A Geometric Rotation-Dispersion-Guided Adaptive Co-Rotational Formulation for Flexible Multibody Dynamics

Implicit flexible multibody dynamics using three-dimensional continuum finite elements remains computationally demanding due to repeated nonlinear iterations in large-scale simulations. The component-level co-rotational (CR) continuum finite element approach alleviates this cost by assigning a shared rotating reference frame to all elements within a flexible component. However, this shared-frame assumption introduces a kinematic approximation when significant rotation differences exist among elements, and reliable criteria for reference-frame construction and approximation assessments are still lacking. This paper presents a systematic framework to control the approximation in component-level CR formulations. A deterministic quality-controlled algorithm is first developed to automatically select three reference nodes from arbitrary three-dimensional component meshes, considering component membership, geometric non-degeneracy, and mesh quality. Furthermore, a mass-weighted rotation-dispersion metric based on the geodesic difference between the relative component and element rotations is proposed to quantitatively evaluate the validity of the shared-frame assumption. The proposed framework is integrated into component-level CR flexible multibody analysis and validated through several numerical examples, including a three-dimensional pendulum, spinning top, flexible cantilever system, and slider-crank mechanism. Results demonstrate that the proposed reference-frame construction and rotation-dispersion assessment effectively characterize and control the kinematic approximation while retaining the computational advantages of component-level CR formulations. This work provides a practical criterion for the reliable application of component-level CR continuum finite elements in large-scale flexible multibody dynamics.