Optimizing Fracture Propagation Using a Phase-Field Approach

在这一建议中，我们考虑了由准静态脆性断裂扩展模型控制的控制问题的数值逼近和解。作为中心建模组件，考虑了裂缝形成和扩展的相场公式。裂纹扩展问题本身可以表示为一个带有不等式约束的极小化问题，该问题受多个相关边条件的约束，如裂纹扩展的不可逆性或材料在裂纹面上的非自渗透。这些都导致了作为一阶必要条件的变分不等式。因此，断裂过程控制的最优化问题产生了函数空间中具有互补约束的数学规划(MPCC)。在这个项目中，我们打算专注于数学挑战，这些挑战也是由应用程序所驱动的，例如控制变分不等式的系数，或者外部优化中的非光滑和/或非凸的成本泛函，例如最大化断裂释放的能量。我们将为所得到的MPCC以及其他障碍式开发一阶和二阶最优性条件。此外，我们还将考虑有限元的离散化，并证明离散逼近的收敛到连续极限。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.