The finite element formulation of geometrically exact rod models depends crucially on the interpolation of the rotation field from the nodes to the integration points where the internal forces and tangent stiffness are evaluated. Since the rotational group is a nonlinear space, standard (isoparametric) interpolation of these degrees of freedom does not guarantee the orthogonality of the interpolated field hence, more sophisticated interpolation strategies have to be devised. We review and classify the rotation interpolation techniques most commonly used in the context of nonlinear rod models and suggest new ones. All of them are compared and their advantages and disadvantages discussed. In particular, their effect on the frame invariance of the resulting discrete models is analyzed.
IntroductionExisting nonlinear rod theories allow to formulate problems involving arbitrarily large displacements, rotations and strains (see e.g.[1] for a review). By deriving the rod equations from the three-dimensional nonlinear theory of deformable bodies with a projection onto a restricted class of rod-like geometries and motions, the former inherit much of the mathematical structure of the fully dimensional problem. This includes the conservation laws of the dynamic problem and the objectivity of the equations under superposed rigid body motions.In the Computational Mechanics literature, the first attempts to formulate finite elements for these rod theories go back to [26] and [27] where a plane rod model due to Reissner [21] was implemented. A three dimensional extension of this theory was later presented in [24] by Simo, and its finite element formulation in [28]. Many works followed this pioneering one: [29,9,30,15,18,5,31], to name a few, contributing to a rapid development of the topic.A distinctive feature of nonlinear rod theories of the type mentioned above is that the kinematic variables at every point of the model include both displacements and rotations, and as such they can be classified as Cosserat theories. Apart from its geometric implications, this aspect has important consequences in the formulation of numerical models and in particular finite element ones. The rotation variables belong to a nonlinear manifold and thus their treatment requires special attention which has prompted several authors study this subject in many publications (e.g. [2,6,8]). An issue which is particularly important in the context of the finite element method is the interpolation of rotation variables, which is needed to obtain the values of the rotation field at integration points of the rod, given its values only at the nodes of the finite element mesh. Since a standard interpolation of the nodal rotations does not give, in general, an orthogonal field of rotations, several different solutions have been proposed in the literature. See, the articles mentioned in the previous paragraph and [10, 23].As mentioned above, a fundamental property of nonlinear rod models is the frame invariance of the equations. It is thus desirable that nume...