ACTA AERONAUTICAET ASTRONAUTICA SINICA >
Uncertainty analysis and gradient optimization design of airfoil based on polynomial chaos expansion method
Received date: 2022-05-15
Revised date: 2022-06-06
Accepted date: 2022-06-28
Online published: 2022-07-08
Supported by
National Natural Science Foundation of China(1200021118)
An efficient and reliable uncertainty gradient optimization design method is developed by coupling the adjoint method and the non-intrusive polynomial chaos method. Using the adjoint equation method to solve the derivative of the objective function with respect to the uncertain variable， we develop a gradient-enhanced polynomial chaos expansion method. Various examples at subsonic and transonic speeds prove that this method can improve the efficiency and accuracy of uncertainty analysis. Meanwhile， the sensitivity of uncertain variables is quantified using a global sensitivity analysis method based on variance decomposition. A statistical moment gradient solution method for the coupled adjoint equation of polynomial chaos is established， and an uncertain gradient optimization design system built by combining the gradient-enhanced polynomial chaos expansion method. Based on the optimization design system， the deterministic and uncertain optimization design research of two-dimensional low subsonic and transonic airfoils is conducted. The optimization results show that， compared with the deterministic optimal design， the uncertain optimal design can improve the ability to resist the uncertainty perturbation of Mach number and angle of attack by reasonably balancing the deterministic performance and the uncertain performance， and optimize the average performance and performance robustness. The mean value and the standard deviation of the drag coefficient can be reduced by up to 17% and 80%， respectively. The deterministic optimization design may lead to a decrease in performance robustness.
Yifu CHEN , Yuhang MA , Qingsheng LAN , Weiping SUN , Yayun SHI , Tihao YANG , Junqiang BAI . Uncertainty analysis and gradient optimization design of airfoil based on polynomial chaos expansion method[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2023 , 44(8) : 127446 -127446 . DOI: 10.7527/S1000-6893.2022.27446
| 1 | GIESING J， BARTHELEMY J F. A summary of industry MDO applications and needs［C］∥7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. Reston： AIAA， 1998. |
| 2 | ZANG T A， HEMSCH M J， HILBURGER M W，et al. Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles［R］. Washington， D. C.：NASA Langley Research Center， 2002. |
| 3 | HUYSE L， LEWIS R M. Aerodynamic shape optimization of two-dimensional airfoils under uncertain conditions［M］. Washington， D.C.： Institute for Computer Applications in Science and Engineering， NASA Langley Research Center 2001. |
| 4 | WALTERS R W， HUYSE L. Uncertainty analysis for fluid mechanics with applications［R］.Washington， D. C.：NASA Langley Research Center， 2002. |
| 5 | CHEN W， JIN R C， SUDJIANTO A. Analytical variance-based global sensitivity analysis in simulation-based design under uncertainty［J］. Journal of Mechanical Design， 2005， 127（5）： 875-886. |
| 6 | CAFLISCH R E. Monte Carlo and quasi-Monte Carlo methods［J］. Acta Numerica， 1998， 7： 1-49. |
| 7 | PELLETIER D， TURGEON E， LACASSE D， et al. Adaptivity， sensitivity， and uncertainty： Toward standards of good practice in computational fluid dynamics［J］. AIAA Journal， 2003， 41（10）： 1925-1933. |
| 8 | LUCKRING J， HEMSCH M， MORRISON J. Uncertainty in computational aerodynamics［C］∥41st Aerospace Sciences Meeting and Exhibit. Reston： AIAA， 2003. |
| 9 | SCHAEFER J A， ROMERO V J， SCHAFER S R， et al. Approaches for quantifying uncertainties in computational modeling for aerospace applications［C］∥AIAA Scitech 2020 Forum. Reston： AIAA， 2020. |
| 10 | ROELOFS M， VOS R. Technology evaluation and uncertainty-based design optimization： A review［C］∥ 2018 AIAA Aerospace Sciences Meeting. Reston：AIAA，2018. |
| 11 | KNIO O M， LE MA?TRE O P. Uncertainty propagation in CFD using polynomial chaos decomposition［J］. Fluid Dynamics Research， 2006， 38（9）： 616-640. |
| 12 | NAJM H N. Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics［J］. Annual Review of Fluid Mechanics， 2009， 41： 35-52. |
