In this paper, an adaptive phase-field approach is proposed for three-dimensional fracture modeling in brittle materials. The scaled boundary finite element method (SBFEM) is used to solve the phase-field and elasticity equations in a staggered scheme. This is motivated by the ability of the SBFEM to handle polyhedral elements with hanging nodes straightforwardly. Such elements occur in octree meshes, which facilitate rapid transitions in element size and lend themselves to adaptive refinement. The SBFEM also provides an energy-based error indicator, which follows directly from the semi-analytical solution approach and does not require stress recovery. The error indicator is used to ensure the solution accuracy for both fields and combined with a criterion based on phase-field evolution. The computational cost associated with 3D fracture modeling is further reduced by exploiting the geometric similarity of cells occurring in balanced octree meshes. To validate the proposed method, several classical benchmark problems are solved, resulting in efficient and robust solutions compared to the literature.

在本文中，提出了一种自适应相场方法用于脆性材料的三维断裂建模。缩放边界有限元法(SBFEM) 用于求解交错方案中的相场和弹性方程。这是因为 SBFEM 能够直接处理带有悬挂节点的多面体元素。这些元素出现在八叉树中网格，这有助于单元大小的快速转换并有助于自适应细化。SBFEM 还提供了一个基于能量的误差指示器，它直接来自半解析解法，不需要应力恢复。误差指标用于确保两个场的求解精度，并结合基于相场演化的标准。通过利用平衡八叉树网格中出现的单元的几何相似性，进一步降低了与 3D 裂缝建模相关的计算成本。为了验证所提出的方法，解决了几个经典的基准问题，与文献相比，得到了有效且稳健的解决方案。