Concurrent wing and highlift system aerostructural optimization
Abstract
A method is presented for concurrent aerostructural optimization of wing planform, airfoil and high lift devices. The optimization is defined to minimize the aircraft fuel consumption for cruise, while satisfying the field performance requirements. A coupled adjoint aerostructural tool, that couples a quasithreedimensional aerodynamic analysis method with a finite beam element structural analysis is used for this optimization. The Pressure Difference Rule is implemented in the quasithreedimensional analysis and is coupled to the aerostructural analysis tool in order to compute the maximum lift coefficient of an elastic wing. The proposed method is able to compute the maximum wing lift coefficient with reasonable accuracy compared to highfidelity CFD tools that require much higher computational cost. The coupled aerostructural system is solved using the Newton method. The sensitivities of the outputs of the developed tool with respect to the input variables are computed through combined use of the chain rule of differentiation, automatic differentiation and coupledadjoint method. The results of a sequential optimization, where the wing shape and high lift device shape are optimized sequentially, is compared to the results of simultaneous wing and high lift device optimization.
Keywords
Aerostructural optimization High lift devices Coupled adjoint sensitivity analysis1 Introduction
Although knowledge of the physics of highlift devices (HLD) has come a long way since the fundamental paper of A.M.O Smith on highlift aerodynamics in 1975 (Smith 1975), analysis and optimization of highlift devices still proves to be a difficult subject. Through the use of Computational Fluid Dynamics (CFD) and increased computing capabilities, extensive research on the subject has become possible. In early days, this research mainly focused on achieving highlift requirements to satisfy takeoff and landing performance requirements. However, over the past years the focus has switched to reducing weight and complexity (van Dam 2002) as aircraft manufacturers tend to use less complex highlift devices (Reckzeh 2003). The importance of weight and aerodynamic performance of highlift devices in aircraft design is illustrated by Meredith (1993). According to Meredith, an increase of 0.1 in lift coefficient at constant angle of attack results in a reduction of approach attitude by about one degree, reducing landing gear length and thereby saving up to 1400 lb. Moreover, an increase of 1.5% in maximum lift coefficient (\(C_{L_{\max }}\)) may result in an extra 6600 lb payload at fixed approach speed while an 1% increase in takeoff lift over drag ratio (L/D) is equal to a 2800 lb increase in payload or a 150 nm range increase.
Even though numerous semiempirical methods exist to predict the wing weight, drag and lift of multielement wings (Raymer 2012; Torenbeek 1982; Roskam 2000; Pepper et al. 1996), the accuracy of these methods does not yield the level of accuracy required by the industry, requiring e.g. a drag prediction accuracy of one drag count (van Dam 2003). To achieve the required accuracy, more physics based methods are required such as Computational Fluid Dynamics (CFD) and Finite Element Methods (FEM) tools. Example of application of such highfidelity analysis for wing optimization can be found in the work of Martins et al. (2004), Kennedy and Martins (2014) and Barcelos and Maute (2008). The downside of these tools is that they require the use of high performance computational resources, making optimization problems in some cases too costly to solve. An alternative to the highfidelity 3D aerodynamic solvers is the quasithreedimensional (Q3D) analysis methodology, which combines twodimensional viscous airfoil data with inviscid threedimensional wing aerodynamic data. This methodology requires only a portion of the computational power required for highfidelity tools while generating sufficiently accurate results. Examples of using the Q3D method for aerodynamic analysis was presented by van Dam (2002), Elham (2015) and Mariens et al. (2014). Elham and Van Tooren developed a coupledadjoint aerostructural analysis and optimization tool by coupling a Q3D method to a FEM (Elham and van Tooren 2016a). This tool has been validated for drag prediction and twist deformation. Using the coupled adjoint method, the tool is able to compute the derivatives of the outputs with respect to the inputs analytically enabling gradient based optimization.
In the traditional design methodology, the design of wing shape and HLD is done sequentially. The wing planform and airfoil shapes are designed (or optimized) first and then the HLD shape is determined (Flaig and Hilbig 1993; Nield 1995). It is known that sequential design and optimization may result in a suboptimal design. In this paper a method for concurrent aerostructural optimization of wing and HLD is presented. In such a method the shape of the wing planform, airfoil, HLD as well as the wingbox structure is optimized simultaneously to minimize the aircraft mission fuel weight and satisfy the aircraft field performance requirements, that are the main drivers for HLD design.
The structure of this paper is as follows: First, the basic framework of the aerostructural analysis and optimization is described. Then the modifications applied to the aerostructural tool are explained, followed by a description of the method for predicting maximum lift. Then the method of coupling the modified methods is explained, followed by a validation of the extended model. Finally, a test case optimization is presented for a Fokker 100 class wing.
2 Aerostructural analysis and optimization framework
For a complete description of this coupledadjoint aerostructural analysis and optimization method and derivation of the coupledadjoint method, the reader is referred to Elham and van Tooren (2016a).
3 Maximum lift prediction
Since MSES more often than not fails to converge at high angles of attack, the Pressure Difference Rule (PDR), developed by Valarezo and Chin (1994) is used for the estimation of \(C_{L_{\max }}\). The PDR states that for a given chord Reynolds number and free stream Mach number, there exist a relation between the wing stall and the pressure difference between the suction peak and trailing edge pressure. While Valarezo and Chin made use of a higherorder panel method to obtain the pressure difference at several spanwise section, any other reliable method may be used such as the Q3D method described in this paper, due to the fact that empirical data is used in the analysis which takes viscous effects into account. Furthermore it was identified that this rule can be used for 3D wing analysis even though it relies on 2D sectional data. This is due to the fact that at the critical stall section, the suction peak of the 3D wing will be equal to that of the 2D flow for the respective airfoil section.
The effective pressure distribution over each specified spanwise section is then computed from a 2D linear strength vortex panel method, based on the method of Katz and Plotkin (1991), using the effective flow properties as described in Section 2. To analyze the airfoil using the panel method code the effective angle of attack and the effective Mach number are required. These values are obtained from the global angle of attack and free stream Mach number by adjusting for sweep effects and downwash (see Elham and van Tooren (2016a) for more details).
Besides producing the given outputs, the panel method is able to produce the derivatives of the outputs with respect to the inputs using a combination of the chain rule of differentiation and Automatic Differentiation (AD) in reverse mode using the Matlab AD toolbox Intlab (Rump 1999).
4 Aerostructural coupling
The fourth equation in (5) states that the inviscid 2D lift computed by the panel method is equal to the VLM lift distribution, corrected for sweep. The system is solved using the same Newton method for iteration described in Section 2.
5 Sensitivity analysis
In order to use the Newton method for iteration, the partial derivatives of the governing equations with respect to the state variables (matrix J in (2)) are required. Additionally, to perform gradient based optimization, the sensitivities of any function of interest with respect to the design variables such as the wing planform or airfoil shape are required. The present tool computes all of the required derivatives through a combination of AD, chain rule of differentiation and the aforementioned coupledadjoint method.
Partial derivatives of aerostructural PDR system
Γ  U  α  α _{ i }  

