  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.
