Open Access
Issue
JNWPU
Volume 43, Number 5, October 2025
Page(s) 1041 - 1054
DOI https://doi.org/10.1051/jnwpu/20254351041
Published online 05 December 2025
  1. BARNHILL R E, RIESENFELD R F. Computer aided geometric design[J]. Salt Lake City: Academic Press, 2014 [Google Scholar]
  2. SHI Fazhong. Computer aided geometric design and non-uniform rational B-splines[M]. Beijing: Higher Education Press, 2013 (in Chinese) [Google Scholar]
  3. FERGUSON J. Multivariable curve interpolation[J]. Journal of the ACM, 1964, 11(2): 221–228. [Article] [Google Scholar]
  4. COONS S A. Surfaces for computer-aided design of space figures[M]. Cambridge: MIT Press, 1967 [Google Scholar]
  5. BEZIER P. Mathematical and practical possibilities of UNISURF[M]. Paris: Academic Press, 1974: 127–152 [Google Scholar]
  6. DE BOOR C. On calculating with B-splines[J]. Journal of Approximation Theory, 1972, 6(1): 50–62. [Article] [CrossRef] [Google Scholar]
  7. COX M G. The numerical evaluation of B-splines[J]. IMA Journal of Applied Mathematics, 1972, 10(2): 134–149. [Article] [Google Scholar]
  8. GORDON W J, RIESENFELD R F. B-spline curves and surfaces[M]. Berlin: Springer-Verlag, 1974: 95–126 [Google Scholar]
  9. PIEGL L, TILLER W. The NURBS book[M]. Berlin: Springer-Verlag, 2012 [Google Scholar]
  10. PRATT M J. Introduction to ISO 10303-the step standard for product data exchange[J]. Journal of Computing and Information Science in Engineering, 2001, 1(1): 102–103. [Article] [Google Scholar]
  11. ZHU L, YAN B, WANG Y, et al. Inspection of blade profile and machining deviation analysis based on sample points optimization and NURBS knot insertion[J]. Thin-Walled Structures, 2021(162): 107540 [Google Scholar]
  12. XU Z, WANG A, HOU F, et al. Three-dimensional reconstruction of industrial parts from a single image[J]. Visual Computing for Industry, Biomedicine, and Art, 2024, 7(1): 7. [Article] [Google Scholar]
  13. 合肥九韶智能科技有限公司. PowerCAD 1.0发布, 开启中国高端CAD新篇章[EB/OL]. (2024-12-31)[2025-01-20]. https://amcax.net/nd.jsp?id=1270 [Google Scholar]
  14. SEDERBERG W T, FINNIGAN T G, LI X, et al. Watertight trimmed nurbs[J]. ACM Transactions on Graphics, 2008, 27(3): 1–8 [Google Scholar]
  15. SEDERBERG W T, ZHENG J, BAKENOV A, et al. T-splines and T-NURCCs[J]. ACM Trans on Graphics, 2003, 22(3): 477–484. [Article] [Google Scholar]
  16. SEDERBERG W T, CARDON L D, FINNIGAN T G, et al. T-spline simplification and local refinement[J]. ACM Trans on Graphics, 2004, 23(3): 276–283. [Article] [Google Scholar]
  17. HU Wenkai, MA Hongyu, LIU Yazui, et al. T-splines a new representation for CAD, CAE and CAM[J]. Journal of Graphics, 2022, 43(6): 1018–1033 (in Chinese) [Google Scholar]
  18. ZHANG J, LI X. Local refinement for analysis-suitable++ T-splines[J]. Computer Methods in Applied Mechanics and Engineering, 2018(342): 32–45 [Google Scholar]
  19. FENG C, TAGUCHI Y. FasTFit: a fast T-spline fitting algorithm[J]. Computer-Aided Design, 2017, 92: 11–21. [Article] [Google Scholar]
  20. CASQUERO H, WEI X, TOSHNIWAL D, et al. Seamless integration of design and Kirchhoff-Love shell analysis using analysis-suitable unstructured T-splines[J]. Computer Methods in Applied Mechanics and Engineering, 2020, 360: 112765. [Article] [Google Scholar]
  21. SCOTT M, LI X, SEDERBERG T, et al. Local refinement of analysis-suitable T-splines[J]. Computer Methods in Applied Mechanics and Engineering, 2012, 213: 206–222 [Google Scholar]
  22. KANG H, LI X. DE Boor-like evaluation algorithm for analysis-suitable T-splines[J]. Graphical Models, 2019, 106: 101042. [Article] [Google Scholar]
  23. LI X, ZHANG J. AS++ T-splines: linear independence and approximation[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 333: 462–474. [Article] [Google Scholar]
  24. LI X, SEDERBERG W T. S-splines: a simple surface solution for IGA and CAD[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 350: 664–678. [Article] [Google Scholar]