R _{1}  AIC  \(\frac {\partial AIC}{\partial U}{\Gamma }  \frac {\partial RHS}{\partial U} \)  \(  \frac {\partial RHS}{\partial \alpha } \)  0 
R _{2}  \( \frac {\partial F}{\partial {\Gamma }}\)  K  0  0 
R _{3}  0  0  \(\frac {\partial KS}{\partial \alpha }\)  \(\frac {\partial KS}{\partial \alpha _{i}}\) 
R _{4}  \(\frac {\partial C_{l_{\perp }}}{\partial {\Gamma }}\)  \(\frac {\partial C_{l_{{2d}_{\text {inv}}}}}{\partial U}\)  \(\frac {\partial C_{l_{{2d}_{\text {inv}}}}}{\partial \alpha }\)  \(\frac {\partial C_{l_{{2d}_{\text {inv}}}}}{\partial \alpha _{i}}\) 
6 Verification and validation
While the aerostructural tool developed by Elham and Van Tooren has been validated for wing drag and wing deformation (Elham and van Tooren 2016a), the enhanced method needs to be validated for maximum wing lift coefficient prediction and computation of wing lift over drag ratios in highlift conditions. Finally, the sensitivities are verified through Finite Differencing (FD).
6.1 Maximum lift coefficient
The maximum error between the computed and experimental results is 4.38% at a flap deflection of 10^{∘}. A second test case has been performed to validate \(C_{L_{\max }}\) of the Fokker 100 class wing, for which the geometry and flow parameters are described in Section 7. Using the PDR, the clean \(C_{L_{\max }}\) was predicted to be 1.71 (See Fig. 7b). Compared to the actual value of 1.72 (Obert 2009), this is an error of 0.58%. The largest error is 9.17% at a flap deflection of 24^{∘}. Considering the fact that in Valarezo and Chin (1994) the PDR has been validated against experimental data for numerous multielement wing combinations up to flap angles of 40 degrees, the inaccuracy of the Fokker 100 \(C_{L_{\max }}\) at high flap deflections may well be attributed to the fact that the exact flap geometry of the Fokker 100 wing was unavailable for this research.
6.2 Wing weight
6.3 Airfield performance
Landing distance is computed analogously to takeoff distance, taking into account that approach speed V _{ A } must be at least 1.23 times higher than V _{s1g}, the approach angle should not be steeper than 3 deg and the touch down velocity V _{ T D } is assumed to be 1.15 times V _{s1g} according to Raymer. This results in an average flaring velocity V _{ F L } of 1.19 times \(V_{S_{0}}\). The load factor during landing can be taken as 1.2 and the rolling friction coefficient due to deployed brakes during the ground run can be taken to be 10 times higher than during takeoff. It should be noted that typically, the aircraft rolls free for 1 to 3 seconds before the pilot applies the brakes.
Fokker 100 Airfield performance
Actual  Computed  𝜖  

