Fourth-order isogeometric phase-field modeling of dynamic brittle fracture: Numerical study and comparison with second-order models

We provide a thorough numerical investigation on phase-field simulations for dynamic brittle fracture; our analysis focuses on numerical performance, in terms of computational time and accuracy, measured by quantitative physical properties: energy conservation, crack speed, and critical stress. Simulations are performed by means of a semi-implicit staggered scheme, combining a Newmark method with the Projected Successive Over-Relaxation algorithm. This coupling ensures both an accurate solution of the elasto-dynamic equation and an effective solution of the linear complementarity problem, arising in the evolution of the phase-field parameter. In particular, with the Newmark method numerical dissipation is almost negligible; as a result, energy is conserved and simulations feature a remarkable representation of P and S waves, which irradiate from the boundary condition and from the tip, reflect on the boundary and interact with each other. While accuracy requires both small time increments and small mesh sizes, which easily lead to unaffordable computational time, we employ a fourth order AT1 functional that allows (as a rule of thumb) to halve the mesh size, and then halve the time increment. This multiplicative effect produces a remarkable saving of computational time (up to the order of 90 % ) without compromising the quality of the results. Moreover, AT1 functionals, compared with AT2, feature a sharp value of the critical stress and sharper damage profiles and showcase in all results a gain in accuracy. We employ an isogeometric framework that yields a straightforward discretization of the higher-order operators. First, we provide a calibration of the weight in front of the higher-order term, balancing the trade-off between accuracy and computational cost. Then, we compare the results obtained with second- and fourth-order AT1 and AT2 energies, using a few benchmark setups designed to focus the analysis on specific computational and physical aspects.