Quadtree-polygonal smoothed finite element method for adaptive brittle fracture problems

Abstract

In this work, a quadtree-polygonal smoothed finite element method is proposed for adaptive consistent framework of phase field model on brittle fracture problems. A Smoothed Galerkin Weak form aided with the gradient smoothing technique is formulated to construct the variational formulations for both displacement and phase field. Staggered scheme is employed to solve the coupled phase and displacement field, in which the displacement field is obtained by Newton iterating and central difference method for implicit and explicit dynamic, respectively, while the phase field is solved directly with a linear equation. The critical history energy in the phase field fracture model is obtained with spectral decomposition. In order to acquire high efficiency without accuracy loss, a novel quadtree-based adaptive algorithm is developed for phase field fracture model, and the nodal phase field value is adopted as the direct indicator for mesh refinement. In this way, mesh local refinement is implemented with quadtree subdivision when the nodal phase field value is achieved the given threshold. Meanwhile, arbitrary sided polygonal elements provide an effective way to connect different mesh regions with different sizes. In other words, there is no hanging node but the connecting node on polygonal elements. Several numerical examples are performed for validating the feasibility of the proposed approach, in which the adaptive quadtree-polygonal method can save much computational costs without accuracy loss.