×

Truncated hierarchical Catmull-Clark subdivision with local refinement. (English) Zbl 1425.65028

Summary: In this paper we present a new method termed Truncated Hierarchical Catmull-Clark Subdivision (THCCS), which generalizes truncated hierarchical B-splines to control grids of arbitrary topology. THCCS basis functions satisfy partition of unity, are linearly independent, and are locally refinable. THCCS also preserves geometry during adaptive \(h\)-refinement and thus inherits the surface continuity of Catmull-Clark subdivision, namely \(C^2\)-continuous everywhere except at the local region surrounding extraordinary nodes, where the surface continuity is \(C^1\). Adaptive isogeometric analysis is performed with THCCS basis functions on a benchmark problem with extraordinary nodes. Local refinement on complex surfaces is also studied to show potential wide application of the proposed method.

MSC:

65D07 Numerical computation using splines
65D17 Computer-aided design (modeling of curves and surfaces)

Software:

ISOGAT; CHARMS
PDFBibTeX XMLCite
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., Isogemetric Analysis: Toward Integration of CAD and FEA (2009), John Wiley & Sons · Zbl 1378.65009
[3] Sederberg, T. W.; Zheng, J.; Bakenov, A.; Nasri, A., T-splines and T-NURCCs, ACM Trans. Graphics, 22, 477-484 (2003)
[4] Sederberg, T. W.; Cardon, D. L.; Finnigan, G. T.; North, N. S.; Zheng, J.; Lyche, T., T-spline simplification and local refinement, ACM Transactions on Graphics, 23, 276-283 (2004)
[5] Deng, J.; Chen, F.; Li, X.; Hu, C.; Tong, W.; Yang, Z.; Feng, Y., Polynomial splines over hierarchical T-meshes, Graphical Models, 70, 76-86 (2008)
[6] Vuong, A.-V.; Giannelli, C.; Jüttler, B.; Simeon, B., A hierarchical approach to adaptive local refinement in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 200, 3554-3567 (2011) · Zbl 1239.65013
[7] Bornermann, P. B.; Cirak, F., A subdivision-based implementation of the hierarchical B-spline finite element method, Comput. Methods Appl. Mech. Engrg., 253, 584-598 (2013) · Zbl 1297.65147
[8] Dokken, T.; Lyche, T.; Pettersen, K. F., Polynomial splines over locally refined box-partitions, Comput. Aided Geom. Design, 30, 331-356 (2013) · Zbl 1264.41011
[9] Johannessen, K. A.; Kvamsdal, T.; Dokken, T., Isogeometric analysis using LR B-splines, Comput. Methods Appl. Mech. Engrg., 269, 0, 471-514 (2014) · Zbl 1296.65021
[10] Zhang, Y.; Bazilevs, Y.; Goswami, S.; Bajaj, C.; 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
[11] Wang, W.; Zhang, Y.; Xu, G.; Hughes, T. J.R., Converting an unstructured quadrilateral/hexahedral mesh to a rational T-spline, Comput. Mech., 50, 65-84 (2012) · Zbl 1312.65197
[12] Wang, W.; Zhang, Y.; Liu, L.; Hughes, T. J.R., Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology, (A Special Issue of Solid and Physical Modeling 2012. A Special Issue of Solid and Physical Modeling 2012, Comput. Aided Design, 45 (2013)), 351-360
[13] 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
[14] Scott, M. A.; Borden, M. J.; Verhoosel, C. V.; Sederberg, T. W.; Hughes, T. J.R., Isogeometric finite element data structures based on Bézier extraction of T-splines, Internat. J. Numer. Methods Engrg., 88, 126-156 (2011) · Zbl 1242.65243
[15] 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, 1437-1445 (2010) · Zbl 1231.65027
[16] Li, X.; Zheng, J.; Sederberg, T. W.; Hughes, T. J.R.; Scott, M. A., On the linear independence of T-spline blending functions, Comput. Aided Geom. Design, 29, 63-76 (2012) · Zbl 1251.65012
[17] Scott, M. A.; Li, X.; Sederberg, T. W.; Hughes, T. J.R., Local refinement of analysis-suitable T-splines, Comput. Methods Appl. Mech. Engrg., 213-216, 206-222 (2012) · Zbl 1243.65030
