HIGHER-ORDER NUMERICAL MANIFOLD METHOD BASED ON ANALYSIS-SUITABLE T-SPLINE
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摘要: 数值流形方法是一种非常灵活的数值计算方法，连续体的有限单元方法和块体系统的非连续变形分析方法只是这一数值方法的特例.数值流形方法中高阶位移函数的构造可通过提高权函数的阶次来实现，这种方法往往需要沿单元边界配置适当的边内节点，这些结点的出现增加了前处理的复杂性，特别是对于大型复杂的空间问题.另一方面，在数值流形方法中可通过缩小单元尺寸（h加密）来提高求解精度.当模拟裂纹扩展时，这种细化策略可用来克服裂纹尖端的奇异性.一个传统的解决方案是细化整个网格，但这会导致计算效率的显著降低.将适合分析的T样条（analysis-suitable T-spline，AST）引入数值流形方法中来建立高阶数值流形方法的分析格式，有效的避免了该问题的出现.AST样条基函数具有线性无关，单位分解，局部加密等许多重要性质，使得其非常适合用于工程设计及分析.在引入AST样条后，可通过改变数学覆盖的构造形式建立不同阶次的数值流形方法分析格式；AST样条自身的局部加密性质也使得数值流形方法中的数学网格局部加密更容易实现.算例结果表明：随着AST样条基函数阶次的提高，数值流形方法的计算结果有了明显的改善；基于AST样条基函数的数值流形方法在保持计算精度的前提下降低了自由度的数量.Abstract: Numerical manifold method (NMM) is a very flexible numerical method which contains and combines finite element method (FEM) and discontinuous deformation analysis (DDA).High-order numerical manifold method can be constructed by increasing the order of the weight function.This method often needs to configure the appropriate edge nodes along the element boundary, the emergence of these nodes increase the complexity of pre-processing, especially for large and complex spatial problems.On the other hand, the level of approximation of NMM can be improved by splitting the elements into smaller ones (known as h-refinement).With regard to the h-refinement, a cover refinement strategy is necessary to overcome the singularity of the stress when simulating crack propagation in NMM.One traditional solution is to refine the entire mesh which can lead to a significant decrease in the computational efficiency.In this paper analysis-suitable T-spline is introduced into NMM and regular rectangular meshes are used as the mathematical cover system.Specifically, analysis-suitable T-spline is linearly independent, forms a partition of unity, and can be locally refined which make it meet the demands of both design and analysis.The basis function of analysis-suitable T-spline is adopted as the weight function in NMM to construct high-order NMM and make the local refinement for feasible adaptive procedure.Two numerical examples are given to demonstrate the accuracy and efficiency of the proposed method and the results show that the higher order AS T-spline based NMM shows higher accuracies when solving both continuous and discontinuous problems.Furthermore, the local mesh refinement using analysis-suitable T-spline reduces the number of degrees of freedom while maintaining calculation accuracy at the same time.
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图 6 扩展T网格
Figure 6. The extended T-mesh. Face extensions are represented by dotted arrows and edge extensions are represented by dashed arrows. The T-junctions are denoted by black dots
图 7 (a)适合分析的T网格(b)扩展T网格
Figure 7. (a) An analysis-suitable T-mesh (b) the extended T-mesh
图 10 $p =1$时权值在一个数学覆盖中的分布 \$\{x_{1},x_{2},x_{3}\}=\{0,1, 2\}$,$\{ y_{1},y_{2},y_{3}\}=\{0,1,2\}$
Figure 10. Distribution of weights in a mathematical cover when $p =1$ with $\{x_{1},x_{2},x_{3}\}=\{0,1, 2\}$,$\{ y_{1},y_{2},y_{3}\}=\{0,1,2\}$
图 11 $p =2$时权值在一个数学覆盖中的分布$ \{x_{1},x_{2},x_{3}, x_{4}\}=\{0,1,2,3\}$,$\{y_{1},y_{2},y_{3},y_{4}\}=\{0,1,2,3\}$
Figure 11. Distribution of weights in a mathematical cover when $p =2$ with $\{x_{1},x_{2},x_{3}, x_{4}\}=\{0,1,2,3\}$,$\{y_{1},y_{2},y_{3}, y_{4}\}=\{0,1,2,3\}$
图 12 $p =3$时权值在一个数学覆盖中的分布 $\{x_{1},x_{2},x_{3},x_{4},x_{5}\}=\{0, 1,2,3,4\}$,$\{y_{1},y_{2},y_{3},y_{4}, y_{5}\}=\{0,1,2,3,4\}$
Figure 12. Distribution of weights in a mathematical cover when $p=3$ with $\{x_{1},x_{2},x_{3},x_{4},x_{5}\}=\{0,1,2,3,4\}$,$\{y_{1},y_{2},y_{3},y_{4},y_{5}\}=\{0,1,2,3,4\}$
表 1 采用一次AST样条数值流形方法计算结果
Table 1. SIFs results for NMM_T1
表 2 采用二次AST样条数值流形方法计算结果
Table 2. SIFs results for NMM_T2
表 3 采用三次AST样条数值流形方法计算结果
Table 3. SIFs results for NMM_T3
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