On the parametrization of finite rotations in computational mechanics: A classification of concepts with application to smooth shells

Abstract

This paper concerns the computational treatment of large rotations with application to the finite element discretisation of smooth shells. Formulations based on various rotational parametrizations are reviewed and classified with respect to their update structure: category (II) is based on total rotational degrees of freedom leading to an additive update structure, whereas category (I) relies on linearized rotational degrees of freedom leading to a multiplicative update structure. Based on this classification, new formulations are developed. Among them are the Rodrigues formula applied within category (II) with additive update structure for the two components of the rotation vector and two successive elementary rotations (in the sense of Euler angles) applied within category (I) yielding a singularity-free formulation. The numerical examples confirm that every rotational parametrization when applied within category (II) leads to singularities, whereas the application within category (I) enables the calculation of overall rotations unrestricted in size without any singularity.