Comparison of two-dimensional- one-dimensional coupling methods for time-harmonic elasticity

The coupling of two-dimensional (2D) and one-dimensional (1D) models in time-harmonic elasticity is considered. The hybrid 2D-1D model is justified in the case where some regions in the 2D computational domain behave approximately in a 1D way. This hybrid model, if designed properly, is much more efficient than the standard 2D model taken for the entire problem. Two important issues related to such hybrid 2D-1D models are (a) the design of the hybrid model and its validation (with respect to the original problem), and (b) the way the 2D-1D coupling is done, and the coupling error generated. The present paper focuses on the second issue. Three numerical methods are adapted to the 2D-1D coupling scenario, for elastic time-harmonic waves: the Panasenko method, the Dirichlet-to-Neumann (DtN) method, and the Nitsche method. All three are existing methods that deal with interfaces; however, none of them has previously been adopted and applied to the type of problem under study here. The accuracy of the 2D-1D coupling by the three methods is compared numerically for a specially designed benchmark problem, and conclusions are drawn on their relative performances.