Section: New Results
Optimum design and control in fluid dynamics and its couplings
In computational sciences for physics and engineering, Computational Fluid Dynamics (CFD) are playing one of the major roles in the scientific community to foster innovative developments of numerical methodologies. Very naturally, our expertise in compressible CFD has led us to give our research on numerical strategies for optimum design a particular, but not exclusive focus on fluids.
The framework of our research aims to contribute to numerical strategies for PDEconstrained multiobjective optimization, with a particular emphasis on CPUdemanding computational applications in which the different criteria to be minimized (or reduced) originate from different physical disciplines that share the same set of design variables. These disciplines are often fluids, as a primary focus, coupled with some other disciplines, such as structural mechanics.
Our approach to competitive optimization is focused on the twodiscipline problem. It is based on a particular construction of Nash games, relying on a split of territory in the assignment of individual strategies. A methodology has been proposed for the treatment of twodiscipline optimization problems in which one discipline, the primary discipline, is preponderant, or fragile. Then, it is recommended to identify, in a first step, the optimum of this discipline alone using the whole set of design variables. Then, an orthogonal basis is constructed based on the evaluation at convergence of the Hessian matrix of the primary criterion and constraint gradients. This basis is used to split the working design space into two supplementary subspaces to be assigned, in a second step, to two virtual players in competition in an adapted Nash game, devised to reduce a secondary criterion while causing the least degradation to the first. The formulation has been proved to potentially provide a set of Nash equilibrium solutions originating from the original singlediscipline optimum point by smooth continuation, thus introducing competition gradually [53] . (see also subsectionsubsect:helico).
Our approach to cooperative optimization, in theory, is not limited in number of objective functions. It is based on a result of convex analysis established for a general unconstrained multiobjective problem in which all the gradients are assumed to be known. The theorem [16] states that in the convex hull of the gradients, there exists a unique vector of minimal norm, $\omega $; if it is nonzero, the vector $\omega $ is a descent direction common to all criteria; otherwise, the current design point is Paretostationary. This result led us to generalize the classical steepestdescent algorithm by using the vector $\omega $ as search direction. We refer to the new algorithm as the multiplegradient descent algorithm (MGDA). The MGDA yields to a Paretostationary point, and actual Paretooptimality is then tested [54] (see also subsection 6.2.4 ).
The two approaches have been combined to explore the Pareto front segmentwise as illustrated on Figure 2 .

MultipleGradient Descent Algorithm (MGDA)
Participants : JeanAntoine Désidéri, Régis Duvigneau, Matteo Giacomini, Abderrahmane Habbal, Adrien Zerbinati.
Theory and numerical experimentation of the MGDA construction
In multiobjective optimization, the knowledge of the Pareto set provides valuable information on the reachable optimal performance. A number of evolutionary strategies (PAES, NSGAII, etc), have been proposed in the literature and proved to be successful to identify the Pareto set. However, these derivativefree algorithms are very demanding in terms of computational time. Today, in many areas of computational sciences, codes are developed that include the calculation of the gradient, cautiously validated and calibrated.
The notion of Paretostationarity, originally established to be a necessary condition of optimality in differentiable multiobjective optimization of unconstrained problems, has been extended to problems subject to equality constraints. On this basis, we were able to establish that by augmenting, in a classical manner, the objectivefunctions of a penalty term equal to the square of the constraint violation, and applying the standard MGDA to it, would result in converged solutions that are Paretostationary in the extended sense. Numerical experimentation on this is ongoing.
Metamodelassisted CFD optimization by MGDA
Using MGDA in a multi objective optimization problem requires the evaluation of a large number of points with regard to criteria, and their gradients. In the particular case of a CFD problems, each point evaluation is very costly since it involves a flow computation, possibly the solution of an adjointequation. To alleviate this difficulty, we have proposed to construct metamodels of the functionals of interest (lift, drag, etc) and to calculate approximate gradients by local finite differences. These metamodels are updated throughout the convergence process to the evaluation of the new design points by the highfidelity model, here the 3D compressible Euler equations.
This variant of MGDA has been tested successfully over a problem of external aerodynamic optimumshape design of an aircraft wing consisting of reducing wavedrag, and augmenting lift. After only a few cycles of database updates, the Pareto front visibly forms, and this result is achieved at a very moderate computational cost [68] . This variant has been extended successfully to an internal flow optimization problem related to an automobile airconditioning system and governed by the NavierStokes equations. This more difficult problem has been proposed by Renault within the OMD2 ANR project. These studies have been reported in A. Zerbinati's doctoral thesis [38] .
Exact shape gradients
MGDA has successfully been tested over a twoobjective optimization problem governed by twodimensional elasticity. The deformation of a plate is calculated using an isogeometric approximation (see 6.3 ) and compliance derived from it. The exact parametric shape gradient is calculated, yielding the gradient of the objective function in two antagonistic situations differing by the loading. Paretofronts are thus identified.
Perspectives
MGDA offers the possibility to handle in a rational way several objectivefunctions for which gradients are known or approximated concurrently. This potential opens methodological paths to several themes of interest in highfidelity simulationbased optimization: optimization of complex systems whose performance is evaluated w.r.t. several criteria originating from different, coupled disciplines; optimization under uncertainties, by introducing sensitivities as additional objectives; optimization of timedependent systems, such as optimization of flowcontrol devices that generate a periodic flow (see next subsection), by converting the problem into a multipoint problem by timediscretization of the time and parameterdependent functional; etc.
Flow control
Participants : Régis Duvigneau, Jérémie Labroquère, Emmanuel Guilmineau [Ecole Centrale de Nantes] .
Shape optimization methods are not efficient to improve the performance of fluid systems, when the flow is characterized by a strong unsteadiness related to a massive detachment. This is typically the case for the flow around an automotive body or a wing in stall condition. To overcome this difficulty, flow control strategies are developed, that aim at manipulating vortex dynamics by introducing some active actuators, such as periodic blowing/suction jets. In this context, the choice of the control parameters (location, amplitude, frequency) is critical and not straightforward. Therefore, we develop a methodology to determine optimal control parameters by coupling the simulation of unsteady actuated flows with optimization algorithms. Two research axes have been considered :

