Fourth-order hybrid phase field analysis with non-equal order elements and dual meshes for simulating crack propagation

High computational cost handicaps the practical application of the phase field approach for rock-like materials. The aim of this study was to investigate the feasibility of saving computing resources, especially computing time, by reducing the number and order of displacement elements. Specifically, we developed a novel scheme (i.e., TMNE) based on dual meshes and non-equal order elements using NURBS-based IGA, and then combined it with the hybrid formulation of the fourth-order model to tackle the high computing cost. The scheme was shown to save more than 50% of the computing time and the number of DOFs compared with the traditional scheme in most cases. This indicates that there is a possibility of wasting computing resources in traditional discretization of displacement field. Additionally, the scheme simplifies the data transfer between meshes and only one case need to be considered, even when combined with adaptive remeshing. Last but not least, the scheme can also be extended to a variety of NURBS elements (C⁰-C⁴) as required.