Aerodynamic aircraft design for mission performance by multipoint optimization

Transcription

1 Aerodynamic aircraft design for mission performance by multipoint optimization François Gallard Airbus Opérations SAS, 316 route de Bayonne, TOULOUSE CEDEX 9, France. Matthieu Meaux EADS - Innovation works, Campus Engineering, BP 90112, BLAGNAC CEDEX, France. Marc Montagnac Centre Europén de Recherche et de Formation en Calcul Scientifique, Computational Fluid Dynamics, 42, avenue Gaspard Coriolis, TOULOUSE CEDEX 1, France Bijan Mohammadi Université Montpellier II Mathématiques, CC51, MONTPELLIER, France. This paper addresses the problem of multiple operating conditions design, in the context of optimal aircraft mission performance and from the aerodynamic point of view. In particular, the choice of the flight conditions to be included in the optimization problem from desired condition ranges has to be done such that the optimization problem is wellpoised with a minimal computational cost. To this aim, a new method called Gradient Span Analysis is given, based on the analysis of the linear dependencies provided by adjoint sensitivities. A wing-body configuration is presented as test case and optimization results show the interest in terms of computational time savings. Physical interpretation of the method is also given. Nomenclature AoA C s Cd Cd p Cd w Cd ind Cd vp Cl g H k J j LoD M m Angle of Attack Jet engines fuel specific consumption Drag coefficient Pressure drag coefficient Wave drag coefficient Induced drag coefficient Viscous pressure drag coefficient Lift coefficient Earth gravity acceleration Hessian matrix approximation by BFGS formula Optimization objective function Any post processed function such as Cl or Cd. Cl/Cd : Lift over drag ratio Initial sampling size of the operating condition ranges Minimal sampling size of the operating condition ranges PhD student Research Engineer Research Engineer Professor 1 of 17

2 n p R S s s k S ref T t t 0 t f V W w X y k O ad I Subscripts k Symbols α χ w λ χ ω Design variables number Operating condition ranges number Flow state equation, typically Reynolds Averaged Navier-Stokes residuals Aircraft skin nodes coordinates Line-search step Design variables vector difference between two iterates Aircraft reference surface Engines thrust Time Take-off time Landing time Aircraft air speed Flow state vector of conservative variables Aircraft mass Computational domain mesh Gradient difference between two iterates Optimization admissible domain Operating condition ranges, α I Optimization iteration index Operating condition, typically a (Mach number, Angle of Attack, Reynolds) tuple Design variables vector Fuel overall consumption Discrete adjoint vector Gradient vector of total derivatives with respect to χ Weights of an aggregate objective function I. Introduction Aircraft design for performance is often viewed as a single point problem, such as minimizing the drag at a typical cruise flight condition. However, during its flight, an aircraft encounters continuous ranges of operating conditions. The Mach number for example varies because of air traffic control constraints. Moreover, the fuel burn reduces the mass, which has an impact on the optimal flight altitude and the lift coefficient, hence the angle of attack. In the end, the block fuel, or mission overall fuel consumption depends on the aircraft performance at every encountered flight condition. Aerodynamic shape design must then take these flight conditions into account since single condition design is known to lead to poor offdesign performance, which has been reported as drag-creep, 1 single-point optimization effect 2 or localized optimization. 3 Aircraft design involves complex and strongly non-linear phenomena; the anticipation of the impact of a shape modification on the flow and the performance at multiple flight conditions seems beyond human comprehension. Optimization techniques are a way to assist the designer in this task. Gradient-based methods are efficient when the variations of aerodynamic forces with respect to the geometry are computed with adjoint formulations. The method was applied to aircraft gradient-based optimization in the pioneering work by Jameson. 4 Recent work by Liem et al. 5 demonstrated the interest of the multipoint approach for fuel efficiency, where the choice of designed operating conditions is based on surrogate models. The aim of this paper is to demonstrate that the aerodynamic design of an aircraft for its optimal mission performance can be obtained using a multipoint optimization based on the Gradient Span Analysis (GSA). 6 In this way, the operating conditions ranges are sampled such that the computation cost is minimal and the problem is well-poised. 2 of 17

