Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-10T14:13:17.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Evaluation Algorithm of PHT-Spline Surfaces

Published online by Cambridge University Press:  12 September 2017

Zhihua Wang ,
Falai Chen  and
Jiansong Deng
Zhihua Wang
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P. R. China School of Mathematics and Computer Science, Anqing Normal University, Anqing 246011, P. R. China
Falai Chen*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P. R. China
Jiansong Deng
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, P. R. China
*
*Corresponding author. Email address:chenfl@ustc.edu.cn (F. L. Chen)
Get access

Abstract

PHT-splines are a type of polynomial splines over hierarchical T-meshes which posses perfect local refinement property. This property makes PHT-splines useful in geometric modeling and iso-geometric analysis. Current implementation of PHT-splines stores the basis functions in Bézier forms, which saves some computational costs but consumes a lot of memories. In this paper, we propose a de Boor like algorithm to evaluate PHT-splines provided that only the information about the control coefficients and the hierarchical mesh structure is given. The basic idea is to represent a PHT-spline locally in a tensor product B-spline, and then apply the de-Boor algorithm to evaluate the PHT-spline at a certain parameter pair. We perform analysis about computational complexity and memory costs. The results show that our algorithm takes about the same order of computational costs while requires much less amount of memory compared with the Bézier representations. We give an example to illustrate the effectiveness of our algorithm.

