A Quasi Energy and Momentum Conservative Algorithm Implemented With a Co‐Rotational Quadrilateral Shell Element Formulation Using Vectorial Rotational Variables - Li, Zhongxue; Lin, Xunda; Vu‐Quoc, Loc; Izzuddin, Bassam A.; Wei, Haoyan; Xu, Jin; Qian, Hongtao; Zhuo, Xin (International journal for numerical methods in engineering, 2025)
ABSTRACT This paper proposes a flexible multi‐body dynamics approach for elastic smooth and non‐smooth shells undergoing large deformations and large overall motions. The formulation is based on a co‐rotational curved quadrilateral shell element employing vectorial rotational variables and a quasi energy and momentum conservation algorithm. Hamilton's principle is adopted to derive the system's dynamic differential equations. When differentiating the kinetic energy functional with respect to time, the part involving the first‐order differentiation of vectorial rotational variables with respect to time is integrated into an equivalent load vector, yielding a symmetric equivalent mass matrix. Accelerations, velocities, displacements, body, and surface loads of the generalized midpoints are generated by convex functions. The Newmark scheme is applied to transform the dynamic differential equations of the system into a set of nonlinear equations. Instead of using strains and their first/second derivatives with respect to local nodal variables at the midpoint configuration, the formulation employs time‐averaged assumed strains (computed via the MITC method) and their corresponding derivatives evaluated at both ends of the time step for calculating the internal force vector and element tangent stiffness matrix in the local coordinate system. The transformation matrix from local to global coordinates, however, remains computed at the midpoint configuration. This approach ensures near‐exact conservation of total energy and exact conservation of linear and angular momenta once the external loads vanish, while also yielding symmetric tangent stiffness matrices in both local and global coordinate systems. Finally, four examples of three smooth shells and one non‐smooth shell problems subjected to impulse loads are solved to verify the proposed formulation for flexible multi‐body dynamics of shells. It is shown that the results exhibit excellent agreement with those from other references, demonstrating the reliability, accuracy, and long‐term stability of the proposed quasi energy and momentum conserving algorithm.