  25. LI X, LI X. AS++ T-splines: arbitrary degree, nestedness and approximation[J]. Numerische Mathematik, 2021, 148: 795–816. [Article] [Google Scholar]
  26. LU Yu, WANG Jian, PENG Lihua, et al. Analysis-suitable T-spline fitting for sculptured surface reconstruction[J]. Aeronautical Manufacturing Technology, 2020, 63(19): 96–102 (in Chinese) [Google Scholar]
  27. ARAPAKOPOULOS A, POLICHSHUK R, SEGIZBAYEV Z, et al. Parametric models for marine propellers[J]. Ocean Engineering, 2019, 192: 106595. [Article] [Google Scholar]
  28. YANG J, ZHAO G, WANG W, et al. Surface blending using T-splines in semi-nurbs form[J]. Computer-Aided Design, 2022, 146: 103210. [Article] [Google Scholar]
  29. WANG A, LI L, CHANG H, et al. T-spline surface smoothing based on L-ring neighborhood space angle[J]. Journal of Computational Design and Engineering, 2022, 9(4): 1246–1257. [Article] [Google Scholar]
  30. LU Z, JIANG X, HUO G, et al. A fast T-spline fitting method based on efficient region segmentation[J]. Computational and Applied Mathematics, 2020, 39(2): 55. [Article] [Google Scholar]
  31. XIAO Wenlei, LIU Yazui, OLEKSANDR Zavalnyi, et al. The three-layer data structure of open source kernel T-spline and its algorithms[J]. Journal of Computer-Aided Design & Computer Graphics, 2017, 29(11): 2023–2036 (in Chinese) [Google Scholar]
  32. WANG W, ZHANG Y, DU X, et al. An efficient data structure for calculation of unstructured T-spline surfaces[J]. Visual Computing for Industry, Biomedicine, and Art, 2019, 2(1): 1–9. [Article] [Google Scholar]
  33. SAVIO G, MENEGHELLO R, CONCHERI G. Geometric modeling of lattice structures for additive manufacturing[J]. Rapid Prototyping Journal, 2018, 24(2): 351–360. [Article] [Google Scholar]
  34. XU X, YE X, ZHANG S. A macro BDM H-div mixed finite element on polygonal and polyhedral meshes[J]. Applied Numerical Mathematics, 2024, 206: 283–297. [Article] [Google Scholar]
  35. HE X, WANG R, FENG C, et al. A novel type of boundary extraction method and its statistical improvement for unorganized point clouds based on concurrent delaunay triangular meshes[J]. Sensors, 2023, 23(4): 1915. [Article] [Google Scholar]
  36. LI S X, JERARD R B. 5-axis machining of sculptured surfaces with a flat-end cutter[J]. Computer-Aided Design, 1994, 26(3): 165–178. [Article] [Google Scholar]
  37. MA Z, CHEN Y, LI J, et al. Method of triangular mesh modeling in numerical control machining simulation[J]. Journal of Central South University of Technology, 2010, 17(5): 1021–1027. [Article] [Google Scholar]
  38. XUN G, FENG H. Cutter-workpiece engagement determination for general milling using triangle mesh modeling[J]. Journal of Computational Design and Engineering, 2016, 3(2): 151–160. [Article] [Google Scholar]
  39. XU Chenyang, LI Jingrong, WANG Qinghui, et al. Tool path planning using angle-based flattening for mesh surfaces machining[J]. Journal of Computer-Aided Design & Computer Graphics, 2017, 29(4): 728–733 (in Chinese) [Google Scholar]
  40. XU C, LI J, WANG Q, et al. Contour parallel tool path planning based on conformal parameterisation utilising mapping stretch factors[J]. International Journal of Production Research, 2019, 57(1): 1–15. [Article] [Google Scholar]
  41. LIANG F, KANG C, FANG F. Tool path planning on triangular mesh surfaces based on the shortest boundary path graph[J]. International Journal of Production Research, 2022, 60(9): 2683–2702. [Article] [Google Scholar]
  42. KIM H. Optimum tool path generation for 2.5d direction-parallel milling with incomplete mesh model[J]. Journal of Mechanical Science and Technology, 2010, 24(5): 1019–1027. [Article] [Google Scholar]
  43. KIM H. Tool path generation for contour parallel milling with incomplete mesh model[J]. The International Journal of Advanced Manufacturing Technology, 2010, 48(5/6/7/8): 443–454. [Google Scholar]
  44. GAN W, FU J, SHEN H, et al. A morphing machining strategy for artificial bone[J]. Journal of Zhejiang University-Science A, 2014, 15(3): 157–171. [Article] [Google Scholar]
