5. Conclusions
This research study consists of the development of an aeroelastic optimization framework which with appropriate configuration is useful for either mass or similarity optimization between a subscale model and a reference wing. The core of the framework consists of the aeroelastic analysis (static and dynamic) combined with the appropriate functions for the objectives and constraints and an ACO algorithm (MIDACO). The parameterization of the wing includes geometric parameters and thicknesses for the stiffness distribution and 10 lumped masses for the mass distribution (for the scaling problem).
A mass optimization problem was initially set up for the uCRM wing. In this case, there were only geometric and thickness parameters (e.g., spar positions, number of ribs, number of stringers and thicknesses at specified control points). Stiffness, stress, and flutter constraints were included with target values being set based on the literature and the Civil Aviation Regulations FAR25. The optimizer converged at a structural mass of $6902kg$ while all the constraints were satisfied. Of interest are the optimized variables’ values. The thickness parameters decrease gradually along the wing, except of the Yehudi break where maximum stress usually arises. The spars’ position, as expected, converged around the position of the airfoil’s maximum thickness in order to optimize the stiffness in the x direction while the underestimated drag (because of the potential flow implemented by DLM does not require so much stiffness in the z direction. Finally, the ribs’ number converged to 51, which is close to the optimal solution presented in [37].
As far as the scaling problem is concerned, the objectives and constraints were modified in order to meet similarity requirements. In fact the problem’s objectives were the RMS error of inflight shape difference and the average MAC of the N modes that were selected to optimize. The constraints included were the equality of each eigenfrequency, the equality of the model’s mass to a target value defined by the scaling parameters, and the avoidance of maximum stresses greater than the maximum allowable. Also, a constraint to avoid flutter at a certain speed for the optimal solution was considered.
The predescribed optimization problem was first run with a singlestep approach, for N=10 modes to optimize and the material was aluminium. The results were deemed unacceptable. The RMS error of the inflight shape difference converged to the value of 0.45, indicating bad matching of the static aeroelastic response. The constraints of the modal response were satisfied but the modal similarity was quite low indicating no similarity at all. These problems were addressed by changing the material to magnesium, changing the modes to match to N=5 and the onestep approach to a twostep (serial) approach in order to reduce the computing time and get acceptable results quicker. After those alterations, the optimizer converged to acceptable values indicating a good matching of both the static aeroelastic and modal response. In fact the RMS error of the inflight shape difference of the optimal solution is $0.0549m$ and the mean MAC value of the first 5 modes is $1{f}_{min}=0.934$. The constraints of the problem were also satisfied but there is room for improvement mainly in terms of frequency error. As far as the flutter is concerned, the flutter speed of the optimal solution is way beyond the diving speed, indicating no flutter.
Of interest is the location of the spars, which are quite close to the leading and trailing edges, in contrast to the reference wing. This is likely due to the increased stiffness presented by the subscale structure given the reduced aerodynamic loads. Placing the spars towards the edges, reduces the stiffness around the xaxis and thus the structure becomes more flexible thus avoiding the very low values of the thicknesses and thus the construction difficulties. The number of ribs was also reduced to 15 in order to reduce the torsional stiffness of the wing indicating the significance of the parameterization of ribs’ number. Future implementations in the framework are directed towards the following:
The investigation of other types of materials (such as wood or composites) and combination of them would greatly increase the design space and ensure the presence (or absence) of the effects of the full model (such as skin buckling).
The calculation of derivatives in order to include a gradientbased optimizer in the framework. It will give the opportunity for a more efficient optimization since we take advantage of a gradientfree optimizer.
Introduction of a high fidelity aerodynamic solver such as Euler or RANS. As already mentioned, the potential theory upon which the DLM is based, underestimates the drag distribution while overestimating the lift. A highfidelity solver will avoid such problems and provide more accurate load prediction.
The parameterization of the number of spars will give the opportunity to design wings with more than two spars and it will greatly increase the design space. Also, it will allow the design of other types of wings such as military aircraft wings that incorporate multispar configurations.
Figure 1.
Commercial aircraft wings aspect ratio trend [
4].
Figure 1.
Commercial aircraft wings aspect ratio trend [
4].
Figure 2.
Modern approach for aeroelastic effects investigation.
Figure 2.
Modern approach for aeroelastic effects investigation.
Figure 3.
Aerodynamic models used in aeroelastic calculations.
Figure 3.
Aerodynamic models used in aeroelastic calculations.
Figure 4.
Structural models used in aeroelastic calculations.
Figure 4.
Structural models used in aeroelastic calculations.
Figure 5.
The American experimental aircraft Grumman X29.
Figure 5.
The American experimental aircraft Grumman X29.
Figure 6.
High subsonic flutter model of Grumman F6F in NACA Langley Memorial Aeronautical Laboratory [19].
Figure 6.
High subsonic flutter model of Grumman F6F in NACA Langley Memorial Aeronautical Laboratory [19].
