Section: New Results

Numerical schemes and algorithms for fluid mechanics.

Participants : Rémi Abgrall [Corresponding member] , Guillaume Baurin, Pietro Marco Congedo, Cécile Dobrzynski, Marc Duruflé, Dante De Santis, Algiane Froehly, Gianluca Geraci, Robin Huart, Arnaud Krust, Cédric Lachat, Mario Ricchiuto, Birte Schmidtman, Héloïse Beaugendre, Sébastien Blaise.

Residual distribution schemes

This year, many developments have been conducted and implemented in the RealfluiDS and SLOWS software after the initial ideas discussed in [55] and in [64] , [61] , [65] , which have opened up many doors.

First of all, the parallel three dimensional high order extension of the scheme of [55] has been finally validated on several external aerodynamics configurations[3] , including the classical ONERA M6 wing case on a large mesh containing $5.5{10}^6$ vertices (simulation run on 256 processors), and a business jet configuration in supersonic conditions, on a mesh obtained by the GAMMA3 EPI.

Meanwhile, the improvement of the treatment of viscous terms has been investigated within the PhD theses of G. Baurin and D. DeSantis [46] , [19] . The validation on laminar flows of a classical formulation based on a Petrov-Galerkin approach [60] [17] has shown its limitations. An improved formulation, based on a recovery of the solution gradient, has been proposed and tested. In both the second and third order cases, while showing the improvement in accuracy for steady state laminar flows, the results also show a slow iterative convergence, and a systematic small accuracy drop when the cell Reynolds number is of order one. These issues are currently under investigations, while the current formulation is being enhanced by adding a Spalart Almaras turbulent model. Contributions to these activities come from the PhD of Guillaume Baurin, who has implemented the third order version of our methods in a real industrial platform (N3S Natur of SAFRAN developped by Incka), and from the PhD of Dante DeSantis who is developing the turbulent implementation in RealfluiDS within the EU project IDIHOM.

Meanwhile, we are refining and validating the extension of the schemes to elements using improved approximations based on Bézier and NURBS polynomials. the initial two-dimensional implementation [56] , [59] of third and fourth order schemes on curved meshes is now being enhanced by adding a local mesh refinement procedure and is also currently being extended to three space dimensions. Contributions to this topic come from the PhD of Algiane Froehly.

R. Abgrall has extended the RDS formalism to Lagrangian hydrodynamics. The results are comparable to what can be obtained for more standard methods, a publication is in preparation.

Concerning time dependent flows, the ideas of [61] , [64] have led to two main lines of developments. On one hand, the unconditionally second order and stable space-time approach of [61] has been further validated [14] and extended to higher orders of accuracy [51] . The main advantage of this technique is its ability to preserve monotonicity unconditionally w.r.t. the time step. This has interesting applications in shallow water flows [36] in which the schemes previously developed [65] did allow to preserve the positivity of the water depth, however with an inefficient implicit procedure constrained by an explicit-type time step restriction.

In parallel, the genuinely explicit formulation of [64] has been combined with the positivity preserving approach of [65] to obtain a genuinely explicit positivity preserving scheme for shallow water simulations [16] . With a time step restriction quite close to that necessary for the scheme of [65] , the approach proposed allows a very efficient explicit time stepping with a tenfold reduction of the computational time for the same accuracy level.

These developments are implemented in the SLOWS platform and are thoroughly summarized in the manuscript [2] . We now dispose of a spectrum of numerical tools allowing either classical temporal integration based on implicit multistep schemes, or on unconditionally stable and positivity preserving space-time schemes, or on a genuinely explicit approach. Current developments aim at extending these tools to arbitrary accuracy, and at developing hybrid implicit/explicit approaches.

A Stabilized Finite Element Method for Compressible Turbulent Flows.

