[CVAE-GAN] An inverse design method for supercritical airfoil based on conditional generative models

桃扇骨 2022-09-02 12:54 135阅读 0赞

# An inverse design method for supercritical airfoil based on conditional generative models #

Author: Jing WANG, Runze LI, Cheng HE et al. [\[PDF\]][PDF]

> supercritical airfoil: 超临界机翼

## Introducation ##

### Inverse Designed ###

Usually proceeds in two steps.

First, target distributions are generated to reflect the design goals. (be specified by an experienced aerodynamicist)

Second, optimization methods are applied to optimize the parameterized pressure distributions to obtain the target distributions.

### CGAN ###

GAN: generator competes against discriminator

CGAN: Conditional Generative Adversarial Networks

based on a vector set of conditional data

### CVAE ###

VAE: Variational AutoEncoder

encodes the input data to a probabilistic latent space

CVAE: Conditional Variational AutoEncoder

## Method ##

### Dataset ###

1.  x-y coordinates

tuple(x,y) starting from the trailing edge.

only the y coordinates are involved in training

which can reduce the dimensionality of the input or output of the deep learning network to accelerate the training procedure

1.  Wall Mach number distributions

壁面马系数分布

This paper only use airfoils with single shock wave concluding the weak shock pressure distribution was evaluated as the best.

1.  Features of wall Mach number distributions

<table> 
 <thead> 
 <tr> 
 <th>Features</th> 
 <th>Symbols</th> 
 <th>Implication</th> 
 </tr> 
 </thead> 
 <tbody> 
 <tr> 
 <td>Suction peak</td> 
 <td> F s p F_{sp} Fsp</td> 
 <td>The x x x and M a w Ma_{w} Maw values at the point with the highest wall Mach number.</td> 
 </tr> 
 <tr> 
 <td>Start of the shock wave</td> 
 <td> F s w 0 F_{sw0} Fsw0</td> 
 <td>The x x x and M a w Ma_{w} Maw values at the start of the shock wave.</td> 
 </tr> 
 <tr> 
 <td>End of the shock wave</td> 
 <td> F s w 1 F_{sw1} Fsw1</td> 
 <td>The x x x and M a w Ma_{w} Maw values at the end of the shock wave.</td> 
 </tr> 
 <tr> 
 <td>Aft loading</td> 
 <td> F a l F_{al} Fal</td> 
 <td>The x x x and M a w Ma_{w} Maw values at the point with the maximum difference in the wall number between the upper and lower surfaces near the trailing edge.</td> 
 </tr> 
 <tr> 
 <td>Maximum wall Mach number of lower surface</td> 
 <td> F l m F_{lm} Flm</td> 
 <td>The x x x and M a w Ma_{w} Maw values at the point with the maximum wall Mach number on the lower surface.</td> 
 </tr> 
 </tbody> 
</table>

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### Generative model ###

compare the CVAE and CVAE-GAN

use WGAN instead of traditional GANs. ***improves the stability of learning***

The VAE decoder and the GAN generator are collapsed into one by sharing their parameters and training them jointly.

Encoder network  E E E: maps the data sample  x x x to a latent representation  z z z through a learned distribution  P ( z ∣ x , c ) P(z|x,c) P(z∣x,c), where  c c c is the given condition that the data satisfy.

Generative network  G G G: generates the  x ′ x^\{'\} x′ under the given latent vector  z z z and conditions  c c c by sampling from a learned distribution  P ( x ∣ z , c ) P(x|z,c) P(x∣z,c).

Loss(training stage):  L generator  = MSE ⁡ ( x ′ , x ) + KL ⁡ ( q ( z ∣ x , c ) ∣ ∣ p ( z ∣ x ) ) + D ( G ( z ) ) L\_\{\\text \{generator \}\}=\\operatorname\{MSE\}\\left(x^\{\\prime\}, x\\right)+\\operatorname\{KL\}(q(z \\mid x, c)|| p(z \\mid x))+D(G(z)) Lgenerator =MSE(x′,x)\+KL(q(z∣x,c)∣∣p(z∣x))\+D(G(z))

Discriminative network  D D D: This network is the same as that in the WGAN.

Loss(final loss):  L discriminator  = D ( x ) − D ( G ( z ) ) L\_\{\\text \{discriminator \}\}=D(x)-D(G(z)) Ldiscriminator =D(x)−D(G(z))

The entire dataset is used for training instead of being split into a training subset and a validation subset.

Validation can be achieved by verifying the consistency between the labels of the generated data and the input.

### Mapping model ###

Neural network + Smoothness measurement of airfoil shape

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## Result ##

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## Conclusion ##

(1) A designed mapping network based on the deep neural network for the inverse design process is conducted, and the model is capable of estimating the corresponding airfoil shape to the given wall Mach distribution. The predicted results show that the present approach has reasonable accuracy for both training and testing samples.

(2) In order to measure the quality of the airfoil shape based on the generated wall Mach number distributions, a smoothness measurement for the airfoil shape is proposed. The validity and accuracy of the measurement were demonstrated on five datasets. A threshold value is given based on the statistical results to determine whether the generated airfoil shape is smooth and the target wall Mach number distribution is further realistic.

(3) Two generative models, the CVAE and CVAE-GAN, for the inverse design process are proposed. The effectiveness of the two models is demonstrated and compared. The results show that both models can generate diverse and realistic wall Mach number distributions satisfying the given features, while the CVAE-GAN model outperforms the CVAE model and achieves better reconstruction accuracies for all the samples in the dataset.

[PDF]: https://www.sciencedirect.com/science/article/pii/S1000936121000662
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[img-SO0p3MLS-1627974060086_Users_coffechoz_Documents_Markdown_Aerodynamic Shape_An inverse design method for supercritical airfoil 4 based on conditional generative models.imgs_image-20210803144854089.png]: /images/20220829/114ca810b02340958dc0f1f07953a2d7.png
[img-Ia2QQ0Cm-1627974060087_Users_coffechoz_Documents_Markdown_Aerodynamic Shape_An inverse design method for supercritical airfoil 4 based on conditional generative models.imgs_image-20210803145157146.png]: /images/20220829/8079e75e0beb444ebb8a9ea1b9edc81c.png
[img-N9yhybuY-1627974060088_Users_coffechoz_Documents_Markdown_Aerodynamic Shape_An inverse design method for supercritical airfoil 4 based on conditional generative models.imgs_image-20210803145222406.png]: /images/20220829/4ff25949a2c14826821972c7f00804dd.png