基于潜在扩散模型的翼型参数化方法

doi:10.7527/S1000-6893.2025.31180

Abstract

Abstract:

To alleviate the curse of dimensionality problem in aerodynamic shape optimization and improve the representation capability as well as optimization efficiency of parameterization method， this paper proposes a new airfoil parameterization method named Latent Diffusion Model （LDM）， which combines Class-Shape Transformation （CST）， Autoencoder （AE）， and Denoising Diffusion Implicit Model （DDIM）. The geometric quality of the airfoils generated by the proposed method is analyzed. Then， the effect of different latent dimensions on the distribution of the samples is examined. Next， the fitting accuracy and the representational capability of LDM is compared with those of four different parameterization methods： CST-AE， Principal Component Analysis （PCA）， Free Form Deformation （FFD）， and CST. Finally， airfoil aerodynamic optimization is conducted to verify the performance of the LDM method. The results show that the LDM can generate smooth and acceptable airfoil samples. Compared with other parameterization methods， this method offers a more accurate description and stronger representation capability for airfoils. Additionally， the LDM demonstrates faster convergence and shorter optimization time. The optimized airfoil exhibits better aerodynamic performance and a more stable optimization process. In the future， this method has the potential to be extended to aerodynamic optimization for more complex shapes， such as wing segments， nacelles and fans.

Key words: diffusion model, autoencoder, class-shape transformation (CST), airfoil parameterization, aerodynamic optimization design

CLC Number:

V211

Ruitao ZHANG, Cong WANG, Jun TAO, Liyue WANG, Gang SUN. Airfoil parameterization method based on latent diffusion model[J]. Acta Aeronautica et Astronautica Sinica, 2025, 46(10): 631180.

