Mixed-dimensional coupling via an extended Dirichlet-to-Neumann method

Recently, a Dirichlet-to-Neumann (DtN) coupling method was proposed for mixed-dimensional modeling of timeharmonic wave problems. The original two-dimensional (2D) problem’s domain in this multiscale scenario is assumed to consist of two regions: a bulky one and a slender one. In a previous publication on the DtN coupling method, the problems considered were such that in the slender region, the exact solution approximately behaved in a one-dimensional (1D) way, namely its lateral variation decayed rapidly away from the wave source. In the present paper, a more general class of problems is considered. The computational domain still includes a slender region ("a long tail" or "a tree"), but the solution in that region does not necessarily behave in a 1D way. Such a persistent 2D behavior occurs for sufficiently large wave numbers, as is shown here. The DtN coupling method is extended for this more general situation. The problem in the slender part is reduced to a sequence of 1D problems. In the hybrid model, the bulky and slender regions are discretized by using 2D and 1D finite element formulations, respectively, which are then coupled together by employing on the interface the numerically calculated DtN maps associated with the 1D problems. To enhance the accuracy of the calculated DtN map, a boundary flux recovery technique is applied on the interface. The hybrid model is more efficient than the standard 2D model taken for the entire problem, yet its accuracy is not significantly lower. The performance of the method is demonstrated via numerical examples.