Arbitrary Order Virtual Element Methods for High-Order Phase-Field Modeling of Dynamic Fracture

Accurate modeling of fracture nucleation and propagation in brittle and ductile materials subjected to dynamic loading is important in predicting material damage and failure under extreme conditions. Phase-field fracture models have garnered a lot of attention in recent years due to their success in representing damage and fracture processes in a wide class of materials and under a variety of loading conditions. Second-order phase-field fracture models are by far the most popular among researchers (and increasingly, among practitioners), but fourth-order models have started to gain broader acceptance since their more recent introduction. The exact solution corresponding to these high-order phase-field fracture models has higher regularity. Thus, numerical solutions of the model equations can achieve improved accuracy and higher spatial convergence rates. In this work, we develop a virtual element framework for the high-order phase-field model of dynamic fracture. The virtual element method (VEM) can be regarded as a generalization of the classical finite element method. In addition to many other desirable characteristics, the VEM allows computing on polytopal meshes. Here, we use $H^1$-conforming virtual elements and the generalized-$\alpha$ time integration method for the momentum balance equation, and adopt $H^2$-conforming virtual elements for the high-order phase-field equation. We verify our virtual element framework using classical quasi-static benchmark problems and demonstrate its capabilities with the aid of numerical simulations of dynamic fracture in brittle materials.