计算机科学

计算机科学 ›› 2026, Vol. 53 ›› Issue (9): 219-227.doi: 10.11896/jsjkx.260600135

• 计算机图形学&多媒体 • 上一篇    下一篇

基于余切拉普拉斯算子的细分等几何配点法

徐海伦, 康红梅   

  1. 苏州大学数学科学学院 江苏 苏州 215006
  • 收稿日期:2026-04-02 修回日期:2026-07-07 出版日期:2026-09-15 发布日期:2026-09-10
  • 通讯作者: 康红梅(khm@suda.edu.cn)
  • 作者简介:(hlxu0102@stu.suda.edu.cn)
  • 基金资助:
    国家重点研发计划(2024YFA1016300);国家自然科学基金(12571409);数学科学全国重点实验室开放课题合作团队项目(SLMS-2025-KFKT-TD-04)

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 Published:2026-09-15 Online: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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摘要： 细分基函数在规则区域内具有高阶连续性,并在整个定义域内保持光滑,因此非常适合复杂域的等几何配点法。针对细分曲面的等几何配点中面临的配点集选择与边界条件施加挑战,提出了一种配点方案,以实现配点与基函数的一一对应,从而避免最小二乘求解。具体来说,所有内部顶点作为内部配点,用于满足偏微分方程,边界的边中点及角点作为边界配点,用于施加边界条件。另外,针对细分曲面奇异点处曲率无界的问题,余切拉普拉斯算子被应用于奇异点处的方程组装。该方法实现简单,计算量小,可以获得稳定的配点解。数值实验结果表明,通过设置细分矩阵的次大特征值λ为0.50,采用调整的Loop细分的等几何配点法可以达到与规则区域一致的二阶收敛,而Loop细分则无法达到这一点。

关键词: 等几何配点法, 细分曲面, 余切拉普拉斯算子, Loop细分, 调整的Loop细分

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

中图分类号: 

  • TP391
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