论文标题
相位场断裂最佳控制问题的时空配方，离散化和计算绩效研究
Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems
论文作者
论文摘要
这项工作的目的是开发用于相位最佳控制问题的时空离散方案。特别是在最佳控制最小化问题中，将跟踪类型的成本功能最小化以通过相位场变量将裂纹转向所需的模式。为了获得此类最佳解决方案，需要确定Neumann类型边界条件。首先，使用不连续的Galerkin公式得出了向前问题的时间离散化。在这里，一个挑战是包括正规化术语和破解性不可逆性约束。最佳控制设置是通过拉格朗日方法来制定的，而拉格朗日方法则从中得出了原始部分，伴随，切线和伴随的Hessian。本文中的整个牛顿算法是通过消除状态约束（即位移和相位未知数）来基于减少方法的，但将控制变量保留为唯一未知的变量。从低阶不连续的盖尔金离散化中，最终获得了伴随的时间步变方案。我们的配方和算法发展都得到了证实，并通过六个数值实验进行了说明。
The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.
