A Geometrical domain decomposition method in computational fluid dynamics

The domain decomposition (DD) method we present in this work aims solving incompressible flows around objects in relative motion. The DD algorithm is based on a Dirichlet/Neumann (Robin) coupling applied to overlapping subdomains. Hence, it is an extension of the classical Dirichlet/Neumann (Robin) method which uses disjoint suddomains. Actually, the field of application of this work is wider as it proposes to set up a possible theoretical framework for studying the overlapping extensions of classical mixed methods: the Dirichlet/ Robin,Dirichlet/Neumann, Robin/Neumann and Robin/Robin DD methods.
We observe that mixed DD methods inherit some properties of the Schwarz method while they keep the behavior of the classical mixed DD methods when the overlap tends to zero. As a main result, we show that the overlap makes the proposed methods more robust than disjoint mixed DD methods.
The DD method we propose is geometric and algorithmic. It is geometric because the partition of the computational domain is performed before the meshing, and in accordance to the DD coupling. It is also algorithmic because the solution of each subdomain is carried out by a Master code. This strategy is very flexible as it requires almost no modification to the original numerical code. Therefore, only the Master code has to be adapted to the numerical code and strategies used on each subdomain.
We present a detailed description of the implementation of the DD methods in the numerical framework of finite elements. We present interpolation techniques for Derichlet and Neumann data as well as conversation algorithms. Once the domain decomposition coupling and interpolation techniques are defined, we set up a Chimera method for the solution of the flow over objets in relative movements. Tensorial transformations are introduced to be able to express variables measures in one subdomain.
Finally, the DD algorithm is applied to an implicit finite element code for the solution of the Navier-Stokes equations and also of the Reynolds Averaged Navier-Stokes equations together with a one-equation turbulence model.