An Energy–Momentum Conserving Algorithm for Co-Rotational Quadrilateral Shell Elements in Nonlinear Multibody Dynamics

A new computational framework for nonlinear dynamic analysis of smooth shell structures is presented in this paper. The new framework is based on Simo & Tarnow’s energy–momentum conservation algorithm. A novel co-rotational nine-node quadrilateral shell element is embedded in the new framework. The dynamic equilibrium differential equations are derived using the Hamilton principle and solved by the Newmark algorithm. At each step, midpoint interpolation is applied to both nodal variables and their time derivatives. The average value of strains at the beginning and the end of each step is used to evaluate strain energy to obtain a symmetric tangent stiffness matrix. When deriving the kinetic energy functional, the first-order derivatives of vectorial rotational variables are embedded into equivalent nodal forces. Therefore, a symmetric equivalent mass matrix is generated. The symmetric stiffness and mass matrices significantly reduce the workload in solving the nonlinear governing equations. Benchmark validations reveal close agreement with results in the existing literature. The proposed algorithm is applicable for solving smooth shell structures undergoing large displacements and rotations within spatial domains, while maintaining unconditional stability and geometric exactness.