Computer Science

Computer Science ›› 2026, Vol. 53 ›› Issue (9): 219-227.doi: 10.11896/jsjkx.260600135

• Computer Graphics & Multimedia • Previous Articles     Next Articles

Subdivision Isogeometric Collocation Based on Cotangent Laplacian

XU Hailun, KANG Hongmei   

  1. School of Mathematical Science,Soochow University,Suzhou,Jiangsu 215006,China
  • Received:2026-04-02 Revised:2026-07-07 Online:2026-09-15 Published:2026-09-10
  • About author:XU Hailun,born in 2001,Ph.D candidate,is a member of CCF(No.V3307G).His main research interest is computer aided geometric design.
    KANG Hongmei,born in 1987,Ph.D,associate professor,is a member of CCF(No.H7999M).Her main research interests include computer aided geome-tric design and isogeometric analysis.
  • Supported by:
    National Key R&D Program of China(2024YFA1016300),National Natural Science Foundation of China(12571409) and Open Research Fund Collaborative Team Project of the State Key Laboratory of Mathematical Sciences(SLMS-2025-KFKT-TD-04).

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Abstract: Subdivision basis functions possess high-order continuity within regular regions and remain smooth across the entire domain,making them highly suitable for isogeometric collocation in complex domains.To address the challenges of selecting an appropriate set of collocation points in isogeometric collocation for subdivision surfaces,a collocation scheme is proposed to establish a bijection between collocation points and subdivision basis functions,thereby avoiding least-squares solving.Specifically,all internal vertices are used to satisfy partial differential equations,while the edge midpoints and corner points of the boundary are used to apply boundary conditions.Considering the problem of unbounded curvature at extraordinary points on subdivision surfaces,the cotangent Laplacian is applied to assembly at extraordinary points.This method is simple to implement,requires low computational cost,and yields stable collocation solutions.Numerical experimental results demonstrate that the isogeometric collocation method using the proposed tuned Loop subdivision,with the subdominant eigenvalue λ of the subdivision matrix set to 0.50,achieves the same second-order convergence rate as in regular regions,whereas Loop subdivision fails to do so.

Key words: Isogeometric collocation, Subdivision surfaces, Cotangent Laplacian, Loop subdivision, Tuned Loop subdivision

CLC Number: 

