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A NURBS enhanced extended finite element approach for unfitted CAD analysis. (English) Zbl 1311.65017
Summary: 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.

65D17 Computer-aided design (modeling of curves and surfaces)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
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[1] Babuška, I; Banerjee, U, Stable generalized finite element method (SGFEM), Comput Methods Appl Mech Eng, 201-204, 91-111, (2012) · Zbl 1239.74093
[2] 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
[3] 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 · Zbl 1227.74123
[4] Béchet, E; Minnebo, H; Moës, N; Burgardt, B, Improved implementation and robustness study of the X-FEM for stress analysis around cracks, Int J Numer Methods Eng, 64, 1033-1056, (2005) · Zbl 1122.74499
[5] Béchet, E; Moës, N; Wohlmuth, B, A stable Lagrange multiplier space for the stiff interface conditions within the extended finite element method, Int J Numer Methods Eng, 78, 931-954, (2009) · Zbl 1183.74259
[6] Belytschko, T; Parimi, C; Moës, N; Usui, S; Sukumar, N, Structured extended finite element methods of solids defined by implicit surfaces, Int J Numer Methods Eng, 56, 609-635, (2003) · Zbl 1038.74041
[7] Benson, DJ; Bazilevs, Y; Luycker, E; Hsu, MC; Scott, M; Hughes, TJR; Belytschko, T, A generalized finite element formulation for arbitrary basis functions: from isogeometric analysis to XFEM, Int J Numer Methods Eng, 83, 765-785, (2010) · Zbl 1197.74177
[8] Boor, CD, On calculation with \(B\)-splines, J Approx Theory, 6, 50-62, (1972) · Zbl 0239.41006
[9] 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 · Zbl 1195.74173
[10] Chessa, J; Wang, H; Belytschko, T, On the construction of blending elements for local partition of unity enriched finite elements, Int J Numer Methods Eng, 57, 1015-1038, (2003) · Zbl 1035.65122
[11] 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 · Zbl 0261.65079
[12] Cottrell J, Hughes TJ, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FE. Wiley, New York · Zbl 1378.65009
[13] Cowper, G, Gaussian quadrature formulas for triangles, Int J Numer Methods Eng, 7, 405-408, (1973) · Zbl 0265.65013
[14] Cox M (1971) The numerical evaluation of B-splines. Tech. Rep. DNAC 4, National Physical Laboratory, Teddington
[15] Luycker, E; Benson, DJ; Belytschko, T; Bazilevs, Y; Hsu, MC, X-FEM in isogeometric analysis for linear fracture mechanics, Int J Numer Methods Eng, 87, 541-565, (2011) · Zbl 1242.74105
[16] Dolbow, J; Harari, I, An efficient finite element method for embedded interface problems, Int J Numer Methods Eng, 78, 229-252, (2009) · Zbl 1183.76803
[17] 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 · Zbl 1183.76803
[18] 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 · Zbl 1194.74517
[19] 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 · Zbl 0152.42802
[20] 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 · Zbl 1242.74105
[21] Fries, T, A corrected XFEM approximation without problems in blending elements, Int J Numer Methods Eng, 75, 503-532, (2008) · Zbl 1195.74173
[22] Fries, TP; Belytschko, T, The extended/generalized finite element method: an overview of the method and its applications, Int J Numer Methods Eng, 84, 253-304, (2010) · Zbl 1202.74169
[23] Ghorashi, SS; Valizadeh, N; Mohammadi, S, Extended isogeometric analysis for simulation of stationary and propagating cracks, Int J Numer Methods Eng, 89, 1069-1101, (2012) · Zbl 1242.74119
[24] 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 · Zbl 1202.74169
[25] Haasemann, G; Kästner, M; Prüger, S; Ulbricht, V, Development of a quadratic finite element formulation based on the XFEM and NURBS, Int J Numer Methods Eng, 86, 598-617, (2011) · Zbl 1216.74022
[26] 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 · Zbl 1068.74076
[27] 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
[28] 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 · Zbl 1151.74419
[29] 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 · Zbl 1216.74022
[30] 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 · Zbl 0881.65099
[31] 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
[32] 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 · Zbl 1353.74071
[33] 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 · Zbl 0605.65071
[34] 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 · Zbl 1227.74112
[35] Melenk, J; Babuška, I, The partition of unity finite element method: basic theory and applications, Comput Methods Appl Mech Eng, 139, 289-314, (1996) · Zbl 0881.65099
[36] Moës, N; Dolbow, J; Belytschko, T, A finite element method for crack growth without remeshing, Int J Numer Methods Eng, 46, 131-150, (1999) · Zbl 0955.74066
[37] 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 · Zbl 1054.74056
[38] Moës, N; Béchet, E; Tourbier, M, Imposing Dirichlet boundary conditions in the extended finite element method, Int J Numer Methods Eng, 67, 1641-1669, (2006) · Zbl 1113.74072
[39] 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 · Zbl 1225.65111
[40] Nitsche, J, 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, (1971) · Zbl 0229.65079
[41] 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 · Zbl 1162.74506
[42] 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 · Zbl 1348.74340
[43] Sala-Lardies E, Huerta A (2012) Optimally convergent high-order X-FEM for problems with voids and inclusions. In: ECCOMAS 2012, Vienna · Zbl 1197.74177
[44] 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 · Zbl 1230.74197
[45] 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 · Zbl 1242.74161
[46] 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 · Zbl 1348.65055
[47] 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
[48] 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
[49] Sevilla, R; Fernández-Méndez, S; Huerta, A, NURBS-enhanced finite element method (NEFEM), Int J Numer Methods Eng, 76, 56-83, (2008) · Zbl 1162.65389
[50] Sevilla, R; Fernández-Méndez, S; Huerta, A, 3D NURBS-enhanced finite element method (NEFEM), Int J Numer Methods Eng, 88, 103-125, (2011) · Zbl 1242.78032
[51] 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 · Zbl 1242.65244
[52] 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 · Zbl 1183.74259
[53] Strouboulis, T; Babuška, I; Copps, K, The design and analysis of the generalized finite element method, Comput Methods Appl Mech Eng, 181, 43-69, (2000) · Zbl 0983.65127
[54] 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
[55] 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 · Zbl 1029.74049
[56] Szabó B, Babuška I (1991) Finite element analysis. Wiley, New York · Zbl 0792.73003
[57] 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 · Zbl 1239.74093
[58] 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 · Zbl 1172.74050
[59] Zienkiewicz OC, Taylor R (1991) The finite element method, Vols. 1, 2, 3. McGraw-Hill, London · Zbl 1122.74499
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