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，其中裂纹不可逆约束通过原-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。首要目标是从互补系统中严谨推导出原-对偶积极集公式，该公式已在文献中被广泛使用，但针对相场断裂问题尚无其详细的数学推导。基于此，我们提出了一种修正的组合积极集牛顿方法，与先前准单块设置中的类似算法相比，显著降低了计算成本。对于许多实际问题，牛顿法收敛迅速，但积极集法需要大量迭代，为此本文提出了三种不同的效率改进策略。随后，我们对线性化过程进行迭代，以将问题推进至单块极限。新算法在编程框架pfm-cracks [T. Heister, T. Wick; pfm-cracks: A parallel-adaptive framework for phase-field fracture propagation, Software Impacts, Vol. 6 (2020), 100045] 中实现。数值算例中，我们进行了性能研究并考察了效率提升效果，重点分析了在保持数值解和目标泛函精度条件下的计算复杂度。我们的算法建议通过多个二维和三维空间基准测试得到验证，其中还探讨了预测-校正自适应性和并行性能研究。