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Xin Li   

  1. School of Mathematical Science, University of Science and Technology of China, Hefei, Anhui, China
  • Received:2014-02-18 Revised:2015-04-08 Online:2015-07-15 Published:2015-07-15
  • Supported by:

    This work was supported by the Chinese Universities Scientific Fund, the NSF of China (No.11031007, No.60903148), SRF for ROCS SE, and the CAS Startup Scientific Research Foundation and NBRPC 2011CB302400.

Xin Li. SOME PROPERTIES FOR ANALYSIS-SUITABLE T-SPLINES[J]. Journal of Computational Mathematics, 2015, 33(4): 428-442.

Analysis-suitable T-splines (AS T-splines) are a mildly topological restricted subset of T-splines which are linear independent regardless of knot values [1-3]. The present paper provides some more iso-geometric analysis (IGA) oriented properties for AS T-splines and generalizes them to arbitrary topology AS T-splines. First, we prove that the blending functions for analysis-suitable T-splines are locally linear independent, which is the key property for localized multi-resolution and linear independence for non-tensorproduct domain. And then, we prove that the number of T-spline control points contribute each Bézier element is optimal, which is very important to obtain a bound for the number of non zero entries in the mass and stiffness matrices for IGA with T-splines. Moreover, it is found that the elegant labeling tool for B-splines, blossom, can also be applied for analysis-suitable T-splines.

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[1] X. Li, J. Zheng, T. W. Sederberg, T.J.R. Hughes, M.A. Scott, On the linear independence of T-splines blending functions, Computer Aided Geometric Design, 29 (2012), 63-76.

[2] L.B. Veiga, A. Buffa, D.C.G. Sangalli, Analysis-suitable T-splines are dual-compatible, Comput. Methods Appl. Mech. Engrg, (2012), 249-252.

[3] J. Zhang, X. Li, On the linear independence and partition of unity of arbitrary degree analysissuitable T-splines, Communications in Mathematics and Statistics, submitted.

[4] T.W. Sederberg, J. Zheng, A. Bakenov, A. Nasri, T-splines and T-NURCCSs, ACM Transactions on Graphics, 22:3 (2003), 477-484.

[5] T.W. Sederberg, D.L. Cardon, G.T. Finnigan, N.S. North, J. Zheng, T. Lyche, T-spline simplification and local refinement, ACM Transactions on Graphics, 23 (2004), 276-283.

[6] H. Ipson, T-spline merging, MSc. Thesis, Brigham Young University (April 2005).

[7] T.W. Sederberg, G.T. Finnigan, X. Li, H. Lin, H. Ipson, Watertight trimmed NURBS, ACM Transactions on Graphics, 27, (2008), Article no. 79.

[8] M.A. Scott, X. Li, T.W. Sederberg, T.J.R. Hughes, Local refinement of analysis-suitable T-splines, Computer Methods in Applied Mechanics and Engineering, 213-216, (2012), 206-222.

[9] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, (2005), 4135-4195.

[10] J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, Wiley, Chichester, 2009.

[11] J.A. Cottrell, A. Reali, Y. Bazilevs, T.J.R. Hughes, Isogeometric analysis of structural vibrations, Computer Methods in Applied Mechanics and Engineering, 195, (2006), 5257-5296.

[12] J.A. Evans, Y. Bazilevs, I. Babuška, T.J.R. Hughes, n-widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method, Computer Methods in Applied Mechanics and Engineering, 198, (2009), 1726-1741.

[13] S. Lipton, J.A. Evans, Y. Bazilevs, T. Elguedj, T.J.R. Hughes, Robustness of isogeometric structural discretizations under severe mesh distortion, Computer Methods in Applied Mechanics and Engineering, 199, (2010), 357-373.

[14] Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott, T.W. Sederberg, Isogeometric analysis using T-splines, Computer Methods in Applied Mechanics and Engineering, 199, (2010), 229-263.

[15] M.J. Borden, C.V. Verhoosel, M.A. Scott, T.J.R. Hughes, C.M. Landis, A phase-field description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering, 217-220, (2012), 77-95.

[16] D.J. Benson, Y. Bazilevs, E. De Luycker, M. C. Hsu, M.A. Scott, T.J.R. Hughes, T. Belytschko, A generalized finite element formulation for arbitrary basis functions: from isogeometric analysis to XFEM, International Journal for Numerical Methods in Engineering, 83 (2010), 765-785.

[17] A. Buffa, D. Cho, G. Sangalli, Linear independence of the T-spline blending functions associated with some particular T-meshes, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1437-1445.

[18] X. Li, M.A. Scott, Analysis-suitable T-splines: Characterization, refinablility and approximation, Mathematical Models and Methods in Applied Sciences, 24 (2014), 1141C1164.

[19] L.B. Veiga, A. Buffa, G. Sangalli, R. Vazquez, Analysis-suitable T-splines of arbitrary degree: definition and properties, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1979-2003.

[20] L. Ramshaw, Blossoming: A connect-the-dots approach to splines, Technical Report 19, Digital Equipment Corporation, Systems Research Center.

[21] L. Ramshaw, Blossoms are polar forms, Computer Aided Geometric Design, (1989), 323-358.

[22] Z. Chen, Y. Feng, The blossom appproach to the dimension of the bivariate spline space, Journal of Computational Mathematics, 18 (2000), 501-514.

[23] J. Deng, F. Chen, X. Li, C. Hu, W. Tong, Z. Yang, Y. Feng, Polynomial splines over hierarchical T-meshes, Graphical Models, 74 (2008), 76-86.

[24] X. Li, J. Deng, F. Chen, Surface modeling with polynomial splines over hierarchical T-meshes, The Visual Computer, 23 (2007), 1027-1033.

[25] X. Li, J. Deng, F. Chen, Polynomial splines over general T-meshes, The Visual Computer, 26 (2010), 277-286.

[26] Z. Huang, J. Deng, Y. Feng, F. Chen, New proof of dimension formula of spline spaces over T-meshes via smoothing cofactors, Journal of Computational Mathematics, 24 (2006), 501-514.

[27] D. Forsey, R. Bartels, Hierarchical b-spline refinement, Comput. Graph, 22 (1988), 205-212.

[28] A.-V. Vuong, C. Giannelli, B. Jüttler, B. Simeon, A hierarchical approach to adaptive local refinement in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 200 (2011), 3554-3567.

[29] T. Dokken, T. Lyche, K.F. Pettersen, Polynomial splines over locally refined box-partitions, CAGD 30, (2013), 331-356.

[30] G.T. Finnigan, Arbitrary degree T-splines, MsC. Thesis, Brigham Young University (August 2008).
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