## Section: New Results

### Numerical schemes and algorithms for fluid mechanics.

Participants : Rémi Abgrall [ Corresponding member ] , Guilaume Baurin, Cécile Dobrzynski, Marc Duruflé, Algyane Froehly, Robin Huart, Arnaud Krust, Pascal Jacq, Cédric Lachat, Luc Mieussens (IMB/IPB), Pierre-Henri Maire (CELIA/CEA), Adam Larat, Mario Ricchiuto, Jirka Treffilick.

#### Residual distribution schemes

This year, many developments have been conducted and implemented in
the `FluidBox` software after [28] which has
opened up many doors.

We have extended the 3rd order RD scheme to Navier Stokes problems, as
well as to unsteady problems. Pascal Jacq and Cédric Lachat have
extended the communication scheme in `FluidBox` so as to handle high
order schemes.

Some difficulties have appeared due to the non positive nature of the Lagrange basis functions, in particular for unsteady problems. In order to overcome this problem, we have shown how to extend the method to non Lagrange basis, for example by means of Bézier approximation. Some very first results for unsteady problems have been given by J. Trefilik and confirm our expectations.

Mario Ricchiuto is conducting an analysis of the mass matrix for second order unsteady problems. The aim is to lower the number of operations by constructing an approximation of the scheme, which remains second order, but with a diagonal mass matrix. This should provide a much more efficient method than the one which was used before.

Guillaume Baurin has started his PhD. The goal is to extend our current (3rd order) methodology to multicomponent flows for SNECMA. Arnaud Krust has started his PhD, and, with G. Baurin, is examining several algorithmic solutions to the discretisation of the viscous terms vis RD scheme. Algyane Froehly has started her PhD and is studying other than Lagrange elements in the context of RD schemes for compressible fluid flow problems.

Adam Larat has finished and defended his PhD on high order RD schemes with the first applications to 3D and Navier Stokes applications. This has been done in the context of the ADIGMA project.

Algyane Froehly has started her PhD wich topic is the conception of RD schemes using non lagrange elements like Bezier elements or NURBS. Arnaud Krust has started his PhD in studying the approximation of the navier Stokes equations using solution-dependant elements.

Mario Ricchiuto and Luc Mieussens have started to study how to combine Residual Distribution schemes and Asymptotic Preserving scheme methodologies.

#### Uncertainty quantification

R. Abgrall has started to develop a strategy for computing some statistical parameters that need to be introduced because some elements of a physical model are unknown. For example, the boundary might be uncertain because of imperfections, or the inflow boundary conditions, or some parameters describing the equation of state or a turbulent model. In the approach we are working on, the main parameters will be the conditional expectancy of the fluid description (density, velocity, pressure). The approach is non intrusive. For now, some encouraging but preliminary results have been obtained for scalar hyperbolic and parabolic models.

#### Discontinuous Galerkin schemes, New elements in DG schemes

The earlier versions of `Montjoie ` could only handle hex and quadrangles.
However the generation of purely hexahedral meshes is still
challenging, that's why we have studied, in collaboration with Morgane Bergot, finite elements methods able
to handle hybrid meshes, including hexahedra, prisms, pyramids and
tetrahedra. Contrary to the other elements, finite element space for
pyramids is non-polynomial, and in [29] , the optimal
finite element space is discussed for pyramids with an exhaustive
comparison of all previous works about this issue.

These elements
have been implemented in `Montjoie ` and applied to wave equation and
time-domain Maxwell equations. The results showed the advantage to
use hybrid meshes when no nice hexahedral mesh was
available. These results have been conducted for continuous finite
elements and with discontinuous Galerkin method as well.

Rémi Abgrall and Pierre-Henri Maire, with François Vilar (PhD at CELIA funded by a CEA grant started in october 2009), have started to work on Lagrangian schemes within the Discontinuous Galerkin schemes.

#### Mesh adaptation

C. Dobrzynski has worked on fully parallel mesh adaptation procedure that uses standard sequential mesh adaptation codes. The idea is to adapt the mesh on each processor without changing the interfaces, after which interfaces are modified. The main advantage is simplicity, because there is no need to parallelize mesh generation tools (insert/delete, swap, etc). The main techniques are described in [31] , [32] .

C Dobrzynski has also developed an efficient tool for handling moving
2D and 3D meshes. Here, contrarily to most ALE methods, the
connectivity of the mesh is changing in time as the objects within the
computational domain are moving. The objective is to guaranty a high
quality mesh in term of minimum angle for example. Other criteria,
which depend on the physical problem under consideration, can also
been handled. Currently this meshing tool is being coupled with
`FluidBox` in order to produce CFD applications. One target example is
the simulation of the 3D flow over helicopter blades.

We also have started to work on the definition of an anisotropic metric which is computed from the output of a Residual distribution code. Once this will be done, standard mesh adaptation method will be used so that the numerical error of the solution is controlled.

Moreover, a work on high order mesh generation has begun. We are modifying the classic mesh operators to take into account the curve edges. Beginning with a derefined valid curve mesh, we would to be able to generate an uniform refined curve mesh and also to adapt the mesh density in certain region (boundary layer).