A new planar triangular element based on the absolute nodal coordinate formulation

In this paper, an improved three-node incomplete cubic planar triangular element is proposed based on the two recently reported absolute nodal coordinate formulation triangular elements. Compared with the existing absolute nodal coordinate formulation elements, a different set of polynomial basis is used to develop the new element using the method analogous to the one used in the conventional Zienkiewicz triangular element. Concise shape functions are obtained by employing both Cartesian and area coordinate sets and the concept of independent area gradient coordinate vector. From the view of the order of the polynomial basis, the criterion for developing incomplete cubic absolute nodal coordinate formulation triangular element that captures the quadratic accuracy is presented. Additionally, the algebraic constraint method used in developing the incomplete cubic triangular element is discussed. Based on the criterion, the proposed element is compared analytically with the previous incomplete cubic element. On the other hand, the proposed element is evaluated using both the static and dynamic numerical examples. The element successfully passes the patch test. The results obtained by the proposed element in this paper agree well with analytical solutions or those given by the full cubic element/general commercial finite element software. The higher accuracy, better convergence of the proposed element and the criterion are verified.