A Study on Nodal and Isogeometric Formulations for Nonlinear Dynamics of Shear- and Torsion-Free Rods

Abstract

In this work, we compare the nodal and isogeometric spatial discretization schemes for the nonlinear formulation of shear- and torsion-free rods introduced in Gebhardt and Romero (see Reference no. 31). We investigate the resulting discrete solution space, the accuracy, and the computational cost of these spatial discretization schemes. To fulfill the required $C^1$ continuity of the rod formulation, the nodal scheme discretizes the rod in terms of its nodal positions and directors using cubic Hermite splines. Isogeometric discretizations naturally fulfill this with smooth spline basis functions and discretize the rod only in terms of the positions of the control points (see Nguyen et al. in Reference no. 41), which leads to a discrete solution in multiple copies of the Euclidean space ${ℝ}^3$. They enable the employment of basis functions of one degree lower, that is, quadratic $C^1$ splines, and possibly reduce the number of degrees of freedom (dofs). When using the nodal scheme, since the defined director field is in the unit sphere $S^2$, preserving this for the nodal director variable field requires an additional constraint of unit nodal directors. This leads to a discrete solution in multiple copies of the manifold ${ℝ}^3\times S^2$; however, it results in zero nodal axial stress values. Allowing arbitrary length for the nodal directors, that is a nodal director field in ${ℝ}^3$ instead of $S^2$ as within discrete rod elements, eliminates the constrained nodal axial stresses and leads to a discrete solution in multiple copies of ${ℝ}^3$. To enforce the unit nodal director constraint, we discuss two approaches using the Lagrange multiplier and penalty methods. We compare the resulting semi-discrete formulations and the computational cost of these discretization variants. We numerically demonstrate our findings via examples of a planar roll-up, a catenary, and a mooring line.