A major computational issue rises in the study of the properties of complex fluids generally involving a microstructure (e.g. red blood cells) interacting with a suspending fluid. Solving this fluid-structure interaction problem in our applications often involves mathematical challenges related to the constraint of inextensibility of the structure (e.g. RBC membrane). Different methods are deployed, like Boundary Integral (BI)a, Lattice Boltzmann (LB)b, Level Set (LS) methods coupled with finite elements, or fluid particle dynamics (FPD)d. Each method has its own advantages and drawbacks, so that a given problem requires the adoption of an appropriate approach. For example, the BI is very precise but limited to Stokes flow and simple geometries, while LB is appropriate for complex geometries (e.g. complex microvasculature) and lends itself to massive parallel computing, while it may lead to artefacts due finite compressibility. FPD is adopted for rigid particle suspensions, especially in the study of microswimmers. Finally, LS is quite versatile and allows for treating inertial effects, but the diffuse-like nature of the interface makes the results often sensitive to the width over which the interface is represented. Cross checking of our results using different methods is a common practice that enables ruling out spurious effects associated with a given technique.
Other theoretical issues include nonlinear rheology problems, based on homogenization approaches, and inverse problems allowing extraction of traction forces of cells from the knowledge of displacements determined experimentally.
Finally, we take part in the development of two large scientific computing libraries for partial differential equation (PDE) problems in 1D, 2D and 3D : Feel++g, which features high performance computing (HPC) with a massively parallel hybrid architecture, and Rheolefh, which emphasizes complex rheology flow problems. They can be exploited on our in-house PC cluster and on those of mesocentre Ciment.
One of the main advantage of level set methods is that it handle implicitly topological changes of the domains. Moreover, one can add many interfaces with only one level set and no special care has to be taken for the breaking / collapsing of the two fluids. We took the previous parameters and added many bubble or many drops in the same domain.