Parameterized method for standard rational absolute nodal coordinate formulation circular arc element

In engineering applications, connecting free-form curves with circular arcs is often necessary, emphasizing the significance of a standardized representation for circular arcs. This study proposes a novel method for constructing a parametric standard Rational absolute nodal coordinate formulation (RANCF) circular arc element based on the positional coordinates of its geometric configuration. The proposed method derives parameterized expressions for the nodal coordinates and weight vectors of the element. By adjusting the parameter value, the nodal coordinates and corresponding weights of the element can be flexibly modified, allowing the same geometric configuration of RANCF circular arc elements to be represented with different parameters. The primary distinction between RANCF circular arc elements with different parameters lies in the varying mapping relationships between the parameter space and the physical space. This mapping relationship can be adjusted through rational parameter transformations, enabling the conversion of different parameterized RANCF circular arc elements that represent the same geometric configuration. To evaluate the quality of different parameterized standard RANCF elements, we propose a criterion based on minimizing the cumulative gradient difference. The findings show that when the parameter value equals 1, corresponding to chord length parameterization, the standard RANCF circular arc element exhibits optimal performance. Numerical examples are provided to compare the performance of various parametric standard RANCF circular arc elements, demonstrating that convergence can be achieved using fewer optimized parametric elements.