A well-behaved grid-size-insensitive projection immersed boundary method for fluid-structure interaction problems

The projection immersed boundary method (IBM) has been successfully adopted in the simulations of compressible and incompressible flow over stationary and moving objects. The standard approach transfers the boundary conditions into the Navier-Stokes equations directly (as body forces) resulting in differential-algebraic equation (DAE). However, the method requires that the size of the surface (Lagrangian) grid be comparable to the size of the flow (Eulerian) grid. The coupled equations are otherwise ill-conditioned if surface grid is much finer, rendering fluid-structure interaction simulations with large deformation almost impossible. Here, we propose a modified projection method that removes this limitation and allows the surface grid to be much finer than the fluid grid while keeping the system of equations well-posed. The equations for the body force are modified using Tikhonov regularization, with the graph Laplacian used to formulate the penalty term. The regularized equations are better conditioned and produce significantly smoother solutions. This improvement makes the projection IBM applicable to more complex problems without manual regriding or user intervention. Various examples will be presented to demonstrate the versatility and accuracy of the new method.