3 II. Aircraft mission performance The aircraft mission performance can be defined by its fuel overall consumption. The lift-weight and drag-thrust equilibrium in cruise gives aircraft balance equations: { 1 2 ρcls ref V 2 = wg, 1 2 ρcds (1) ref V 2 = T Combining the two equations and writing that for a jet engine the fuel consumption is proportional to the thrust, the Breguet equation is obtained: dw(t) dt = C s(α(t))g(t) w(t) (2) LoD(χ, α(t), w(t)) This first order differential equation can be integrated once on the time variable, on the range [t 0, t f ]. The fuel overall consumption is the difference between take-off and landing mass w = w(t 0 ) w(t f ), and with Eq. (2): t=tf w (χ) = t=t 0 dw(t) t=tf dt = dt t=t 0 C s (α(t))w(t)g(t) dt (3) LoD(χ, α(t), w(t)) One can minimize this quantity, paying attention to the dependencies between terms, in particular the mass drives the lift coefficient through Eq. (1), and the lift over drag is a direct function of lift. On the other hand, when a high number of design variables are involved such as in 3D aircraft configurations, gradient-based algorithms prove to be the most efficient approach, so the derivative of this function with respect to the design variables has to be computed. Assuming that the functions are sufficiently regular to swap derivation and integration operators and that the design variables only impacts the aerodynamic performance and not the engine performance or the structural mass: d w (χ) dχ t=tf = t=t 0 C s (α(t))w(t)g(t) LoD(χ, α(t), w(t)) LoD(χ, α(t), w(t)) 2 dt (4) χ In a numerical optimization process based on CFD, flow calculations are performed at a given angle of attack, mass and altitude, so a discretization of this equation in time is required. d w (χ) dχ = n C s (α(t i ))w(t i )g(t i ) LoD(χ, α(t i ), w(t i )) LoD(χ, α(t i ), w(t i )) 2 δt i (5) χ i=1 Finally, a set of scalar positive functions ω i (χ) can be defined such that: d w (χ) dχ = n i=1 ω i (χ) LoD(χ, α(t i), w(t i )) χ (6) Eq. (6) can be viewed as an aggregate objective function at selected operating conditions with nonconstant weights. From the aerodynamic optimization point of view, the question that rises is how to choose the operating conditions and weights so that the optimization process converges to an interesting engineering solution. As a consequence, the multipoint formulation does not necessarily require mission and aircraft performance models in order to explicitly compute the weighting function ω i (χ). The same reasoning can be made for multi-mission optimization. A larger integration domain in Eq. (3) results in an additional sum on top of Eq. (5) and in the end in a modification of ω i (χ) in Eq. (6). The formulation is therefore multi-mission, as improving the lift-over-drag polar α LoD(χ, α) has a positive influence on all the missions of the aircraft. From an algorithmic point of view, gradient-based algorithms with hessian matrix approximation, as bounds constrained limited memory BFGS 7 generate the next design variables using the gradient of the function but not the function value itself. Their iteration scheme is similar to Eq. (7). χ k+1 = χ k sh 1 k χj(χ), (7) 3 of 17

