Iterative convergence in phase-field brittle fracture computations: exact line search is all you need

Variational phase-field models of brittle fracture pose a local constrained minimization problem of a non-convex energy functional. In the discrete setting, the problem is most often solved by alternate minimization, exploiting the separate convexity of the energy with respect to the two unknowns. This approach is theoretically guaranteed to converge, provided each of the individual subproblems is solved successfully. However, strong non-linearities of the energy functional may lead to failure of iterative convergence within one or both subproblems. We analyze and visualize the energy along Newton directions to illustrate why Newton's method without line search fails. Motivated by this, we propose to employ an exact line search algorithm based on bisection, which (under certain conditions) can guarantee global convergence of Newton's method for each subproblem and consequently the successful determination of critical points of the energy through the alternate minimization scheme. Through several benchmark tests computed with various strain energy decompositions and two strategies for the enforcement of the irreversibility constraint in two and three dimensions, we demonstrate the robustness of the approach and assess its efficiency in comparison with other commonly used line search algorithms. With the outlined approach, we are able to compute the especially demanding Brazilian test featuring contact in 3D with the star-convex model.