Modified Newton methods and mesh adaptivity for variational phase-field fracture propagation Thomas Wick ([email protected]) Institut für Angewandte Mathematik, Leibniz Universität Hannover Welfengarten 1, 30167 Hannover, Germany Centre de Mathématiques Appliquées, École Polytechnique, Université Paris-Saclay 91128 Palaiseau, France

In this presentation, the purpose is on the development of a fully monolithic solution algorithm for quasi-static phase-field fracture propagation. Phase-field fracture is a variational approach to fracture [1] and consists of two coupled partial differential equations and it is well known that the underlying energy functional is non-convex and sophisticated algorithms are required; see [3] for recent results. The incremental, spatially-discretized problem is treated with adaptive finite elements and predictor-corrector mesh adaptivity [2] that allows for a very small regularization parameter in the crack region. The nonlinear problem is solved with adaptive modified Newton algorithms, which work as inner loop within an inexact augmented Lagrangian iteration for relaxing the crack irreversibility constraint [3]. Specifically, the fully monolithic approach is compared to a quasi-monolithic approach in which phase-field is approximated through extrapolation in the displacement equation. These comparisons are done in terms of certain quantities of interest and computational cost. Moreover, features such as crack nucleation, joining, branching and fracture networks are addressed. All findings are critically commented pointing to open questions and future improvements. Short Bio From October 2017, Thomas Wick joins the Leibniz University Hannover in Germany as a Professor for Scientific Computing. From September 2016 to September 2017, he was maitre de conferences (Assistant Professor) at Ecole Polytechnique in Palaiseau in France. In 2015 he was visiting professor (6 months) at the Technische Universtität München in Germany. He was two years (2014-2016) a research scientist at RICAM Linz in Austria and, prior to that, two years (2012-2014) a Postdoctoral fellow at ICES Austin in Texas/USA. Thomas Wick earned his PhD in December 2011 under supervision of Professor Rolf Rannacher at Heidelberg University in Germany. His main research interests are multiphysics problems (fluid-structure interaction, reactive flow, porous media), phase-field / variational fracture, adaptive finite elements and goal-oriented a posteriori error estimation, robust and efficient solvers, and finally optimization and variational inequalities.

References [1] G.A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46 (1998), pp. 1319–1342. [2] T. Heister, M. F. Wheeler, and T. Wick, A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phasefield approach, Comp. Meth. Appl. Mech. Engrg., 290 (2015), pp. 466 – 495. [3] T. Wick, Modified Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation, Comp. Meth. Appl. Mech. Engrg., 325 (2017), pp. 577 – 611.

RAM3 Workshop

Rome, 22-24 November 2017