Rotation vector and its complement parameterization for singularity-free corotational shell element formulations

Theoretical and computational aspects of rotation vector and its complement parameterization are examined in detail in this paper. The mutual relationships between the variations of rotation vector and its complement and the spin variable are presented. It shows that the switch of rotation parameter between rotation vector and its complement preserves not only the strains but also the angular velocity and acceleration, and the force vectors and tangent matrices of an element. Two singularity-free corotational shell element formulations are presented. The first formulation is a non-consistent one, in which a simple way only using rotation vector is proposed to modify the exact rotation update approach via the Baker–Campbell–Hausdorff formula, while the second formulation is a consistent one. The novelty is that, following the presented method, the existing singular spatial beam and shell element formulations based on rotation vector parameterization could be modified to be the singularity-free ones with only a few changes. Finally, three numerical examples involving large rotations are analyzed to demonstrate the capability of the presented formulations.