DtN-based mixed-dimensional coupling using a Boundary Stress Recovery technique

A new computational approach for the mixed-dimensional modeling of time-harmonic waves in elastic structures is proposed. A two-dimensional (2D) structure is considered, which includes a part that is assumed to behave in a one-dimensional (1D) way. The 2D and 1D structural regions are discretized by using 2D and 1D Finite Element (FE) formulations. The hybrid model, if designed properly, is much more efficient than the standard 2D model taken for the entire problem. One important issue related to such hybrid 2D-1D models is the way the 2D-1D coupling is done, and the coupling error generated. Here, three closely-related coupling methods are considered. They are all based on the Dirichlet-to-Neumann (DtN) map associated with the 1D problem, on the interface. The first coupling method is the DtN method in its usual form, where the DtN map is calculated numerically, and where the 1D and 2D problems are solved separately. The second coupling method is the one devised by Carka, Mear and Landis (CML), which is equivalent to the former one, but in which the 2D and 1D interface solutions are solved for simultaneously. In the third coupling method, the DtN map is enforced on the interface iteratively. Direct application of these three methods results in low accuracy, as a recent study shows. Therefore, these methods are used here in conjunction with a Boundary Stress Recovery (BSR) technique, originally proposed by Hughes, which provides the same order of accuracy for the stress as for the primary variable. The performance of the three methods is demonstrated and they are compared via numerical examples. Conclusions are drawn on their relative merit.