In this work, we examine a numerical phase-field fracture framework in which the crack irreversibility constraint is treated with a primal-dual active set method and a linearization is used in the degradation function to enhance the numerical stability. The first goal is to carefully derive from a complementarity system our primal-dual active set formulation, which has been used in the literature in numerous studies, but for phase-field fracture without its detailed mathematical derivation yet. Based on the latter, we formulate a modified combined active-set Newton approach that significantly reduces the computational cost in comparison to comparable prior algorithms for quasi-monolithic settings. For many practical problems, Newton converges fast, but active set needs many iterations, for which three different efficiency improvements are suggested in this paper. Afterwards, we design an iteration on the linearization in order to iterate the problem to the monolithic limit. Our new algorithms are implemented in the programming framework pfm-cracks [T. Heister, T. Wick; pfm-cracks: A parallel-adaptive framework for phase-field fracture propagation, Software Impacts, Vol. 6 (2020), 100045]. In the numerical examples, we conduct performance studies and investigate efficiency enhancements. The main emphasis is on the cost complexity by keeping the accuracy of numerical solutions and goal functionals. Our algorithmic suggestions are substantiated with the help of several benchmarks in two and three spatial dimensions. Therein, predictor-corrector adaptivity and parallel performance studies are explored as well.

翻译：本文研究了一种数值相场断裂框架，其中裂缝不可逆性约束通过原始-对偶积极集方法处理，并在退化函数中采用线性化以增强数值稳定性。第一个目标是仔细推导出基于互补系统的原始-对偶积极集公式，该公式已在文献的众多研究中被使用，但针对相场断裂问题尚缺乏详细的数学推导。基于此，我们提出了一种改进的组合积极集牛顿方法，与先前用于准单块设置的类似算法相比，该方法显著降低了计算成本。对于许多实际问题，牛顿法收敛迅速，但积极集需要多次迭代，为此本文提出了三种不同的效率改进策略。随后，我们设计了一种基于线性化的迭代方法，以求将问题迭代至单块极限。我们的新算法在编程框架pfm-cracks [T. Heister, T. Wick; pfm-cracks: 一种用于相场断裂传播的并行自适应框架，Software Impacts, Vol. 6 (2020), 100045] 中实现。在数值算例中，我们进行了性能研究并探讨了效率提升，主要关注在保持数值解和目标泛函精度的前提下降低计算复杂度。我们的算法建议通过二维和三维空间中的多个基准测试得到了验证，其中还探讨了预测-校正自适应性和并行性能研究。