4 where H k is an approximation of the Hessain matrix Goldfarb Shanno) formula Eq. (10) at the k th iterate. [ ] d 2 J(χ) d 2 χ given by the BFGS (Broyden Fletcher y k = χ J k+1 (χ) χ J k (χ), (8) s k = χ k+1 χ k, (9) H k+1 = I + k T y i y i H is i s T i H i. (10) y it s i s it H i s i i=1 The BFGS formula also does not depend on the function value itself but only on the gradients and design variables history. Only line-search or trust region updates are used to ensure a sufficient decrease of the function value by avoiding too large steps s in Eq. (7) and depend on the function value. As w (χ) is strictly decreasing with every LoD(χ, t i ), if an optimization strategy decreases each of them, then w will decrease. Finally, the aim is to control the continuous function t LoD(χ, α(t), w(t)), and the quantity w in Eq. (3) is a scalar criteria that makes a link between this goal and the optimizer. Because the optimization strategy is based on the gradients of the objective more than the function value itself, the focus has to be made on the gradients LoD(χ,α(ti),w(ti)) χ when α(t) varies. One can also accept a modification of the criteria w, more precisely of the weights ω i (χ) if the approach saves computation time. This leads us to the importance of controlling the continuous function t LoD(χ, α(t), w(t)) through its gradients with a finite set of instants t i and at a minimal cost. III. The GSA method Zing and Elias 8 proposed a technique to automatically select the sampling points in the operating condition ranges. In that approach, the necessity to add an operating condition in the multipoint problem is detected by computing the drag over the whole operating condition ranges at checkpoints during the optimization problem. When a local maximum of the drag is detected between the sampling points, an additional one is added. Li et al 3 made a link between this local maximum and the fact that the gradient of the drag at this point is a new descent direction. In the GSA approach, the linear dependency of the gradients in the operating conditions ranges are used to select the sampling set whereas Li et al. implicitly supposed that the gradients were linearly independent which leads to the conclusion that the operating condition ranges have to be finely sampled. But the physics modeled by fluid state equations at two different operating conditions can be very similar, and that can be exploited from a design point of view. The purpose of the GSA algorithm is to adequately sample the operating conditions ranges to setup a multipoint optimization problem so that the local maximum behavior noticed by Zing and Elias is anticipated and avoided. In a preprocessing step, a relatively fine sampling of the operating conditions is performed on the initial shape and the gradient of the drag with respect to the design variables are computed at those conditions. On this initial set, the linear dependencies of the gradients are calculated and a minimal set of linearly independent conditions are selected by the GSA algorithm Alg. (1), based on a Gram-Schmidt decomposition. This last set will be used to formulate a weighted sum optimization problem. Because the gradient gives the first order variations of the drag, if no descent direction is missed by the algorithm then the local maximum effect will be avoided. The minimal number of sampling points ensuring the control of the drag on the continuous range is another result of the approach. It is equal to the dimension of the vector space spanned by the drag gradients when the operating conditions vary in the ranges. 6 A dynamic process that checks the dimension of the gradient span can be setup, similarly to, 8 but in practice, when the geometrical changes are not too important, this sampling is kept constant. Theorem 1 in 6 expresses that the zero gradient optimality condition on a set of m well chosen conditions implies the optimality of derived problems built on the same m first conditions and any additional one taken 4 of 17

5 in the operating condition ranges I, but with modified weights, as shown in Eq. (11). [( m ) ( )] m+1 α m+1 I ω k χ j(χ, α k ) = 0 = ω k R m R\{0} ω k χ j(χ, α k ) = 0 k=1 k=1 (11) The spanned vector space of a set of vectors is composed of all the possible linear combinations of this set. If the operating conditions are selected such that the condition of Eq. (12) is ensured, it is not possible to find any operating condition α m+1 selected in the ranges I at which the drag j(x, α m+1 ) can be decreased at the first order. Corollary 1 in 6 proves this statement formulated in Eq. (13). Finding an operating condition at which the drag gradient is not linearly dependent of the design operating conditions gradients would mean that the problem is ill-poised and the performance at this point can be improved without degrading the others, as shown by Li et al. 3 Span{ χ j(χ, α k ), k [1... m]} = Span{ χ j(χ, α), α I} (12) (ω 1,..., ω m ) R m m k=1 ω k x j(x, α k ) = 0 = α m+1 I, ( ω 1,.., ω m+1 ) (R m R ), D R n, t R m+1 k=1 ω k j(x + t D, α k ) = m+1 k=1 ω k j(x, α k ) + O(t 2 ) (13) The dimension of the gradient span Span{ χ j(χ, α), α I} defines the minimal number of operating conditions required to control the function j(χ, α) on the continuous set α I. As a consequence, it can be defined as the control dimension of the polar α j(χ, α). The GSA algorithm, selects a minimal set I m that ensures the condition Eq. (12), from an initial fine sampling I M I that is used to estimate Span{ χ j(χ, α), α I}. In the end, the GSA approach avoids the local maximum effect, without taking more operating conditions in the optimization problem than the number of design variables, 3 which is extremely costly. Moreover, the theorem shows why such a condition is over restrictive in practice. Eq. (11) also has a counter-intuitive consequence: adding an additional operating condition to a well-poised problem is equivalent to modifying the weights of the aggregate objective function. IV. The optimization problem The multipoint optimization problem using an aggregate objective function is defined in Eq. (14). I IR p, O ad IR n, (α 1,..., α m ) I m, (ω 1,..., ω m ) R m, Cd C 1 : R n I R Cl C 1 : R n I R m minimize J(χ) = ω k Cd(χ, α k ) χ O ad k=1 subject to {R(W k, χ, α k ) = 0, k [1... m]}, and {Cl(χ, α k ) = Cl 0 (α k ), k [1... m]}. (14) Dedicated Computational Fluid Dynamics solvers compute the flow solution W by solving the fluid state equations R(W, χ, α) = 0 for every α. The discrete adjoint method enables to compute the sensitivities of the drag and the lift at moderate cost. First, the adjoint vector λ j for the function j on the mesh X and for the state vector W at the operating condition α is the solution of the linear system of Eq. (15), j(w, X, α) W + λ T j R(W, X, α) W = 0. (15) 5 of 17

