A survey on the local refinable splines

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• 作者：Xin Li ; FaLai Chen ; HongMei Kang ; JianSong Deng
• 关键词：geometric modeling ; isogeometric analysis ; B ; splines ; hierarchical B ; splines ; T ; splines ; polynomial splines over hierarchical T ; meshes (PHT ; splines) ; locally refined (LR) B ; splines ; local refinement
• 刊名：SCIENCE CHINA Mathematics
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：59
• 期：4
• 页码：617-644
• 全文大小：1,827 KB
• 参考文献：1.Ainsworth M, Oden J T. A posteriori error estimation in finite element analysis. Comput Methods Appl Engrg, 1997, 142: 1–88MathSciNet CrossRef MATH
  2.Babu˜ska I, Vogelius M. Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer Math, 1984, 44: 75–102MathSciNet CrossRef
  3.Bazilevs Y, Calo V M, Cottrell J A, et al. Isogeometric analysis using T-splines. Comput Methods Appl Mech Engrg, 2010, 199: 229–263MathSciNet CrossRef MATH
  4.Bazilevs Y, Hsu M C, Scott M A. Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Comput Methods Appl Mech Engrg, 2012, 249: 28–41MathSciNet CrossRef
  5.Berdinskya D, Oh M, Kim T, et al. On the problem of instability in the dimension of a spline space over a T-mesh. Comput Graph, 2012, 36: 507–513CrossRef
  6.Borden M J, Scott M A, Verhoosel C V, et al. A phase-field description of dynamic brittle fracture. Comput Methods Appl Mech Engrg, 2012, 217: 77–95MathSciNet CrossRef MATH
  7.Bressan A. Some properties of LR-splines. Comput Aided Geom Design, 2013, 30: 778–794MathSciNet CrossRef MATH
  8.Buffa A, Cho D, Kumar M. Characterization of T-splines with reduced continuity order on T-meshes. Comput Methods Appl Mech Engrg, 2012, 201: 112–126MathSciNet CrossRef MATH
  9.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, 2010, 199: 1437–1445MathSciNet CrossRef MATH
  10.Cardon D L. T-spline simplification. Master’s thesis. Provo: Brigham Young University, 2007
  11.Cottrell J A, Hughes T J R, Reali A. Studies of refinement and continuity in isogeometric analysis. Comput Methods Appl Mech Engrg, 2007, 196: 4160–4183MathSciNet CrossRef MATH
  12.Cottrell J A, Hughes T J R, Bazilevs Y. Isogeometric Analysis: Toward Integration of CAD and FEA. Chichester: Wiley, 2009CrossRef
  13.Cottrell J A, Reali A, Bazilevs Y, et al. Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Engrg, 2006, 195: 5257–5296MathSciNet CrossRef MATH
  14.Deng J, Chen F, Feng Y. Dimensions of spline spaces over T-meshes. J Comput Appl Math, 2006, 194: 267–283MathSciNet CrossRef MATH
  15.Deng J, Chen F, Li X, et al. Polynomial splines over hierarchical T-meshes. Graph Models, 2008, 74: 76–86CrossRef
  16.Deng J, Chen F, Jin L. Dimensions of biquadratic spline spaces over T-meshes. J Comput Appl Math, 2013, 238: 68–94MathSciNet CrossRef MATH
  17.Dokken T, Lyche T, Pettersen K F. Polynomial splines over locally refined box-partitions. Comput Aided Geom Design, 2013, 30: 331–356MathSciNet CrossRef MATH
  18.Dörfel M, Jüttler B, Simeon B. T-spline finite element method for the analysis of shell structures. Internat J Numer Methods Engrg, 2009, 80: 507–536MathSciNet CrossRef
  19.Dörfel M, Jüttler B, Simeon B. Adaptive isogeometric analysis by local h-refinement with T-splines. Comput Methods Appl Mech Engrg, 2009, 199: 264–275MathSciNet CrossRef MATH
