Numerical simulations of brittle fracture using phase-field approaches often employ a discrete approximation framework that applies the same order of interpolation for the displacement and phase-field variables. Most common is to use linear finite elements to discretize the linear momentum and phase-field equations. However the use of $P_1$ Lagrange shape functions to model the phase-field is not optimal, since the latter develops cusps for fully developed cracks that in turn occur at locations correspoding to Gauss points of the associated FE model for the mechanics. Such feature is challenging to reproduce accurately with low order elements, and consequently element sizes must be made very small relative to the phase-field regularization parameter in order to achieve convergence of results with respect to the mesh. In this paper, we combine the standard $P_1$ FE discretization of stress equilibrium with a cell-centered finite volume approximation of the phase-field evolution equation based on the two-point flux approximation that is constructed on the same simplex mesh. Compared to a pure FE formulation utilizing linear elements, the proposed framework results in looser restrictions on mesh refinement with respect to the phase-field length scale. Furthermore, initialization of the history field is straightforward and accomplished through a local procedure. The ability to employ a coarser mesh relative to the traditional implementation is shown for several numerical examples, demonstrating savings in computational cost on the order of 50 to 80 percent for the studied cases.

中文翻译：

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相场断裂的组合有限元-有限体积框架

使用相场方法对脆性断裂进行数值模拟通常采用离散近似框架，该框架对位移和相场变量应用相同的插值顺序。最常见的是使用线性有限元来离散线性动量和相场方程。然而，使用 $P_1$ 拉格朗日形状函数来模拟相场并不是最佳的，因为后者为完全发展的裂纹产生尖点，而裂纹又发生在对应于力学相关有限元模型的高斯点的位置。这种特征很难用低阶单元准确再现，因此单元尺寸必须相对于相场正则化参数非常小，以便实现关于网格的结果收敛。在本文中，我们将应力平衡的标准 $P_1$ FE 离散化与基于在同一单纯形网格上构建的两点通量近似的相场演化方程的单元中心有限体积近似相结合。与使用线性元素的纯 FE 公式相比，所提出的框架对相场长度尺度的网格细化产生了更宽松的限制。此外，历史字段的初始化很简单，并通过本地过程完成。几个数值示例显示了使用相对于传统实现更粗的网格的能力，表明所研究案例的计算成本节省了 50% 到 80%。