Section: New Results
Sensitivity analysis for unsteady flows
Participants : Régis Duvigneau, Maxime Stauffert, Camilla Fiorini [UVST] , Christophe Chalons [UVST] .
The adjoint equation method, classically employed in design optimization to compute functional gradients, is not well suited to complex unsteady problems, because of the necessity to solve it backward in time. Therefore, we investigate the use of the sensitivity equation method, which is integrated forward in time, in the context of compressible flows. More specifically, the following research axes are considered:

Sensitivity analysis in presence of shocks
While the sensitivity equation method is a common approach for parabolic systems, its use for hyperbolic ones is still tedious, because of the generation of discontinuities in the state solution, yielding Dirac distributions in the sensitivity solution. To overcome this difficulty, we investigate a modified sensitivity equation, that includes an additional source term when the state solution exhibits discontinuities, to avoid the generation of deltapeaks in the sensitivity solution. We consider as typical example the 1D compressible Euler equations. Different approaches are tested to integrate the additional source term: a Roe solver, a Godunov method and a moving cells approach. Applications to uncertainty quantification in presence of shocks are demonstrated and compared to the classical MonteCarlo method [26]. This study is achieved in collaboration with C. Chalons and C. Fiorini from University of Versailles.

For problems with regular solution, we investigate the recursive use of the sensitivity equation method to estimate highorder derivatives of the solution with respect to parameters of interest. Such derivatives provide useful information for optimization or uncertainty quantification. More precisely, the thirdorder derivatives of flow solutions governed by 2D compressible NavierStokes equations are estimated with a satisfactory accuracy.

When shape parameters are considered, the evaluation of flow sensitivities is more difficult, because equations include an additional term, involving flow gradient, due to the fact that the parameter affects the boundary condition location. To overcome this difficulty, we propose to solve sensitivity equations using an isogeometric Discontinuous Galerkin (DG) method, which allows to estimate accurately flow gradients at boundary and consider boundary control points as shape parameters. First results obtained for 2D compressible Euler equations exhibit a suboptimal convergence rate, as expected, but a better accuracy with respect to a classical DG method [40].