Multi-model and multi-scale optimizationstrategies
Application to sonic boom reduction
F. Alauzet* S. Borel-Sandou** L. Daumas*** A.Dervieux**** Q. Dinh*** S. Kleinveld*** A. Loseille* Y.Mesri**** G. Rog***
* INRIA, Projet Gamma, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LeChesnay Cedex, France
** Universit Pierre et Marie Curie, Laboratoire J.L. Lions, 175 rue du Chevaleret,75013 Paris, France
*** Dassault Aviation, DGT, 78 quai Marcel Dassault, 92552 St-Cloud Cedex 300,France
**** INRIA, Projet Tropics, 2004 Route des Lucioles, BP 93, 06902 Sophia-AntipolisCedex, France
RSUM. Loptimisation de la forme dun avion supersonique ncessite un modle compositecomportant une composante 3D haute fidlit en mcanique des fluides et un modle simplifide propagation du bang. La prise en compte de cette complexit est tudie dans le cadredune boucle doptimization, avec des adjoints discrets exacts de lcoulement 3D et un systmede dformation de maillage. Lintroduction dune mthode dadaptation de maillage est aussiconsidre.
ABSTRACT. The shape optimization of a supersonic aircraft need a composite model combininga 3D CFD high-fidelity model and a simplified boom propagation model. The management ofthis complexity is studied in an optimization loop, with exact discrete adjoints of 3D flow andmesh deformation system. The introduction of a mesh adaptation algorithm is also considered.
MOTS-CLS : Optimisation, forme optimale, paramtrisation, diffrentiation automatique, adap-tation de maillage anisotrope, bang sonique
KEYWORDS: Optimization, shape design, parameterization, automatic differentiation, anisotropicmesh adaptation, sonic boom.
European Journal of Computational Mechanics. Volume 10 n 1/2007, pages 1 20
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When passing to real life applications, optimal design algorithms have their com-plexity increased from many standpoints. We consider in this paper the treatment ofmultiple scales. By multiple scales, we mean that the modeled physics is not easilyput in a unique PDE model, due to the very large and very small scales involved inthe problem. A classical approach in that case consists in using specialized modelsfor the different range of scales and considering the system as a multi-model one.Then the Multi-disciplinary Design Optimization (MDO) methods can be applied.These methods are able to couple different physics and also different level of fide-lity for the same physics, see for example (Sobieszczanski-Sobieski et al., 1997) and(Alexandrov, 1997).
In our particular application, the sonic boom reduction, we address a part of themulti-scale issue by applying a mesh adaptation loop. The three main ingredients ofmesh adaptation are :
(i) the choice of a criterion governing the local repartition of nodes and the align-ments of elements,
(ii) the model of mesh,(iii) the way the mesh generator is coupled with the rest of the algorithm.
Concerning (i), the user has to known what is the priority between working with ac-curate flow field evaluation or only with an accurate final answer, the optimal controlapproximation. Accuracy of the whole flow can be set in general and is solved by lo-cal error estimators, as interpolation estimators, see (Alauzet et al., 2007) and (Zhuet al., 1997). When focussing on functional value accuracy, goal-oriented a posterioriestimators are then considered, see (Giles et al., 2001) and (Venditti et al., 2002).Functional error can be also minimized as in (Koobus et al., 2007). The issue of accu-racy in optimal control itself is addressed for example in (Polak, 1997) and (Becker etal., 2001).The model of mesh, (ii), states for any kind of mesh modification algorithm, goingfrom basic scheme such as mesh enrichment with patterns to global mesh modifica-tions or generation with advanced meshing techniques. In our case, the volume meshis adapted by local mesh modifications of the previous mesh using mesh operationssuch as vertex insertion, edge and face swap, collapse and node displacement. Thevertex insertion procedure uses an anisotropic generalization of the Delaunay kernel.Coupling mesh adaptation with optimization algorithm, (iii), is not a trivial issue. Ac-curacy of adaptation should not be deteriorated by the optimization and robustnessof optimization should not be lost by mesh updates. Our proposed approach is ins-pired by an analogy with the fixed point mesh adaptation method for unsteady flowsproposed in (Alauzet et al., 2007).