6 Algorithm 1 Gradient Span Analysis Algorithm (GSA) indices {1... M} for j = 1 M do c m 0 for n indices do q j χ j(χ, α j ) for i = 1 j 1 do q j q j π qi (q j ) end for q j qj q j c 0 for i = 1 M do v χ j(χ, α i ) for k = 1 j do v v π qk (v) end for if v < ɛ χ j(χ, α i ) then c c + 1 end if if c > c m then q m q j c m c n m n end if end for q j q m indices indices\{n m } end for if c m = M then END end if end for 6 of 17

7 Total derivatives of functionals with respect to mesh node coordinates is assembled in Eq. (16), dj(x, α) dx = j(w, X, α) dx + λ T j R(W, X, α). (16) X Volume sensitivities can be composed with surface sensitivities to compute the derivative of functions with respect to surface coordinates S, dj(s, α) ds = dj(x, α) dx(s) dx ds. (17) Finally the surface is parametrized with design variables χ and the gradient of the function is calculated, χ j(χ, α) = dj(s, α) ds(χ) ds dχ. (18) From Eq. (18), the gradients of the objective function at various operating conditions α is obtained, and from this information, the GSA algorithm 1 provides the operating conditions set {α k, k (1..m)} to be used in the multipoint optimization problem. The weights ω i, i (1..m) of the objective function are an estimation of the utopia point distance. 9 As a consequence, an estimation of the potential gains expected by the optimization process has to be estimated. To this aim, the pressure drag objective is decomposed in physical components: induced drag Cd ind, wave drag Cd w and viscous pressure drag Cd vp using the ONERA FFD code. 10 The potential gain at each condition is estimated as a fraction of these decompositions as shown in Eq. (19). ω = 1 0.9Cd w + 0.2Cd ind + 0.1Cd vp (19) The gains obtained after a single point optimization can provide a good estimation for the potential gain fractions in Eq. (19), but a rough estimate can be sufficient. In this way, the optimization formulation (14) is fully determined by an automated physical analysis of the problem. V. Numerical methods 11, 12 Numerical computations were performed with the elsa Onera software. This code manages both the flow analysis and the flow sensitivity aspects. It solves the 3D compressible Reynolds-Averaged Navier- Stokes equations using a cell-centered finite-volume method on structured grids as well as the associated discrete adjoint equations. The turbulence model chosen is the one-equation Spalart-Allmaras 13 model. V.A. Flow solver The spatial convective fluxes of the mean flow are discretized with the upwind Roe scheme 14 with the Harten s entropic correction. A MUSCL scheme (Monotone Upstream-centered Schemes for Conservation Laws) 15 associated with a Van Albada limiter 16 provides a second-order accurate scheme. The spatial convective fluxes of the turbulent flow are discretized with the first-order upwind Roe scheme. Spatial diffusive fluxes are approximated with a second-order central scheme. The turbulent equations are solved separately from the mean flow equations at each time step with the same time-marching method. The backward Euler implicit scheme drives the time integration. The resulting linear systems are solved with the scalar Lower-Upper Symmetric Successive Over-Relaxation (LU-SSOR) method. 17 A standard nonlinear multigrid algorithm 18 combined with local time stepping accelerates the convergence to steady-state solutions. V.B. Adjoint solver All of the methods previously described in the flow solver are differentiated by hand. The turbulent eddy viscosity and thermal conductivity are assumed constant 19, 20 during the differentiation. 7 of 17

