Efficient BFGS quasi-Newton method for large deformation phase-field modeling of fracture in hyperelastic materials

Abstract

The prediction of crack propagation in materials is a crucial problem in solid mechanics, with many practical applications ranging from structural integrity assessment to the design of advanced materials. The phase-field method has emerged as a powerful tool for modeling crack propagation in materials, due to its ability to accurately capture the propagation of cracks. However, current phase-field algorithms suffer from the elevated computational cost associated with the so-called staggered solution scheme, which requires extremely small time increments to advance the crack due to its inherent conditional stability. In this paper, we present, for the first time, a quantitative analysis detailing the numerical implementation and comparison of two common solution strategies for the coupled large-deformation solid-mechanics-phase-field problem, namely the quasi-Newton based Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) and the full Newton based alternating minimization (or staggered) (AM/staggered) algorithm. We demonstrate that the BFGS algorithm is a more efficient and advantageous alternative to the traditional AM/staggered approach for solving coupled large-deformation solid-mechanics-phase-field problems. Our results highlight the potential of the quasi-Newton BFGS algorithm to significantly reduce the computational cost of predicting crack propagation in hyperelastic materials while maintaining the accuracy and robustness of the phase-field method.