A monolithic DAE formulation for dynamic phase-field fracture: BDF adaptive time integration and 2D/3D benchmarks

We consider dynamic brittle fracture in the framework of phase-field models and revisit its structure from the viewpoint of differential–algebraic equations (DAEs). Building on variational formulations of phase-field fracture, we treat the displacement field and the phase-field variable as primary unknowns of a coupled evolution system. Irreversibility is enforced through a history variable, so that the continuous problem takes the form of an index-1 DAE that preserves the thermodynamic structure of the underlying fracture model. On this basis, we propose a monolithic time-integration strategy in which the full DAE system is advanced by an implicit, variable-order backward differentiation formula (BDF) scheme. This approach provides adaptive time stepping and automatic control of local truncation errors, while avoiding the additional splitting error and algorithmic dissipation associated with staggered displacement/phase-field updates. The formulation is implemented in a finite element code that supports both small- and large-displacement kinematics, mixed finite elements, and distributed-memory parallelism, enabling large-scale two- and three-dimensional dynamic fracture simulations. The method is assessed on a set of benchmark problems, including single-edge-notched tension and impact configurations such as the Kalthoff–Winkler test and pre-cracked plates. The numerical results demonstrate that the proposed DAE-based formulation accurately reproduces experimentally observed crack paths and branching patterns, while providing improved robustness and controllable algorithmic dissipation.