  20.Evans E J, Scott M A, Li X, et al. Hierarchical analysis-suitable T-splines: Formulation, Bézier extraction, and application as an adaptive basis for isogeometric analysis. Compu Methods Appl Mech Engrg, 2015, 284: 1–20MathSciNet CrossRef
  21.Evans J A, Bazilevs Y, Babuška I, et al. n-widths, sup-infs, and optimality ratios for the k-version of the isogeometric finite element method. Comput Methods Appl Mech Engrg, 2009, 198: 1726–1741MathSciNet CrossRef MATH
  22.Farin G E. Curves and Surfaces for CAGD, A Practical Guide. 5th ed. San Francisco: Morgan Kaufmann Publishers, 1999
  23.Farin G E. NURBS Curves and Surfaces: From Projective Geometry to Practical Use. 4th ed. Natick: AK Peters Ltd, 2002
  24.Feichtinger R, Fuchs M, Jüttler B, et al. Dual evolution of planar parametric spline curves and T-spline level sets. Comput Aided Design, 2008, 40: 13–24CrossRef MATH
  25.Finnigan G T. Arbitrary degree T-splines. Master’s thesis. Provo: Brigham Young University, 2008
  26.Forsey D, Bartels R. Hierarchical B-spline refinement. ACM SIGGRAPH Comput Graph, 1988, 22: 205–212CrossRef
  27.Forsey D, Bartels R. Surface fitting with hierarchical splines. ACM Trans Graph, 1995, 14: 134–161CrossRef
  28.Giannelli C, Jüttler B. Bases and dimensions of bivariate hierarchical tensor-product splines. J Comput Appl Math, 2013, 239: 162–178MathSciNet CrossRef MATH
  29.Giannelli C, Jüttler B, Speleers H. THB-splines: The truncated basis for hierarchical splines. Comput Aided Geom Design, 2012, 29: 485–498MathSciNet CrossRef MATH
  30.Greiner G, Hormann K. Interpolating and approximating scattered 3D-data with hierarchical tensor product Bsplines. In: Méhauté A L, Rabut C, Schumaker L L, eds. Surface Fitting and Multiresolution Methods. Innovations in Applied Mathematics. Nashville: Vanderbilt University Press, 1997, 163–172
  31.He Y, Wang K, Wang H, et al. Manifold T-spline. In: Geometric Modeling and Processing, vol. 4077. Lecture Notes in Computer Science. Berlin-Heidelberg: Springer, 2006, 409–422
  32.Huang H. Heterogeneous object scalar modeling and visualization using trivariate T-splines. PhD thesis. Stony Brook: Stony Brook University, 2010
  33.Huang Z, Deng J, Feng Y, et al. New proof of dimension formula of spline spaces over T-meshes via smoothing cofactors. J Comput Math, 2006, 24: 501–514MathSciNet MATH
  34.Huang Z, Deng J, Li X. Dimensions of spline spaces over general T-meshes. J Univ Sci Technol China, 2006, 36: 573–581MathSciNet
  35.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, 2005, 194: 4135–4195MathSciNet CrossRef MATH
  36.Ipson H. T-spline Merging. Master’s thesis. Provo: Brigham Young University, 2006
  37.Jin L. A Lower Bound on the Dimension of Bicubic Spline Spaces over T-meshes. In: 2010 International Forum on Information Technology and Applications, vol. 1. New York: IEEE, 2010, 109–112CrossRef
  38.Kang H, Chen F, Deng J. Modified T-splines. Comput Aided Geom Design, 2013, 30: 827–843MathSciNet CrossRef MATH
  39.Kang H, Chen F, Deng J. Hierarchical B-splines on regular triangular partitions. Graph Models, 2014, 76: 289–300CrossRef
  40.Kiss G, Giannelli C, Jüttler B. Algorithms and data structures for truncated hierarchical B-splines. In: Mathematical Methods for Curves and Surfaces. Berlin-Heidelberg: Springer, 2014, 304–323CrossRef
