Eng 89(9):1069–1101. doi:10.1002/nme.3277
Gomez H, Calo VM, Bazilevs Y, Hughes TJ (2008) Isogeometric analysis of the Cahn-Hilliard phase-field model. Comput Methods Appl Mech Eng 197(49–50):4333–4352. doi:10.1016/j.cma.2008.05.003, http://www.sciencedirect.com/science/article/pii/S0045782508001953
Haasemann G, Kästner M, Prüger S, Ulbricht V (2011) Development of a quadratic finite element formulation based on the XFEM and NURBS. Int J Numer Methods Eng 86(4–5):598–617. doi:10.1002/nme.3120
Hansbo A, Hansbo P (2004) A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput Methods Appl Mech Eng 193(33–35):3523–3540. doi:10.1016/j.cma.2003.12.041, http://www.sciencedirect.com/science/article/B6V29-4BRSGX0-4/2/0a5f9d036b9ef16b2a0e571b247ff04d
Huerta A, Casoni E, Sala-Lardies E, Fernandez-Mendez S, Peraire J (2010) Modeling discontinuities with high-order elements. In: ECCM 2010. Palais des Congres, Paris
Hughes T, Cottrell J, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39–41):4135–4195. doi:10.1016/j.cma.2004.10.008, http://www.sciencedirect.com/science/article/pii/S0045782504005171
Kim HJ, Seo YD, Youn SK (2009) Isogeometric analysis for trimmed CAD surfaces. Comput Methods Appl Mech Eng 198(37–40):2982–2995. doi:10.1016/j.cma.2009.05.004, http://www.sciencedirect.com/science/article/pii/S0045782509001856
Kim HJ, Seo YD, Youn SK (2010) Isogeometric analysis with trimming technique for problems of arbitrary complex topology. Comput Methods Appl Mech Eng 199(45–48):2796–2812. doi:10.1016/j.cma.2010.04.015, http://www.sciencedirect.com/science/article/pii/S0045782510001325
Királyfalvi G, Szabó B (1997) Quasi-regional mapping for the p-version of the finite element method. Finite Elem Anal Des 27(1):85–97. doi:10.1016/S0168-874X(97)00006-1, http://linkinghub.elsevier.com/retrieve/pii/S0168874X97000061
Legrain G, Chevaugeon N, Dréau K (2012) High order X-FEM and levelsets for complex microstructures: uncoupling geometry and approximation. Comput Methods Appl Mech Eng 241–244(0):172–189. doi:10.1016/j.cma.2012.06.001, http://www.sciencedirect.com/science/article/pii/S0045782512001880
Lenoir M (1986) Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J Numer Anal 23(3):562–580. doi:10.1137/0723036, http://link.aip.org/link/?SNA/23/562/1
Lipton S, Evans J, Bazilevs Y, Elguedj T, Hughes T (2010) Robustness of isogeometric structural discretizations under severe mesh distortion. Comput Methods Appl Mech Eng 199(5–8):357–373. doi:10.1016/j.cma.2009.01.022, http://www.sciencedirect.com/science/article/pii/S0045782509000346
Melenk J, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150
Moës N, Cloirec M, Cartraud P, Remacle JF (2003) A computational approach to handle complex microstructure geometries. Comp Methods Appl Mech Eng 192:3163–3177. http://dx.doi.org/doi:10.1016/S0045-7825(03)00346-3
Moës N, Béchet E, Tourbier M (2006) Imposing Dirichlet boundary conditions in the extended finite element method. Int J Numer Methods Eng 67(12):1641–1669
Moumnassi M, Belouettar S, Béchet É, Bordas SP, Quoirin D, Potier-Ferry M (2011) Finite element analysis on implicitly defined domains: an accurate representation based on arbitrary parametric surfaces. Comput Methods Appl Mech Eng 200(5–8):774–796. doi:10.1016/j.cma.2010.10.002, http://www.sciencedirect.com/science/article/pii/S004578251000280X
Nitsche J (1971) Über ein Variationprinzip zur lösung von Dirichlet-Problem bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh Math Sem Univ Hamburg 36:9–15
Parvizian J, Duster A, Rank E (2007) Finite cell method - h and p extension for embedded domain problems in solid mechanics. Comput Mech 41(1):121–133. doi:10.1007/s00466-007-0173-y, http://www.springerlink.com/index/10.1007/s00466-007-0173-y