Figures/Tables 23

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Table 1

Fig.6

Fig.7

Fig.8

Fig.9

Fig.10

Fig.11

Fig.12

Table 2

Fig.13

Fig.14

Table 3

Fig.15

Table 4

Fig.16

Table 5

Fig.17

Table 6

References 35

1	BÉZIER P. Mathematical and practical possibilities of unisurf［M］∥Computer Aided Geometric Design. Amsterdam： Elsevier， 1974： 127-152.
2	GORDON W J， RIESENFELD R F. B-spline curves and surfaces［M］∥Computer Aided Geometric Design. Amsterdam： Elsevier， 1974： 95-126.
3	PIEGL L， TILLER W. The NURBS book［M］. Germany： Springer， 1997.
4	HICKS R M， HENNE P A. Wing design by numerical optimization［J］. Journal of Aircraft， 1978， 15（7）： 407-412.
5	SEDERBERG T W， PARRY S R. Free-form deformation of solid geometric models［C］∥Proceedings of the 13th Annual Conference on Computer Graphics and Interactive Techniques. New York： ACM， 1986： 151-160.
6	KULFAN B. A universal parametric geometry representation method-“CST”［C］∥45th AIAA Aerospace Sciences Meeting and Exhibit. Reston： AIAA， 2007.
7	SHAN S Q， WANG G G. Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions［J］. Structural and Multidisciplinary Optimization， 2010， 41（2）： 219-241.
8	FORRESTER A I J， KEANE A J. Recent advances in surrogate-based optimization［J］. Progress in Aerospace Sciences， 2009， 45（1-3）： 50-79.
9	ABDI H， WILLIAMS L J. Principal component analysis［J］. Wiley Interdisciplinary Reviews， 2010， 2（4）： 433-459.
10	TAO J， SUN G， GUO L Q， et al. Application of a PCA-DBN-based surrogate model to robust aerodynamic design optimization［J］. Chinese Journal of Aeronautics， 2020， 33（6）： 1573-1588.
11	SCHÖLKOPF B， SMOLA A， MÜLLER K R. Nonlinear component analysis as a kernel eigenvalue problem［J］. Neural Computation， 1998， 10（5）： 1299-1319.
12	KRAMER M A. Nonlinear principal component analysis using autoassociative neural networks［J］. AIChE Journal， 1991， 37（2）： 233-243.
13	KAPSOULIS D， TSIAKAS K， ASOUTI V， et al. The use of Kernel PCA in evolutionary optimization for computationally demanding engineering applications［C］∥ 2016 IEEE Symposium Series on Computational Intelligence （SSCI）. Piscataway： IEEE Press， 2016： 1-8.
14	吴则良，叶建川，王江，等. 基于深度自动编码器神经网络的飞行器翼型参数降维与优化设计［J］. 兵工学报， 2022， 43（6）： 1326-1336.
	WU Z L， YE J C， WANG J， et al. Parameter dimensionality reduction and optimal design of aircraft airfoil based on deep autoencoder neural network［J］. Acta Armamentarii， 2022， 43（6）： 1326-1336 （in Chinese）.
15	邬晓敬. 气动外形优化设计中的不确定性及高维问题研究［D］. 西安：西北工业大学， 2018.
	WU X J. Research on uncertainty and high dimensional problems in aerodynamic shape optimization design［D］. Xi’an： Northwestern Polytechnical University， 2018 （in Chinese）.
16	余婧，蒋安林，刘亮，等. 基于PCA降维的气动外形参数化方法［J］. 航空学报， 2024， 45（10）： 129125.
	YU J， JIANG A L， LIU L， et al. PCA aerodynamic geometry parametrization method［J］. Acta Aeronautica et Astronautica Sinica， 2024， 45（10）： 129125 （in Chinese）.
17	WANG Y Y， SHIMADA K， BARATI FARIMANI A. Airfoil GAN： encoding and synthesizing airfoils for aerodynamic shape optimization［J］. Journal of Computational Design and Engineering， 2023， 10（4）： 1350-1362.
18	HO J， JAIN A， ABBEEL P， et al. Denoising diffusion probabilistic models［C］∥Proceedings of the 34th International Conference on Neural Information Processing Systems. New York： ACM， 2020： 6840-6851.
19	GOODFELLOW I， POUGET-ABADIE J， MIRZA M， et al. Generative adversarial networks［J］. Communications of the ACM， 2020， 63（11）： 139-144.
20	DHARIWAL P， NICHOL A. Diffusion models beat GANs on image synthesis［C］∥Proceedings of the 35th International Conference on Neural Information Processing Systems. New York： ACM， 2021： 8780-8794.
21	ARJOVSKY M， BOTTOU L. Towards Principled Methods for Training Generative Adversarial Networks［DB/OL］. arXiv preprint： 1701.0486， 2017.
22	WANG Y， BLEI D， CUNNINGHAM J P. Posterior collapse and latent variable non-identifiability［J］. Advances in Neural Information Processing Systems， 2021， 34： 5443-5455.
23	MAZÉ F， AHMED F， MAZÉ F， et al. Diffusion models beat GANs on topology optimization［C］∥Proceedings of the Thirty-Seventh AAAI Conference on Artificial Intelligence and Thirty-Fifth Conference on Innovative Applications of Artificial Intelligence and Thirteenth Symposium on Educational Advances in Artificial Intelligence. New York： ACM， 2023： 9108-9116.
24	SONG J， MENG C， ERMON S. Denoising diffusion implicit models［DB/OL］. arXiv preprint： 2010.0250， 2020.
25	KULFAN B， BUSSOLETTI J. “Fundamental” Parameteric geometry representations for aircraft component shapes［C］∥11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Reston： AIAA， 2006.
26	关晓辉，宋笔锋，李占科. CSRT与CST气动外形参数化方法对比［J］. 空气动力学学报， 2014， 32（2）： 228-234.
	GUAN X H， SONG B F， LI Z K. Comparison of the CSRT and the CST parameterization methods［J］. Acta Aerodynamica Sinica， 2014， 32（2）： 228-234 （in Chinese）.
27	张德虎，席胜，田鼎. 典型翼型参数化方法的翼型外形控制能力评估［J］. 航空工程进展， 2014， 5（3）： 281-288， 295.
	ZHANG D H， XI S， TIAN D. Geometry control ability evaluation of classical airfoil parametric methods［J］. Advances in Aeronautical Science and Engineering， 2014， 5（3）： 281-288， 295 （in Chinese）.
28	关晓辉，李占科，宋笔锋. CST气动外形参数化方法研究［J］. 航空学报， 2012（4）： 625-633.
	GUAN X H， LI Z K， SONG B F. A study on CST aerodynamic shape parameterization method［J］. Acta Aeronauticaet Astronautica Sinica， 2012（4）： 625-633 （in Chinese）.
29	尹强，高正红. 外形参数化方法对气动优化过程的影响［J］. 科学技术与工程， 2012， 12（14）： 3394-3398.
	YIN Q， GAO Z H. Effect of shape parameterization on aerodynamic shape optimization［J］. Science Technology and Engineering， 2012， 12（14）： 3394-3398 （in Chinese）.
30	TENENBAUM J B， DE SILVA V， LANGFORD J C. A global geometric framework for nonlinear dimensionality reduction［J］. Science， 2000， 290（5500）： 2319-2323.
31	RONNEBERGER O， FISCHER P， BROX T. U-Net： convolutional networks for biomedical image segmentation［M］∥Medical Image Computing and Computer-Assisted Intervention-MICCAI 2015. Cham： Springer International Publishing， 2015： 234-241.
32	NICHOL A Q， DHARIWAL P. Improved denoising diffusion probabilistic models［C］∥International conference on machine learning. Virtual Event： PMLR， 2021： 8162-8171.
33	CHEN W， CHIU K， FUGE M D. Airfoil design parameterization and optimization using Bézier generative adversarial networks［J］. AIAA Journal， 2020， 58（11）： 4723-4735.
34	GRETTON A， BORGWARDT K M， RASCH M J， et al. A kernel two-sample test［J］. The Journal of Machine Learning Research， 2012， 13（1）： 723-773.
35	MCKAY M D， BECKMAN R J， CONOVER W J. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code［J］. Technometrics， 2000， 42（1）： 55-61.

