In this study, an effective numerical framework for fracture simulations is proposed using the edge-based smoothed finite element method (ES-FEM) and isotropic damage model. The duality between the Delaunay triangulation and Voronoi tessellation is utilized for the mesh construction and the compatible use of the finite element solution with the Voronoi-cell lattice geometry. The mesh irregularity is introduced to avoid calculating the biased crack path by adding random variation in the nodal coordinates, and the ES-FEM elements are defined along the Delaunay edges. With the Voronoi tessellation, each nodal mass is calculated and the fractured surfaces are visualized along the Voronoi edges. The rotational degrees of freedom are implemented for each node by introducing the elemental formulation of the Voronoi-cell lattice model, and the accurate visualizations of the rotational motions in the Voronoi diagram are achieved. An isotropic damage model is newly incorporated into the ES-FEM formulation, and the equivalent elemental length is introduced with an additional geometric factor to simulate the consistent softening behaviors with reducing the mesh sensitivity. The full matrix form of the smoothed strain-displacement matrix is constructed for optimal use in the element-wise computations during explicit time integration, and parallel computing is implemented for the enhancement of the computational efficiency. The simulated results are compared with the theoretical solutions or experimental results, which demonstrates the effectiveness of the proposed methodology in the simulations of the quasi-brittle fractures.

在这项研究中，使用基于边缘的平滑有限元方法 (ES-FEM) 和各向同性损伤模型提出了一种有效的断裂模拟数值框架。Delaunay 三角剖分和 Voronoi 细分之间的对偶性用于网格构造和有限元解与 Voronoi 单元格几何的兼容使用。引入网格不规则性以避免通过在节点坐标中添加随机变化来计算有偏差的裂纹路径，并且沿 Delaunay 边缘定义 ES-FEM 元素。使用 Voronoi 镶嵌，计算每个节点质量，并沿着 Voronoi 边缘可视化断裂表面。通过引入 Voronoi-cell 晶格模型的基本公式，为每个节点实现了旋转自由度，并且实现了 Voronoi 图中旋转运动的准确可视化。ES-FEM 公式中新加入了各向同性损伤模型，并引入了等效元素长度和附加几何因子，以模拟一致的软化行为，同时降低网格敏感性。构建平滑应变-位移矩阵的完整矩阵形式，以便在显式时间积分期间的元素计算中最佳使用，并实施并行计算以提高计算效率。将模拟结果与理论解或实验结果进行了比较，证明了该方法在准脆性断裂模拟中的有效性。