s _{TO}[m]  1760  1827  3.6% 
s _{LNG}[m]  1345  1436  6.3% 
6.4 Sensitivities Verification
Sensitivity verification
Function  Variable  FD  Coupledadjoint  Relative difference (%)  FD step size 

s _{LNG} [m]  Thickness of wing upper panel at root section [m]  −58.5482  −58.6677  2.03 × 10^{−1}  1e9 
...  Thickness of wing lower panel at root section [m]  −38.5779  −38.8700  7.57 × 10^{−1}  1e9 
...  First Chebyshev mode amplitude at root section []  −299.5294  −298.8453  2.28 × 10^{−1}  1e6 
...  Inboard leading edge sweep [rad]  942.4716  944.4626  2.11 × 10^{−1}  1e9 
...  Span up to wing kink [m]  −168.8636  −168.8063  3.39 × 10^{−2}  1e9 
...  Flap span [m]  −1304.3331  −1300.6584  2.81 × 10^{−1}  1e9 
...  Flap overlap [%c]  17567.2443  17567.8219  3.28 × 10^{−3}  1e9 
...  Flap gap [%c]  13016.4608  13017.0928  4.85 × 10^{−3}  1e9 
...  Flap deflection [rad]  −2984.6424  −2984.3981  1.87 × 10^{−3}  1e9 
7 Test case application
The fifth group of design variables is used to perturb the flap’s position using the mode amplitudes \(G_{t_{k}}\) (see Fig. 9). The mode amplitudes consist of two translational modes. \(G_{t_{1}}\) controls the horizontal translation of the flap and \(G_{t_{2}}\) the vertical translation. The third mode amplitude \(G_{t_{3}}\) controls the flap deflection.
The final group of variables are used to avoid unnecessary iterations for aeroelastic analysis. The optimization problem is subject to a number of constraints including constraints on structural failure and aileron effectiveness as described in Elham and van Tooren (2016a). In the same research, Elham and Van Tooren included a constraint on wing loading to take airfield performance into account. In the present research, the wing loading constraint is replaced by constraints on takeoff and landing distance.
Optimization variables and constraints
Variable group  Symbol  # 

