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arXiv · 2510.05407

A convergent adaptive finite element method for a phase-field model of dynamic fracture

Ram Manohar·S. M. Mallikarjuaniah
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Abstract

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.

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Ram Manohar, S. M. Mallikarjuaniah. 2025-10-06. A convergent adaptive finite element method for a phase-field model of dynamic fracture. https://arxiv.org/abs/2510.05407

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