相场断裂最优控制问题的时空公式化、离散化和计算性能研究
Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems
摘要：
本工作的目的是发展相场最优控制问题的时空离散格式。具体地说，在最优控制最小化问题中，跟踪型成本泛函被最小化，以通过相场变量将裂纹引导到期望的图案中。为了获得这样的最优解，需要确定Neumann型边界条件。首先，时间离散的前向问题推导出使用不连续的伽辽金公式。在这里，一个挑战是包括正则化项和裂纹不可逆性约束。最优控制设置制定的拉格朗日方法的原始部分，伴随，切线和伴随海森。在此，整个牛顿算法是基于通过消除状态约束（即位移和相场未知数）但保持控制变量作为唯一未知数的简化方法。通过低阶间断Galerkin离散，得到了相应的伴随时间步格式。我们的配方和算法的发展得到证实，并说明了六个数值实验。
英文摘要：
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.
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