## Section: New Results

### Particular applications of simulation methods

#### Hermitian interpolation under uncertainties

Participants : Jean-Antoine Désideri, Manuel Bompard [Doctoral Student, ONERA/DSNA until December 2011; currently post-doctoral fellow in Toulouse] , Jacques Peter [Research Engineer, ONERA/DSNA] .

In PDE-constrained global optimization, iterative algorithms are commonly efficiently accelerated by techniques relying on approximate evaluations of the functional to be minimized by an economical, but lower-fidelity model (meta-model), in a so-called Design of Experiment (DoE). Various types of meta-models exist (interpolation polynomials, neural networks, Kriging models, etc). Such meta-models are constructed by pre-calculation of a database of functional values by the costly high-fidelity model. In adjoint-based numerical methods, derivatives of the functional are also available at the same cost, although usually with poorer accuracy. Thus, a question arises : should the derivative information, available but known to be less accurate, be used to construct the meta-model or ignored ? As a first step to investigate this issue, we have considered the case of the Hermitian interpolation of a function of a single variable, when the function values are known exactly, and the derivatives only approximately, assuming a uniform upper bound $\epsilon $ on this approximation is known. The classical notion of best approximation has been revisited in this context, and a criterion introduced to define the best set of interpolation points. This set was identified by either analytical or numerical means. If $n+1$ is the number of interpolation points, it is advantageous to account for the derivative information when $\epsilon \le {\epsilon}_{0}$, where ${\epsilon}_{0}$ decreases with $n$, and this is in favor of piecewise, low-degree Hermitian interpolants. In all our numerical tests, we have found that the distribution of Chebyshev points is always close to optimal, and provides bounded approximants with close-to-least sensitivity to the uncertainties [56] .

#### Mesh qualification

Participants : Jean-Antoine Désideri, Maxime Nguyen, Jacques Peter [Research Engineer, ONERA/DSNA] .

M. Nguyen Dinh is conducting a CIFRE thesis at ONERA supported by AIRBUS France. The thesis topic is the qualification of CFD simulations by anisotropic mesh adaption. Methods for refining the 2D or 3D structured mesh by node movement have been examined closely. Secondly, it is investigated how could the local information on the functional gradient $\u2225dJ/dX\u2225$ be exploited in a multi-block mesh context. This raises particular questions related to conservation at the interfaces.

Several criteria have been assessed for mesh qualification in the context of inviscid-flow simulation and are currently being extended to the RANS context. These results have been presented internationally in the communication [54] and the publication [44] .

#### Hybrid meshes

Participants : Sébastien Bourasseau, Jean-Antoine Désideri, Jacques Peter [Research Engineer, ONERA/DSNA] , Pierre Trontin [Research Engineer, ONERA/DSNA] .

S. Bourasseau has started a CIFRE thesis at ONERA supported by SNECMA. The thesis is on mesh adaption in the context of hybrid meshes, that is, made of both structured and unstructured regions. Again, the aim is to exploit at best the function gradient provided by the adjoint-equation approach. Preliminary experiments have been conducted on geometries of stator blade yielding the sensitivities to global shape parameters.

The on-going developments are related to the extension to the hybrid-mesh context of the full shape gradient in a 3D Eulerian flow computation.

#### Data Completion Problems Solved as Nash Games

Participants : Abderrahmane Habbal, Moez Kallel [University of Tunis] .

The Cauchy problem for an elliptic operator is formulated as a two-player Nash game.

Player (1) is given the known Dirichlet data, and

*uses as strategy variable the Neumann condition*prescribed over the inaccessible part of the boundary.Player (2) is given the known Neumann data, and

*plays with the Dirichlet condition*prescribed over the inaccessible boundary.The two players solve in parallel the associated Boundary Value Problems. Their respective objectives involve the

*gap between the non used Neumann/Dirichlet known data and the traces of the BVP's solutions*over the accessible boundary, and are*coupled through a difference term*.

We prove the existence of a unique Nash equilibrium, which turns out to be the reconstructed data when the Cauchy problem has a solution. We also prove that the completion algorithm is stable with respect to noise. Many 3D experiments were performed which illustrate the efficiency and stability of our algorithm [42] .