Immersed Boundary methods are an active field of research in CFD since its introduction by C. Peskin [ADD REF HERE]. This kind of approach deals with problems involving interfaces (fluid/solid or free surface for instance). So as to simplify the mesh generation in case of complex geometry and/or moving boundaries, the interface is known implicitly on the grid (structured or unstructured) that is thus generated independantly of the geometry.
During my PhD at INRIA Bordeaux Sud Ouest (FRANCE), I worked with a immersed boundary method called Penalization in the context of rigid bodies in motion. The aim was to perform simulations of imposed moving motion (oscillating naca airfoil for instance) as well as fully fluid structure interaction problems where the motions of the solid is ruled by gravity and the force exerted by the fluid on the solid. Although the imposition of the boundary condition is only first order, the strategy was to compensate this loss of accuracy by using mesh adaptation startegy.
My current research at Duke University (USA) brings contributions to the development of the Shifted Boundary Methods [ADD REF HERE]. I apply this technique for free surface flow problems and Poisson problem with a specific emphasize of the treatment of Neumann type boundary conditions.
During my PhD, so as to perform the fluid structure interaction simulations with the immersed boundary method, mesh adaptation has been employed so as to increase the solid definition (that is known implicilty) and improve the solution.
For steady simulations, a h-adaptation has been employed. This kind of adaptation consists in remeshing. The aim is to provide to the remesher a metric field for the willing refinement. Those metrics are computed according to the level set and the solution so as to remove extra unecessary nodes where the solution is almost constant and to perform refinement close to the solid boundary with two aims. First, refining in close to the solid boundary so as to recover at best the solid geometry and impose properly the boundary conditions. Secondly, remove extra unecessary nodes where the solution is almost constant and refine in areas of large physical variations.
For unsteady simulations, an r-adaptation strategy that allows to refine the mesh with constant connectivity can be employed. This kind of adaptation combined with ALE scheme allows to perform adaptive simulation without any kind of remeshing/interpolations. Two approaches to adapt the mesh have been studied. With the first one, the mesh is assimilated to a material ruled by the elasiticy equations. Defining properly the forces applied on this "material" provides refinement close to the solid boundary and to the physics of the flow. The second approach consists in considering a monitor function that would equidistribute the node of the mesh. The problems solved in this situation is a "Laplacian like" model.
Darcy flow problem is of great interest for several reasons. First it is itself a problem of value for its application of flow in porous media. Secondly, every development for Darcy flow can easily be applied to problems involving second order derivative. Indeed, those problems can be solved in a mixed form, meaning that the gradient of the solution is considered as an unknown and the problem becomes a first order system of equations. For instance, the Laplace problem becomes (with some sign difference) the Darcy flow problem.
I am currently working at Duke on the development of several numerical schemes for solving this problem. First applying a simple Galerkin method, the aim is to recover a second order convergence rate on both pressure and flux by performing some pressure enrichment. This allows to apply the Shifited Boundary Method with second order accuracy on the pressure for both Dirichlet and Neumann boundary conditions.
As it is intersting to study the flow in media of different porousity, meaning a discontinuity in the permeability, it is interesting to propose discontinuous solution. For that, Discontinuous Galerkin or Enriched Galerkin schemes can be employed. However, the drawback of such approach is their high increase in the number of degrees of freedom and thus in the size of the system to solve. One of the work proposed at Duke is to perform a genuily mapping from a discontinuous discretization to a continuous one allowing to solve a system for only continuous discretization degrees of freedom. The discontinuous solution is then recovered through the mapping.