## 1 Introduction
To resolve this limitation, [PERSON] and coworkers [7] introduced an energy-momentum conserving algorithm, which ensures exact conservation of both energy and momentum within the Newmark algorithm framework by incorporating temporal averaging of stress tensors to define midpoint stress states. This advancement not only demonstrates unconditional stability in strongly nonlinear problems but also reveals design principles for conservative algorithms addressing geometric nonlinearity and establishes a theoretical foundation for further development of higher-order conservation algorithms. In 2019, [PERSON] et al. took energy conservation as a constraint condition and selected more than two interpolation points in the calculation of internal forces to im-prove calculation accuracy [8]. Their computational framework is similar to [PERSON]'s generalized energy-momentum conservation algorithm. [PERSON] et al. took the geometrically exact shell model as the basis [9], introduced [PERSON]'s energy-momentum conserving algorithm, and successfully calculated the mechanical responses of shell structures under dynamic loads. Recent research in flexible multibody system dynamics has shown enhancements to classical numerical algorithms, specifically addressing the geometric nonlinearities in governing dynamic equations. Among them, heterogenous asynchronous time integrators [10],composite time integration algorithms [11], and symplectic structure-preserving Hamilton algorithms [12] are the most successful ones. However, progressive stability deterioration in long-duration simulations of dynamic systems with geometric nonlinearity is still a challenge in flexible multibody system dynamics. The balance between computational accuracy and energy dissipation needs to be explored. To address the challenge of progressive stability deterioration, this study develops a numerical framework for large displacement dynamic analysis of thin-wall structures, which integrates a novel co-rotational nine-node quadrilateral element and the energy-momentum conserving algorithm. The proposed framework exhibits the following key features:  
1. The formulation achieves symmetric tangent stiffness matrices in both global and local coordinate systems by introducing vectorial rotational variables, which ensures direct additivity and updates [13]. 2. The nonlinear equation solving is significantly simplified by an incremental iterative algorithm, where nodal variable increments are directly superimposed onto the accumulated variables through algebraic summation operations. Therefore, complex iterative updates typically involved in conventional incremental approaches are avoided. 3. Extending Simo's energy-momentum conservation algorithm, a midpoint interpolation strategy is rigorously implemented for all strain measures in the strain energy functional. Conservation of energy and momentum is achieved with symmetric element tangent stiffness matrices. 4. The algorithm incorporates two parameters--spectral radius \(\rho_{\infty}\) and dissipation coefficient \(\xi\)--to control energy dissipation in generalized midpoint variables and their time derivatives. It prevents high-frequency energy oscillations while maintaining numerical stability through energy dissipation controls.