Proximal Galerkin for Phase Field Fracture
Abstract
The phase-field method has emerged as a powerful tool for simulating fracture mechanics, yet it presents significant numerical challenges, particularly regarding the enforcement of physical constraints such as irreversibility and boundedness of the phase-field variable. This work proposes the proximal Galerkin (PG) methodology as a robust and efficient framework for solving phase-field fracture problems. By reformulating the inequality-constrained optimization problem into a sequence of saddle-point problems involving latent variables, the PG method rigorously enforces the physical bounds of the phase-field variable and naturally handles the irreversibility condition. This approach is directly applicable to both static and dynamic phase-field fracture problems. The numerical results demonstrate that the PG framework accurately reproduces theoretical predictions and experimental observations, while offering a unified, mathematically consistent treatment of the constraints inherent to phase-field fracture modeling.
Cite
@article{arxiv.2604.26210,
title = {Proximal Galerkin for Phase Field Fracture},
author = {Miguel Castillón and Biswajit Khara and Jørgen S. Dokken and Thomas M. Surowiec and Brendan Keith and Yuri Bazilevs},
journal= {arXiv preprint arXiv:2604.26210},
year = {2026}
}