[18] Forsey, D. R.; Bartels, R. H., Hierarchical B-spline refinement, Comput. Graphics, 22, 205-212 (1988)
[19] Kraft, R., Adaptive and linearly independent multilevel B-splines, (Méhauté, A. L.; Rabut, C.; Schumaker, L. L., Surface Fitting and Multiresolution Methods (1997), Vanderbilt University Press), 209-218 · Zbl 0937.65014
[20] Giannelli, C.; Jüttler, B.; Speleers, H., THB-splines: The truncated basis for hierarchical splines, Comp. Aided Geom. Design, 29, 485-498 (2012) · Zbl 1252.65030
[21] Cashman, T. J.; Augsdörfer, U. H.; Dodgson, N. A.; Sabin, M. A., NURBS with extraordinary points: high-degree, non-uniform, rational subdivision schemes, ACM Transactions on Graphics, 28, 46, 1-46 (2009), 9, July
[22] Reif, U., A unified approach to subdivision algorithms near extraordinary vertices, Comp. Aided Geom. Design, 12, 153-174 (1995) · Zbl 0872.65007
[23] Pan, Q.; Xu, G.; Zhang, Y., Unified method for hybrid subdivision surface design using geometric partial differential equations, (A Special Issue of Solid and Physical Modeling 2013. A Special Issue of Solid and Physical Modeling 2013, Comput. Aided Design, 46 (2014)), 110-119
[24] Zorin, D.; Schröder, P., Subdivision for modeling and animation, (ACM Siggraph Course Notes (2000))
[25] Sabin, M., Recent progress in subdivision: a survey, (Advances in Multiresolution for Geometric Modelling (2005)), 203-230 · Zbl 1065.65038
[26] Catmull, E.; Clark, J., Recursively generated B-spline surfaces on arbitrary topological meshes, Comput. Aided Design, 10, 350-355 (1978)
[29] Cirak, F.; Ortiz, M.; Schröder, P., Subdivision surfaces: a new paradigm for thin shell analysis, Internat. J. Numer. Methods Engrg., 47, 2039-2072 (2000) · Zbl 0983.74063
[30] Cirak, F.; Scott, M. J.; Antonsson, E. K.; Ortiz, M.; Schröder, P., Integrated modeling, finite-element analysis, and engineering design for thin shell structures using subdivision, Comput. Aided Design, 34, 137-148 (2002)
[31] Burkhart, D.; Hamann, B.; Umlauf, G., Iso-geometric finite element analysis based on Catmull-Clark subdivision solids, Comput. Graph. Forum, 29, 1575-1584 (2010)
[32] Grinspun, E.; Krysl, P.; Schröder, P., CHARMS: A simple framework for adaptive simulation, ACM Transactions on Graphics, 21, 281-290 (2002) · Zbl 1396.65043
[33] Scott, M. A.; Thomas, D. C.; Evans, E. J., Isogeometric spline forests, Comput. Methods Appl. Mech. Engrg., 269, 0, 222-264 (2014) · Zbl 1296.65023
[34] Giannelli, C.; Jüttler, B.; Speleers, H., Strongly stable bases for adaptively refined multilevel spline spaces, Adv. Comput. Math., 1-32 (2013)
[35] Liu, L.; Zhang, Y.; Hughes, T. J.R.; Scott, M. A.; Sederberg, T. W., Volumetric T-spline construction using Boolean operations, Eng. Comput. (2014)
[36] Scott, M. A., T-splines as a Design-Through-Analysis technology (2011), The University of Texas at Austin, (Ph.D. thesis)
[37] Reif, U.; Schröder, P., Curvature integrability of subdivision surfaces, Adv. Comput. Math., 14, 2, 157-174 (2001) · Zbl 0986.65009
[38] Arden, G., Approximation properties of subdivision surfaces (2001), University of Washington, (Ph.D. thesis)
[39] Nguyen, T.; Karčiauskas, K.; Peters, J., A comparative study of several classical, discrete differential and isogeometric methods for solving Poisson’s equation on the disk, Axioms, 3, 280-300 (2014) · Zbl 1311.68175
[40] Zore, U.; Jüttler, B.; Kosinka, J., On the linear independence of (truncated) hierarchical subdivision splines, Geometry + Simulation Report, 17 (2014)
[41] Peters, J.; Wu, X., On the local linear independence of generalized subdivision functions, SIAM J. Numer. Anal., 44, 6, 2389-2407 (2006) · Zbl 1131.65020
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.