  41.Kraft R. Adaptive and linearly independent multilevel B-splines. In: Méhauté A L, Rabut C, Schumaker L L, eds. Surface Fitting and Multiresolution Methods. Innovations in Applied Mathematics. Nashville: Vanderbilt University Press, 1997, 209–218
  42.Lang F-G, Xu X-P. Dimension formulae for tensor-product spline spaces with homogeneous boundary conditions over regular T-meshes. Thai J Math, 2012, 10: 693–701MathSciNet MATH
  43.Li C J, Wang R H, Zhang F. Improvement on the dimensions of spline spaces on T-mesh. J Inf Comput Sci, 2006, 3: 235–244
  44.Li T. On the dimenson of spline space over T-meshes. Master’s thesis. Dalian: Dalian University of Technology, 2007
  45.Li W-C, Ray N, Lévy B. Automatic and interactive mesh to T-spline conversion. In: Proceedings of the fourth Eurographics symposium on Geometry processing. Switzerland: Eurographics Association, 2006, 191–200
  46.Li X. T-splines and spline over T-meshes. PhD thesis. Hefei: University of Science and Technology of China, 2008
  47.Li X, Chen F. On the instability in the dimension of spline space over particular T-meshes. Comput Aided Geom Design, 2011, 28: 420–426MathSciNet CrossRef MATH
  48.Li X, Deng J. On the dimension of splines spaces over T-meshes with smoothing cofactor-conformality method. Comput Aided Geom Design, 2015, doi:10.1016/j.cagd.2015.12.002
  49.Li X, Deng J, Chen F. Dimensions of spline spaces over 3D hierarchical T-meshes. J Inf Comput Sci, 2006, 3: 487–501
  50.Li X, Deng J, Chen F. Surface modeling with polynomial splines over hierarchical T-meshes. Vis Comput, 2007, 23: 1027–1033CrossRef
  51.Li X, Deng J, Chen F. Polynomial splines over general T-meshes. Vis Comput, 2010, 26: 277–286CrossRef
  52.Li X, Scott M A. Analysis-suitable T-splines: Characterization, refinablility and approximation. Math Models Methods Appl Sci, 2014, 24: 1141–1164MathSciNet CrossRef MATH
  53.Li X, Zheng J, Sederberg T W, et al. On the linear independence of T-splines blending functions. Comput Aided Geom Design, 2012, 29: 63–76MathSciNet CrossRef MATH
  54.Lipton S, Evans J A, Bazilevs Y, et al. Robustness of isogeometric structural discretizations under severe mesh distortion. Comput Methods Appl Mech Engrg, 2010, 199: 357–373MathSciNet CrossRef MATH
  55.Myles A, Pietroni N, Kovacs D, et al. Feature-aligned T-meshes. ACM Trans Graph, 2010, 29: 117CrossRef
  56.Mourrain B. On the dimension of spline spaces on planar T-meshes. Math Comp, 2014, 83: 847–871MathSciNet CrossRef MATH
  57.Nasri A, Sinno K, Zheng J. Local T-spline surface skinning. Vis Comput, 2012, 28: 787–797CrossRef
  58.Nguyen-Thanh N, Kiendl J, Nguyen-Xuan H, et al. Rotation free isogeometric thin shell analysis using PHT-splines. Comput Methods Appl Mech Engrg, 2011, 200: 3410–3424MathSciNet CrossRef MATH
  59.Nguyen-Thanh N, Muthu J, Zhuang X, et al. An adaptive three-dimensional RHT-splines formulation in linear elasto-statics and elasto-dynamics. Comput Mech, 2013, 53: 369–385MathSciNet CrossRef MATH
  60.Nguyen-Thanh N, Nguyen-Xuan H, Bordas S P A, et al. Isogeometric analysis using polynomial splines over hierarchical T-meshes for two-dimensional elastic solids. Comput Methods Appl Mech Engrg, 2011, 200: 1892–1908MathSciNet CrossRef MATH
