A Mass Conservative Scheme for Fluid–Structure Interaction Problems by the Staggered Discontinuous Galerkin Method

In this paper, we develop a new mass conservative numerical scheme for the simulations of a class of fluid–structure interaction problems. We will use the immersed boundary method to model the fluid–structure interaction, while the fluid flow is governed by the incompressible Navier–Stokes equations. The immersed boundary method is proven to be a successful scheme to model fluid–structure interactions. To ensure mass conservation, we will use the staggered discontinuous Galerkin method to discretize the incompressible Navier–Stokes equations. The staggered discontinuous Galerkin method is able to preserve the skew-symmetry of the convection term. In addition, by using a local postprocessing technique, the weakly divergence free velocity can be used to compute a new postprocessed velocity, which is exactly divergence free and has a superconvergence property. This strongly divergence free velocity field is the key to the mass conservation. Furthermore, energy stability is improved by the skew-symmetric discretization of the convection term. We will present several numerical results to show the performance of the method.     

A Monolithic Approach to Fluid–Composite Structure Interaction

We study a nonlinear fluid–structure interaction (FSI) problem between an incompressible, viscous fluid and a composite elastic structure consisting of two layers: a thin layer (membrane) in direct contact with the fluid, and a thick layer (3D linearly elastic structure) sitting on top of the thin layer. The coupling between the fluid and structure, and the coupling between the two structures is achieved via the kinematic and dynamic coupling conditions modeling no-slip and balance of forces, respectively. The coupling is evaluated at the moving fluid–structure interface with mass, i.e., the thin structure. To solve this nonlinear moving-boundary problem in 3D, a monolithic, fully implicit method was developed, and combined with an arbitrary Lagrangian–Eulerian approach to deal with the motion of the fluid domain. This class of problems and its generalizations are important in e.g., modeling FSI between blood flow and arterial walls, which are known to be composed of several different layers, each with different mechanical characteristics and thickness. By using this model we show how multi-layered structure of arterial walls influences the pressure wave propagation in arterial walls, and how the presence of atheroma and the presence of a vascular device called stent, influence intramural strain distribution throughout different layers of the arterial wall. The detailed intramural strain distribution provided by this model can be used in conjunction with ultrasound B-mode scans as a predictive tool for an early detection of atherosclerosis (Zahnd et al. in IEEE international on ultrasonics symposium (IUS), pp 1770–1773, 2011).