A Variable-Length Rational Finite Element Based on the Absolute Nodal Coordinate Formulation

The variable-length arbitrary Lagrange-Euler absolute nodal coordinate formulation (ALE-ANCF) finite element, which employs nonrational interpolating polynomials, cannot exactly describe rational cubic Bezier curves such as conic and circular curves. The rational absolute nodal coordinate formulation (RANCF) finite element, whose reference length (undeformed length) is constant, can exactly represent rational cubic Bezier curves. A new variable-length finite element called the ALE-RANCF finite element, which is capable of accurately describing rational cubic Bezier curves, is proposed and was formed by combining the desirable features of the ALE-ANCF and RANCF finite elements. To control the reference length of the ALE-RANCF element within a suitable range, element segmentation and merging schemes are proposed. It is demonstrated that the exact geometry and mechanics are maintained after the ALE-RANCF element is divided into two shorter ones, and compared with the ALE-ANCF elements, there are smaller deviations and oscillations after two ALE-RANCF elements are merged into a longer one. Numerical examples are presented, and the feasibility and advantages of the ALE-RANCF finite element are demonstrated.