Equivalent panel thickness  T  40 
Wing planform  P  8 
Flap planform  P_{ f }  1 
Airfoil shape  G_{ j }  160 
Flap position  \(\text {G}_{t_{k}}\)  3 
Surrogate variables  X ^{∗}  2 
Constraint  Equation  
Compression upper panel  F _{compression} ≤ 0  104 
Compression lower panel  F _{compression} ≤ 0  52 
Tension upper panel  F _{tension} ≤ 0  52 
Tension lower panel  F _{tension} ≤ 0  104 
Buckling upper panel  F _{buckling} ≤ 0  104 
Buckling lower panel  F _{buckling} ≤ 0  52 
Shear front spar  F _{shear} ≤ 0  78 
Buckling front spar  F _{buckling} ≤ 0  78 
Shear rear spar  F _{shear} ≤ 0  78 
Buckling rear spar  F _{buckling} ≤ 0  78 
Fatigue  F _{fatigue} ≤ 0  52 
Aileron Effectiveness  \(1  \frac {M_{a}}{M_{a_{\min }}}\leq 0\)  1 
Takeoff distance  \( \frac {s_{\text {TO}}}{s_{\text {TO}_{0}}} 1\leq 0\)  1 
Landing distance  \(\frac {s_{\text {LNG}}}{s_{\text {LNG}_{0}}}  1\leq 0\)  1 
Fuel weight  \(\frac {W_{\text {fuel}}}{W_{\text {fuel}}^{*}}  1 = 0\)  1 
Maximum Take Off Weight  \(\frac {\text {MTOW}}{\text {MTOW}^{*}}  1 = 0\)  1 
Load case  type  H [m]  M  n [g] 

1  pull up, M _{ D }  7500  0.84  2.5 
2  pull up, V _{ D }  0  0.57  2.5 
3  push down, M _{ D }  7500  0.84  1 
4  gust, M _{ D }  7500  0.84  1.3 
5  roll, 1.15V _{ D }  4000  0.81  1 
6  cruise, M _{cruise}  10670  0.77  1 
7  takeoff, V _{2}  0    1 
8  landing, V _{A}  0    1 
The structural analysis is performed for the load cases listed in Table 5. Fatigue is simulated by limiting the stress in the wing box lower panel to 42% of the maximum allowable stress of the material in a 1.3g gust load case (Hürlimann et al. 2011). The aircraft rolling moment due to aileron deflection (L _{ δ }) in the critical roll case is limited to be higher or equal to L _{ δ } of the initial aircraft.
The mission fuel weight (W _{fuel}) is computed based on the method of Roskam (2003). The required fuel use for cruise is computed using the Bréguet range equation, while statistical factors are used to determine the fuel weight of the remaining segments of the mission. The total aircraft drag is assumed to be the sum of the wing drag and the drag of the rest of the aircraft based on Fokker 100 aircraft data (Obert 2009). The drag of the aircraft minus wing is kept constant during the optimization. The aircraft range, cruise Mach number, altitude and engine parameters are determined based on aircraft data. The aircraft Maximum takeoff Weight (MTOW) is assumed to be equal to the payload weight, the aircraft fuel weight, the wing structural weight and a weight components that is called the rest weight. which is the operational empty weight minus the wing structural weight. The rest weight of the aircraft is computed from the Fokker 100 weight data and is kept constant during optimization.

Wing A (Concurrent optimization): The wing planform and airfoil shape and the high lift device geometry are optimized simultaneously for minimizing the aircraft mission fuel weight and satisfying the field performance constraints, (7).

Wing B (Sequential optimization): Wing planform and airfoils are first optimized for minimum fuel weight (based on the cruise condition) without any airfield performance constraint. Then the highlift devices are optimized for satisfying the field performance requirements.
Initial and optimized wing geometry variables
Parameter  Initial  Wing A  Wing B 

c _{ r } [m]  5.97  5.52  5.49 
λ []  0.18  0.10  0.12 
b _{1} [m]  4.70  3.97  4.39 
b _{2} [m]  9.34  11.73  10.42 
Λ_{1} [^{∘}]  25.5  21.26  21.46 
Λ_{2} [^{∘}]  21.5  18.23  20.91 
𝜖 _{1} [^{∘}]  −0.65  −0.80  −0.76 
𝜖 _{2} [^{∘}]  −5.40  −4.32  −4.34 
b _{ f }[m]  6.50  5.06  5.50 
Characteristics of the initial and the optimized aircraft
MTOW [kg]  W _{fuel} [kg]  W _{wing} [kg]  S _{wing} [m ^{2}]  \(C_{L_{\text {cruise}}}\)  \(C_{D_{\text {cruise}}}\)  \(C_{D_{i}}\)  \(C_{D_{p}}\)  \(C_{D_{f}}\)  