In this work, Héloïse Beaugendre, Boniface Nkonga and Christelle Wervaecke proposed a strongly coupled numerical formulation for the Spalart-Allmaras model, in the framework of Stabilized Finite Element Methods. Computations are performed for compressible Newtonian fluids (2D and 3D) on unstructured grids of high aspect ratio. Results are compared with experimental data and also with solutions obtained by different numerical strategies. The additional transport equations for subscale model are often numerically weakly coupled to Navier-Stokes equations through operator splitting. These variables are strongly coupled for the transport process within a stabilized finite element formulation. The stabilization tensor is defined, such as to reduce mesh dependencies and to still be consistent at the asymptotic of highly anisotropic meshes. Indeed, this tensor involves a measure of the local length scale $h$ which can be difficult to define in the case of a stretched element. In this work, the local length scale is implicitly given by the inverse of the absolute flux Jacobian matrix as proposed in Barth (1998) and more recently in Abgrall (2006). The stabilized finite element strategy is also suitable for complex geometries and the resulting schemes have a compact stencil which we exploit for efficient parallel strategies combining domain decomposition and message passing tools (MPI).

Uncertainty quantification

R. Abgrall and P.M. Congedo have made a detailed comparison between the semi-intrusive method developed last year with more classical non intrusive polynomial chaos methods, and Monte Carlo results. The effectiveness of this method is illustrated for a modified version of Kraichnan-Orszag three-mode problem where a discontinuous pdf is associated to the stochastic variable, and for a nozzle flow with shocks. The results have been analyzed in terms of accuracy and probability measure flexibility. Finally, the importance of the probabilistic reconstruction in the stochastic space is shown up on an example where the exact solution is computable, the viscous Burgers equation. These results have been reported in [25] , [47]

Following this studies, two contributions have been obtained within the context of Gianluca Geraci's thesis. First one is an adaptive strategy, inspired by the Harten multi-resolution framework that has been developed in order to compute efficiently statistics. This preliminary work aims to show the potentialities of this approach in order to evaluate the possibility to include this strategy in the semi-intrusive method developed in the recent years. We obtained [34] well-converged results with a lower computational cost due to a reduction of the numerical evaluations.

Second contribution [48] is a study concerning the Sparse Grid techniques coupled with Polynomial Chaos for multi-dimensional stochastic problems. Sparse grid techniques have been used to compute the multi-dimensional integrals needed to evaluate the coefficients of the polynomial expansion. We also investigated the possibility to reduce the number of random variables by means of an ANOVA analysis.

P.M. Congedo investigated the possibility to perform a stochastic inverse analysis by using an hybrid method within a Polynomial Chaos/Genetic Algorithms framework. This strategy has been applied on the numerical simulation of a dense gas shock-tube. Previous theoretical and numerical studies have shown that a rarefaction shock wave (RSW) is relatively weak and that the prediction of its occurrence and intensity are highly sensitive to uncertainties on the initial flow conditions and on the fluid thermodynamic model. The objective of this work has been to introduce an innovative, flexible and efficient algorithm combining computational fluid dynamics (CFD), uncertainty quantification (UQ) tools and metamodel-based optimization in order to obtain a reliable estimate for the RSW probability of occurrence and to prescribe the experimental accuracy requirements ensuring the reproducibility of the measurements with sufficient confidence.

Uncertainty quantification tools have been used to perform some applicative studies on epistemic uncertainties, in particular on some complex equations of state [8] , and some turbulence [12] models. We have also started considering the influence of model parameters uncertainties in free surface models for long-waves such as tsunamis [52] , coupling the numerics developed in the team for shallow-water flows 5.3 and the tools available for uncertainty quantifications. This is certainly a field of application where these developments will demonstrate very useful.

Within the associated team AQUARIUS activities (collaboration with Stanford University), two efficient global strategy for robust optimization have been developed. First one is based on the extension of simplex stochastic collocation to the optimization space, while the second one consists in an hybrid strategy using ANOVA decomposition. The Simplex Stochastic Collocation (SSC) method has been developed for adaptive uncertainty quantification (UQ) in computational problems with random inputs. In this work [30] , we showed how this formulation based on Simplex space representation, discretization of non-hypercube probability spaces and adaptive refinements can be easily coupled with a well-known optimization method, i.e. Nelder-Mead algorithm, also known as Downhill Simplex Method. Numerical results showed that this method is very efficient for mono-objective optimization and minimizes global number of deterministic evaluations in order to determine optimal design. This method has been then applied to a realistic problem of robust optimization of a two-component race-car airfoil.