Figure 7.
Recent aeroelastic scaling practises ([20,21,22]).
Figure 7.
Recent aeroelastic scaling practises ([20,21,22]).
Figure 8.
Example of FEM mesh with shell and beam elements.
Figure 8.
Example of FEM mesh with shell and beam elements.
Figure 9.
3D wing surface modeled by flat panels.
Figure 9.
3D wing surface modeled by flat panels.
Figure 10.
Mean camber line construction method.
Figure 10.
Mean camber line construction method.
Figure 11.
W2GJ matrix influence on boxes’ inclination.
Figure 11.
W2GJ matrix influence on boxes’ inclination.
Figure 12.
Example of coupling nodes in a wing.
Figure 12.
Example of coupling nodes in a wing.
Figure 13.
Static aeroelastic framework flowchart.
Figure 13.
Static aeroelastic framework flowchart.
Figure 14.
Modal analysis framework flowchart.
Figure 14.
Modal analysis framework flowchart.
Figure 15.
Dynamic aeroelastic framework flowchart.
Figure 15.
Dynamic aeroelastic framework flowchart.
Figure 16.
Aeroelastic optimization framework flowchart.
Figure 16.
Aeroelastic optimization framework flowchart.
Figure 17.
Aeroelastic scaling framework flowchart.
Figure 17.
Aeroelastic scaling framework flowchart.
Figure 18.
uCRM wing OML and internal structure.
Figure 18.
uCRM wing OML and internal structure.
Figure 19.
uCRM wing twist distribution.
Figure 19.
uCRM wing twist distribution.
Figure 20.
Objective and constraints violation during the 1^{st} run.
Figure 20.
Objective and constraints violation during the 1^{st} run.
Figure 21.
Violation of each constraint during the 1^{st} run.
Figure 21.
Violation of each constraint during the 1^{st} run.
Figure 22.
Objective and constraints violation during the 2^{nd} run.
Figure 22.
Objective and constraints violation during the 2^{nd} run.
Figure 23.
Violation of each constraint during the 2^{nd} run.
Figure 23.
Violation of each constraint during the 2^{nd} run.
Figure 24.
3D view of the optimized geometry.
Figure 24.
3D view of the optimized geometry.
Figure 25.
Optimized thickness distribution of the internal geometry, upper and lower skins along with the corresponding control point values.
Figure 25.
Optimized thickness distribution of the internal geometry, upper and lower skins along with the corresponding control point values.
Figure 26.
Displacement distribution of the uCRM wing.
Figure 26.
Displacement distribution of the uCRM wing.
Figure 27.
Wing static aeroelastic displacements.
Figure 27.
Wing static aeroelastic displacements.
Figure 28.
Von Mises stress distribution of the uCRM wing.
Figure 28.
Von Mises stress distribution of the uCRM wing.
Figure 29.
Coefficient of pressure distribution.
Figure 29.
Coefficient of pressure distribution.
Figure 30.
Vg and Vf plots.
Figure 30.
Vg and Vf plots.
Figure 32.
Objectives during the parallel optimization run.
Figure 32.
Objectives during the parallel optimization run.
Figure 33.
Violation of each constraint during the parallel optimization run.
Figure 33.
Violation of each constraint during the parallel optimization run.
Figure 34.
Objectives during 1^{st} run of static aeroelastic response similarity optimization.
Figure 34.
Objectives during 1^{st} run of static aeroelastic response similarity optimization.
Figure 35.
Objectives during 2^{nd} run of static aeroelastic response similarity optimization.
Figure 35.
Objectives during 2^{nd} run of static aeroelastic response similarity optimization.
Figure 36.
Optimized thickness distribution of each component and its corresponding control point values for the scaling problem.
Figure 36.
Optimized thickness distribution of each component and its corresponding control point values for the scaling problem.
Figure 37.
Inflight aerodynamic surface of the optimized design compared to the target shape.
Figure 37.
Inflight aerodynamic surface of the optimized design compared to the target shape.
Figure 38.
Comparison of the scaled and reference wing.
Figure 38.
Comparison of the scaled and reference wing.
Figure 39.
Von Mises stress distribution of the scaled model.
Figure 39.
Von Mises stress distribution of the scaled model.
Figure 40.
Objectives during the run of modal similarity optimization for N=5 modes.
Figure 40.
Objectives during the run of modal similarity optimization for N=5 modes.
Figure 41.
Violation of each constraint during the run of modal similarity optimization for N=5 modes.
Figure 41.
Violation of each constraint during the run of modal similarity optimization for N=5 modes.
Figure 42.
Optimized lumped masses of the scaled model.
Figure 42.
Optimized lumped masses of the scaled model.
Figure 43.
First eight eigenmodes of the scaled model compared to the modes of the reference wing (red indicates undeformed shape and blue indicates the deformed shape).
Figure 43.
First eight eigenmodes of the scaled model compared to the modes of the reference wing (red indicates undeformed shape and blue indicates the deformed shape).
Figure 44.
Objectives during the run of modal similarity optimization for N=10 modes.
Figure 44.
Objectives during the run of modal similarity optimization for N=10 modes.
Figure 45.
Violation of each constraint during the run of modal similarity optimization for N=10 modes.
Figure 45.
Violation of each constraint during the run of modal similarity optimization for N=10 modes.
Figure 46.
Flutter results of the optimized scaled wing.
Figure 46.
Flutter results of the optimized scaled wing.
Figure 47.
Spars’ position during the 1^{st} run of static aeroelastic response optimization.
Figure 47.
Spars’ position during the 1^{st} run of static aeroelastic response optimization.
Figure 48.
Number of ribs during the 1^{st} run of static aeroelastic response optimization.
Figure 48.
Number of ribs during the 1^{st} run of static aeroelastic response optimization.
Table 1.
A summary of the features of the publications referenced in this work.
Table 1.
A summary of the features of the publications referenced in this work.
Author 
Aerodynamic 
Structural 
Optimization 
Thicknesses 
Topology 
French and Eastep 
DLM 
Beam FEM 
Gradient 
✓ 
✗ 
Richards et al.