  61.Peng X, Tang Y. Automatic reconstruction of T-spline surfaces. J Image Graph, 2010, 15: 1818–1825
  62.Schillinger D, Dedé L, Scott M A, et al. An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline cad surfaces. Comput Methods Appl Mech Engrg, 2012, 249: 116–150MathSciNet CrossRef
  63.Schumaker L L, Wang L. Spline spaces on tr-meshes with hanging vertices. Numer Math, 2011, 118: 531–548MathSciNet CrossRef MATH
  64.Schumaker L L, Wang L. Approximation power of polynomial splines on T-meshes. Comput Aided Geom Design, 2012, 29: 599–612MathSciNet CrossRef MATH
  65.Schumaker L L, Wang L. Splines on triangulations with hanging vertices. Constr Approx, 2012, 36: 487–511MathSciNet CrossRef MATH
  66.Schumaker L L, Wang L. On hermite interpolation with polynomial splines on T-meshes. J Comput Appl Math, 2013, 240: 42–50MathSciNet CrossRef MATH
  67.Scott M A, Borden M J, Verhoosel C V, et al. Isogeometric finite element data structures based on Bézier extraction of T-splines. Internat J Numer Methods Engrg, 2011, 88: 126–156MathSciNet CrossRef MATH
  68.Scott M A, Li X, Sederberg T W, et al. Local refinement of analysis-suitable T-splines. Comput Methods Appl Mech Engrg, 2012, 213: 206–222MathSciNet CrossRef MATH
  69.Scott M, Simpson R, Evans J, et al. Isogeometric boundary element analysis using unstructured T-splines. Comput Methods Appl Mech Engrg, 2013, 254: 197–221MathSciNet CrossRef MATH
  70.Sederberg T W, Cardon D L, Finnigan G T, et al. T-spline simplification and local refinement. ACM Trans Graph, 2004, 23: 276–283CrossRef
  71.Sederberg T W, Finnigan G T, Li X, et al. Watertight trimmed NURBS. ACM Trans Graph, 2008, 27: 79CrossRef
  72.Sederberg T W, Zheng J, Bakenov A, et al. T-splines and T-NURCCSs. ACM Trans Graph, 2003, 22: 477–484CrossRef
  73.Sederberg T W, Zheng J, Sewell D, et al. Non-uniform recursive subdivision surfaces. Proce Siggraph, 1999, 43: 387–394
  74.Song W, Yang X. Freeform deformation with weighted T-splin. Vis Comput, 2005, 21: 139–151MathSciNet CrossRef
  75.Tang Y, Li Y, Chen Z, et al. Implicit T-spline surface reconstruction to achieve closure. J Comput Aided Design Comput Graph, 2011, 23: 270–275
  76.Tian L, Chen F, Du Q. Adaptive finite element methods for elliptic equations over hierarchical T-meshes. J Comput Appl Math, 2011, 236: 878–891
  77.Tong W, Feng Y, Chen F. A surface reconstruction based on implicit T-spline surfaces. J Comput Aided Design Comput Graph, 2006, 18: 358–365
  78. T-splines Inc. http://​www.​tsplines.​com/​rhino, 2011
  79.Veiga L B, Buffa A, Sangalli D C G. Isogeometric analysis using T-splines on two-patch geometries. Comput Methods Appl Mech Engrg, 2011, 200: 1787–1803MathSciNet CrossRef MATH
  80.Veiga L B, Buffa A, Sangalli D C G. Analysis-suitable T-splines are dual-compatible. Comput Methods Appl Mech Engrg, 2012, 249: 42–51MathSciNet CrossRef
  81.Veiga L B, Buffa A, Sangalli D C G, et al. Analysis-suitable T-splines of arbitrary degree: Definition and properties. Math Models Methods Appl Sci, 2013, 23: 1979–2003MathSciNet CrossRef MATH
  82.Verfürth R. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. New York: John Wiley & Sons Inc, 1996MATH
