Geometrically Explicit Cosserat-Rod Modeling with Piecewise Linear Strain for Complex Rod Systems

Abstract

This paper presents a geometrically explicit formulation for Cosserat rods that unifies configuration-space and strain-based representations within a single modeling framework. The proposed method uses nodal configurations on the Lie group SE(3) as generalized coordinates, while internal strains are reconstructed via a piecewise-linear parameterization. This hybrid design preserves the geometric rigor of Lie-group formulations and retains the locality, simplicity, and computational efficiency characteristic of strain-parameterized rod models. The formulation naturally avoids shear and membrane locking without additional stabilization techniques, and it accommodates arbitrary rod networks, closed-loop architectures, and gridshell-like structures through element-wise assembly. A Riemannian Newton solver is further developed to solve the equilibrium equations directly on SE(3), providing rapid convergence and consistent treatment of rotations. Numerical examples demonstrate that the method achieves high accuracy with only a few elements and generalizes seamlessly from single rods to complex multi-rod systems. These properties highlight the potential of the proposed formulation as a fast, robust, and scalable simulation tool for slender mechanisms and compliant structures.

Figures (16)

• Figure 1: Topology of a rod network with $6$ nodes and $7$ elements. The network consists of nodes and elements; each element connects two nodes and carries one independent strain-slope parameter. All incident elements share one node.
• Figure 2: Riemannian optimization scheme. An update is computed in the tangent space $T_{x_k}\mathcal{M}$ and mapped back to the manifold via the retraction $\mathrm{Ret}_{x_k}(\delta x)$, producing the next iterate $x_{k+1}$ toward the solution $x^\ast$.
• Figure 3: Comparison of deformation results between the proposed two elements (CSE and LSE) and the ground truth (GT).
• Figure 4: Cantilever 45-degree bend subjected to a fixed load.
• Figure 5: Tip-displacement error $e_p$ versus the total DoFs for CSE and LSE (reference: dense mesh with $1000$ elements).

Theorems & Definitions (8)

• Definition 1: Homogeneous transformation matrix
• Definition 2: Strain of Cosserat rod
• Definition 3: Retraction map
• Definition 4: Riemannian gradient
• Remark 1
• Remark 2
• Remark 3
• Remark 4