8 The resolution method to solve the discrete adjoint equation is the preconditioned first-degree iterative method, similar to an approximate Newton method, of Eq. (20). Ã(λ (l+1) λ (l) ) = Aλ (l) b, (20) R where the matrix à is an approximation of the jacobian matrix A = W. The approximation matrix à derives from the linearization of a first-order Steger-Warming flux-vector splitting scheme for the convective flux and the linearization of the diffusive flux neglecting the spatial cross derivatives. The resulting linear systems are solved with a few steps of a block LU-SSOR algorithm at each iteration of the iterative method. VI. The XRF-1 test-case The XRF-1 model is a wing-body aerodynamic configuration representative of modern civil transport aircrafts. In particular, shocks on the wings in transsonic conditions are known to raise problems for multiple Mach numbers optimizations. 1 VI.A. Parametrization An in-house parametric and differentiated CAD engine called PADGE (Parametric And Differentiated Geometrical Engine) is used to build a parametric model. Eleven wing sections are parametrized with airfoils and parametric B-splines to link them together. Finally Coons patches are used to create the surfaces. An analytical reverse mode of the tool is available to compute functional gradients of Eq. (18) from discrete adjoint skin sensitivities. Decreasing the wing thickness would decrease the drag, but it would also increase the mass of the internal structure. The compromise is multidisciplinary. Because no structural sizing is involved in the present aerodynamic optimization process, the two structural beams are built in the wing CAD model and their thickness are geometrical constraints. The whole model, displayed in Fig. 1, contains 91 design variables. Figure 1: Parametric CAD model of the XRF-1 wings VI.B. The initial mesh The Reynolds-Averaged-Navier-Stokes solution and its discrete adjoint are computed on the structured mesh shown in Fig. 2 made of 140 blocks with 22 millions cells. The height of the first layer of cells is such that y + is approximately equal to one, and the mesh is sufficiently fine to capture the physics of the flow in cruise conditions according to best practices for the used numerical schemes, when compared to flight and wind tunnel tests. 8 of 17

9 Figure 2: Multiblocks structured mesh of the XRF-1 model VI.C. Mesh update A mesh deformation approach is used to update the mesh when the CAD model is updated by new design parameters. Subtraction of the initial CAD to the updated one provides a surface deformation field that has to be propagated into the volume mesh. Here, the method relies on an algebraic integral formulation. 21 It is analytically differentiated to provide a reverse mode for the computation of the surface mesh sensitivity from the adjoint volume mesh sensitivity in Eq. (17). In this way, the impact of the mesh deformation on the solution is derived. VII. Operating conditions choice by GSA The operating conditions range is 0.6 Mach 0.87 and 2.3 AoA 2.9, discretized in 38 initial discrete samples. At each of these, the flow, its adjoint and the drag gradient are computed, and the GSA algorithm identifies 5 conditions shown in Fig. 3. Because at the end of the optimization process, the weighted sum of the drag gradients is null, they are linearly dependent. GSA selects linearly independent gradients; so one arbitrary operating condition is added to the problem, it can be operating condition of the single point optimization Original sampling Selected by GSA AoA Figure 3: Flight conditions selected by GSA for the optimization problem Mach Figure 4 shows the drag sensitivity with respect to the surface of Eq. (17) at the 5 operating conditions selected by GSA on the XRF-1 model. 9 of 17

