A simple hybrid linear and nonlinear interpolation finite element for the adaptive Cracking Elements Method

The Cracking Elements Method (CEM) is a numerical tool for simulation of quasi-brittle fracture. It neither needs remeshing, nor nodal enrichment, or a complicated crack-tracking strategy. The cracking elements used in the CEM can be considered as a special type of Galerkin finite elements. A disadvantage of the CEM is that it uses nonlinear interpolation of the displacement field (e.g. Q8 and T6 elements for 2D problems), introducing more nodes and consequently requiring greater computing efforts than in case of elements based on linear interpolation of the displacement field. With the aim to solve this problem we propose a hybrid linear and nonlinear interpolation finite element for the adaptive CEM presented in this work. A simple strategy is proposed for treating elements with $p$ edge nodes, where $p\in \left(0,n\right)$, with $n$ as the edge number of the considered element. Only a few program codes are needed. Then, by just adding edge and center nodes to the elements experiencing cracking, while keeping linear interpolation of the displacement field for the elements outside the cracking domain, the number of total nodes is reduced to almost one half of the number in case of using the conventional CEM. Numerical investigations have shown that the new approach not only preserves all of the advantages of the CEM, but also results in a significantly enhanced computing efficiency.