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Efficient quadrature for NURBS-based isogeometric analysis. (English) Zbl 1227.65029
Summary: We initiate the study of efficient quadrature rules for NURBS-based isogeometric analysis. A rule of thumb emerges, the ”half-point rule”, indicating that optimal rules involve a number of points roughly equal to half the number of degrees-of-freedom, or equivalently half the number of basis functions of the space under consideration. The half-point rule is independent of the polynomial order of the basis. Efficient rules require taking into account the precise smoothness of basis functions across element boundaries. Several rules of practical interest are obtained, and a numerical procedure for determining efficient rules is presented.
We compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics. The new rules represent improvements over those used previously in isogeometric analysis.

MSC:
65D32 Numerical quadrature and cubature formulas
65D07 Numerical computation using splines
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[1] Akkermann, 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
[2] Auricchio, F.; Beirão de Veiga, L.; Buffa, A.; Lovadina, C.; Reali, A.; Sangalli, G., A fully “locking-free” isogeometric approach for plane linear elasticity problems: a stream function formulation, Comput. methods appl. mech. engrg., 197, 160-172, (2007) · Zbl 1169.74643
[3] Bazilevs, Y.; Beirão de Veiga, L.; Cottrell, J.A.; Hughes, T.J.R.; Sangalli, G., Isogeometric analysis: approximation stability and error estimates for h-refined meshes, Math. models methods appl. sci., 16, 1-60, (2006) · Zbl 1103.65113
[4] Bazilevs, Y.; Michler, C.; Calo, V.M.; Hughes, T.J.R., Weak Dirichlet boundary conditions for wall-bounded turbulent flows, Comput. methods appl. mech. engrg., 196, 4853-4862, (2007) · Zbl 1173.76397
[5] Bazilevs, Y.; Calo, V.M.; Cottrell, J.A.; Hughes, T.J.R.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. methods appl. mech. engrg., 197, 173-201, (2007) · Zbl 1169.76352
[6] E. Cohen, R.F. Riesenfeld, G. Elber, Geometric Modeling with Splines, A.K. Peters, 2001. · Zbl 0980.65016
[7] Cottrell, J.A.; Reali, A.; Bazilevs, Y.; Hughes, T.J.R., Isogeometric analysis of structural vibrations, Comput. methods appl. mech. engrg., 195, 5257-5296, (2006) · Zbl 1119.74024
[8] Cottrell, J.A.; Hughes, T.J.R.; Reali, A., Studies of refinement and continuity in isogeometric structural analysis, Comput. methods appl. mech. engrg., 196, 4160-4183, (2007) · Zbl 1173.74407
[9] Dörfel, M.R.; Jüttler, B.; Simeon, B., Adaptive isogeometric analysis by local h-refinement with T-splines, Comput. methods appl. mech. engrg., 199, 264-275, (2010) · Zbl 1227.74125
[10] G.E. Farin, NURBS curves and surfaces: from projective geometry to practical use, A.K. Peters, 1995.
[11] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications
[12] 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
[13] 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
[14] Irons, B.M., Engineering application of numerical integration in stiffness method, J. amer. inst. aeronaut. astronaut., 14, 2035-2037, (1966) · Zbl 0208.53303
[15] The MathWorks, Optimization Toolbox User’s Guide, 2008. <http://www.mathworks.com/access/helpdesk/help/pdf_doc/optim/optim_tb.pdf>.
[16] Piegl, L.; Tiller, W., The NURBS book, (1997), Springer-Verlag · Zbl 0868.68106
[17] Reali, A., An isogeometric analysis approach for the study of structural vibrations, J. earthquake engrg., 10, s.i. 1, 1-30, (2006)
[18] Rogers, D.F., An introduction to NURBS with historical perspective, (2001), Academic Press
[19] Schumaker, L.L., Spline functions: basic theory, (1993), Krieger · Zbl 0449.41004
[20] Sederberg, T.W.; Zheng, J.; Bakenov, A.; Nasri, A., T-splines and T-NURCCS, ACM trans. graph., 22, 477-484, (2003)
[21] Zhang, Y.; Bazilevs, Y.; Goswami, S.; Bajaj, C.L.; Hughes, T.J.R., Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow, Comput. methods appl. mech. engrg., 196, 2943-2959, (2007) · Zbl 1121.76076
[22] Zienkiewicz, O.C.; Cheung, Y.K., The finite element method in structural and continuum mechanics, (1967), McGraw-Hill · Zbl 0189.24902
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