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图学学报

图学学报 ›› 2022, Vol. 43 ›› Issue (1): 110-117.DOI: 10.11996/JG.j.2095-302X.2022010110

• 计算机图形学与虚拟现实 • 上一篇    下一篇

基于改进的 PHT-样条自适应等几何配点法

  

  1. 1. 西北工业大学力学与土木建筑学院，陕西 西安 710129；  2. Institute of Structure Mechanics, Bauhaus University Weimar, Thuringia Weimar 99423
  • 出版日期:2022-02-28 发布日期:2022-02-16
  • 基金资助:
    国家自然科学基金项目(11902263)；陕西省自然科学基金项目(2019JQ-623) 

An adaptive isogeometric collocation method with improved PHT-splines 

  1. 1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an Shaanxi 710129, China; 2. Institute of Structure Mechanics, Bauhaus University Weimar, Weimar Thuringia 99423, Germany 
  • Online:2022-02-28 Published:2022-02-16
  • Supported by:
    National Natural Science Foundation of China (11902263); Shaanxi Province Science Foundation (2019JQ-623)

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摘要： 将传统等几何配点法扩展至任意高阶单元并且满足自适应局部细分功能，提出一种基于改进的 PHT 样条单元的自适应等几何配点法。改进的 PHT 样条单元依然具有传统 PHT 样条单元局部细分功能，但因 为传统 PHT 样条函数在层级网格划分后需要对部分基函数的定义域进行截断处理，所以在层级细分过于频繁 区域，部分函数可能因为严重变形而影响计算稳定性，而改进的 PHT 样条函数无需截断处理，定义域内基函 数始终具有稳定形态，这使得改进的 PHT 样条单元更适合高阶连续性计算及多层网格细分。该算法结合 PHT 样条单元的特点，选取高斯点作为配置点。为了简化边界施加条件，采用了耦合线性方程组的方法，在问题域 内采用高斯配点法，在问题域边界采用传统伽辽金方法，最终耦合 2 组线性方程组。本算法的局部细分准则基 于复原解和复原解误差。实例计算结果表明，基于改进的 PHT 样条的自适应等几何配点法可以扩展至任意高 阶单元计算，并满足最佳收敛率，且与理论值吻合。

关键词: 等几何分析方法, 配点法, PHT 样条函数, 高斯配置点, 高阶单元, 局部细分

Abstract: The Gaussian isogeometric analysis (IGA) collocation method was extended to arbitrary higher order polynomial degrees. The current IGA collocation method applied a new hierarchical basis over T-meshes (PHT-splines), which took advantage of the tensor product structure to prevent the decay phenomenon from happening in the original PHT basis. The improved method collocated at Gaussian points as the superconvergent points for the new PHT elements. Based on the new PHT basis, the current collocation method can be extended to arbitrary higher order approximation. In order to simplify the collocation boundary condition, a hybrid method was adopted to impose the boundary condition, using the Galerkin method for the boundary part and combing with the collocation solving system. The local refinement strategy was driven by a recovery-based error estimator that invoked computing an improved approximation without knowledge of the exact solution. The proposed collocation method can obtain the optimal convergent rates, compared with the IGA Galerkin method. 

Key words: isogeometric analysis, collocation method, PHT-splines, Gaussian collocation points, higher order elements, adaptive refinement 

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