Keywords

PHT-splineshierarchical T-meshknot insertevaluation algorithm
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 760 - 774
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Forsey, D. R. and Bartels, R. H., Hierarchical B-spline refinement, Comp. Graph., 22(4) (1988), pp. 205–212.CrossRefGoogle Scholar
[2] Forsey, D. R. and Bartels, R. H., Surface fitting with hierarchical splines, ACM T. Graphic., 14(2) (1995), pp. 134–161.Google Scholar
[3] Sederberg, T., Zheng, J., Bakenov, A. and Nasri, A., T-splines and T-NURCCSs, ACM T. Graphic., 22(3) (2003), pp. 477–484.Google Scholar
[4] Sederberg, T., Cardon, D., Finnigan, G., North, N., Zheng, J. and Lyche, T., T-spline simplification and local refinement, ACM T. Graphic., 23(3) (2004), pp. 276–283.CrossRefGoogle Scholar
[5] Deng, J. S., Chen, F. L., Li, X., Hu, C. Q., Tong, W. H., Yang, Z. W. and Feng, Y. Y., Polynomial splines over hierarchical T-meshes, Graph. Models, 70 (2008), pp. 76–86.Google Scholar
[6] Li, X., Deng, J. and Chen, F., Surface modeling with polynomial splines over hierarchical T-meshes, Visual Comput., 23(12) (2007), pp. 1027–1033.CrossRefGoogle Scholar
[7] Li, X., Deng, J. S. and Chen, F. L., Polynomial splines over general T-meshes, Visual Comput., 26(4) (2010), pp. 277–286.Google Scholar
[8] Li, X. and Scott, M. A., ANALYSIS-SUITABLE T-SPLINES: CHARACTERIZATION, REFINEABILITY, and APPROXIMATION, Math. Mod. Meth. Appl. S., 24(06) (2014), pp. 1141–1164.Google Scholar
[9] Scott, M. A., Li, X., Sederberg, T. W. and Hughes, T. J. R., Local refinement of analysis-suitable T-splines, Comput. Method. Appl. M., 4 (2012), pp. 206–222.CrossRefGoogle Scholar
[10] Li, X., Zheng, J. M., Sederberg, T. W., Hughes, T. J. R. and Scott, M. A., On the linear independence of T-spline blending functions, Comput. Aided Geom. D., 29 (1) (2012), pp. 63–76.CrossRefGoogle Scholar
[11] Giannelli, C., Juttler, B., Speleers, H., THB-splines: the truncated basis for hierarchical splines, Comput. Aided Geom. D., 29 (2012), pp. 485–498.Google Scholar
[12] Dokken, T., Lyche, T., Pettersen, K. F., Polynomial splines over locally refined box-partitions, Comput. Aided Geom. D., 30 (2013), pp. 331–356.Google Scholar
[13] Pan, M., Tong, W. and Chen, F., Compact implicit surface reconstruction via low-rank tensor approximation, Computer-Aided Design, 78 (2016), pp. 158–167.Google Scholar
[14] Tian, L., Chen, F. and Du, Q., Adaptive finite element methods for elliptic equations over hierarchical T-meshes, J. Comput. Appl. Math., 236 (2011), pp. 878–891.Google Scholar
[15] Nguyen-Thanh, N., Kiendl, J., Nguyen-Xuan, H., Wuchner, R., Bletzinger, K. U., Bazilevs, Y. and Rabczuk, T., Rotation free isogeometric thin shell analysis using PHT-splines, Comput. Method. Appl. M., 2011, pp. 3410–3424.Google Scholar
[16] Nguyen-Thanh, N., Nguyen-Xuan, H., Bordas, S. P. A. and Rabczuk, T., Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids, Comput. Method. Appl. M., 200 (2011), pp. 1892–1908.Google Scholar
[17] Nguyen-Thanh, N., Muthu, J., Zhuang, X. and Rabczuk, T., An adaptive three dimensional RHT-spline formulation in linear elasto-statics and elasto-dynamics, Comput. Mech., 3(2) (2014), pp. 369–385.CrossRefGoogle Scholar
[18] Nguyen-Thanh, N., Valizadeh, N., Nguyen, M. N., Nguyen-Xuan, H., Zhuang, X., Areias, P., Zi, G., Bazilevs, Y., De Lorenzis, L. and Rabczuk, T., An extended isogeometric thin shell analysis based on Kirchhoff-Love theory, Comput. Method. Appl. M., 284 2015, pp. 265–291.Google Scholar
[19] Wang, P., Xu, J. L., Deng, J. S. and Chen, F. L., Adaptive isogeometric analysis using rational PHT-splines, Computer-Aided Design, 43 (2011), pp. 1438–1448.Google Scholar
[20] Wang, J., Yang, Z., Jin, L., Deng, J. and Chen, F., Parallel and adaptive surface reconstruction based on implicit PHT-splines, Comput. Aided Geom. D., 28(8) (2011), pp. 463–474.Google Scholar
[21] Deng, J., Chen, F. and Feng, Y., Dimensions of spline spaces over T-meshes, J. Comput. Appl. Math., 194 (2006), pp. 267–283.Google Scholar
[22] Mourrain, B., On the dimension of spline spaces on planar T-meshes, Math. Comput., 83 (2014), pp. 847–871.Google Scholar
[23] Li, X. and Chen, F., On the instability in the dimension of splines spaces over T-meshes, Comput. Aided Geom. D., 28 (2011), pp. 420–426.Google Scholar
[24] Deng, J., Chen, F. and Jin, L., Dimensions of biquadratic spline spaces over T-meshes, J. Comput. Appl. Math., 238 (2013), pp. 68–94.Google Scholar
[25] Wu, M., Deng, J. and Chen, F., Dimension of spline spaces with highest order smoothness over hierarchical T-meshes, Comput. Aided Geom. D., 30 (2013), pp. 20–34.CrossRefGoogle Scholar
[26] Berdinsky, D., Oh, M., Kim, T. and Mourrain, B., On the problem of instability in the dimension of a spline space over a T-mesh, Comput. Graph., 36 (2012), pp. 507–513.Google Scholar
[27] Zeng, C., Deng, F., Li, X. and Deng, J., Dimensions of biquadratic and bicubic spline spaces over hierarchical T-meshes, J. Comput. Appl. Math., 287 (2015), pp. 162–178.Google Scholar
[28] Zeng, C., Deng, F. and Deng, J., Bicubic hierarchical B-splines: Dimensions, completeness, and bases, Comput. Aided Geom. D., 38 (2015), pp. 1–23.Google Scholar
[29] Farin, G., Curves and Surfaces for CAGD–A Practical Guide, 5th ed., Morgan Kaufmann Publishers, 2002.Google Scholar