Efficient numerical computational strategies for solving phase-field models in fracture analysis - Ramos, Caio Silva (2025)
Phase-field models for fracture have emerged as powerful computational frameworks that provide a variational formulation of Griffith-type fracture mechanics by describing the competition between elastic and fracture energy potentials. These models naturally incorporate fracture evolution into the governing equations, allowing cracks to propagate along paths of least energy resistance without requiring explicit crack tracking algorithms or ad hoc propagation criteria. However, phase-field fracture simulations are computationally expensive due to the non-convex, highly nonlinear nature of the energy functional, which leads to poor convergence in standard Newton-Raphson solvers. This work addresses these computational challenges through two algorithmic innovations: an enhanced Limitedmemory BroydenFletcherGoldfarbShanno (L-BFGS) method with a novel quasi-Newton line search strategy, and the BORAM algorithm that combines L-BFGS with Over-Relaxed Alternating Minimization (ORAM). The enhanced L-BFGS incorporates a gradientbased line search method that ensures algorithmic robustness by dynamically adjusting search directions with adaptive step sizes, preventing divergence during critical crack propagation events. The BORAM algorithm provides an adaptive solution strategy that employs convergence rate detection to automatically switch between L-BFGS and ORAM methodologies based on real-time assessment of crack evolution dynamics. Comprehensive numerical experiments covering brittle and quasi-brittle fracture scenarios demonstrate that both algorithms achieve substantial computational efficiency gains, with average performance improvements of approximately five-fold compared to traditional alternating minimization approaches, while maintaining high accuracy and robustness across diverse fracture patterns and mixed-mode loading conditions.