A spatio-temporal adaptive phase-field fracture method

We present an energy-preserving mechanic formulation for dynamic quasi-brittle fracture in an Eulerian-Lagrangian formulation, where a second-order phase-field equation controls the damage evolution. The numerical formulation adapts in space and time to bound the errors, solving the mesh-bias issues these models typically suffer. The time-step adaptivity estimates the temporal truncation error of the partial differential equation that governs the solid equilibrium. The second-order generalized-$\alpha$ time-marching scheme evolves the dynamic system. We estimate the temporal error by extrapolating a first-order approximation of the present time-step solution using previous ones with backward difference formulas; the estimate compares the extrapolation with the time-marching solution. We use an adaptive scheme built on a residual minimization formulation in space. We estimate the spatial error by enriching the discretization with elemental bubbles; then, we localize an error indicator norm to guide the mesh refinement as the fracture propagates. The combined space and time adaptivity allows us to use low-order linear elements in problems involving complex stress paths. We efficiently and robustly use low-order spatial discretizations while avoiding mesh bias in structured and unstructured meshes. We demonstrate the method's efficiency with numerical experiments that feature dynamic crack branching, where the capacity of the adaptive space-time scheme is apparent. The adaptive method delivers accurate and reproducible crack paths on meshes with fewer elements.