Mixed-Dimensional Modeling for Time-Dependent Waves

Dan Givoli

Technion – Israel Institute of Technology

Monday 14th April 2025

14:00:00

Aula Saleri

In recent years, along with the increase in the size and complexity of computational models, the scenario of mixed-dimensional (e.g., 2D-1D or 3D-2D, and in general HighD-LowD) coupling has drawn a lot of attention. Fields of application where this scenario is of special interest include, among others, (a) blood-flow analysis, (b) hydrological and geophysical flow models, and (c) elastic structures. In this talk we will review and discuss the application of several coupling schemes to time-dependent wave problems governed by the scalar wave equation and by the equations of elastodynamics. After providing the motivation for mixed-dimensional modeling, we will consider the hybrid problem at the continuous level. We will discuss the appropriate interface conditions and prove the well-posedness of the hybrid problem. Among the coupling methods for mixed-dimensional models, we will discuss the Panasenko method, the Nitsche method and two versions of the Dirichlet-to-Neumann (DtN) method. The various methods differ in the way they implement the coupling conditions. We will also discuss the appropriate placement of the HighD-LowD interface, and how this can be estimated a priori and a posteriori. The hybrid model is discretized by the finite element method. A special challenge is the attack of elastic problems involving out-of-plane bending. The difficulty lies in the fact that there is a mismatch in the type of unknown variables between the HighD and LowD models, and that there is a significant difference between the types of differential equations and between the types of finite elements (C0 vs. C1) used in each part of the problem. We will present an appropriate coupling approach for such problems. We will show the results of some numerical experiments using the various methods, and will compare between them in accuracy and efficiency. Contatti: luca.formaggia@polimi.it