One way of dealing with fluid-structure interaction problems with large deformations is to use fictitious domain method. A fixed fluid mesh is used while the one for the structure is deforming. The two meshes are thus independent. One of the major drawback of these two non-fitted meshes is that the numerical fluid solution of a fluid-structure interaction problem can not be accurately approximated. Indeed, these kind of solutions have strong discontinuities in the pressure across the structure and weak ones in the fluid velocity that can not be well approximated with classic affine finite element. To circumvent this issue and to recover the optimal order obtained with two fitted-meshes, one can seek a smooth extension of the solution within the fictitious part of the mesh. This smooth extension is non-physical in the fictitious domain but approximates well the exact solution in the physical domain.
In my PhD, to simulate the motion of immersed rigid bodies, we developed a method on cartesian meshes to find a smooth extension of the solution. The fictitious domain is a cube embedding the system of the fluid and the rigid bodies. The smooth extension is found by solving the fluid problem in the whole domain with a well-chosen extension of the right-hand side. By minimizing a cost function, it is indeed possible to find an extension of the right-hand side such that the fluid solution in the whole domain (which is smooth) is also the exact solution in the physical domain. I have written, from scratch, in collaboration with Loic Gouarin, a parallel C++ code that implement this method.
In my current post-doc, the problems are different. We are aiming at simulating the motion of the heart valves in the blood flow. The structure is a thin-walled membrane and the fluid has discontinuities in the pressure (strong) and the fluid (weak) across this structure. The fictitious part here comes from the XFEM method. The elements of the fluid mesh that are intersected by the structure are duplicated, extending the solution from each side of the structure. The coupling conditions of the fluid-structure problem are enforced in a weak way with Nitsche's formulation. Thus, the variational formulation requires to compute integrals over cut-elements. A mesh intersection algorithm, developed by Frédéric Alauzet at Inria Paris-Rocquencourt, is used to sub-triangulate the elements intersected by the structure. I am implementing this method in the C++ library FELiScE developed at Inria.
I worked on the nasal airflow during my first post-doc at Inria. The aim is to study the valve region and the effect of a surgery on the airflow. From medical imaging, a 3D mesh of the nasal cavity is computed. Navier-Stokes equations are solved in this domain with projection methods that we have implemented in the FELiScE library. From this first solution, the mesh is adapted with respect to the direction of the flow. This is done with the library feflo developed by Adrien Loseille at Inria.
I worked with Sanjay Pant and Irène Vignon-Clementel on parameter estimation for the analysis of haemodynamic in patient-specific data. Our interest is to estimate the pressure drop across a coarctation of an aorta. The geometry of the aorta comes from medical imaging and experimental data are available. The estimation of the parameter is performed on a reduced order (0D) representation of the model with the Unscented Kalman Filter (UKF). The Navier-Stokes equations are solved in the 3D geometry with the library FELiScE. An iterative procedure between the reduced order model and the 3D model is performed and suitable parameters are found.