×

zbMATH — the first resource for mathematics

Variationally consistent domain integration for isogeometric analysis. (English) Zbl 1425.65161
Summary: Spline-type approximations for solving partial differential equations are the basis of isogeometric analysis. While the common approach of using integration cells defined by single knot spans using standard (e.g., Gaussian) quadrature rules is sufficient for accuracy, more efficient domain integration is still in high demand. The recently introduced concept of variational consistency provides a guideline for constructing accurate and convergent methods requiring fewer quadrature points than standard integration techniques. In this work, variationally consistent domain integration is proposed for isogeometric analysis. Test function gradients are constructed to meet the consistency conditions, which only requires solving small linear systems of equations. The proposed approach allows for significant reduction in the number of quadrature points employed while maintaining the stability, accuracy, and optimal convergence properties of higher-order quadrature rules. Several numerical examples are provided to illustrate the performance of the proposed domain integration technique.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D17 Computer-aided design (modeling of curves and surfaces)
PDF BibTeX XML Cite
Full Text: DOI
References:
[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] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric analysis: toward integration of CAD and FEA, (2009), John Wiley & Sons, Ltd. New York, NY · Zbl 1378.65009
[3] Beirão da Veiga, L.; Buffa, A.; Rivas, J.; Sangalli, G., Some estimates for \(h - p - k\) refinement in isogeometric analysis, Numer. Math., 118, 2, 271-305, (2011) · Zbl 1222.41010
[4] 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
[5] Hughes, T. J.R.; Reali, A.; Sangalli, G., Efficient quadrature for NURBS- based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 199, 301-313, (2010) · Zbl 1227.65029
[6] Auricchio, F.; Calabrò, F.; Hughes, T. J.R.; Reali, A.; Sangalli, G., A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 249, 15-27, (2012) · Zbl 1348.65059
[7] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Evans, J.; Hughes, T. J.R.; Lipton, S.; Scott, M. A.; Sederberg, T. W., Isogeometric analysis using T-splines, Comput. Methods Appl. Mech. Engrg., 199, 229-263, (2010) · Zbl 1227.74123
[8] Nguyen-Thanh, N.; Kiendl, J.; Nguyen-Xuan, H.; Wüchner, R.; Bletzinger, K. U.; Bazilevs, Y.; Rabczuk, T., Rotation free isogeometric shell analysis using PHT-splines, Comput. Methods Appl. Mech. Engrg., 200, 47-48, 3410-3424, (2011) · Zbl 1230.74230
[9] Johannessen, K. A.; Kvamsdal, T.; Dokken, T., Isogeometric analysis using LR B-splines, Comput. Methods Appl. Mech. Engrg., 269, 471-514, (2014) · Zbl 1296.65021
[10] Chen, J. S.; Wu, C. T.; Yoon, S.; You, Y., A stabilized conforming nodal integration for Galerkin mesh-free methods, Internat. J. Numer. Methods Engrg., 50, 435-466, (2001) · Zbl 1011.74081
[11] Duan, Q.; Li, X.; Zhang, H.; Belytschko, T., Second-order accurate derivatives and integration schemes for meshfree methods, Internat. J. Numer. Methods Engrg., 92, 399-424, (2012) · Zbl 1352.65390
[12] Chen, J. S.; Hillman, M.; Rüter, M., An arbitrary order variationally consistent integration method for Galerkin meshfree methods, Internat. J. Numer. Methods Engrg., 95, 387-418, (2013) · Zbl 1352.65481
[13] Chen, J. S.; Yoon, S.; Wu, C. T., Nonlinear version of stabilized conforming nodal integration for Galerkin meshfree methods, Internat. J. Numer. Methods Engrg., 53, 2587-2615, (2002) · Zbl 1098.74732
[14] Wang, D.; Chen, J. S., Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation, Comput. Methods Appl. Mech. Engrg., 193, 1065-1083, (2004) · Zbl 1060.74675
[15] Wang, D.; Chen, J. S., A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration, Internat. J. Numer. Methods Engrg., 74, 368-390, (2008) · Zbl 1159.74460
[16] Chen, J. S.; Wang, D., A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinates, Internat. J. Numer. Methods Engrg., 68, 151-172, (2006) · Zbl 1130.74055
[17] Yoo, J. W.; Moran, B.; Chen, J. S., Stabilized conforming nodal integration in the natural-element method, Internat. J. Numer. Methods Engrg., 60, 861-890, (2004) · Zbl 1060.74677
[18] Hu, H. Y.; Chen, J. S.; Hu, W., Weighted radial basis collocation method for boundary value problems, Internat. J. Numer. Methods Engrg., 69, 2736-2757, (2007) · Zbl 1194.74525
[19] Chen, J. S.; Hu, W.; Hu, H. Y., Reproducing kernel enhanced local radial basis collocation method, Internat. J. Numer. Methods Engrg., 75, 600-627, (2008) · Zbl 1195.74278
[20] Schillinger, D.; Evans, J. A.; Reali, A.; Scott, M. A.; Hughes, T. J.R., Isogeometric collocation: cost comparison with Galerkin methods and extension to adaptive hierarchical NURBS discretizations, Comput. Methods Appl. Mech. Engrg., 267, 170-232, (2013) · Zbl 1286.65174
[21] Wang, D.; Zhang, H., A consistently coupled isogeometric-meshfree method, Comput. Methods Appl. Mech. Engrg., 268, 843-870, (2014) · Zbl 1295.65015
[22] Bazilevs, Y.; Beirao da 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, 1031-1090, (2006) · Zbl 1103.65113
[23] J, Nitsche, Über ein variationsprinzip zur Lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hambg., 36, 9-15, (1971) · Zbl 0229.65079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.