Section: New Results

Control of approximation errors

Participants : Frédéric Alauzet [GAMMA team, Inria-Rocquencourt] , Estelle Mbinky [GAMMA team, Inria-Rocquencourt] , Olivier Allain [Lemma] , Alexandre Carabias, Hubert Alcin, Alain Dervieux.

This is a joint research between Inria teams Gamma (Rocquencourt), Sciport, Castor and the Lemma company. Gamma brings mesh and approximation expertise, Sciport brings adjoint methods, and CFD applications are developed by CASTOR and Lemma.

The resolution of the optimum problem using adjoint-mode AD can be used in a slightly different context than optimal shape design, namely mesh adaptation. This will be possible if we can map the mesh adaptation problem into a differentiable optimal control problem. To this end, we express the mesh adaptation problem in a purely functional form: the mesh is reduced to a continuous property of the computational domain named the continuous metric. We minimize a continuous model of the error resulting from that metric. Thus the search of an adapted mesh is transformed into the search of an optimal metric.

In 2012, this activity is amplifying. A work on goal-oriented mesh adaptation for unsteady Euler flows submitted to the journal JCP has been accepted and published [13] . Its extension to the compressible Navier-Stokes model has been developed in 2D  [22] and in 3D [11] . A further extension to Large Eddy Simulation has been defined and developed in the WOLF demonstrator. A communication at ECCOMAS (Vienna) has been presented and papers are being written for publication in journal.

The method is being extended to a third-order approximation, the Vertex-CENO. This approximation was defined collaboratively between university of Montpellier, IMM-Moscow and Sciport. A more accurate version is studied by Alexandre Carabias. A new mesh adaptation theory involving error estimates and criteria has been developed by Gamma and Sciport. The extension of the multiscale adaptation method is considered by Estelle Mbinky at Rocquencourt and has been presented at ECCOMAS (Vienna). The extension of the goal-oriented method is considered by Alexandre Carabias and first results were presented at ECCOMAS (Vienna). A cooperation with CEMEF and university of Nice is considered and a ERC common proposal, CMILE, has been built. Anisotropic mesh adaptation allows for better convergence towards continuous solutions, and in particular more accurate a posteriori error estimates and correctors. The synergy between correctors and mesh adaptation is the subject of a joint contribution (Gamma and Sciport) for the FP7 UMRIDA proposal (nov. 2012).