This work is concerned with computing the effective crack energy of periodic and random media which arises in mathematical homogenization results for the Francfort–Marigo model of brittle fracture. A previous solver based on the fast Fourier transform (FFT) led to solution fields with ringing or checkerboard artifacts and was limited in terms of the achievable accuracy. As computing the effective crack energy may be recast as a continuous maximum flow problem, we suggest using the combinatorial continuous maximum flow discretization introduced by Couprie et al. The latter is devoid of artifacts, but lacks an efficient large-scale solution method. We fill this gap and introduce a novel solver which relies upon the FFT and a doubling of the local degrees of freedom which is resolved by the alternating direction method of multipliers (ADMM). Last but not least we provide an adaptive strategy for choosing the ADMM penalty parameter, further speeding up the solution procedure. We demonstrate the salient features of the proposed approach on problems of industrial scale.

中文翻译：

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一种基于快速傅里叶变换的计算组合一致网格上异质材料有效裂纹能的方法

这项工作涉及计算在 Francfort-Marigo 脆性断裂模型的数学均匀化结果中出现的周期性和随机介质的有效裂纹能量。先前基于快速傅里叶变换 (FFT) 的求解器导致解域具有振铃或棋盘伪影，并且在可实现的精度方面受到限制。由于计算有效裂纹能量可能会被改写为连续最大流问题，我们建议使用组合连续最大流Couprie 等人引入的离散化。后者没有工件，但缺乏有效的大规模解决方法。我们填补了这一空白并引入了一种新颖的求解器，该求解器依赖于 FFT 和乘法器交替方向法 (ADMM) 解决的局部自由度加倍。最后但并非最不重要的是，我们提供了一种选择 ADMM 惩罚参数的自适应策略，进一步加快了求解过程。我们展示了所提出的方法在工业规模问题上的显着特征。