10 Figure 4: Total derivative of the drag with respect to the surface z-coordinate at the five linearly independent flight conditions given by GSA First of all, the design space being of dimension 91, it is noteworthy that only 5 conditions are sufficient to build a basis of the spanned 38 initial conditions drag gradients. It shows the interest of the GSA approach in terms of computation time saving. Then, it is observed that the design conditions in Fig. 4 displays different patterns of surface sensitivities defined in Eq. (17). Shock positions have a particularly large influence on that phenomenon and this position changes when the Mach number changes. Each of these patterns lead to a linearly independent drag gradient when assembled with the CAD sensitivity in Eq. (18). This links the non-linear physical phenomena of shocks displacement when the operating condition varies, to the gradient span dimension and in the end the choice of the operating conditions for the optimization problem. VIII. Single point optimization A single point optimization of the configuration is run at Mach 0.83 and Cl This enables to check the behavior of the optimization process, and to compare the results with the multipoint optimization. The L-BFGS-B code from Zhu et al. 7 is used as optimizer. The lift constraint Cl Cl 0 = 0 is handled in the flow solver by a target lift approach. Newton iterations with finite differences on the function Cl(AoA) are performed during the convergence of the flow and the cost is about 20% more than a fixed AoA calculation. A lagrangian is derived to compute the derivative of the drag at constant lift in Eq. (21), the derivatives being calculated by discrete adjoints of Cl and Cd. This lagrangian is provided to the l-bfgs-b algorithm. χ L(Cd, Cl Cl 0 )(χ) = Cd(χ, AoA) χ Cd(χ,AoA) AoA Cl(χ,AoA) AoA Cl(χ, AoA). (21) χ 10 of 17

11 105 Cdp Mach 0.83 Cl 0.53 Clp Mach 0.83 Cl 0.53 Cd pressure % Cl Iteration Figure 5: Single point optimization convergence history Figure 5 shows that the target lift approach succeeds in maintaining the lift during the optimization by automatic adjustment of the angle of attack. The overall drag reduction of 11% is mainly due to the shock smoothing as shown in Fig. 7, which also leads to a viscous pressure drag reduction. Fig. 6 also shows smooth pressure coefficient iso-lines and the global view of the configuration. Figure 6: Pressure coefficient at single point optimum (a) 15 % Span (b) 40 % Span (c) 80 % Span Figure 7: Cp distributions and airfoil deformations at single point convergence, Mach 0.83 Cl 0.53 In the Fig. 7 the displayed deformations are coherent with the pressure coefficients modifications: material 11 of 17

12 is added behind the upper-wing shock position to lower the re-compression, or removed in the subsonic area to lower the flow acceleration. The resulting wave drag is almost zero at the optimum. The lower and upper deformation curves cross at four points: leading and trailing edge, and the two structural beams intersections, which means that the geometrical constraint of constant beam thickness are respected. The lift repartition across the wingspan is also slightly moved outboard to reduce the induced drag. Resulting gains are summarized in Fig Delta Cd (%) CDw CDi CDvp CDf CDsp Figure 8: Drag gain as percentages of total initial drag for the single point XRF-1 optimization case IX. Multipoint optimization A multipoint optimization is run, with the required operating conditions. The target lift approach is employed, and, as in the single point case, succeeds in ensuring a constant lift during the process. In Fig. 9, the pressure drag history for each operating condition is displayed. Each condition contributes to the overall drag reduction of Eq. (5). As a consequence, the mission fuel burn has been reduced. It also means that the initial design was not on the Pareto front of these 6 operating conditions. Cd pressure % Mach 0.8 Cl 0.46 Mach 0.83 Cl 0.53 Mach 0.86 Cl 0.5 Mach 0.6 Cl 0.36 Mach 0.7 Cl 0.39 Mach 0.75 Cl Iteration Figure 9: Optimizations convergence history In figs. 10 to 16 the pressure coefficient plots give a global view of the flow at the optimum. As expected, the multipoint optimum being a compromise, the shocks in transonic conditions are much stronger than at the single point optimum, which almost achieves a shock-free flow. However, the wave drag is reduced in a significant way as shown in Fig. 17. The other components of the pressure drag are also reduced; the induced drag is decreased by moving outboard the center of lift, as in the single point optimization. The reduction of the shock intensity lowers the boundary layers thicknesses behind the shocks, that reduces the viscous pressure drag. Therefore the viscous pressure drag reduction is a consequence of the wave drag reduction, even if the gradient of this component is not exact in the adjoint solver that assumes frozen turbulence. 12 of 17