  83.Verhoosel C V, Scott M A, Borden M J, et al. Discretization of higher-order gradient damage models using isogeometric finite elements. In: ICES report 11-12, The Institute for Computational Engineering and Sciences. Texas: The University of Texas at Austin, 2011, 89–120
  84.Verhoosel C V, Scott M A, de Borst R, et al. An isogeometric approach to cohesive zone modeling. Internat J Numer Methods Engrg, 2011, 87: 336–360MathSciNet CrossRef MATH
  85.Verhoosel C V, Scott M A, Hughes T J R, et al. An isogeometric analysis approach to gradient damage models. Internat J Numer Methods Engrg, 2011, 86: 115–134MathSciNet CrossRef MATH
  86.Villamizar N, Mourrain B. Bounds on the dimension of splines spaces. Effective Methods Algebra Geom, http://​www.​ math.kth.se/mega2011/MEGA/Nelly.pdf, 2011MATH
  87.Vuong A-V, Giannelli C, Jüttler B, et al. A hierarchical approach to adaptive local refinement in isogeometric analysis. Comput Methods Appl Mech Engrg, 2011, 200: 3554–3567MathSciNet CrossRef MATH
  88.Wang H, He Y, Li X, et al. Geometry-aware domain decomposition for T-spline based manifold modeling. Comput Graph, 2009, 33: 359–368CrossRef
  89.Wang J, Yang Z, Jin L, et al. Adaptive surface reconstruction based on implicit PHT-splines. In: SPM’ 10 Proceedings of the 14th ACM. Symposium on Solid and Physical Modeling. New York: ACM, 2010, 101–110CrossRef
  90.Wang L. Trivariate polynomial splines on 3D T-meshes. PhD thesis. Nashville: The Vanderbilt University, 2012
  91.Wang P, Xu J, Deng J, et al. Adaptive isogeometrican alysis using rational PHT-splines. Comput Aided Design, 2011, 43: 1438–1448CrossRef
  92.Wang W, Zhang Y, Lie L, et al. Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology. Comput Aided Design, 2013, 45: 351–360MathSciNet CrossRef
  93.Wang W, Zhang Y, Xu G, et al. Converting an unstructured quadrilateral/hexahedral mesh to a rational T-spline. Comput Mech, 2012, 50: 65–84MathSciNet CrossRef MATH
  94.Wang Y, Zheng J. Curvature-guided adaptive T-spline surface fitting. Compu Aided Geom Design, 2013, 45: 1095–1107MathSciNet CrossRef
  95.Wang Y, Zheng J, Seah H S. Conversion between T-splines and hierarchical B-splines. In: Proceedings of the 8th IASTED International Conference Computer Graphics and Imaging. Calgary: Acta Press, 2005, 8–13
  96.Weller F, Hagen H. Tensor product spline spaces with knot segments. In: Daehlen M, Lyche T, Schumaker L, eds. Mathematical Methods for Curves and Surfaces. Nashville: Vanderbilt University Press, 1995, 563–572
  97.Weng B, Pan R J, Yao Z Q, et al. Watermarking T-spline surfaces. In: 11th IEEE International Conference on Communication Technology. New York: IEEE, 2008: 773–776
  98.Weng B, Pan R J, Yao Z Q, et al. A fragile watermarking algorithm for T-spline surfaces. In: IEEE Youth Conference on Information. Computing and Telecommunication. New York: IEEE, 2009, 546–549CrossRef
  99.Weng B, Pan R J, Yao Z Q, et al. Robust watermarking for T-spline surfaces based on nonuniform B-spline wavelets transformation. In: IEEE Youth Conference on Information Computing and Telecommunications. New York: IEEE, 2010, 335–338CrossRef
  100.Wu M, Deng J, Chen F. The dimension of spline spaces with highest order smoothness over hierarchical T-meshes. Comput Aided Geom Design, 2013, 30: 20–34MathSciNet CrossRef MATH