the resolution of the unsteady sensitivity equations derived from the state equations, to exhibit the dependency of the flow dynamics with respect to the control ;

the optimization of control parameters using a statistical metamodelbased strategy.
In this perspective, unsteady Reynolds Averaged NavierStokes equations are solved, with some turbulence closures. Different models for synthetic jet have been implemented to simulate the actuation, and then validated for different turbulence closures [70] .
Specific developments have be carried out in the metamodelbased optimizer to include a noise term into Gaussian Process model, which is used to filter errors arising from unsteady simulations. A systematic assessment of modeling and numerical errors has been archived [57] , for a backward facing step testcase, with the objective of controlling the reattachment point location.
This activity is conducted in collaboration with the CFD team of Ecole Centrale de Nantes.
Robust design
Participants : JeanAntoine Désidéri, Régis Duvigneau, Daïgo Maruyama.
This work aims at developing robust design tools for aircraft w.r.t. aerodynamic performance subject to uncertainties, arising from geometrical features and fluctuations of inflow conditions. The robust design process is considered as a multiobjective optimization problem, which consists in minimizing or maximizing statistical moments of the cost function.
In the context of airfoil design, MGDA is used to improve simultaneously the mean and variance of the lift and drag coefficients, yielding a fourobjective optimization problem [71] .
Sonic boom reduction
Participants : Gérald Carrier [Research Engineer, ONERA/DAAP] , JeanAntoine Désideri, Andrea Minelli, Itham Salah El Din [Research Engineer, ONERA/DAAP] .
When an aircraft flies at supersonic speed, it generates at ground level an Nshaped shock structure which can cause serious environmental damage (“sonic boom”). Thus a problem of interest in aerodynamic optimization is to design such an aircraft to reduce the intensity of the sonic boom while maintaining the aerodynamic performance (drag minimization under lift constraint). Andrea Minelli aimed at contributing to this twodiscipline optimization problem. In the first part of his work, an inverse problem has been formulated and solved for “shaped sonic boom” and found in excellent agreement with the GeorgeSeebassDarden theory [82] for the calculation of the Whitham function corresponding to the lowestboom (axisymmetric) shape. Method and results have been generalized to more general geometries and have been presented internationally in [58] .
Besides, aeroacoustic optimizations have been realized successfully by coupling the aerodynamic optimizer (based on Euler calculations by the elsA software) with the sonicboom computation in a Nash game formulation. These experiments, conducted with our optimization platform FAMOSA, have demonstrated that starting from the shape optimized aerodynamically, one could retrieve smoothly a shape corresponding to nearlyoptimal sonicboom reduction [36] . and [54] .
Helicopter rotor blade optimization in both situations of hovering and forward flight
Participants : Michel Costes [Research Engineer, ONERA/DAAP] , JeanAntoine Désideri, Arnaud Le Pape [Research Engineer, ONERA/DAAP] , Enric Roca Leon.
E. Roca Leon is conducting a CIFRE thesis supported by EUROCOPTER (Marignane) at ONERA DAAP. This thesis follows the doctoral thesis of A. Dumont in which the adjointequation approach was used to optimize a rotor blade in hovering flight. The goal of this new thesis is to solve a twoobjective optimization problem in which the hoveringflight criterion is considered preponderant, but a new criterion that takes into account the forwardflight situation is also introduced, concurrently. The second criterion is the power necessary to maintain the forward motion. The first phase of thesis work has been devoted to the set up of a hierarchy of models from low to high fidelity, in order to calibrate appropriate functional criteria. Then, actual twoobjective optimizations are conducted via our Nash game approach to competitive optimization with territory splitting based on reduced Hessian diagonalization. A first successful experiment has been realized in which 16 geometrical parameters have been optimized to reduce the power in forward motion while maintaining suboptimality of the drag in hover. These results have been accepted for presentation at the American Helicopter Society Forum [62] , and [53] .
Optimum design in naval hydrodynamics
Participants : Régis Duvigneau, Louis Blanchard, Elisa Berini [KEpsilon company] .
Naval hydrodynamics field has recently shown a growing interest for optimum design methods. The computational context is especially complex because it implies unsteady twophase turbulent flows, with possibly very high Reynolds number (up to ${10}^{9}$). The use of automated design optimization methods for such problems requires new developments to take into account the large CPU time necessary for each simulation and the specificity of the geometries considered.
Some developments have been initiated on the geometrical modelling of hull shapes by parametric surfaces. The objective was to be able to modify existing hull shapes by controlling a small number of parameters, that are meaningful for naval architects. We have considered as testcase the bow shape for trawler ships. As a second step, an optimum shape procedure has been set up, based on a metamodelbased optimizer, the developed CAD model and the simulation tool for freesurface flows provided by KEpsilon company. The objective was to reduce the wave drag of a trawler ship by adding a bow, whose parameters are optimized [50] .