A spacetime variational integration approach to the full discretization of flexible beams based on absolute nodal coordinate formulation

A spacetime variational integration approach for flexible beams is developed based on the absolute nodal coordinate formulation (ANCF), where the discretization is accomplished in a consistent way that regards displacements in both space and time. Specifically, the Hermite interpolation functions are used for spatial discretization, considering that the nodal position and its axial gradient are essential to interpolate displacements for ANCF beam elements. The quadratic polynomials with the use of variables at three consecutive time steps are adopted for temporal discretization. The discrete equations of motion are derived by a discrete variational principle based on the full spacetime discretization, which ultimately renders a variational integrator. Compared to variational integrators for the large rotation vector formulations and geometrically exact beam formulations, the present approach has the advantage of avoiding nonlinearity due to spatial rotational parametrizations as well as the complex expressions for inertia forces, such that computation complexity is significantly reduced. Numerical experiments are performed to validate the capabilities of the present spacetime variational integrator for flexible beams with large displacements, rotations, and deformations. The results reveal its good energy-momentum preserving behavior for conservative systems.