An Arbitrary Lagrangian–Eulerian formulation of a geometrically exact Timoshenko beam running through a tube

When a beam moves through a curved tube, there exists a large number of contacts between the beam and the inner wall of the tube. Usually, it is necessary to use fine Lagrangian meshes to discretize the beam for possible contact area in order to obtain sufficiently accurate results, even if the contact is only once for a very short time. Accordingly, the computation process is very time-consuming due to the large number of degrees of freedom (DOF) of the discretized system. To solve this problem, an Arbitrary Lagrangian–Eulerian (ALE) formulation of a Timoshenko beam based on geometrically exact beam theory, which considers the rotation around the axis of the beam, is presented in this paper. In this formulation, the mesh nodes of the beam are not associated with the material points, which provides the flexibility to mesh the beam freely. For this reason, the ALE mesh nodes can be arranged along the tube and constrained to move just within the corresponding tube cross section. The axial movement of the beam can be described by the material points flowing through the axially constrained mesh nodes. In so doing, only the beam in the high-curvature section of the tube needs to be finely meshed, resulting in a significant reduction of the DOF of the whole system. Several examples are presented and discussed to demonstrate the correctness and efficiency of the proposed method.