In this paper, a new four-node incomplete cubic tetrahedral element (ICTE) based on the absolute nodal coordinate formulation (ANCF) is developed employing both volume coordinate and Cartesian coordinate parameter set. From the view of the order of interpolation polynomial basis, a criterion to develop the incomplete cubic ANCF tetrahedral element that can guarantee the quadratic accuracy is proposed. Based on the criterion, the new element and the other two existing incomplete cubic ANCF tetrahedral elements are compared analytically. The three elements are evaluated by both the static and dynamic numerical simulations. The new element successfully passes the patch test. The solutions of the proposed element in this paper agree well with analytical solutions or those given by the full cubic tetrahedral element/general commercial FE software. The higher accuracy and better convergence of the new element are verified. In addition, the method to develop incomplete cubic element by applying position constraints on the face center points and other corresponding material points of the full cubic element is discussed.

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