Multiphase problems are characterized by the interaction of two or more phases which are separated by deformable interfaces. The interface deformation introduces geometric nonlinearities and topological changes which make the numerical solution of these problems a very challenging task. The traditional approach consists in formulating the model equations first, and then to discretize them using, for example, finite differences, finite elements, or finite volumes. Available methods are either efficient or accurate, but unfortunately not both. A reason may be that the discrete equations inevitably include various approximations and they do not always preserve the essential mathematical properties of the original continuous system.
The main idea of discrete differential geometry is to reformulate the original continuous equations in such a way that its discretized version maintains the essential mathematical features of the continuous problem. This area of research is still at its early stages. Some work has been done for the solution of classical fluid flow problems and elasticity problems, but the treatment of problems with interfaces (e.g. fluid-solid interactions) still remains open.
This workshop will offer a unique opportunity to explore the capabilities of discrete differential geometry for the solution of multiphase flow problems which, hopefully, will lead to the development of a new generation of both accurate and efficient numerical methods. The workshop will gather scientists with different expertises (engineers, computer scientists and mathematicians) from both IUPUI and other US institutions. In particular we will be discussing fluid-structure interaction problems arising in Medicine and Life Science, such as the interaction between blood flow and compliant vessel walls, and the deformation of red blood cells caused by blood flow.