Rank E, Ruess M, Kollmannsberger S, Schillinger D, Düster A (2012) Geometric modeling, isogeometric analysis and the finite cell method. Comput Methods Appl Mech Eng 249–252:104–115. doi:10.1016/j.cma.2012.05.022, http://linkinghub.elsevier.com/retrieve/pii/S0045782512001855
Sala-Lardies E, Huerta A (2012) Optimally convergent high-order X-FEM for problems with voids and inclusions. In: ECCOMAS 2012, Vienna
Schillinger D, Rank E (2011) An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry. Comput Methods Appl Mech Eng 200(47–48):3358–3380, doi:10.1016/j.cma.2011.08.002, http://www.sciencedirect.com/science/article/pii/S004578251100257X
Schillinger D, Düster A, Rank E (2011) The hp-d-adaptive finite cell method for geometrically nonlinear problems of solid mechanics. Int J Numer Methods Eng 89(9): 1711–1202. doi:10.1002/nme.3289, http://dx.doi.org/10.1002/nme.3289
Schillinger D, Dedè L, Scott MA, Evans JA, Borden MJ, Rank E, Hughes TJ (2012) An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Comput Methods Appl Mech Eng 249–252: 116–150. doi:10.1016/j.cma.2012.03.017, http://www.sciencedirect.com/science/article/pii/S004578251200093X
Sederberg TW, Zheng J, Bakenov A, Nasri A (2003) T-splines and T-NURCCs. ACM Trans Graph 22(3):477–484, doi:10.1145/882262.882295, http://doi.acm.org.gate6.inist.fr/10.1145/882262.882295
Sevilla R, Fernández-Méndez S (2011) Numerical integration over 2D NURBS-shaped domains with applications to NURBS-enhanced FEM. Finite Elem Anal Des 47(10):1209–1220. doi:10.1016/j.finel.2011.05.011, http://www.sciencedirect.com/science/article/pii/S0168874X1100117X
Sevilla R, Fernández-Méndez S, Huerta A (2008) NURBS-enhanced finite element method (NEFEM). Int J Numer Methods Eng 76(1):56–83. doi:10.1002/nme.2311
Sevilla R, Fernández-Méndez S, Huerta A (2011a) 3D NURBS-enhanced finite element method (NEFEM). Int J Numer Methods Eng 88(2):103–125. doi:10.1002/nme.3164
Sevilla R, Fernández-Méndez S, Huerta A (2011b) Comparison of high-order curved finite elements. Int J Numer Methods Eng 87(8):719–734. doi:10.1002/nme.3129
Seweryn A, Molski K (1996) Elastic stress singularities and corresponding generalized stress intensity factors for angular corners under various boundary conditions. Eng Fract Mech 55(4):529–556. doi:10.1016/S0013-7944(96)00035-5
Strouboulis T, Babuška I, Copps K (2000) The design and analysis of the generalized finite element method. Comput Methods Appl Mech Eng 181:43–69
Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190(32–33):4081–4193. doi:10.1016/S0045-7825(01)00188-8, http://www.sciencedirect.com/science/article/pii/S0045782501001888
Sukumar N, Chopp DL, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite element method. Comput Method Appl Mech Eng 190:6183–6200. http://dx.doi.org/10.1016/S0045-7825(01)00215-8
Szabó B, Babuška I (1991) Finite element analysis. Wiley, New York
Szabó B, Düster A, Rank E (2004) The p-version of the finite element method, Chapt. 5. In: Encyclopedia of computational mechanics. Wiley, New York, pp 120–140
Yazid A, Abdelkader N, Abdelmadjid H (2009) A state-of-the-art review of the X-FEM for computational fracture mechanics. Appl Math Model 33(12):4269–4282. doi:10.1016/j.apm.2009.02.010, http://www.sciencedirect.com/science/article/pii/S0307904X09000560
Zienkiewicz OC, Taylor R (1991) The finite element method, Vols. 1, 2, 3. McGraw-Hill, London
Acknowledgments
The support of the ERC Advanced Grant XLS no. 291102 is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Legrain, G. A NURBS enhanced extended finite element approach for unfitted CAD analysis. Comput Mech 52, 913–929 (2013). https://doi.org/10.1007/s00466-013-0854-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-013-0854-7