Efficiency improvement on the ANCF cable element by using the dot product form of curvature

Abstract

The adoption of the continuum mechanics concept has enabled the absolute nodal coordinate formulation (ANCF) cable element to handle large deformation and large rotation problems. Generally, the generalized elastic forces (GEFs) and tangent stiffness matrices are formulated by differentiating the curvature of the cross product form, which needs labour-intensive construction process, leads to complicated expressions and therefore impairs the computational efficiency of the element. This work aims to improve the computational efficiency of the ANCF cable element by simplifying the GEFs and tangent stiffness matrices based on the dot product form of curvature. The element is interpolated by the curvature constrained interpolation method and defined by following the decoupling elastic line approach, which ensures the second-order accuracy. The material curvature is presented with just dot product terms through two approaches: the original curvature concept approach (OCCA) and the Lagrange's identity approach (LIA). Based on the OCCA and the LIA, concise formulations of GEFs and tangent stiffness matrices are derived through matrix manipulation. In contrast, the GEFs and tangent stiffness matrices derived from the cross product form of curvature are traditionally constructed by deducing every single item individually, which is summarized as the itemized traversal approach (ITA) in this work. To verify the correctness and efficiency of the proposed approaches, some geometrically nonlinear problems of the straight beam and the curved beam are solved and thoroughly analysed. It is found that far fewer number of the second-order elements are needed than that of the first-order element to get the desired accuracy solution. Besides, the convergence rate of the OCCA, LIA and ITA is identical and the solutions of these three approaches coincide with the results obtained by ABAQUS for five-digits because the definitions of the GEFs and tangent stiffness matrices are essentially identical despite different curvature expressions. However, with the same number of second-order elements, the OCCA and LIA are both more than ten times more efficient than the ITA in terms of CPU time, in which the OCCA is the most efficient one. In future, the proposed approaches can be applied in finite element practices for saving computational time without loss of accuracy.