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Section: New Results

Control of approximation errors

Participants : Gautier Brèthes, Eléonore Gauci, Alain Dervieux, Adrien Loseille [Gamma3 team, Inria-Rocquencourt] , Frédéric Alauzet [Gamma3 team, Inria-Rocquencourt] , Loïc Frazza [Gamma3 team, Inria-Saclay] , Stephen Wornom, Anca Belme [university of Paris 6] .

Reducing approximation errors as much as possible is a particular kind of optimal control problem. We formulate it exactly this way when we look for the optimal metric of the mesh, which minimizes a user-specified functional (goal-oriented mesh adaptation). In that case, the usual methods of optimal control apply, using adjoint states that can be produced by Algorithmic Differentiation.

Our theoretical studies in mesh adaptation are supported by the ANR project MAIDESC coordinated by ECUADOR and Gamma3, which deals with meshes for interfaces, third-order accuracy, meshes for boundary layers, and curved meshes.

The thesis of Éléonore Gauci on the goal-oriented criteria for CFD and coupled CSM-CFD systems is continuing. Éléonore Gauci gave a presentation at ECCOMAS in Crete.

Further studies of mesh adaptation for viscous flows are currently performed and a paper in collaboration with Gamma3 and university of Paris 6 (Anca Belme) is being written for a Journal.

An important novelty in mesh adaption is the norm-oriented AA method. The method relies on the definition of ad hoc correctors. It has been developed in the academic platform “FMG” for elliptic problems. Gautier Brèthes gave several presentations in conferences, a journal article has been published [12]. The introduction of the norm-oriented idea considerably amplifies the impact of adjoint-based AA. The applied mathematician and the engineer now have methods when faced to mesh adaptation for the simulation of a complex PDE system, since they can specify which error norm level they wish, and for which norm. Another version is developed jointly with Inria team Gamma3 for the compressible Euler model.

A work of extension of a different standpoint, the tensorial metric method was started during the thesis of Gautier Brèthes and has been been submitted to a journal.

CFD application are supported by the European FP7 project UMRIDA which deals with the application of AA to approximation error modelling and control.

This involves an extensive work on a series of RANS (Reynolds Averaged Navier-Stokes) adaptative computations relying on the multi-scale method on the one hand, and on the other hand on further development by Gamma3 and Ecuador of the novel norm-oriented method for the compressible Euler model. This will be first published as a chapter contributed to the UMRIDA monography  [24]: II.1.4 Numerical uncertainties estimation and mitigation by mesh adaption Frédéric Alauzet, Alain Dervieux, Loïc Frazza and Adrien Loseille.