A multisymplectic Lie algebra variational integrator for flexible multibody dynamics on the special Euclidean group SE (3)

A multisymplectic Lie algebra variational integrator (MLAVI) for simulating the dynamics of a flexible multibody system is proposed. The flexible bodies are described by the geometrically exact beam elements established on the special Euclidean Lie group SE(3). A lemma is also provided to demonstrate that the used discrete compatibility equation can make temporal and spatial interpolation independent. Then, the reduced Lagrangian density on the Lie algebra is derived, and the discrete dynamics of motions are established by the discretization of the covariant variational principle. Furthermore, another novel closed-form equality relation lemma on the right tangent maps of the associated exponential map and Cayley map is originally proved, which greatly helps to simplify the Jacobi matrix of the proposed MLAVI in the Newton iteration process. The accuracy and efficiency of the proposed MLAVI are validated by six static and dynamic numerical examples. Numerical results show that the proposed MLAVI can preserve the system's symplectic structure, momentum, and energy for long-time simulation.