Cracking Elements Method (CEM) is a numerical tool to simulate quasi-brittle fractures, which does not need remeshing, nodal enrichment, or complicated crack tracking strategy. The cracking elements used in the CEM can be considered as a special type of finite element implemented in the standard finite element frameworks. One disadvantage of CEM is that it uses nonlinear interpolation of the displacement field (Q8 or T6 elements), introducing more nodes and consequent computing efforts than the cases with elements using linear interpolation of the displacement field. Aiming at solving this problem, we propose a simple hybrid linear and non-linear interpolation finite element for adaptive cracking elements method in this work. A simple strategy is proposed for treating the elements with $p$ edge nodes $p\in\left[0,n\right]$ and $n$ being the edge number of the element. Only a few codes are needed. Then, by only adding edge and center nodes on the elements experiencing cracking and keeping linear interpolation of the displacement field for the elements outside the cracking domain, the number of total nodes was reduced almost to half of the case using the conventional cracking elements. Numerical investigations prove that the new approach inherits all the advantages of CEM with greatly improved computing efficiency.

翻译：开裂单元法（CEM）是一种模拟准脆性断裂的数值工具，该方法无需重新划分网格、节点富集或复杂的裂纹追踪策略。CEM中使用的开裂单元可视为在标准有限元框架中实现的一种特殊类型有限元。CEM的一个缺点在于其采用位移场的非线性插值（Q8或T6单元），相较于采用位移场线性插值的单元情况，会引入更多节点及相应的计算量。为解决此问题，本文提出一种用于自适应开裂单元法的简单混合线性与非线性插值有限元。我们提出一种简单策略来处理具有$p$个边节点（$p\in\left[0,n\right]$，$n$为单元边数）的单元，仅需少量代码即可实现。随后，仅对发生开裂的单元添加边节点和中心节点，并对开裂区域外的单元保持位移场的线性插值，使得总节点数相较于使用传统开裂单元的情况减少近半。数值研究表明，新方法继承了CEM的所有优点，并显著提升了计算效率。