A numerically efficient scheme for elastic immersed boundaries

The main approaches to simulate fluid flows in complex moving geometries, use either moving-grid or immersed boundary techniques [5, 6, 7]. This former type of methods imply re-meshing, which are expensive computationally in the fluid/elastic-structure interaction cases that involve large structure deformations. In contrast, in the immersed boundary techniques, the effect of the boundary is applied remotely to the fluid by a constraint/penalty on the governing equations or a locally modified discretization/stencil: the fluid mesh is then globally independent of the moving interface, described by Lagrangian coordinates, and the effect of the interaction is introduced into the fluid variables at the Eulerian grid points next to the interface. Many applications of fluid/flexible-body interaction simulations with large deformation are in bio-engineering. The accuracy of the input data in such a problem is not very high and one may prefer to emphasis the robustness of the numerical method over high accuracy of the solution process. A major advantage of the Immersed Boundary Method (IBM), pioneered by C.S. Peskin [10], is the high level of uniformity of mesh and stencil, avoiding the critical interpolation processes of the cut-cell/direct methods. Based on the standard finite-difference method, the IBM allows highly efficient domain decomposition techniques to be implemented. In other words, the difficulty of simulating dynamical interaction phenomena with complex geometries can be overcome by implementing, in a fast and easy way, large fine grid parallel computations that takes full advantage of a uniform stencil on an extended regular domain, as described in [3, 4], for blood flow applications.We are first going to recall the IBM formulation.