  101.Yang H, Fuchs M, Juettler B, et al. Evolution of T-spline level-sets for 2D/3D shape reconstruction. Tech Rep, http://​www.​industrial-geometry.​at/​events/​strobl2006/​2006-06huaiping.​pdf, 2006
  102.Yang H, Fuchs M, Jüttler B, et al. Evolution of T-spline level sets with distance field constraints for geometry reconstruction and image segmentation. Proceedings of the IEEE International Conference on Shape Modeling and Applications. Linz: Johannes Kepler University, 2006, 247–252
  103.Yang H, Jüttler B. 3D shape metamorphosis based on T-spline level sets. Vis Comput, 2007, 23: 1015–1025CrossRef
  104.Yang H, Jüttler B. Evolution of T-spline level sets for meshing non-uniformly sampled and incomplete data. Vis Comput, 2008, 24: 435–448CrossRef
  105.Yang H, Kepler J, Juttler B. Meshing non-uniformly sampled and incomplete data based on displaced T-spline level sets. In: IEEE International Conference on Shape Modeling and Applications. Linz: Johannes Kepler University, 2007, 251–260
  106.Yang X, Zheng J. Approximate T-spline surface skinning. Comput Aided Design, 2012, 44: 1269–1276CrossRef
  107.Zhang F. On the dimension of spline space over several special type of T-meshes. Master’s thesis. Dalian: Dalian Technologic School, 2005
  108.Zhang J, Li X. On the linear independency and partition of unity of arbitrary degree analysis-suitable T-splines. Comm Math Stat, 2015, 2: 353–364CrossRef MATH
  109.Zhang M. On the properties of T-splines. Master’s thesis. Jinlin: Jinlin University, 2005
  110.Zhang Y, Wang W, Hughes T J R. Solid T-spline construction from boundary representations for genus-zero geometry. Comput Methods Appl Mech Engrg, 2012, 249: 185–197MathSciNet CrossRef
  111.Zhang Y, Wang W, Hughes T J R. Conformal solid T-spline construction from boundary T-spline representations. Comput Mech, 2013, 51: 1051–1059MathSciNet CrossRef MATH
  112.Zhao X J, Lu M, Zhang H X. Patch reconstruction with T-splines iterated fitting for mesh surface. Appl Mech Mater, 2011, 88: 491–496CrossRef
  113.Zheng J, Wang Y. Periodic T-splines and tubular surface fitting: Curves and surfaces. Lecture Notes in Comput Sci, 2012, 6920: 731–746CrossRef MATH
  114.Zheng J, Wang Y, Seah H S. Adaptive T-spline surface fitting to z-map models. In: Proceeding GRAPHITE’ 05 Proceedings of the 3rd International Conference on Computer Graphics and Interactive Techniques in Australasia and South East Asia. Providence, RI: Amer Math Soc, 2005, 405–411CrossRef
  115.Zhuang X. Improvement on the local refinement algorithm for T-splines. Master’s thesis. Dalian: Dalian University of Technology, 2008
  
• 作者单位：Xin Li (1)
  FaLai Chen (1)
  HongMei Kang (1)
  JianSong Deng (1)
  
  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Chinese Library of Science
  Applications of Mathematics
  
• 出版者：Science China Press, co-published with Springer
• ISSN：1869-1862

文摘

This paper provides a survey of local refinable splines, including hierarchical B-splines, T-splines, polynomial splines over T-meshes, etc., with a view to applications in geometric modeling and iso-geometric analysis. We will identify the strengths and weaknesses of these methods and also offer suggestions for their using in geometric modeling and iso-geometric analysis.