Elastodynamic 2D-1D coupling using the DtN method

Coupling of a 2D model and a 1D model, to form a hybrid mixed-dimensional model, is considered in the context of elastic wave propagation. A Dirichlet-to-Neumann (DtN) method is used to perform this coupling. This approach is an extension of previous work (which was applied to steady-state wave problems) to the time-dependent regime. It is based on enforcing the continuity of the DtN map, relating the displacements to the tractions, on the 2D-1D interface. To apply the DtN map, the approach of discretization in time first (the Rothe method) is adopted, resulting in an elliptic problem at each time step. The more typical case, where longitudinal waves dominate in the 1D sub-domain, is considered first. Then the more general case is considered, where transverse waves are present as well, and several ways to handle it are discussed. The proposed DtN approach is compared to the simpler Panasenko semi-weak approach, and is shown to be advantageous, in particular in the presence of transverse waves.