13 (a) Mach 0.8 Cl 0.46 (b) Mach 0.83 Cl 0.53 (c) Mach 0.86 Cl 0.5 (d) Mach 0.6 Cl 0.36 (e) Mach 0.7 Cl 0.39 (f) Mach 0.75 Cl 0.48 Figure 10: Pressure coefficient at multiple operating conditions optimum 13 of 17

14 (a) 15 % Span (b) 40 % Span (c) 80 % Span Figure 11: Cp distributions and airfoil deformations at multipoint convergence, Mach 0.8 Cl 0.46 (a) 15 % Span (b) 40 % Span (c) 80 % Span Figure 12: Cp distributions and airfoil deformations at multipoint convergence, Mach 0.83 Cl 0.53 (a) 15 % Span (b) 40 % Span (c) 80 % Span Figure 13: Cp distributions and airfoil deformations at multipoint convergence, Mach 0.86 Cl 0.5 (a) 15 % Span (b) 40 % Span (c) 80 % Span Figure 14: Cp distributions and airfoil deformations at multipoint convergence, Mach 0.6 Cl of 17

15 (a) 15 % Span (b) 40 % Span (c) 80 % Span Figure 15: Cp distributions and airfoil deformations at multipoint convergence, Mach 0.7 Cl 0.39 (a) 15 % Span (b) 40 % Span (c) 80 % Span Figure 16: Cp distributions and airfoil deformations at multipoint convergence, Mach 0.75 Cl 0.48 Delta Cd (%) Mach 0.8 Cl 0.46 Mach 0.83 Cl 0.53 Mach 0.86 Cl 0.5 Mach 0.6 Cl 0.36 Mach 0.7 Cl 0.39 Mach 0.75 Cl CDw CDi CDvp CDf CDsp Figure 17: Drag gain as percentages of total initial drag on the multipoint case The CPU cost of the multipoint optimization was hours on Intel(R) Xeon(R) 2.93GHz processors, taking into account the initial sampling for the GSA algorithm and the optimization run itself. The restitution time is 8 days on 400 CPU. It represents a 90% cost cut compared to an approach where n + 1 samples are selected in the operating condition ranges, 3 without missing any descent direction, so any potential drag gain in the operating condition ranges. The robust optimization cost 8 times more than single point one. X. Conclusion Aircraft mission performance optimization raises the question of improving the aerodynamic lift-overdrag polar under an infinite set of operating conditions such as Mach number, Reynolds and aircraft mass. 15 of 17

