Efficient aerodynamic shape optimization by using unsupervised manifold learning to filter geometric features

Figures & data

Figure 1. Geometric shapes of class and shape transformation (CST) method. (a) Class function; (b) Shape function

Table 1. Convergence of the aerodynamic performance of the mesh for the RAE2822 airfoil.

Grid size	$C_l$	$C_d\left(\mathrm{counts}\right)$
$128\times 64$	0.823	208.6
$192\times 96$	0.824	202.3
$256\times 120$	0.824	195.3
$320\times 160$	0.824	194.9
$448\times 256$	0.824	194.8

Figure 2. Computational grid for the RAE2822 airfoil. (a) Global grid; (b) Local grid

Figure 3. Process of ISOMAP, and comparison of feature extraction.

Figure 4. Correlation between features and geometric parameters.

Figure 5. Variation of correlation and time with $\text{K}$ value.

Figure 6. Extracting features of the airfoil by using ISOMAP. (a) Visualization of the feature space; (b) Shape of the airfoil

Figure 7. Visualization of geometric features and distributed forces. (a) Airfoils and the distribution of their pressure coefficients; (b) Contours of the pressure coefficient

Figure 8. Visualization of integrated forces in the feature space. (a) Mapping of ${\text{C}}_{\text{l}}$ and the probability distribution of ${\text{C}}_{\text{l}}>0.875$ samples; (b) Mapping of ${\text{C}}_{\text{d}}$ and the probability distribution of ${\text{C}}_{\text{d}}<0.015$ samples; (c) Mapping of ${\text{C}}_{\text{m}}$ and the probability distribution of ${\text{C}}_{\text{m}}<-0.075$ samples

Figure 9. Overall framework for the optimization of the aerodynamic shape of the airfoils by using geometric feature filtering.

Figure 10. Detailed process of optimization of the aerodynamic shape of the airfoil.

Figure 11. Uniform initial sampling method for feature space

Figure 12. The process of screening samples using geometric features

Figure 13. Convergence histories

Figure 14. Comparison of the results of unconstrained optimization. (a) Comparison of optimized and initial airfoils; (b) Comparison of the distributions of wall pressure between the optimized and initial airfoils

Figure 15. Results of optimized feature projection by TLBO-ISOMAP-${\lambda }_{0.15}$ and TLBO. (a) First generation of optimization by TLBO-ISOMAP-${\lambda }_{0.15};$(b) First generation of optimization by TLBO; (c) 10th generation of optimization by TLBO-ISOMAP-${\lambda }_{0.15}$; (d) 10th generation of optimization by TLBO; (e) 30th generation of optimization by TLBO-ISOMAP-${\lambda }_{0.15}$; (f) 30th generation of optimization by TLBO; (g) $5$0th generation of optimization by TLBO-ISOMAP-${\lambda }_{0.15}$; (h) 50th generation of optimization by TLBO

Table 2. Optimization problem statement

	Case 1		Case 2		Case 3
	Function	Description	Function	Description	Function	Description
$\mathrm{Min}$	$C_d$	Drag coefficient	$C_d$	Drag coefficient	$C_d$	Drag coefficient
$S.t.$	$C_l\ge 0.824$	Lift constraint	$C_l\ge 0.824$	Lift constraint	$C_l\ge 0.824$	Lift constraint
	$C_m\ge -0.092$	Pitching moment constraint	$C_m\ge -0.092$	Pitching moment constraint	$C_m\ge -0.092$	Pitching moment constraint
	$A\ge A_{\mathrm{initial}}$	Area constraint	$t_{\max }\ge 12\mathrm{\%}$	Maximum relative thickness constraint	$t_{\max }=12\mathrm{\%}$	Maximum relative thickness constraint
					$l_{t_{\max }}=0.38$	Maximum relative thickness position constraint

Table 3. Result of different constrained optimization

Case	Method	$C_d^p$	CFD iterations required to achieve TLBO optimal value	Increment
Original	…	0.0195		…
Case 1	TLBO-ISOMAP-${\lambda }_{0.2}$	0.0116	460	40.5%
	TLBO	0.0120	840	38.5%
	BADS	0.0119	192	39.0%
Case 2	TLBO-ISOMAP-${\lambda }_{0.2}$	0.0115	520	41.0%
	TLBO	0.0119	980	39.0%
	BADS	0.0126	Not reached	38.5%
Case 3	TLBO-ISOMAP-${\lambda }_{0.2}$	0.0160	360	17.9%
	TLBO	0.0172	960	11.8%
	BADS	0.0169	298	13.3%

Figure 16. Convergence histories. (a) Case 1; (b) Case 2; (c) Case 3

Figure 17. Pressure distributions and airfoils of different constrained optimization. (a) Case 1; (b) Case 2; (c) Case 3

Figure 18. Projections of the top 40% of the samples in terms of aerodynamic performance in case 1. (a) Projection of the sample generated by TLBO-ISOMAP-${\lambda }_{0.2}$; (b) Projection of the sample generated by TLBO

Figure 19. Distribution density of samples generated in generations 30–50 for case 1. (a) TLBO-ISOMAP-${\lambda }_{0.2}$; (b) TLBO

Figure 20. Projection of ${\text{C}}_{\text{d}}^{\text{p}}$ of samples generated in generations 30–50 for case 1. (a) TLBO-ISOMAP-${\lambda }_{0.2}$; (b) TLBO

Figure 21. Visualization of the local feature space of samples generated by TLBO-ISOMAP-${\lambda }_{0.2}$ in generations 30–50 for case 1. (a) Visualization of airfoils and their pressure distributions; (b) Comparison of airfoils

Data availability

Data will be made available on request.