We proposed also an efficient strategy [29] for robust optimization when a large number of uncertainties is taken into account. ANOVA analysis is used in order to perform a variance-based decomposition and to reduce stochastic dimension. A massive use of metamodels allows reconstructing response surfaces for sensitivity indexes and fitness function in the design variables plan. Proposed strategy has been applied to the robust optimization of a turbine cascade for thermodynamically complex flows.

Multiphase flows

Starting from [58] , R. Abgrall and H. Kumar are developping a methods that is able to compute multiphase flows when the interfacial area takes any value. In the previous version, either we could take into account infinite interfacial areas or pure interface problems. M.G. Rodio is developping, along similar lines, a scheme for the Navier Stokes equations.

Numerical schemes for advanced materials

Two parallel lines of work on developments of numerical models for advanced materials have seen important developments this year.

On one hand, Rémi Abgrall and Pierre-Henri Maire (CEA Cesta) are extending the Lagrangian method developped a couple years ago and currently implemented in the CHIC code to elastodynamics. The stress tensor is no longer diagonal and here we consider the Wilkins model. The main difficulty is to understand the role of the second principle and how to deal with the von Mises criteria.

In parallel, Mario Ricchiuto and the group led by Gérard Vignoles at LCTS (UMR-5801 LCTS) have been developing a finite element numerical model of the evolution of the liquid oxide evolution during the healing-phase taking place in the silicon-based composite materials similar to those used in SAFRAN's new aero-engines (http://www.safran-group.com/site-safran-en/innovation-429/areas-of-expertise/composite-materials/?432 ) [50] . This micro-model is meant to be used as a numerical closure for the LCTS' structural mechanics solver [31] , allowing to obtain a faithful description of the material's behavior, including the effects of the healing process.

Discontinuous Galerkin schemes, New elements in DG schemes

Rémi Abgrall and Pierre-Henri Maire (CEA CESTA), with François Vilar (PhD at CELIA funded by a CEA grant started in October 2009), are working on fully Lagrangian schemes within the Discontinuous Galerkin schemes. The idea is to start from the formulation of the Euler equation in full Lagrange coordinates: the spatial derivative are written in Lagrangian coordinates. The mesh element are now curved and we are working on the geometrical conservation law. The application to several standard test case indicate the potential of the method.

Penalization techniques with unstructured adapted meshes

Penalization methods are an efficient alternative to explicitly impose boundary conditions but their accuracy is generally of first order. In this work we combine the easiness of penalization techniques with the precision of unstructured anisotropic mesh adaptation. Level sets are used to describe the geometry so that geometrical and topological changes due to physics are straight forward to follow. Navier-Stokes simulations are performed and a new way to impose a slipping wall boundary condition is proposed.

Mesh adaptation

A work on high order mesh generation has been pushed further. Starting with a $P^1$ (triangle) mesh and some information on the boundary (control point), we are able to generate a valid third order curved mesh. The algorithm is based on edge swaps and is similar to a boundary enforcement procedures. This method is very robust but not efficient of the boundary layer. Indeed the edge swaps destroy a part of the boundary layer. To solve this problem, we investigate the use of linear elasticity to curve a $P^1$ mesh.

Moreover, we started to make high order mesh adaptation. That means we are able to refine high order meshes where the error is maximum and so we generate non uniform meshes of order $k$ with $k>2$. We compute Euler compressible simulations on those meshes to validate the mesh adaptation strategy.

In parallel to these developments, we have started work on a generalized formal approach to obtain discrete adjoint equations for residual based and Petrov-Galerkin finite element schemes [49] . We have shown that these discrete adjoint equations can now be used as a local error estimator for mesh refinement, giving to these methods to the same potential for adaptation of Galerkin schemes.