Mixed-dimensional analysis for coupling 2D elastodynamics and Timoshenko beam

Wave propagation is considered in a two-dimensional (2D) elastic structure, which includes a relatively small region whose behavior is fully 2D and a long and slender region whose bending behavior is like that of a Timoshenko beam. To save in computational effort, the latter region is reduced to a genuinely one-dimensional (1D) Timoshenko beam. The mathematical and computational problem posed then involves the coupling of the two regions, such that a well-posed, accurate, numerically stable and efficient hybrid 2D-elastic-Timoshenko-beam model is formed. The appropriate interface conditions are derived, and the well-posedness of the time-dependent problem is proved. A new computational coupling method is proposed, where the shape functions associated with the axial degrees of freedom on the interface of the elastic solid are modified in a special manner, to allow for the rotation continuity. The method results in a symmetric, positive and stable finite element formulation. Numerical examples are presented which demonstrate the performance of the scheme.