论文标题
强大的预处理，以使相位场断裂问题的混合配方
Robust preconditioning for a mixed formulation of phase-field fracture problems
论文作者
论文摘要
在这项工作中，我们考虑使用相田配方中的几乎不可压缩和（完全）不可压缩的材料中的断裂传播。我们使用弹性方程的混合形式克服体积锁定效果，并为所得系统开发出强大的，非线性和线性求解器方案以及预处理。单一求解的耦合变异不等式系统由三个未知数组成：位移，压力和相位场。使用纽顿的联合算法对部分微分方程中的非线性进行了纽顿算法来解决由于耦合，本构定律和裂纹不可逆性而引起的非线性，并采用了裂纹无效性约束的原始二重性主动设置策略。每个牛顿步骤中的线性系统都通过灵活的广义最小残留方法（GMRE）迭代求解。这项工作的关键贡献是开发特定问题的预定器，该预处理利用位移和压力变量的鞍点结构。纯固体和压力驱动裂缝中的四个数值示例是在均匀和局部精制的网格上进行的，以研究求解器与泊松比的鲁棒性以及离散化和正则化参数。
In this work, we consider fracture propagation in nearly incompressible and (fully) incompressible materials using a phase-field formulation. We use a mixed form of the elasticity equation to overcome volume locking effects and develop a robust, nonlinear and linear solver scheme and preconditioner for the resulting system. The coupled variational inequality system, which is solved monolithically, consists of three unknowns: displacements, pressure, and phase-field. Nonlinearities due to coupling, constitutive laws, and crack irreversibility are solved using a combined Newton algorithm for the nonlinearities in the partial differential equation and employing a primal-dual active set strategy for the crack irreverrsibility constraint. The linear system in each Newton step is solved iteratively with a flexible generalized minimal residual method (GMRES). The key contribution of this work is the development of a problem-specific preconditioner that leverages the saddle-point structure of the displacement and pressure variable. Four numerical examples in pure solids and pressure-driven fractures are conducted on uniformly and locally refined meshes to investigate the robustness of the solver concerning the Poisson ratio as well as the discretization and regularization parameters.
