2025/10/06 by Ram Manohar, Manohar, Ram, S. M. Mallikarjuaniah +1 · 1 citation
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2510.05407
openalex publication_date 2025/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose and analyze an adaptive finite element method for a phase-field model of dynamic brittle fracture. The model couples a second-order hyperbolic equation for elastodynamics with the Ambrosio-Tortorelli regularization of the Francfort-Marigo variational fracture energy, which circumvents the need for explicit crack tracking. Our numerical scheme combines a staggered time-stepping algorithm with a variational inequality formulation to strictly enforce the irreversibility of damage. The mesh adaptation is driven by a residual-based a posteriori-type estimator, enabling efficient resolution of the evolving fracture process zone. The main theoretical contribution is a rigorous convergence analysis, where we prove that the sequence of discrete solutions generated by the AFEM converges (up to a tolerance) to a critical point of the governing energy functional. Numerical experiments for a two-dimensional domain containing an edge-crack under dynamic anti-plane shear loading demonstrate our method's capability of autonomously capturing complex phenomena, including crack branching and tortuosity, with significant computational savings over uniform refinement.