Variational phase-field models of brittle fracture pose a local constrained minimization problem of a non-convex energy functional. In the discrete setting, the problem is most often solved by alternate minimization, exploiting the separate convexity of the energy with respect to the two unknowns. This approach is theoretically guaranteed to converge, provided each of the individual subproblems is solved successfully. However, strong non-linearities of the energy functional may lead to failure of iterative convergence within one or both subproblems. We analyze and visualize the energy along Newton directions to illustrate why Newton's method without line search fails. Motivated by this, we propose to employ an exact line search algorithm based on bisection, which (under certain conditions) can guarantee global convergence of Newton's method for each subproblem and consequently the successful determination of critical points of the energy through the alternate minimization scheme. Through several benchmark tests computed with various strain energy decompositions and two strategies for the enforcement of the irreversibility constraint in two and three dimensions, we demonstrate the robustness of the approach and assess its efficiency in comparison with other commonly used line search algorithms. With the outlined approach, we are able to compute the especially demanding Brazilian test featuring contact in 3D with the star-convex model.

翻译：变分相场脆性断裂模型提出了一个非凸能量泛函的局部约束最小化问题。在离散设置中，该问题通常通过交替最小化来求解，利用能量相对于两个未知量的分离凸性。这种方法在理论上保证了收敛性，前提是每个子问题都能成功求解。然而，能量泛函的强非线性可能导致一个或两个子问题中的迭代收敛失败。我们沿着牛顿方向分析并可视化能量，以说明没有线搜索的牛顿法为何失败。基于此，我们提出采用基于二分法的精确线搜索算法，该算法（在一定条件下）可以保证牛顿法在每个子问题上的全局收敛性，从而通过交替最小化方案成功确定能量的临界点。通过使用多种应变能分解以及在二维和三维中实施不可逆约束的两种策略进行的几个基准测试，我们证明了该方法的鲁棒性，并评估了其与其他常用线搜索算法相比的效率。采用所概述的方法，我们能够计算特别具有挑战性的巴西测试，该测试在三维中采用星凸模型并涉及接触。