There is currently an increasing interest in developing efficient solvers for variational phase-field models of brittle fracture. The governing equations for this problem originate from a constrained minimization of a non-convex energy functional, and the most commonly used solver is a staggered solution scheme. This is known to be robust compared to the monolithic Newton method, however, the staggered scheme often requires many iterations to converge when cracks are evolving. The focus of our work is to accelerate the solver through a scheme that sequentially applies Anderson acceleration and over-relaxation, switching back and forth depending on the residual evolution, and thereby ensuring a decreasing tendency. The resulting scheme takes advantage of the complementary strengths of Anderson acceleration and over-relaxation to make a robust and accelerating method for this problem. The new method is applied as a post-processing technique to the increments of the solver. Hence, the implementation merely requires minor modifications to already available software. Moreover, the cost of the acceleration scheme is negligible. The robustness and efficiency of the method are demonstrated through numerical examples.

当前，人们对开发用于脆性断裂的变相场模型的有效求解器的兴趣日益浓厚。该问题的控制方程式是由非凸能量函数的约束极小值引起的，最常用的求解器是交错求解方案。与整体牛顿法相比，这是已知的健壮方法，但是，交错方案通常需要许多迭代才能收敛裂纹。我们的工作重点是通过一种方案来加速求解器，该方案依次应用安德森加速度和过度松弛，并根据残差演化来回切换，从而确保下降趋势。所得方案利用了安德森加速和过度松弛的互补强度，为该问题提供了一种强大而加速的方法。新方法作为后处理技术应用于求解器的增量。因此，该实施仅需要对已经可用的软件进行较小的修改。而且，加速方案的成本可以忽略不计。通过数值算例证明了该方法的鲁棒性和有效性。