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Numerical efficiency, locking and unlocking of NURBS finite elements. (English) Zbl 1227.74068
Summary: The strategy of using finite elements with NURBS shape functions for approximation of both geometry and displacements (“isogeometric approach”) is investigated from the point of view of finite element technology. Convergence rates are compared to those of classical finite element approaches utilizing standard Lagrange shape functions. Moreover, typical locking phenomena are examined. It is found that higher order inter-element continuity within the NURBS approach results in identical convergence rates but smaller absolute errors compared to \(C^{0}\)-continuous approaches. However, NURBS finite elements suffer from the same locking problems as finite elements using Lagrange shape functions. The discrete shear gap (DSG) method, a general framework for formulation of locking-free elements, is applied to develop a new class of NURBS finite elements. The resulting NURBS DSG elements are absolutely free from locking and preserve the property of improved accuracy compared with standard locking-free finite elements. The method is exemplified for the Timoshenko beam model, but may be applied to more general cases.

74S05 Finite element methods applied to problems in solid mechanics
65D07 Numerical computation using splines
Full Text: DOI
[1] Bischoff, M.; Bletzinger, K.-U., Stabilized DSG plate and shell elements, (), 253-263
[2] Bletzinger, K.-U.; Bischoff, M.; Ramm, E., A unified approach for shear-locking free triangular and rectangular shell finite elements, Comput. struct., 75, 321-334, (2000)
[3] Cottrell, J.A.; Hughes, T.J.R.; Reali, A., Studies of refinement and continuity in isogeometric structural analysis, Comput. method appl. mech. engrg., 196, 4160-4183, (2007) · Zbl 1173.74407
[4] Cottrell, J.A.; Reali, A.; Bazilevs, Y.; Hughes, T.J.R., Isogeometric analysis of structural vibrations, Comput. method appl. mech. engrg., 195, 5257-5296, (2006) · Zbl 1119.74024
[5] Höllig, K., Finite element methods with B-splines, (2003), SIAM Philadelphia · Zbl 1020.65085
[6] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications Inc.
[7] Hughes, T.J.R.; Cottrell, J.A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Comput. method appl. mech. engrg., 194, 4135-4195, (2005) · Zbl 1151.74419
[8] 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 mit k-method NURBS, Comput. method appl. mech. engrg., 197, 4104-4124, (2008) · Zbl 1194.74114
[9] Koschnick, F.; Bischoff, M.; Camprubi, N.; Bletzinger, K.-U., The discrete strain gap method and membrane locking, Comput. method appl. mech. engrg., 194, 2444-2463, (2005) · Zbl 1082.74053
[10] Piegl, L.; Tiller, W., The NURBS book (monographs in visual communication), (1997), Springer-Verlag Berlin
[11] Rogers, D.F., An introduction to NURBS with historical perspective, (2001), Academic Press San Diego, CA
[12] Zienkiewicz, O.C.; Taylor, R.L.; Zhu, J.Z., The finite element method: its basis and fundamentals, (2005), Elsevier Ltd. · Zbl 1307.74005
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