  45. CHAIKIN G M. An algorithm for high-speed curve generation[J]. Computer Graphics and Image Processing, 1974, 3(4): 346–349. [Article] [Google Scholar]
  46. CATMULL E, CLARK J. Recursively generated b-spline surfaces on arbitrary topological meshes[J]. Computer-Aided Design, 1978, 10(6): 350–355. [Article] [Google Scholar]
  47. DOO D, SABIN M. Behaviour of recursive division surfaces near extraordinary points[J]. Computer-Aided Design, 1978, 10(6): 356–360. [Article] [CrossRef] [Google Scholar]
  48. LOOP C. Smooth subdivision surfaces based on triangles[D]. Salt Lake City: Univrsity of Utah, 1987 [Google Scholar]
  49. DYN N, LEVINE D, GREGORY J A. A butterfly subdivision scheme for surface interpolation with tension control[J]. ACM Trans on Graphics, 1990, 9(2): 160–169. [Article] [Google Scholar]
  50. ZORIN D, SCHRÖDER P, SWELDENS W. Interpolating Subdivision for meshes with arbitrary topology[C]//Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques Louisiana, 1996: 189–192 [Google Scholar]
  51. KOBBELT L. Formula -subdivision[C]//Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, New York, 2000: 103–112 [Google Scholar]
  52. LABSIK U, GREINER G. Interpolatory Formula -subdivision[J]. Computer Graphics Forum, 2000, 19(3): 131–138. [Article] [Google Scholar]
  53. LI G, MA W, BAO H. Interpolatory/SPL radic/2-subdivision surfaces[C]//Geometric Modeling and Processing, Beijing, 2004: 185–194 [Google Scholar]
  54. MÜLLER K, REUSCHE L, FELLNER D. Extended subdivision surfaces: building a bridge between NURBS and Catmull-Clark surfaces[J]. ACM Transactions on Graphics, 2006, 25(2): 268–292. [Article] [Google Scholar]
  55. WEI X, ZHANG J Y, HUGHES J T, et al. Extended truncated hierarchical catmull-clark subdivision[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 299: 316–336. [Article] [Google Scholar]
  56. LI X, CHANG Y. Non-uniform interpolatory subdivision surface[J]. Applied Mathematics and Computation, 2018, 324: 239–253 [Article] [Google Scholar]
  57. LI X, WEI X, ZHANG J Y. Hybrid non-uniform recursive subdivision with improved convergence rates[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 352: 606–624. [Article] [Google Scholar]
  58. NOUR M Y, BARRERA D, LAMNII A, et al. MTH subdivision scheme with sharp and semi-sharp features[J]. Journal of Computational and Applied Mathematics, 2024, 442: 115725. [Article] [Google Scholar]
  59. LUO F, LI X. Higher-degrees Hybrid Non-uniform Subdivision Surfaces[J]. Computer-Aided Design, 2025, 179: 103822. [Article] [Google Scholar]
  60. ZHANG G, LIU X. Surface reconstruction and application of subdivision[C]//2011 International Conference on Control, Automation and Systems Engineering, 2011 [Google Scholar]
  61. YANG Zhifei, SHI Xiquan, WANG Weiming, et al. A parameterized surface reconstruction method based on powell-sabin subdivision[J]. Journal of Computer-Aided Design & Computer Graphics, 2023, 35(12): 1875–1886 (in Chinese) [Google Scholar]
  62. ZHANG Xiangyu, LI Ming, MA Xiqing. Skeleton-driven mesh deformation technology based on subdivision[J]. Journal of Computer Applications, 2015, 35(3): 811–815 (in Chinese) [Google Scholar]
  63. LIN Z, LI Y, DENG C. Interpolating meshes of arbitrary topology by catmull-clark surfaces with energy constraint[J]. The Visual Computer, 2024, 40(9): 6081–6092. [Article] [Google Scholar]
  64. SHEN J, KOSINKA J, SABIN A M, et al. Conversion of trimmed NURBS surfaces to catmull-clark subdivision surfaces[J]. Computer Aided Geometric Design, 2014, 31(7/8): 486–498 [Google Scholar]
  65. SHEN J, KOSINKA J, SABIN M, et al. Converting a CAD model into a non-uniform subdivision surface[J]. Computer Aided Geometric Design, 2016, 48: 17–35. [Article] [Google Scholar]
  66. HUGHES T J R, COTTRELL J A, BAZILEVS Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194: 4135–4195. [Article] [CrossRef] [Google Scholar]
