Optimizing Fracture Propagation Using a Phase-Field Approach

在这个建议中，我们考虑由准静态脆性断裂扩展模型控制问题的数值逼近和求解。作为中心建模组件，考虑了裂缝形成和扩展的相场公式。断裂扩展问题本身可以被表述为带有不等式约束的最小化问题，该问题由多个相关侧条件施加，例如断裂生长的不可逆性或材料在断口表面的非自穿透性。这些导致变分不等式作为一阶必要条件。因此，断裂过程控制的优化问题产生了函数空间中具有互补约束（MPCC）的数学规划。在这个项目中，我们打算专注于数学上的挑战，这也是由应用驱动的，比如变分不等式系数的控制，或者外部优化中的非光滑和/或非凸成本函数，比如最大化裂缝释放的能量。我们将为所得到的MPCC以及其他类似障碍物的配方开发一阶和二阶最优条件。此外，我们将考虑有限元的离散化，并展示离散逼近到连续极限的收敛性。这些发现得到了原型数值试验的证实。

Within this proposal, we consider the numerical approximation and solution of control problems governed by a quasi-static brittle fracture propagation model. As a central modeling component, a phase-field formulation for the fracture formation and propagation is considered. The fracture propagation problem itself can be formulated as a minimization problem with inequality constraints, imposed by multiple relevant side conditions, such as irreversibility of the fracture-growth or non-selfpenetration of the material across the fracture surface. These lead to variational inequalities as first order necessary conditions. Consequently, optimization problems for the control of the fracture process give rise to a mathematical program with complementarity constraints (MPCC) in function spaces. Within this project, we intend to focus on mathematical challenges, that are also motivated by applications, such as control of the coefficients of the variational inequality, or nonsmooth and/or nonconvex cost functionals in the outer optimization, such as e.g. maximizing the released energy of the fracture. We will develop first and second order optimality conditions for the resulting MPCC as well as other obstacle-like formulations. Additionally, we will consider the discretization by finite elements and show the convergence of the discrete approximations to the continuous limit. These findings are substantiated with prototype numerical tests.