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Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids. (English) Zbl 1228.74091
Summary: Isogeometric analysis has become a powerful alternative to standard finite elements due to its flexibility in handling complex geometries. One of the major drawbacks of NURBS-based isogeometric finite elements is the inefficiency of local refinement. In this study, we present an alternative to NURBS-based isogeometric analysis that allows for local refinement. The idea is based on polynomial splines and exploits the flexibility of T-meshes for local refinement. The shape functions satisfy important properties such as non-negativity, local support and partition of unity. Several numerical examples are used to demonstrate the reliability of the present method.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65D17 Computer-aided design (modeling of curves and surfaces)
65D07 Numerical computation using splines
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[1] Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. methods appl. mech. engrg., 194, 4135-4195, (2005) · Zbl 1151.74419
[2] Echter, R.; Bischoff, M., Numerical efficiency, locking and unlocking of NURBS finite elements, Comput. methods appl. mech. engrg., 199, 374-382, (2010) · Zbl 1227.74068
[3] Höllig, K., Finite element methods with B-splines, (2003), Society for Industrial and Applies Mathematics Philadelphia · Zbl 1020.65085
[4] Sevilla, R.; Fernandez-Mendez, S.; Huerta, A., Nurbs-enhanced finite element method (NEFEM), Int. J. numer. methods engrg., 76, 56-83, (2008) · Zbl 1162.65389
[5] Shaw, A.; Roy, D., NURBS-based parametric mesh-free methods, Comput. methods appl. mech. engrg., 197, 1541-1567, (2008) · Zbl 1194.74538
[6] Szabo, B.; Duester, A.; Rank, E., The p-version of the finite element method, (), 1, (Chapter 5)
[7] Bazilevs, Y.; Calo, V.M.; Cottrell, J.A.; Evans, J.A.; Hughes, T.J.R.; Lipton, S.; Scottand, M.A.; Sederberg, T.W., Isogeometric analysis using T-splines, Comput. methods appl. mech. engrg., 199, 229-263, (2010) · Zbl 1227.74123
[8] Akkerman, I.; Bazilevs, Y.; Calo, V.M.; Hughes, T.J.R.; Hulshoff, S., The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. mech., 41, 371-378, (2008) · Zbl 1162.76355
[9] Bazilevs, Y.; Calo, V.M.; Hughes, T.J.R.; Zhang, Y., Isogeometric fluid – structure interaction: theory, algorithms, and computations, Comput. mech., 43, 3-37, (2008) · Zbl 1169.74015
[10] Cottrell, J.A.; Reali, A.; Bazilevs, Y.; Hughes, T.J.R., Isogeometric analysis of structural vibrations, Comput. methods appl. mech. engrg., 195, 5257-5297, (2006) · Zbl 1119.74024
[11] Wall, W.A.; Frenzel, M.A.; Cyron, C., Isogeometric structural shape optimization, Comput. methods appl. mech. engrg., 197, 1976-1988, (2008) · Zbl 1194.74263
[12] Hughes, T.J.R.; Reali, A.; Sangalli, G., Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS, Comput. methods appl. mech. engrg., 197, 4104-4124, (2008) · Zbl 1194.74114
[13] Benson, D.J.; Hsu, M.C.; Bazilevs, Y.; Hughes, T.J.R., Isogeometric shell analysis: the reissner – mindlin shell, Comput. methods appl. mech. engrg., 199, 276-289, (2010) · Zbl 1227.74107
[14] Kiendl, J.; Bletzinger, K.-U.; Linhard, J.; Wüchner, R., Isogeometric shell analysis with Kirchhoff-love elements, Comput. methods appl. mech. engrg., 198, 3902-3914, (2009) · Zbl 1231.74422
[15] Cottrell, J.A.; Hughes, T.J.R.; Reali, A., Studies of refinement and continuity in geometry and mesh refinement, Comput. methods appl. mech. engrg., 196, 4160-4183, (2007) · Zbl 1173.74407
[16] Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. methods appl. mech. engrg., 194, 4135-4195, (2005) · Zbl 1151.74419
[17] Forsey, D.; Bartels, R., Hierarchical B-spline refinement, Comput. graphics, 22, 205-212, (1998)
[18] Sederberg, T.W.; Zheng, J.; Bakenov, A.; Nasri, A., T-splines and T-nurccs, ACM trans. graphics, 22, 477-484, (2003)
[19] Sederberg, T.W.; Cardon, D.L.; Finnigan, G.T.; North, N.S.; Zheng, J.; Lyche, T., T-splines and local refinement, ACM trans. graphics, 23, 276-283, (2004)
[20] Buffa, A.; Cho, D.; Sangalli, G., Linear independence of the T-spline blending functions associated with some particular T-meshes, Comput. methods appl. mech. engrg., 199, 23-24, 437-1445, (2009) · Zbl 1231.65027