Thin Airfoil 
Shell FEM 
Both 
✓ 
✗ 
Ricciardi et al.

VLM 
Shell FEM 
Gradient 
✓ 
✗ 
Ricciardi et al.

VLM 
Shell FEM 
Gradient 
✓ 
✗ 
Pontillo et al.

Strip Theory 
Beam FEM 
Gradient 
✓ 
✗ 
Spada et al.

DLM 
NL Shell FEM 
Gradient 
✓ 
✗ 
Mas Colomer et al.

Panel 
Shell FEM 
GradientFree 
✓ 
✗ 
Table 2.
Aluminium’s mechanical properties.
Table 2.
Aluminium’s mechanical properties.
Quantity 
Value 
Units 
Density 
2780 
kg/m^{3}

Young’s modulus 
73.1 
GPa 
Poisson ratio 
0.3 
 
Yield strength 
420 
MPa 
Table 3.
CRM geometric specifications.
Table 3.
CRM geometric specifications.
Parameter 
uCRM9 
Units 
Aspect Ratio 
9.0 
 
Span 
58.76 
m 
Side of body chord 
11.92 
m 
Yehudi chord 
7.26 
m 
MAC 
7.01 
m 
Tip chord 
2.736 
m 
Wimpress reference area 
383.78 
${m}^{2}$ 
Gross area 
412.10 
${m}^{2}$ 
Exposed area 
337.05 
${m}^{2}$ 
1/4 chord sweep 
35 
deg 
Taper ratio 
0.275 
 
