Hybrid finite difference/finite element version of the immersed boundary method

The immersed boundary (IB) method is a framework for modeling systems in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems describes the elasticity of the structure in Lagrangian form and describes the momentum, viscosity, and incompressibility of the fluid-structure system in Eulerian form. Interactions between Lagrangian and Eulerian variables are mediated by integral transforms with delta function kernels. When discretized, the Lagrangian equations are approximated on a curvilinear mesh, the Eulerian equations are approximated on a Cartesian grid, and a regularized version of the delta function is used in approximations to the Lagrangian-Eulerian interaction equations. Here, we employ a version of the IB method that allows us to discretize the structure via standard Lagrangian finite element (FE) methods. Unlike most other extensions of the IB method that use FE structural discretizations, however, our approach retains a finite difference discretization of the incompressible Navier-Stokes equations. A key feature of our numerical scheme is that it enables the use of Lagrangian meshes with mesh spacings that are independent of the grid spacing of the background Eulerian grid. Results from computational experiments are included that demonstrate the accuracy and efficiency of our methodology. Copyright c © 0000 John Wiley & Sons, Ltd.