A simple hybrid linear and non-linear interpolation finite element for adaptive cracking elements method

TL;DR

The paper addresses the high computational cost of the Cracking Elements Method (CEM) by introducing a simple hybrid linear/nonlinear interpolation element that upgrades only cracking-region elements with edge and center nodes, leaving the rest linear. This approach preserves the advantages of Global Cracking Elements (GCE) while significantly reducing degrees of freedom and avoiding remeshing or crack-tracking. The method is validated on an L-shaped panel, disks with an initial crack, and a concrete beam, demonstrating similar accuracy to existing GCEM/XFEM results with about half the computational effort and improved handling of curved boundaries. The proposed strategy is generalizable to other discretizations and is supported by a Fortran routine, highlighting practical impact for efficient quasi-brittle fracture simulations in standard finite element frameworks.

Abstract

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.

Figures (21)

• Figure 1: Relationships among $l_c$, $V^{(e)}$ and $A^{(e)}$ of element $e$ (based on a parallel crack passing through the center point of quadrilateral or triangular types of elements)
• Figure 2: The cohesive zone model (redrawn based on the figure provided in Fernando2019)
• Figure 3: Unit vectors $\mathbf{n}^{(e)}$, $\mathbf{t}^{(e)}$, crack openings $\zeta_n^{(e)}$, $\zeta_t^{(e)}$, and traction forces $T_n^{(e)}$, $T_t^{(e)}$
• Figure 4: Traction-separation curve $T^{(e)}_{eq}\left(\zeta^{(e)}_{eq} \right)$
• Figure 5: Classical and adaptive cracking elements: (a) classical cracking elements, (b) adaptive cracking elements and hanging nodes (see the blue arrows)