基于图像四叉树的改进型比例边界有限元法研究

为提高数值计算的精度, 断裂力学问题的数值模拟需要在裂纹扩展的局部区域采用较密的网格, 而远离裂纹扩展的区域可采用较疏的网格, 且对于裂纹扩展问题的数值模拟, 大多数数值方法又存在局部网格重剖分的问题. 论文提出了一种基于图像四叉树的改进型比例边界有限元法用于模拟裂纹扩展问题, 该方法可根据结构域几何外边界的图像全自动进行四叉树网格剖分, 无需任何人工干预, 网格剖分效率极高, 由于比例边界有限元法本身的优势, 四叉树网格的悬挂节点可以直接地视为新的节点, 无需任何特殊处理. 通过引入虚节点的思想, 将裂纹与四叉树单元边界交叉点作为虚节点, 虚节点的自由度作为附加自由度处理, 并采用水平集函数表征材料内部的裂纹面, 含不连续裂纹面的子域可通过节点水平集函数识别, 使得裂纹扩展时无需进行网格重剖分, 界面的几何特征通过比例边界有限元子域的附加自由度表征. 最后, 通过若干算例验证了该方法的性能, 建议的改进型比例边界有限元法在求解复合型应力强度因子和模拟材料内部裂纹扩展路径时均具有较高的精度. 

IMPROVED SCALED BOUNDARY FINITE ELEMENT METHODS BASED ON IMAGE QUADTREE

To improve numerical accuracy, the numerical simulation of fracture mechanics problems needs to use dense meshes in the local area of crack propagation, while sparse meshes can be used in the area far away from the local area where the crack is located. Additionally, for the numerical simulation of crack propagation problems, most numerical methods have the problem of local remeshing in the process of crack growth. In this paper, an improved scaled boundary finite element method based on image quadtree are proposed to simulate crack propagation problems. This method can automatically perform the quadtree mesh generation according to the image of the geometric outer boundary of the structural domain without any artificial intervention, and the mesh generation efficiency is also very high. Due to the inherent advantages of the scaled boundary finite element method, the hanging nodes of the quadtree mesh can be directly regarded as new nodes without any special treatment. By introducing the idea of virtual node, the intersection of crack and the boundary of quadtree element is regarded as virtual nodes, and the degrees of freedom of virtual nodes are treated as additional degrees of freedom. The level set functions are used to characterize the crack surface inside the material. The subdomain with the discontinuous crack surface can be identified by the node level set functions, so that there is no need for remeshing during crack propagation, and the geometric characteristics of the interface can be characterized by the additional degrees of freedom of the scaled boundary finite element method. Finally, the performance of the proposed method is verified by several numerical examples. The results show that the proposed improved scaled boundary finite element method has high accuracy in solving the mode I and II stress intensity factors as well as simulating the internal crack growth path of materials.