A computationally efficient monolithic arc-length method for phase-field models of fracture based on the Efendiev & Mielke scheme

Brittle fracture has been a central topic in computational mechanics for many years and remains highly relevant today. In the context of rate-independent phase-field models of brittle fracture, cracks may evolve discontinuously in time — also referred to as unstable or brutal crack growth. These discontinuities stem from the non-convexity of the underlying energy and often lead to numerical difficulties. For example, many algorithms fail to converge. One common strategy to address such issues is to replace the (pseudo-)physical time by an alternative parametrization, as known from arc-length methods. This article first provides a careful review of existing arc-length methods, then introduces novel energy-based variants, and highlights the analogies and connections between the different approaches. However, the probably most important contribution of this paper is the reinterpretation of the Efendiev & Mielke scheme – specifically developed for approximating Balanced-Viscosity solutions of time-discontinuous problems such as rate-independent fracture – as a non-standard arc-length method. The term ”non-standard” refers to the fact that an additional inequality constraint must be satisfied for the arc-length. By formulating this scheme within an active-set framework, the discretized equations can be solved monolithically, yielding a robust and computationally efficient novel algorithm. This will be demonstrated by several numerical experiments.