An important feature of the study is the option of a gradient-based minimization.It is motivated by the necessity of considering a large number of design variables. Themain cons of gradient-based minimization involve the risk of finding a local mini-
Multi-model multi-scale optimization 3
mum of poor quality, but more importantly, gradient-based methods need a differen-tiable functional. This requirement may interfere with the treatment of multiple physicscales. Two other important ingredients may have to be installed in the optimizationloop. CAD shape parameterization belong to the functional to minimize, together withthe process generating a new mesh for a new shape. Both have to be handled in thegradient evaluation.
Let us go a little further with the physics. Our application, sonic boom reduction,needs to handle the difficult problem of modeling sonic boom propagation. The systemof Euler equations is recognized as a satisfying model for the whole physical process.But when passing to discrete model for numerical simulations, we have to discretizeon a unique mesh small and large scales. Their ratio may be enormous. Indeed, shockwidth is in micron and the computational domain should extend from the aircraft tothe ground, 15 kilometers below. Simpler models than Euler equations may be chosenfor the whole physical process but they may loose some accuracy in the sonic boomprediction. An alternative is to use the Euler 3D model only in the vicinity of the air-craft. Then, a propagation model is utilized from Euler outputs and solves importantfeatures of pressure perturbation propagation down to the ground. It remains to de-termine the place of this extra model into the optimization process. The propagationcould be directly included into the optimization loop as in (Farhat et al., 2002). Or, thepropagation model could be treated separately by specifying the matching betweenthe two models in the objective functional, see (Alonso et al., 2002). Regarding thisstudy, the last option has been considered here.
In this paper, we first introduce the problem under study in terms of the basichigh-fidelity model. Secondly, we present the two extensions of the basic model, i.e.,the propagation model and mesh adaptive model. Thirdly, we describe a numericaloptimization platform and show which kind of results can be obtained in combina-tion with the propagation model. Then, preliminary experiments on an optimizationplatform coupled with mesh adaptation are described.
2. Mathematical design problem
2.1. Continuous standpoint
The flow around a supersonic aircraft is modeled by the Euler equations. Assumingthat the gas is perfect, non viscous and that there is no thermal diffusion, the Eulerequations for mass, momentum and energy conservation read :
t+. (~U) = 0 ,
+. (~U ~U) +p = 0 ,
+. ((E + p)~U) = 0 ,
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where denotes the density, ~U the velocity vector, E = e+ ~U22 the total energy and
p = ( 1)e the pressure with = 1.4 the ratio of specific heats and e the internalenergy. These equations are symbolically rewritten :
t+ F (W ) = 0 , 
where W = t(, u, v, w, E) is the conservative variables vector and the vectorF represents the convective flux. In fact we are interested only in the steady solutionof this system. It has to be evaluated from the region around the aircraft (near field) tothe ground level (far field), so that the computational domain boundary involves thetwo components, viz. the aircraft wall and the ground. Slip conditions are assume forboth components. The pressure perturbation created by the supersonic aircraft flightextends down to the ground and is the source of uncomfortable noise for populations,i.e., the sonic boom. During this propagation to ground, the perturbation signal trans-forms, see Figure 1, left.
The optimization problem under study in this paper is to find an aircraft shapethat would reduce the impact of this noise. This can be expressed as the research ofa minimum of an objective functional measuring the deviation between the groundpressure signature and a target one. We define it in short as :
(p(W ) ptarget)2ds 
where the integral is taken on ground surface and the functional j depends on theaircraft shape through flow state W .
2.2. Numerical issues
Predicting the Euler flow down to the ground needs the resolution by the 3D meshof wide range of scale sizes. Sharp shocks have to be solved with small numericalwidth, not necessary as small as micron, but yet of the order of centimeter, and this in adomain of tens of kilometers. As a consequence, three-dimensional Euler modeling ofthe pressure perturbation is not possible with standard meshes and todays computers.
In the literature, authors starting with an accurate model -as Euler equations- res-trict the 3D Euler computation to a near field sub-region of the domain, usually measu-red with the ratio of the diameter R of the near field domain to the aircraft chord-lengthL. Then, propagation is handled with a simplified model. The accuracy of a compositemodel combining 3D Euler at near field and a propagation model depends on :