Conic Section Elements Based on the Rational Absolute Nodal Coordinate Formulation

The construction of rational absolute nodal coordinate formulation (RANCF) elements is usually based on a linear transformation of non-uniform rational B-spline (NURBS) geometry. However, this linear transformation can lead to property transfer issues, which greatly reduce the modeling efficiency, especially for conic sections. To overcome this limitation, we first analyze the geometric constraints of conic sections and derive a unique defining equation in rational parametric form. A corresponding degree-elevation formula is also obtained. Using these results, we propose a direct definition method for RANCF elements that explicitly exploits the analytic properties of conic sections. The method provides fast and accurate expressions for the nodal coordinates and weights, and thus enables efficient modeling of RANCF elements for conic-section configurations. We also mitigate the arbitrariness in element definition by introducing, for the first time, the concept of a mapping factor K, which characterizes the mapping between the physical space and the parameter space. Based on this mapping factor, we establish a parameterization procedure for RANCF conic-section elements. An evaluation criterion for K is further proposed and used to define the optimal mapping factor Kopt, which yields an optimal parameterization and allows the construction of Kopt elements. Numerical examples demonstrate that, in large-deformation analyses of flexible systems, the proposed elements can achieve a given accuracy with fewer elements than conventional approaches.