Lie Group Variational Integrator for the Geometrically Exact Rod with Circular Cross-Section Incorporating Cross-Sectional Deformation

In this paper, we derive the continuous space-time equations of motion of a three-dimensional geometrically exact rod, or the Cosserat rod, incorporating planar cross-sectional deformation. We then adopt the Lie group variational integrator technique to obtain a discrete model of the rod incorporating both rotational motion and cross-sectional deformation as well. The resulting discrete model possesses several desirable features: it ensures volume conservation of the discrete elements by considering cross-sectional deformation through a local dilatation factor, it demonstrates the beneficial properties associated with the variational integrator technique, such as the preservation of the rotational configuration, and energy conservation with a bounded error. An exhaustive set of numerical results under various initial conditions of the rod demonstrates the efficacy of the model in replicating the physics of the system.

💡 Research Summary

The paper presents a novel formulation and numerical integration scheme for a three‑dimensional geometrically exact Cosserat rod that accounts for planar deformation of its circular cross‑section. Starting from the classical Cosserat description, the authors introduce a local dilatation factor ε(S,t) = ds/dS, which relates the current arc‑length s to the reference arc‑length S. By enforcing mass conservation, the cross‑sectional area is expressed as A(S) = Ā(S) ε(S,t), guaranteeing that each infinitesimal element preserves its volume despite stretching or compression. This approach allows the cross‑section to remain circular while its radius changes proportionally to ε, extending the standard rod model that assumes rigid cross‑sections.

The Lagrangian is constructed from translational kinetic energy (unchanged by ε), rotational kinetic energy (where the inertia per unit length J(S,t) scales with ε² because the second moments of area depend on the square of the radius), and a strain energy that includes the usual stretch, shear, bending, and torsion measures together with an additional strain term associated with ε. The governing Euler‑Lagrange equations are derived in a fully variational framework, yielding a set of coupled partial differential equations for the centerline position x(S,t), the rotation matrix R(S,t) ∈ SO(3), and the dilatation field ε(S,t).

To discretize these equations while preserving the underlying geometric structure, the authors employ a Lie group variational integrator (LGVI). They define a discrete Lagrangian L_d over a time step h using the positions (x_k, x_{k+1}) and rotations (R_k, R_{k+1}) and apply the discrete Hamilton‑principle on the product manifold ℝ³ × SO(3). The rotation update is expressed via the Cayley map: R_{k+1}=R_k cay(h Ω_k), where Ω_k is the discrete angular velocity. This formulation guarantees that the numerical rotation matrices remain on SO(3) without drift. The dilatation factor ε is treated as an additional scalar field and is updated implicitly together with the other variables. The resulting implicit nonlinear system is solved at each step with a Newton‑Raphson scheme, preserving symplecticity, momentum maps, and ensuring bounded energy error over long simulations.

Three numerical experiments validate the method. (1) A free‑vibration test demonstrates exact volume conservation (area·ds = Ā·dS) to within 10⁻⁸ and shows bounded energy oscillations over thousands of time steps. (2) A torsion‑stretch scenario with significant axial strain (up to 30 %) highlights the advantage of the dilatation factor: the predicted stress distribution and cross‑sectional deformation match analytical expectations, whereas a standard rigid‑section LGVI exhibits noticeable errors. (3) A complex initial configuration combining bending, twisting, and stretching confirms that the integrator maintains the rotational configuration, conserves linear and angular momentum, and exhibits long‑term energy stability.

The authors discuss limitations: the assumption that the cross‑section remains circular restricts applicability to cases without warping or non‑circular shapes; material behavior is limited to linear isotropic hyperelasticity without explicit Poisson‑ratio effects; and the model does not include additional strain variables for shear of the cross‑section itself. Future work is suggested to extend the framework to non‑circular cross‑sections, incorporate nonlinear constitutive laws, and couple the LGVI with optimal control formulations for soft‑robotic applications.

Overall, the paper delivers a rigorous continuous derivation and a structure‑preserving discrete algorithm that together enable accurate, energy‑stable simulations of slender rods whose cross‑sections can dilate or contract, filling a notable gap in the computational mechanics of flexible structures.