Computational methods for multiphysics interface problems

Many problems in science and technology includes a fixed or moving boundary over which
two different physical systems are coupled. This situation is particularly common in systems in
medicine and biology, for instance: in the human arteries the fluid dynamics of the blood couples to the solid dynamics of the arterial wall, in rivers and estuaries the free flow couples to the porous media flow in the infiltrated river bed. Making accurate computational predictions of the evolution of such systems remains an important challenge for engineers and the accurate mathematical analysis of the associated methods is even more daunting. Indeed no known methods allow for rigorous mathematical analysis and many suffer from problems of stability or accuracy depending on the orientation of the interface. Numerical computations are most often performed on a computational mesh, that is a decomposition of the computational domain in a large number of small building blocks, so called elements. An important feature of the methods that we propose is that the interface may cut through the elements of the computational mesh, or in other words, the computational mesh does not need to fit the interface.

In multiphysics problems the situation is often complicated by the fact that the computational mesh may not be adapted to fit the interface, but the coupling of the two systems must take place independent of the mesh. This is the type of situation that we aim to study in the present project. New approaches will be designed for multiphysics couplings over moving interfaces. The mathematical methods will be designed so as to be robust and accurate and we will also explore the possibility to decouple the two systems for efficient time advancement. This may lead to very important savings in computational time, in particular for nonlinear problems.

Three important model cases will be considered: the coupling of two fluids of which one or both may be viscoelastic, the coupling of free flow and porous media flow and finally the coupling of a fluid and an elastic structure. All of these applications have important applications in the modeling of the human cardiovascular system, but also in a wide variety of other applications such as ink-jet printers, environmental science, chemical industry and so on.

An important bottle neck in the efforts to make cutting edge computational methods to bear on important problems in applications is the interfacing that has to be done between on the one hand images and experimental data, for instance geometries of arteries or cerebral vascular structure, and on the other hand the computational codes. An important disadvantage here is the need to go through the double interfacing process of first creating a computational mesh, and then passing this mesh to the computational code. Note that the meshing of complex geometries is a complicated process that can fail, or give rise to very poor quality computational meshes. If the meshing step can be circumvented without compromising accuracy or stability this would be a huge step forward in the efforts to bring computation closer to applications. In the present framework where the mesh does not need to be fitted to interfaces and boundaries this meshing problem is circumvented, since it is always possible to work on a simple structured mesh.

The same double interfacing problem exists in engineering applications where the geometry is given by a CAD drawing. CAD-drawings are in general too imprecise to be used directly in computation or production. Moreover they do not use a representation of surfaces that is suitable for mesh design. Both these problems are solved in our framework, since we only need information on where the surfaces are in order to integrate over them. The representation is irrelevant, as is the precision, provided the mismatch between surfaces remain smaller than the mesh size.

There are countless applications where the present technology is of importance, but let us discuss two. Firstly, in the computation of cardiovascular flows, geometries are patient specific and obtained from experimental data (scans). Here meshing is very awkward, in particular since the experimental 3D image does not necessarily represent an outer domain, but an internal boundary separating for instance the fluid domain from the solid domain of the arterial wall. During the simulation the interaction between the fluid and the solid is extremely important and results in a displacement of the wall, requiring a displacement of the computational mesh if it is to remained fitted to the interface. The flow most also be simulated both in the artery and in the arterial wall resulting in the coupling of Navier-Stokes' equations and Darcy's equation across the interface. In our framework no mesh movement is necessary since the underlying approximation space and variational formulation adapts to fit the moving interface.

Another example in the same framework is the accurate simulation of heartvalves. The design and optimization of artificial heartvalves require accurate tools for simulation, here the valves will move with the fluid, but also open and close, leading to complex contact problems. The approach of re-meshing in order to follow the valves movements is practically impossible, due to the closing of the valve. In general remeshing will fail for problems experiencing some topological change or contact in the interface configuration. Computer based simulations of blood flows, in patient-specific geometries, can provide valuable information to physicians in order to enhance therapy planning. Such
simulations can also be a major ingredient in the design/optimization of medical devices. Moreover, the emerging interest in improving clinical diagnosis through model
personalization (i.e., solving inverse problems coupling clinical data and FSI models)
clearly demands further developments of efficient numerical methods.