On the Lagrangian-Eulerian Coupling in the Immersed Finite Element/Difference Method

The immersed boundary (IB) method is a non-body conforming approach to fluid-structure interaction (FSI) that uses an Eulerian description of the momentum, viscosity, and incompressibility of a coupled fluid-structure system and a Lagrangian description of the deformations, stresses, and resultant forces of the immersed structure. Integral transforms with Dirac delta function kernels couple Eulerian and Lagrangian variables. In practice, discretizations of these integral transforms use regularized delta function kernels, and although a number of different types of regularized delta functions have been proposed, there has been limited prior work to investigate the impact of the choice of kernel function on the accuracy of the methodology. This work systematically studies the effect of the choice of regularized delta function in several fluid-structure interaction benchmark tests using the immersed finite element/difference (IFED) method, which is an extension of the IB method that uses finite element structural discretizations combined with a Cartesian grid finite difference method for the incompressible Navier-Stokes equations. Further, many IB-type methods evaluate the delta functions at the nodes of the structural mesh, and this requires the Lagrangian mesh to be relatively fine compared to the background Eulerian grid to avoid leaks. The IFED formulation offers the possibility to avoid leaks with relatively coarse structural meshes by evaluating the delta function on a denser collection of interaction points. This study investigates the effect of varying the relative mesh widths of the Lagrangian and Eulerian discretizations. Although this study is done within the context of the IFED method, the effect of different kernels could be important not just for this method, but also for other IB-type methods more generally.