Table 4.
CRM wing critical loading conditions.
Table 4.
CRM wing critical loading conditions.
Condition 
Lift Constraint 
Mach Number 
Altitude (m) 
2.5G Maneuver 
2.5 MTOW 
0.64 
0 
Table 5.
Objectives and constraints of the mass minimization problem.
Table 5.
Objectives and constraints of the mass minimization problem.
Objective 
Target 
Mass 
Minimization 
under the constraints 
Maximum Deflection 
$\le 0.15\xb7\mathrm{Semi}\mathrm{Span}$ 
Tip Torsion Angle 
$\le 6$ deg 
First Eigenfrequency 
≥ 1 Hz 
Maximum Von Mises Stress 
≤ 280 MPa 
Flutter Speed 
$\ge 1.2\xb7{V}_{D}$ 
Table 6.
Design variables of the optimization problem.
Table 6.
Design variables of the optimization problem.
Variable 
Lower Bound 
Upper Bound 
Dimension 
Ribs Number 
6 
52 
1 
Stringers Number 
4 
12 
1 
Front Spar Position 
0.1 
0.4 
1 
Rear Spar Position 
0.5 
0.9 
1 
Stringer and Spar Cap Thickness 
1 mm 
10 mm 
2 mm 
Thicknesses at Control Points 
2 mm 
40 mm 
66 mm 
Table 7.
MIDACO parameters.
Table 7.
MIDACO parameters.
Run 
Iterations 
Accuracy 
FOCUS 
Initial Point 
1 
250 
0.1 
0 
Random 
2 
100 
0.01 
10 
Run 1 
Table 8.
optimization results summary.
Table 8.
optimization results summary.
Property 
Value 
Mass 
6902 kg 
Minimum Eigenfrequency 
1.1735 Hz 
Maximum Von Mises Stress 
247.5 MPa 
Tip Torsion 
2.7 deg 
Maximum Deflection 
7.89% of span or 2.32 m 
Flutter Speed 
501 m/s 
Table 9.
Optimized geometric design variables.
Table 9.
Optimized geometric design variables.
Variable 
Value 
Ribs Number 
51 
Stringers Number 
4 
Front Spar Position 
0.3904 
Rear Spar Position 
0.5166 
Stringer Thickness 
2 mm 
Spar Cap Thickness 
3 mm 
Table 10.
Objectives and constraints of the scaling problem.
Table 10.
Objectives and constraints of the scaling problem.
Objective 
Target 
InFlight Shape Difference 
Minimization 
Modal Similarity (MAC) 
Maximization 
Table 11.
Objectives and constraints of the mass minimization problem.
Table 11.
Objectives and constraints of the mass minimization problem.
Constraint 
Target 
Frequency Difference 
$=0Hz$ 
Overall Mass 
$494.5kg$ 
Maximum Von Mises Stress 
$\le 280MPa$ 
Table 12.
Target values of frequency for the optimization problem.
Table 12.
Target values of frequency for the optimization problem.
Mode Number 
Reference Wing Frequency, Hz 
Scaled Wing Frequency, Hz 
1 
1.1915 
2.6642 
2 
3.7067 
8.2885 
3 
5.6416 
12.6151 
4th 
7.58 Hz 
16.9494 
5th 
12.8288 
28.6861 
6th 
16.7503 
37.4548 
7th 
17.9140 
40.0570 
8th 
19.4901 
43.5812 
Table 14.
Magnesium’s mechanical properties.
Table 14.
Magnesium’s mechanical properties.
Quantity 
Value 
Units 
Density 
1800 
$kg/{m}^{3}$ 
Young’s modulus 
45 
$GPa$ 
Poisson ratio 
0.35 
 
Yield strength 
150 
$MPa$ 
Table 15.
MIDACO parameters of the static aeroelastic response similarity optimization.
Table 15.
MIDACO parameters of the static aeroelastic response similarity optimization.
Run 
Iterations 
Accuracy 
FOCUS 
Initial Point 
1 
500 
0.1 
0 
Random 
2 
100 
0.01 
10 
Run 1 
Table 16.
Optimized geometric design variables.
Table 16.
Optimized geometric design variables.
Variable 
Value 
Ribs Number 
15 
Stringers Number 
5 
Front Spar Position 
0.239 
Rear Spar Position 
0.839 
Table 17.
Optimized mass values for the modal response similarity optimization problem.
Table 17.
Optimized mass values for the modal response similarity optimization problem.
Mass 
Value [kg] 
1 
79.78 
2 
9.13 
3 
77.79 
4 
79.86 
5 
0.01 
6 
71.07 
7 
58.12 
8 
22.79 
9 
35.20 
10 
11.64 
Table 18.
Target values of frequency for the optimization problem.
Table 18.
Target values of frequency for the optimization problem.
Mode Number 
Target 
Optimized 
Error 
1 
$2.6642Hz$ 
$2.2743Hz$ 
14.63 % 
2 
$8.2885Hz$ 
$8.0904Hz$ 
2.39 % 
3 
$12.6151Hz$ 
$14.4570Hz$ 
14.60 % 
4 
$16.9494Hz$ 
$20.1439Hz$ 
18.84 % 
5 
$28.6861Hz$ 
$34.9225Hz$ 
21.76 % 