用于相场脆性断裂问题的截断非光滑 Newton 多重网格法及分析
Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems, with analysis
AI总结：
针对小应变脆性断裂相场空间问题，提出可直接处理损伤不可逆约束且具线性复杂度的 TNNMG 求解器，证明其全局收敛性，并通过数值对比显示显著加速且保持鲁棒性。
摘要：
我们提出截断非光滑 Newton 多重网格法（Truncated Nonsmooth Newton Multigrid Method，TNNMG），作为小应变脆性断裂相场方程空间问题的求解器。TNNMG 是一种非光滑多重网格方法，能够以线性时间复杂度求解双凸、分块可分非光滑的极小化问题。它利用问题固有的变分结构，直接处理损伤变量上的逐点不可逆约束，无需正则化，也无需引入局部历史场。在本文中，我们介绍该方法，并说明如何将其应用于若干已建立的相场脆性断裂模型。随后，我们证明对于任意载荷和初始迭代，该求解器都收敛到空间问题非光滑 Euler–Lagrange 方程的一个解。在此过程中，我们展示了所考虑模型的若干关键凸性和正则性性质。与一种算子分裂算法的数值比较表明，该方法在不损失鲁棒性的情况下显著提升了速度。
英文摘要：
We propose the Truncated Nonsmooth Newton Multigrid Method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems with linear time complexity. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without regularization or the introduction of a local history field. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler--Lagrange equations of the spatial problem for any load and initial iterate. On the way, we show several crucial convexity and regularity properties of the models considered here. Numerical comparisons to an operator-splitting algorithm show a considerable speed increase, without loss of robustness.
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