16 Robust optimization by weighted sum minimization is a way of addressing it. In that case, the selection of operating conditions to be included in the aggregate objective has a major impact on the cost of the resolution, and the quality of the solution. A new method named Gradient Span Analysis proves to be useful for assisting the optimization problem setup of this class of problem in an efficient way. Based on a mathematically demonstrated approach, it provides substantial computational time savings and the algorithms remain simple. An interpretation is provided to link the polar control dimension with the non-linear physical phenomena occurring on the aircraft such as shock waves. Optimizations of the XRF-1 wing-body configuration, representative of modern civil transport aircrafts, with RANS simulations in single and multiple operating conditions were presented, discussed and compared. The successful usage of the GSA algorithm shows that the adjoint sensitivities can not only enhance the gradient-based optimization effectiveness but also enable a new way of improving the multiple operating conditions optimization procedure. Acknowledgments The Association Nationale de la Recherche et de la Technologie is acknowledged for funding. Airbus is acknowledged for funding and providing computational resources. Renaud Sauvage, Joel Brezillon and Pascal Larrieu from Airbus are acknowledged for their useful advices and support. References 1 Hicks, R. M. and Vanderplaats, G. N., Application of Numerical Optimization to the Design of Supercritical Airfoils without Drag Creep, Business Aircraft Meeting, No. Paper , Society of Automotive Engineers, Wichita, Kansas, March 29 - April Drela, M., Pros and cons of airfoil optimization, chap. Frontiers of Computational Fluid Dynamics, World Scientific, Singapore, 1998, pp Li, W., Huyse, L., and Padula, S., Robust Airfoil Optimization to Achieve Consistent Drag Reduction Over a Mach Range, Structural and Multidisciplinary Optimization, Vol. 24, No. 1, 2002, pp Jameson, A., Aerodynamic design via control theory, J. Sci. Comput., Vol. 3, No. 3, 1988, pp Liem, R. P., Kenway, G. K. W., and Martins, J. R. R. A., Multi-point, multi-mission, high-fidelity aerostructural optimization of a long-range aircraft configuration, 14th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Indianapolis, IN, 09/ Gallard, F., Montagnac, M., Mohammadi, B., and Meaux, M., Robust parametric shape design by multipoint optimization, Tech. Rep. TR/CFD/12/33, CERFACS, September Zhu, C., Byrd, R. H., Lu, P., and Nocedal, J., L-BFGS-B - Fortran Subroutines for Large-Scale Bound Constrained Optimization, Tech. rep., ACM Trans. Math. Software, Zingg, D. and Elias, S., On Aerodynamic Optimization Under a Range of Operating Conditions, AIAA Journal, Vol. 44, No. 11, 2006, pp Koski, J. and Silvennoinen, R., Norm methods and partial weighting in multicriterion optimization of structures, International Journal for Numerical Methods in Engineering, Vol. 24, No. 6, 1987, pp Destarac, D., Far-field/near field drag balance and applications of drag extraction in CFD, VKI Lecture Series, 2003, pp Cambier, L., Gazaix, M., Heib, S., Plot, S., Poinot, M., Veuillot, J.-P., Boussuge, J.-F., and Montagnac, M., CFD Platforms and Coupling : An Overview of the Multi-Purpose elsa Flow Solver, Aerospace Lab, Vol. 2, March Puigt, G., Gazaix, M., Montagnac, M., Pape, M.-C. L., de la Llave Plata, M., Marmignon, M., Boussuge, J.-F., and Couaillier, V., Development of a new hybrid compressible solver inside the CFD elsa software, 20th AIAA Computational Fluid Dynamics Conference, No. AIAA , Honolulu (HI), USA, June Spalart, P. R. and Allmaras, S. R., A One-Equation Turbulence Transport Model for aerodynamic flows, 30th AIAA Aerospace Sciences Meeting and Exhibit, AIAA Paper , Reno, Nevada, Jan Roe, P. L., Approximate Riemann Solvers, Parameter Vectors and Difference Schemes, Journal of Computational Physics, Vol. 43, No. 2, October 1981, pp van Leer, B., Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov s method, Journal of Computational Physics, Vol. 32, No. 1, 1979, pp van Albada, G. D., van Leer, B., and Roberts, W. W., A Comparative Study of Computational Methods in Cosmic Gas Dynamics, Astronomy and Astrophysics, Vol. 108, 1982, pp Yoon, S. and Jameson, A., Lower-Upper Symmetric-Gauss-Seidel Method for the Euler and Navier-Stokes Equations, AIAA Journal, Vol. 26, No. 9, 1988, pp Jameson, A., Steady State Solutions of the Euler Equations for Transonic Flow by a Multigrid Method, chap. Advances in Scientific Computing, Academic press, 1982, pp Jameson, A., Martinelli, L., and Pierce, N. A., Optimum Aerodynamic Design Using the Navier-Stokes Equations, Theoretical and Computational Fluid Dynamics, Vol. 10, No. 1-4, 1998, pp of 17

17 20 Nielsen, E. J. and Anderson, W. K., Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations, AIAA Journal, Vol. 37, No. 11, 1999, pp M. Meaux, M. C. and Voizard, G., Viscous aerodynamic shape optimization based on the discrete adjoint state for 3D industrial configurations, European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS, of 17