  67. HAO P, WANG Y, TANG H, et al. A NURBS-based degenerated stiffener element for isogeometric static and buckling analysis[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 398: 115245. [Article] [Google Scholar]
  68. CAI S, ZHANG H, ZHANG W. An integrated design approach for simultaneous shape and topology optimization of shell structures[J]. Computer Methods in Applied Mechanics and Engineering, 2023, 415: 116218. [Article] [Google Scholar]
  69. ZHANG K, GUO C, LI Y, et al. Contact analysis for cycloid pinwheel mechanism by isogeometric finite element[J]. Coatings, 2023, 13(12): 2019. [Article] [Google Scholar]
  70. BAZILEVS Y, CALO V M, COTTRELL J A, et al. Isogeometric analysis using T-splines[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5/6/7/8): 229–263 [CrossRef] [Google Scholar]
  71. LIU Z, CHENG J, YANG M, et al. Isogeometric analysis of large thin shell structures based on weak coupling of substructures with unstructured T-splines patches[J]. Advances in Engineering Software, 2019, 135: 102692. [Article] [Google Scholar]
  72. WEI X, LI X, QIAN K, et al. Analysis-suitable unstructured T-splines: Multiple extraordinary points per face[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 391: 114494. [Article] [Google Scholar]
  73. WANG Y, LAN P, LU N, et al. T-spline based isogeometric solid element with locally varying mesh in nonlinear dynamics[J]. Acta Mechanica Sinica, 2023, 40(2): 523222 [Google Scholar]
  74. KANG H, HU W, YONG Z, et al. Isogeometric analysis based on modified loop subdivision surface with improved convergence rates[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 398: 115258. [Article] [Google Scholar]
  75. WANG X, MA W. An extended tuned subdivision scheme with optimal convergence for isogeometric analysis[J]. Computer-Aided Design, 2023, 162: 103544. [Article] [Google Scholar]
  76. LIANG F, KANG C, FANG F. A smooth tool path planning method on nurbs surface based on the shortest boundary geodesic map[J]. Journal of Manufacturing Processes, 2020, 58: 646–658. [Article] [Google Scholar]
  77. WEI J, HOU X, XU G, et al. Modeling and machining of integral impeller based on nurbs curve[J]. The International Journal of Advanced Manufacturing Technology, 2021, 113(7/8): 1–13 [Google Scholar]
  78. HE S, XUAN J, DU W, et al. Spiral tool path generation method in a nurbs parameter space for the ultra-precision diamond turning of freeform surfaces[J]. Journal of Manufacturing Processes, 2020, 60: 340–355. [Article] [Google Scholar]
  79. LIU Y, ZHAO G, HAN P. T-spline surface toolpath generation using watershed-based feature recognition[J]. Applied Sciences, 2020, 10(19): 6790. [Article] [Google Scholar]
  80. GAN W F, FU J Z, SHEN H Y, et al. Five-axis tool path generation in CNC machining of T-spline surfaces[J]. Computer-Aided Design, 2014, 52: 51–63. [Article] [Google Scholar]
  81. ZHAO G, LIU Y, XIAO W, et al. Step-compliant CNC with T-spline enabled toolpath generation capability[J]. The International Journal of Advanced Manufacturing Technology, 2018, 94(5/6/7/8): 1799–1810 [Google Scholar]
  82. ZHAO G, ZAVALNYI O, LIU Y, et al. Prospects for using T-splines for the development of the STEP-NC-based manufacturing of freeform surfaces[J]. The International Journal of Advanced Manufacturing Technology, 2019, 102: 1–16. [Article] [Google Scholar]
  83. ZHANG Z, FENG Y, REN B, et al. Exploratory study of spiral NC tool path generation on triangular mesh based on local subdivision[J]. The International Journal of Advanced Manufacturing Technology, 2016, 83: 835–845. [Article] [Google Scholar]
  84. WANG Q, FENG Y, GAO Y, et al. Smooth fillet-end cutter tool path generation method on triangular-mesh surface based on modified butterfly subdivision[J]. The International Journal of Advanced Manufacturing Technology, 2018, 98: 2831–2847. [Article] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.