Previous Articles    

BASES OF BIQUADRATIC POLYNOMIAL SPLINE SPACES OVER HIERARCHICAL T-MESHES

Fang Deng1, Chao Zeng1, Meng Wu2, Jiansong Deng1   

  1. 1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, China;
    2. School of Mathematics, Hefei University of Technology, Hefei, China
  • Received:2015-01-06 Revised:2016-01-14 Online:2017-01-15 Published:2017-01-15
  • About author:Fang Deng,Email:dengfang@mail.ustc.edu.cn;Chao Zeng,Email:zengchao@mail.ustc.edu.cn;Meng Wu,Email:wumeng@mail.ustc.edu.cn;Jiansong Deng,Email:dengjs@ustc.edu.cn
  • Supported by:

    Supported by the NSF of China (No. 11371341, No.11626253, No. 11601114, No. 11526069), the Anhui Provincial Natural Science Foundation (No. 1608085QA14).

Fang Deng, Chao Zeng, Meng Wu, Jiansong Deng. BASES OF BIQUADRATIC POLYNOMIAL SPLINE SPACES OVER HIERARCHICAL T-MESHES[J]. Journal of Computational Mathematics, 2017, 35(1): 91-120.

Basis functions of biquadratic polynomial spline spaces over hierarchical T-meshes are constructed. The basis functions are all tensor-product B-splines, which are linearly independent, nonnegative and complete. To make basis functions more efficient for geometric modeling, we also give out a new basis with the property of unit partition. Two preliminary applications are given to demonstrate that the new basis is efficient.

CLC Number: 

[1] Y. Bazilevs, L.B. Veiga, J.A. Cottrell, T.J.R. Hughes, and G. Sangalli, Isogeometric analysis:approximation, stability and error estimates for h-refined meshes, Math. Models Methods Appl. Sci., 16(2006), 1031-1090.

[2] Y. Bazilevs, V.M. Calo, J.A. Cottrell, J.A. Evans, T.J.R. Hughes, S. Lipton, M.A. Scott, and T.W. Sederberg, Isogeometric analysis using T-splines, Comput. Meth. Appl. Mech. Engrg., 199(2010), 229-263.

[3] D. Berdinsky, M. Oh, T. Kim, and B. Mourrain, On the problem of instability in the dimension of a spline space over a T-mesh, Comput. Graph., 36(2012), 507-513.

[4] A. Buffa, D. Cho, and G. Sangalli, Linear independence of the T-spline blending functions associated with some particular T-meshes, Comput. Meth. Appl. Mech. Engrg., 199(2010), 1437-1445.

[5] M. Dorfel, B. Juttler, and B. Simeon, Adaptive isogeometric analysis by local h-refinement with T-splines, Comput. Meth. Appl. Mech. Engrg., 199(2010), 264-275.

[6] J. Deng, F. Chen, and Y. Feng, Dimensions of spline spaces over T-meshes, J. Comput. Appl. Math., 194(2006), 267-283.

[7] J. Deng, F. Chen, X. Li, C. Hu, W. Tong, Z. Yang, and Y. Feng, Polynomial splines over hierarchical T-meshes, Graph. Models, 70(2008), 76-86.

[8] J. Deng, F. Chen, and L. Jin, Dimensions of biquadratic spline spaces over T-meshes, J. Comput. Appl. Math., 238(2013), 68-94.

[9] T. Dokken, T. Lyche, and K.F. Pettersen, Polynomial splines over locally refined box-partitions, Comput. Aided Geom. Des., 30(2013), 331-356.

[10] Q. Du, L. T, and X. Zhao, A Convergent Adaptive Finite Element Algorithm for Nonlocal Diffusion and Peridynamic Models, SIAM J. Numer. Anal., 51(2013), 1211-1234.

[11] G. Farin, Curves and Surfaces for CAGD-A Practical Guide, 5th ed., Morgan-Kaufmann, 2002.

[12] M.S. Floater, Parameterization and smooth approximation of surface triangulations, Comput. Aided Geom. Des., 14(1997), 231-250.

[13] D.R. Forsey, and R.H. Bartels, Hierarchical B-spline Refinement, Comput. Graph., 22(1988), 205-212.

[14] C. Giannelli, B. Juttler, and H. Speleers, Thb-splines:The truncated basis for hierarchical splines, Comput. Aided Geom. Des., 29(2012), 485-498.

[15] R. Kraft, Adaptive and linearly independent multilevel B-splines, In Surface Fitting and Multiresolution Methods, Vanderbilt University Press, 2(1997), 209-218.