结构层	网络
L1	双卷积层
L2	下采样层
L3	自注意力层
L4	双卷积层
L5	双卷积层
L6	双卷积层
L7	上采样层
L8	自注意力层
L9	输出层

参数化方法	平均优化耗时/s	LDM方法提速百分比/%
LDM	1 160.58
CST-AE	2 286.38	97.00
PCA	2 359.21	103.28
FFD	1 788.59	54.11
CST	1 997.08	72.08

参数化方法	平均优化升阻比	LDM方法升阻比相对提升百分比/%	标准差	LDM方法标准差相对减少百分比/%
LDM	308.68		3.56
CST-AE	299.57	3.04	13.43	73.49
PCA	258.19	19.55	31.05	88.53
FFD	260.47	18.51	52.95	93.28
CST	254.67	21.20	16.14	77.94

参数化方法	平均优化耗时/s	LDM方法提速百分比/%
LDM	944.43
CST-AE	1 485.61	57.30
PCA	1 693.87	79.35
FFD	1 317.73	39.53
CST	1 505.27	59.38

参数化方法	平均优化升阻比	LDM方法升阻比相对提升百分比/%	标准差	LDM方法标准差相对减少百分比/%
LDM	234.15		7.18
CST-AE	222.25	5.35	12.41	42.14
PCA	211.77	10.57	10.94	34.37
FFD	207.75	12.71	30.60	76.54
CST	215.11	8.85	18.22	60.59

Airfoil parameterization method based on latent diffusion model

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 23

References 35

Related Articles 15

Recommended Articles

Metrics

Comments

[1]	Yunxiang FAN, Huanan AI, Mingzhen WANG, Kai CAO, Xuejun LIU, Hongqiang LYU. Unsteady hydrodynamic load reconstruction of seaplane based on deep learning [J]. Acta Aeronautica et Astronautica Sinica, 2024, 45(20): 129882-129882.
[2]	Chaoyu LIU, Feng QU, Di SUN, Chuanzhen LIU, Zhansen QIAN, Junqiang BAI. Discretized adjoint based aerodynamic optimization design for hypersonic osculating-cone waverider [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2023, 44(4): 126664-126664.
[3]	Jiehua TIAN, Di SUN, Feng QU, Junqiang BAI. Airfoil parameterization method based on CST⁃GAN [J]. Acta Aeronautica et Astronautica Sinica, 2023, 44(18): 128280-128280.
[4]	ZHAN Qingliang, BAI Chunjin, ZHANG Ning, GE Yaojun. Feature extraction method of flow around airfoil based on time-history convolutional autoencoder [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2022, 43(11): 526531-526531.
[5]	YAN Tijin, XIA Yuanqing, ZHANG Hongwei, WEI Minfeng, ZHOU Tong. An anomaly detection method for irregularly sampled spacecraft time series data [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2021, 42(4): 525019-525019.
[6]	LIU Fengbo, JIANG Cheng, MA Tuliang, LIANG Yihua. Aerodynamic optimization design of large civil aircraft using pressure distribution inverse design method based on discrete adjoint [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2020, 41(5): 623372-623372.
[7]	ZHANG Xinshuai, LIU Jun, LUO Shibin. An improved multi-objective cuckoo search algorithm for airfoil aerodynamic optimization design [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2019, 40(6): 122550-122550.
[8]	HAN Zhonghua, ZHANG Yu, XU Chenzhou, WANG Kai, WU Mengmeng, ZHU Zhen, SONG Wenping. Aerodynamic optimization design of large civil aircraft wings using surrogate-based model [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2019, 40(1): 522398-522398.
[9]	SUN Xiangcheng, HAN Zhonghua, LIU Fei, SONG Ke, SONG Wenping. Design and analysis of hypersonic vehicle airfoil/wing at wide-range Mach numbers [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2018, 39(6): 121737-121737.
[10]	LI Li, BAI Junqiang, GUO Tongbiao, CHEN Song. Aerodynamic optimization design for civil aircraft considering relaxed static stability [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2017, 38(9): 121112-121112.
[11]	TIAN Xu, LI Jie. An improved fruit fly optimization algorithm and its application in aerodynamic optimization design [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2017, 38(4): 120370-120370.
[12]	YANG Tihao, BAI Junqiang, WANG Dan, CHEN Song, XU Jiakuan, CHEN Yuanyuan. Aerodynamic Optimization Design for After-body of Tail-mounted Engine Layout Considering Interference of Engines [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2014, 35(7): 1836-1844.
[13]	WANG Yuanyuan, ZHANG Binqian, GUO Zhaodian, DONG Qiang. Aerodynamic Optimization Design for Large Upswept Afterbody of Transport Aircraft Based on FFD Technology [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2013, 34(8): 1806-1814.
[14]	YANG Hui, SONG Wenping, HAN Zhonghua, XU Jianhua. Multi-objective and Multi-constrained Optimization Design for a Helicopter Rotor Airfoil [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2012, 33(7): 1218-1226.
[15]	LI Ding, XIA Lu. Application of Improved Particle Swarm Optimization Algorithm to Aerodynamic Design [J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2012, 33(10): 1809-1816.