  • TP391
[1] COTTRELL J A,HUGHES T J,BAZILEVS Y.Isogeometric Analysis:Toward integration of CAD and FEA[M].Bognor Regis:John Wiley & Sons,2009.
[2] HUGHES T J,COTTRELL J A,BAZILEVS Y.Isogeometricanalysis:CAD,finite elements,NURBS,exact geometry and mesh refinement[J].Computer Methods in Applied Mechanics and Engineering,2005,194(39-41):4135-4195.
[3] LI X,CHEN F,KANG H,et al.A survey on the local refinable splines[J].Science China Mathematics,2016,59:617-644.
[4] CIRAK F,ORTIZ M,SCHRÖDER P.Subdivision surfaces:a new paradigm for thin-shell finite-element analysis[J].International Journal for Numerical Methods in Engineering,2000,47(12):2039-2072.
[5] BARENDRECHT P J.Isogeometric analysis for subdivision surfaces[D].Eindhoven:Eindhoven University of Technology,2013.
[6] PAN Q,XU G,XU G,et al.Isogeometric analysis based on extended Loop’s subdivision[J].Journal of Computational Phy-sics,2015,299:731-746.
[7] RIFFNALLER-SCHIEFER A,AUGSDÖRFER U H,FELL-NER D W.Isogeometric shell analysis with NURBS compatible subdivision surfaces[J].Applied Mathematics and Computation,2016,272:139-147.
[8] BURKHART D,HAMANN B,UMLAUF G.Isogeometric finite element analysis based on Catmull-Clark subdivision solids[J].Computer Graphics Forum,2010,29(5):1575-1584.
[9] XIE J,XU J,DONG Z,et al.Interpolatory Catmull-Clark volumetric subdivision over unstructured hexahedral meshes for modeling and simulation applications[J].Computer Aided Geometric Design,2020,80:101867.
[10] ZHANG Q,SABIN M,CIRAK F.Subdivision surfaces with isogeometric analysis adapted refinement weights[J].Computer-Aided Design,2018,102:104-114.
[11] 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.
[12] WANG X,MA W.An extended tuned subdivision scheme with optimal convergence for isogeometric analysis[J].Computer-Aided Design,2023,162:103544.
[13] LI X,WEI X,ZHANG Y J.Hybrid non-uniform recursive subdivision with improved convergence rates[J].Computer Me-thods in Applied Mechanics and Engineering,2019,352:606-624.
[14] WEI X,LI X,ZHANG Y J,et al.Tuned hybrid nonuniform sub-division surfaces with optimal convergence rates[J].International Journal for Numerical Methods in Engineering,2021,122(9):2117-2144.
[15] JÜTTLER B,MANTZAFLARIS A,PERL R,et al.On numerical integration in isogeometric subdivision methods for PDEs on surfaces[J].Computer Methods in Applied Mechanics and Engineering,2016,302:131-146.
[16] BARENDRECHT P J,BARTOŇ M,KOSINKA J.Efficient quadrature rules for subdivision surfaces in isogeometric analysis[J].Computer Methods in Applied Mechanics and Enginee-ring,2018,340:1-23.
[17] AURICCHIO F,BEIRÃO DA VEIGA L,HUGHES T,et al.Isogeometric collocation methods[J].Mathematical Models and Methods in Applied Sciences,2010,20(11):2075-2107.
[18] REALI A,HUGHES T J.An introduction to isogeometric collocation methods[M] //Isogeometric Methods for Numerical Simulation.2015:173-204.
[19] BEIRÃO DA VEIGA L,LOVADINA C,REALI A.Avoidingshear locking for the Timoshenko beam problem via isogeometric collocation methods[J].Computer Methods in Applied Mechanics and Engineering,2012,241:38-51.
[20] KIENDL J,AURICCHIO F,BEIRÃO DA VEIGA L,et al.Isogeometric collocation methods for the Reissner-Mindlin plate problem[J].Computer Methods in Applied Mechanics and En-gineering,2015,284:489-507.
[21] AURICCHIO F,BEIRÃO DA VEIGA L,KIENDL J,et al.Locking-free isogeometric collocation methods for spatial Timoshenko rods[J].Computer Methods in Applied Mechanics and Engineering,2013,263:113-126.
[22] REALI A,GOMEZ H.An isogeometric collocation approach for Bernoulli-Euler beams and Kirchhoff plates[J].Computer Me-thods in Applied Mechanics and Engineering,2015,284:623-636.
[23] SCHILLINGER D,EVANS J A,REALI A,et al.Isogeometric collocation:cost comparison with galerkin methods and extension to adaptive hierarchical NURBS discretizations[J].Computer Methods in Applied Mechanics and Engineering,2013,267:170-232.
[24] ANITESCU C,JIA Y,ZHANG Y J,et al.An isogeometric collocation method using superconvergent points[J].Computer Methods in Applied Mechanics and Engineering,2015,284:1073-1097.
[25] GOMEZ H,DE LORENZIS L.The variational collocation me-thod[J].Computer Methods in Applied Mechanics and Enginee-ring,2016,309:152-181.
[26] MONTARDINI M,SANGALLI G,TAMELLIN L.Optimal order isogeometric collocation at galerkin superconvergent points[J].Computer Methods in Applied Mechanics and Engineering,2017,316:741-757.
[27] WANG D,QI D,LI X.Superconvergent isogeometric collocation method with greville points[J].Computer Methods in Applied Mechanics and Engineering,2021,377:113689.
[28] LIN H,XIONG Y,WANG X,et al.Isogeometric least-squarescollocation method with consistency and convergence analysis[J].Journal of Systems Science and Complexity,2020,33:1656-1693.
[29] CASQUERO H,LIU L,ZHANG Y,et al.Isogeometric collocation using analysis-suitable T-splines of arbitrary degree[J].Computer Methods in Applied Mechanics and Engineering,2016,301:164-186.
[30] GIUST A,JÜTTLER B.Weighted isogeometric collocationbased on spline projectors[J].Computer Methods in Applied Mechanics and Engineering,2022,391:114554.
[31] JIA Y,ANITESCU C,ZHANG Y J,et al.An adaptive isogeometric analysis collocation method with a recovery-based error estimator[J].Computer Methods in Applied Mechanics and Engineering,2019,345:52-74.
[32] LOOP C.Smooth subdivision surfaces based on triangles[D].Salt Lake City:University of Utah,1987.
[33] LI X,XU H,TAN J,et al.Weighted isogeometric collocation based on subdivision surfaces[J].Engineering with Computers,2026,42(3):110.
[34] PETERS J,WU X.On the local linear independence of genera-lized subdivision functions[J].SIAM Journal on Numerical Ana-lysis,2006,44(6):2389-2407.
[35] MEYER M,DESBRUN M,SCHRÖDER P,et al.Discrete dif-ferential-geometry operators for triangulated 2-manifolds[C] //Visualization and Mathematics III.Berlin:Springer,2003:35-57.
[36] XU H,WEN Z,KANG H.Quasi-interpolation projectors forsubdivision function spaces[J].Graphical Models,2025,137:101250.
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