[16] X. Li, and M.A. Scott, Analysis-suitable T-splines:characterization, refineability, and approximation, Math. Models Mehthods Appl., 24(2014), 1141-1164.

[17] X. Li, J. Zheng, T.W. Sederberg, T.J.R. Hughes, and M.A. Scott, On linear independence of T-spline blending functions, Comput. Aided Geom. Des., 29(2012), 63-76.

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

[19] X. Li, and F. Chen, On the instability in the dimension of splines spaces over T-meshes, Comput. Aided Geom. Des., 28(2011), 420-426.

[20] T.W. Sederberg, J.M. Zheng, A. Bakenov, and A. Nasri, T-splines and T-NURCCs, ACM Trans. Graph., 22(2003), 477-484.

[21] T.W. Sederberg, D.L. Cardon, G.T. Finnigan, N.S. North, J.M. Zheng, and T. Lyche, T-spline simplification and local refinement, ACM Trans. Graph., 23(2004), 276-283.

[22] T.W. Sederberg, G.T. Finnigan, X. Li, and H. Lin, Watertight trimmed NURBS, ACM Trans. Graph., 27(2008).

[23] M.A. Scott, X. Li, T.W. Sederberg, and T.J.R. Hughes, Local refinement of analysis-suitable T-splines, Comput. Meth. Appl. Mech. Engrg., 213(2012), 206-222.

[24] M.A. Scott, M.J. Borden, C.V. Verhoosel, T.W. Sederberg, and T.J.R. Hughes, Isogeometric finite element data structures based on Bzier extraction of T-splines, Int. J. Numer. Methods Engrg., 88(2011), 126-156.

[25] N. Nguyen-Thanh, J. Kiendl, H. Nguyen-Xuan, R. Wchner, K.U. Bletzinger, Y. Bazilevs, and T. Rabczuka, Rotation free isogeometric thin shell analysis using PHT-splines, Comput. Meth. Appl. Mech. Engrg., 200(2011), 3410-3424.

[26] N. Nguyen-Thanh, H. Nguyen-Xuan, S.P.A. Bordas, and T. Rabczuk, Isogeometric finite element analysis using polynomial splines over hierarchical T-meshes, IOP Conf. Ser. Mater. Sci. Eng., 10(2010), doi:10.1088/1757-899X/10/1/012238.

[27] N. Nguyen-Thanh, H. Nguyen-Xuan, S.P.A. Bordas, and T. Rabczuk, Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids, Comput. Meth. Appl. Mech. Engrg., 200(2011), 1892-1908.

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

[29] M. Wu, J. Deng, and F. Chen, The dimension of Spline Spaces with Highest Order Smoothness over Hierarchical T-meshes, Comput. Aided Geom. Des., 30(2013), 20-34.

[30] M. Wu, J.-L. Xu, R.-M. Wang, and Z.-W. Yang, Hierarchical bases of spline spaces with highest order smoothness over hierarchical T-subdivisions, Comput. Aided Geom. Des., 29(2012), 499-509.

[31] P. Wang, J. Xu, J. Deng, and F. Chen, Adaptive isogeometric analysis using rational PHT-splines, Comput. Aided Des., 43(2011), 1438-1448.

[32] J. Wang, Z. Yang, L. Jin, J. Deng, and F. Chen, Parallel and adaptive surface reconstruction based on implicit PHT-splines, Comput. Aided Geom. Des., 28(2011), 463-474.

[33] C. Zeng, F. Deng, J.-S. Deng, Bicubic hierarchical B-splines:Dimensions, completeness, and bases, Comput. Aided Geom. Des., 38(2015), 1-23.

[34] C. Zeng, F. Deng, X. Li, J.-S. Deng, Dimensions of biquadratic and bicubic spline spaces over hierarchical T-meshes, J. Comput. Appl. Math., 287(2015), 162-178.
[1] Hongliang Li, Pingbing Ming, Zhongci Shi. THE QUADRATIC SPECHT TRIANGLE [J]. Journal of Computational Mathematics, 2020, 38(1): 103-124.
[2] M.Y. Xia, G.H. Zhang, G.L. Dai, C.H. Chan. STABLE SOLUTION OF TIME DOMAIN INTEGRAL EQUATION METHODSUSING QUADRATIC B-SPLINE TEMPORAL BASIS FUNCTIONS [J]. Journal of Computational Mathematics, 2007, 25(3): 374-384.
Viewed
Full text


Abstract