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Solving the equations of phase-field fracture robustly and efficiently remains a challenging problem. Recently, the BFGS quasi-Newton method has been reported to work well for such equations [Kristensen & Martinez, 2020). However, there is no convergence theory, and traditional BFGS can only handle differentiable functionals. Therefore, constraints like the non-healing condition have to be removed from the model.

In this talk we explore ways to use BFGS directly for the unmodified fracture phase-field equations. This requires extending BFGS to nonsmooth problems. For this, we combine BFGS with the TNNMG method. TNNMG is a preconditioned Newton method for nonsmooth block-separable optimization problems, which by itself has already been shown to work well for phase-field brittle-fracture problems. The combination of the algorithms has the favorable properties of both: A BFGS-like algorithm with guaranteed convergence that can be applied directly to nonsmooth problems. We demonstrate this with some numerical examples.