Navigating with Stability: Local Minima, Patterns, and Evolution in a Gradient Damage Fracture Model
We investigate the computation of stable fracture paths in brittle thin films using one-dimensional damage models with an elastic foundation. The underlying variational formulation is non-convex, making the evolution path sensitive to algorithmic choices. In this paper, we inquire into the effectiveness of quasi-Newton algorithms as an alternative to conventional Newton-Raphson solvers. These algorithms improve convergence by constructing a positive definite approximation of the Hessian, trading improved convergence for the risk of missing bifurcation points and stability thresholds. In the absence of irreversibility constraints, we construct an equilibrium map that represents all stable and unstable equilibrium states as a function of the external load, using well-known branch-following bifurcation techniques. Our main finding is that quasi-Newton algorithms fail to select stable evolution paths without exact second variation information. To overcome this, we introduce a spectral stability criterion based on the full Hessian, which enables the identification of optimal perturbations and improves path-following accuracy. We then extend our analysis to the irreversible case, where admissible perturbations are constrained to a cone. We develop a nonlinear constrained eigenvalue solver to compute the minimal eigenmode within this restricted space and show that it plays a key role in distinguishing physical instabilities from numerical artefacts. Our results provide practical guidance for robust computation of fracture paths in irreversible, non-convex settings.