Initial  43090  7260  4369  95.4  0.41  0.0188  0.0078  0.0063  0.0047 
Concurrent  42487  6559  4468  92.6  0.42  0.0132  0.0059  0.0027  0.0046 
Sequential  42492  6592  4446  90.1  0.43  0.0141  0.0067  0.0025  0.0049 
Characteristics of the initial and the optimized highlift system
W _{flap} [kg]  S _{flap} [m ^{2}]  b _{ f } [m]  h _{ f } [%c]  g _{ f } [%c]  δ _{ f } [^{∘}]  

Initial  576  17.1  6.50  5.00  2.40  20.00 
Concurrent  443  13.0  5.06  5.69  2.75  28.24 
Sequential  473  14.0  5.50  5.78  2.42  27.76 
HighLift characteristics of the initial and the optimized aircraft
\(C_{{L_{\max }}_{_{TO}}}\)  \(C_{{L_{\max }}_{_{LNG}}}\)  \(V_{s_{\text {1g}}}_{_{\text {MTOW}}}\) [m/s]  \(V_{s_{\text {1g}}}_{_{\text {MLW}}}\) [m/s]  \(\frac {L}{D}_{_{V_{2}}}\)  \(\frac {L}{D}_{_{V_{A}}}\)  s _{TO} [m]  s _{LNG} [m]  

Initital  1.71  1.91  65.03  51.91  19.99  12.84  1827  1436 
Concurrent  1.75  2.12  64.69  50.81  23.39  11.15  1767  1432 
Sequential  1.76  2.03  65.68  52.36  22.00  11.24  1835  1431 
While wing A meets both airfield performance requirements, wing B is not able to achieve takeoff performance with flaps retracted. Although this seems to be an infeasible solution, it is a result of the optimization formulation, in which takeoff is always performed with flaps retracted. So the wing planform was not design to satisfy the field performance and the flap optimization was not performed for takeoff condition. When considering the design of wing B, it can be seen that its wing loading has increased due to the small wing area. While this is beneficial for cruise flight, it is undesirable for maintaining airfield performance.
8 Conclusion
An enhanced coupledadjoint aerostructural analysis and optimization tool has been presented which enables the optimization of highlift devices from the start of the design process. The semiempirical Pressure Difference Rule has been implemented in an existing quasithreedimensional aerodynamic analysis and coupled to the structural solver FEMWET to compute the wing \(C_{L_{\max }}\) taking into account aeroelastic effects. The coupled system is solved using the Newton method.
The modified aerostructural tool is able to compute the derivatives of the outputs with respect to the inputs using a combination of the chain rule of differentiation, automatic differentiation and coupledadjoint method. Wing weight is computed using the empirical method of Torenbeek which enables optimization taking into account the specific weight of highlift devices. Airfield performance is determined using the method of Raymer which includes lift and drag terms at takeoff and landing. Validation of the modifications showed good levels of accuracy for L/D, \(C_{L_{\max }}\), wing weight and airfield performance.
The tool was then used for gradient based aerostructural wing and highlift system optimization for a Fokker 100 class wing. Design variables include flap span and settings, wing planform, airfoil geometry and wingbox structure. The optimization was performed for minimizing the fuel weight, while satisfying constraints on structural failure, \(C_{L_{\max }}\) in takeoff and landing configuration and the roll requirement. The flaps were retracted in the takeoff configuration as the Fokker 100 is certified to perform takeoff with flaps retracted. Two types of optimizations were performed. The proposed concurrent optimization scheme where the highlift system was optimized from the start of the process and a more conventional sequential optimization, in which the planform was first optimized for cruise performance after which the highlift system was sized to minimize fuel weight for the fixed optimized planform taking into account airfield performance.
The concurrent optimization resulted in a fuel weight reduction of 9.65%, while sequential optimization reduced fuel weight by 9.20%. The reduced fuel weight was attributed to a reduction in pressure drag resulting from modified airfoil shapes, reducing shock waves over the wing. The optimized wings both have an increased aspect ratio and reduced sweep angle, which resulted in a reduction of induced drag. To counter the weight penalty due to these modifications, the structural stiffness was reduced near the wing tip. The optimizer reduced flap weight of both optimized wings by respectively 22.9% and 17.9% by reducing flap span. It can be concluded that the proposed method of combining the optimization of highlift and cruise wing design is promising and provides ample opportunities for more research.
Notes
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