Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation

Our goal in this work is to develop and compare modified Newton methods for fully monolithic quasi-static brittle phase-field fracture propagation. In variational phase-field fracture, a smoothed phase-field indicator variable denoting the crack path is coupled to elasticity. Moreover, a crack irreversibility condition is incorporated. To develop a fully monolithic scheme is an extremely challenging task since the underlying problem is non-convex and the Jacobian of Newton's method is indefinite. To split the problem using alternating minimization, thus a partitioned approach, is a possible resort. However, there are good reasons to consider monolithic approaches such as for example robustness and efficiency. Although an error-oriented Newton method can cope with a larger variety of configurations, it appears that this method is not always robust and also not always efficient. Inspired by nonlinear flow problems, as alternative, we develop a modified Newton scheme in which globalization is based on a dynamic modification of the Jacobian matrix rather than utilizing line-search or trust-region strategies. This variation switches smoothly between full Newton and Newton-like steps. In several 2D and 3D numerical examples, all of them with different characteristic features, our modified Newton solver is compared to a backtracking line-search Newton method, another line-search method monitoring the global energy and allowing for negative curvatures, and to already published results of an error-oriented version. These computations also include further modifications of Newton's method and detailed discussions why certain schemes either work or fail. Revisiting all findings, the main outcome of this paper is that the modified Newton scheme with Jacobian modification is currently the only method that works in a robust and efficient way for all provided examples, whereas line-search schemes or the error-oriented scheme show deficiencies for certain configurations.