The proximal Galerkin method for phase-field fracture
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The phase-field method has emerged as a powerful tool for simulating fracture mechanics, yet it presents significant numerical challenges, particularly regarding the non-convexity of the energy functional and the enforcement of physical constraints such as crack irreversibility and the boundedness of the phase-field variable. This work uses the proximal Galerkin method as a robust and efficient framework for solving phase-field fracture problems. By reformulating the inequality-constrained optimization problem as a sequence of saddle-point problems via latent variables, the method rigorously enforces the physical bounds of the phase-field and naturally handles the irreversibility condition without the need for ad hoc penalty parameters or history variables. We validate the proposed approach through a series of numerical examples, including one-dimensional benchmarks with analytical solutions, standard two-dimensional quasi-static fracture tests such as the single-edge notched tension/shear tests, and dynamic cases such as crack branching and Kalthoff--Winkler tests. The results demonstrate that the method accurately reproduces theoretical predictions and aligns with standard staggered schemes, while offering a unified and mathematically consistent treatment of the constraints inherent to phase-field fracture modeling.