[21] Deng, Jiansong; Chen, Falai; Li, Xin; Hu, Changqi; Tong, Weihua; Yang, Zhouwang; Feng, Yuyu, Polynomial splines over hierarchical T-meshes, Graph. models, 70, 76-86, (2008)
[22] Li, Xin; Deng, Jiansong; Chen, Falai, Surface modeling with polynomial splines over hierarchical T-meshes, Visual comput., 23, 1027-1033, (2007)
[23] Li, Xin; Deng, Jiansong; Chen, Falai, Polynomial splines over general T-meshes, Visual comput., 26, 277-286, (2010)
[24] Rabczuk, T.; Belytschko, T., Cracking particles: a simplified meshfree methods for arbitrary evolving cracks, Int. J. numer. methods engrg., 61, 2316-2343, (2004) · Zbl 1075.74703
[25] Rabczuk, T.; Belytschko, T., Application of particle methods to static fracture of reinforced concrete structures, Int. J. fract., 137, 19-49, (2006) · Zbl 1197.74175
[26] Rabczuk, T.; Belytschko, T., A three dimensional large deformation meshfree method for arbitrary evolving cracks, Comput. methods appl. mech. engrg., 196, 2777-2799, (2007) · Zbl 1128.74051
[27] Rabczuk, T.; Samaniego, E., Discontinuous modelling of shear bands with adaptive meshfree methods, Comput. methods appl. mech. engrg., 197, 641-658, (2008) · Zbl 1169.74655
[28] Rabczuk, T.; Zi, G.; Bordas, S.; Nguyen-Xuan, H., A geometrically nonlinear three-dimensional cohesive crack method for reinforced concrete structures, Engrg. fract. mech., 75, 4740-4758, (2008)
[29] Piegl, L.; Tiller, W., The NURBS book, (1997), Springer-Verlag Berlin · Zbl 0868.68106
[30] Salomon, David, Curves and surfaces for computer graphics, (2006), Springer New York · Zbl 1083.65022
[31] Sederberg, T.W.; Zheng, J.; Bakenov, A.; Nasri, A., T-splines and tnurccs, ACM trans. graphics, 22, 3, 161-172, (2003)
[32] Deng, Jiansong; Chen, Falai; Feng, Yuyu, Dimensions of spline spaces over T-meshes, J. comput. appl. math., 194, 267-283, (2006) · Zbl 1093.41006
[33] Stogner, R.H.; Carey, G.F.; Murray, B.T., Approximation of cahn – hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements, Int. J. numer. methods engrg., 76, 636-661, (2008) · Zbl 1195.74132
[34] Rabczuk, T.; Belytschko, T., Adaptivity for structured meshfree particle methods in 2D and 3D, Int. J. numer. methods engrg., 63, 11, 1559-1582, (2005) · Zbl 1145.74041
[35] Austin Cottrell, J.; Hughes, Thomas J.R.; Bazilevs, Yuri, Isogeometric analysis: toward integration of CAD and FEA, (2009), John Wiley & Sons, Ltd. New York · Zbl 1378.65009
[36] Rabczuk, T.; Belytschko, T.; Xiao, S.P., Stable particle methods based on Lagrangian kernels, Comput. methods appl. mech. engrg., 193, 1035-1063, (2004) · Zbl 1060.74672
[37] Rabczuk, T.; Xiao, S.P.; Sauer, M., Coupling of meshfree methods with finite elements: basic concepts and test results, Commun. numer. methods engrg., 22, 1031-1065, (2006) · Zbl 1109.65082
[38] Nguyen-Thanh, N.; Rabczuk, T.; Nguyen-Xuan, H.; Bordas, S., An alternative alpha finite element method (AαFEM) free and forced vibration analysis of solids using triangular meshes, J. comput. appl. math., 223, 9, 2112-2135, (2010) · Zbl 1423.74910
[39] Phu, N.V.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, mathematics and computers in simulation, Math. comput. simulat., 79, 763-813, (2008) · Zbl 1152.74055
[40] Timoshenko, S.P.; Goodier, J.N., Theory of elasticity, (1970), McGraw New York · Zbl 0266.73008
[41] Nguyen-Xuan, H.; Rabczuk, T.; Bordas, S.; Debongnie, J.F., A smoothed finite element method for plate analysis, Comput. methods appl. mech. engrg., 197, 1184-1203, (2008) · Zbl 1159.74434
[42] Nguyen-Xuan, H.; Rabczuk, T.; Nguyen-Thanh, N.; Nguyen-Thoi, T.; Bordas, Stephane, A node-based smoothed finite element method (NS-FEM) for analysis of reissner – mindlin plates, Comput. mech., 46, 679-701, (2010) · Zbl 1260.74029
[43] Nguyen-Thanh, N.; Rabczuk, Timon; Nguyen-Xuan, H.; Bordas, S., An alternative alpha finite element method with stabilized discrete shear gap technique for analysis of mindlin – reissner plates, Finite elem. anal. des., 47, 5, 519-535, (2011)
[44] Nguyen-Thanh, N.; Rabczuk, Timon; Nguyen-Xuan, H.; Bordas, S., A smoothed finite element method for shell analysis, Comput. methods appl. mech. engrg., 198, 165-177, (2008) · Zbl 1194.74453
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