A macro tetrahedral element with vertex rotational D.O.F.s

In this study, a macro tetrahedral element with vertex rotations is presented with the systematic macro element fabrication and the inclusion of proper higher-order deformation modes. The individual element test (IET) proved sub-tetrahedral elements are basically combined for the macro element. During the construction of the macro element, a higher-order stiffness is emerged by moving the influence of removed virtual node stiffness into the deformation modes which are related to the vertex rotations. The fundamental decomposition into a basic stiffness and a higher-order stiffness allows scaling the performance of the macro element. To fulfill the accuracy and stability requirement of the element, the transformation between hierarchical rotations and nodal freedoms is employed to separate rotational freedoms into the constant strain related rotation and the higher-order behavior related one. Each parameter in the element stiffness is tuned through beam-type higher order patch test. The numerical performance of the proposed element is compared with other solid elements with vertex rotation in the benchmark test problem.