| 13 | LOEVEN G A， WITTEVEEN J S， BIJL H. Probabilistic collocation： an efficient non-intrusive approach for arbitrarily distributed parametric uncertainties［C］∥45th AIAA Aerospace Sciences Meeting and Exhibit. Reston： AIAA， 2007. |
| 14 | SIMON F， GUILLEN P， SAGAUT P， et al. A gPC-based approach to uncertain transonic aerodynamics［J］. Computer Methods in Applied Mechanics and Engineering， 2010， 199（17-20）： 1091-1099. |
| 15 | CHASSAING J C， LUCOR D. Stochastic investigation of flows about airfoils at transonic speeds［J］. AIAA Journal， 2010， 48（5）： 938-950. |
| 16 | YAMAZAKI W. Stochastic drag analysis via polynomial chaos uncertainty quantification［C］∥51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. Reston： AIAA， 2013. |
| 17 | SUGA Y， YAMAZAKI W. Aerodynamic uncertainty quantification of supersonic biplane airfoil via polynomial chaos approach［C］∥17th AIAA Non-Deterministic Approaches Conference. Reston： AIAA， 2015. |
| 18 | HOSDER S， WALTERS R W， BALCH M. Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics［J］. AIAA Journal， 2010， 48（12）： 2721-2730. |
| 19 | WEST T K， PHILLIPS B D. Multifidelity uncertainty quantification of a commercial supersonic transport［J］. Journal of Aircraft， 2020， 57（3）： 491-500. |
| 20 | GHISU T， PARKS G T， JARRETT J P， et al. Adaptive polynomial chaos for gas turbine compression systems performance analysis［J］. AIAA Journal， 2010， 48（6）： 1156-1170. |
| 21 | ABRAHAM S， TSIRIKOGLOU P， LACOR C， et al. Uncertainty quantification in industrial turbo-machinery design using sparse polynomial chaos expansions［C］∥ 2018 Multidisciplinary Analysis and Optimization Conference. Reston： AIAA， 2018. |
| 22 | LI Z H， LIU Y M， AGARWAL R K. Uncertainty quantification of geometric and flow variables affecting the performance of a transonic axial compressor［C］∥2018 AIAA Aerospace Sciences Meeting. Reston： AIAA， 2018. |
| 23 | LIU Z Y， WANG X D， KANG S. Stochastic performance evaluation of horizontal axis wind turbine blades using non-deterministic CFD simulations［J］. Energy， 2014， 73： 126-136. |
| 24 | DI STEFANO M A， HOSDER S， BAURLE R A. Effect of turbulence model uncertainty on scramjet strut injector flow field analysis［J］. Computers & Fluids， 2021， 229： 105104. |
| 25 | GUO Z T， CHU W L. Stochastic aerodynamic analysis for compressor blades with manufacturing variability based on a mathematical dimensionality reduction method［J］. Proceedings of the Institution of Mechanical Engineers， Part C： Journal of Mechanical Engineering Science， 2022， 236（10）： 5719-5735. |
| 26 | HUANG M， LI Z G， LI J， et al. Uncertainty quantification and sensitivity analysis of aerothermal performance for the turbine blade squealer tip［J］. International Journal of Thermal Sciences， 2022， 175： 107460. |
| 27 | 王晓东， 康顺. 多项式混沌法求解随机Burgers方程［J］. 工程热物理学报， 2010， 31（3）： 393-398. |
| WANG X D， KANG S. Solving stochastic Burgers equation using polynomial chaos decomposition［J］. Journal of Engineering Thermophysics， 2010， 31（3）： 393-398 （in Chinese）. | |
| 28 | 王晓东， 康顺. 多项式混沌方法在随机方腔流动模拟中的应用［J］. 中国科学，2011，41（6）：790-798. |
| WANG X D， KANG S. Application of polynomial chaos on numerical simulation of stochastic cavity flow［J］.Sci China Tech Sci， 2011， 41（6）：790-798 （in Chinese）. | |
| 29 | 邬晓敬， 张伟伟， 宋述芳， 等. 翼型跨声速气动特性的不确定性及全局灵敏度分析［J］. 力学学报， 2015， 47（4）： 587-595. |
| WU X J， ZHANG W W， SONG S F， et al. Uncertainty quantification and global sensitivity analysis of transonic aerodynamics about airfoil［J］. Chinese Journal of Theoretical and Applied Mechanics， 2015， 47（4）： 587-595 （in Chinese）. | |
| 30 | ZHAO H， GAO Z H， XU F， et al. Adaptive multi-fidelity sparse polynomial chaos-Kriging metamodeling for global approximation of aerodynamic data［J］. Structural and Multidisciplinary Optimization， 2021， 64（2）： 829-858. |
| 31 | SCHAEFER J A， CARY A W， MANI M， et al. Uncertainty quantification across design space using spatially accurate polynomial chaos［C］∥AIAA Scitech 2020 Forum. Reston： AIAA， 2020. |
| 32 | ZEIN S. A polynomial chaos expansion trust region method for robust optimization［J］. Communications in Computational Physics， 2013， 14（2）： 412-424. |
| 33 | WANG X D， HIRSCH C， LIU Z Y， et al. Uncertainty-based robust aerodynamic optimization of rotor blades［J］. International Journal for Numerical Methods in Engineering， 2013， 94（2）： 111-127. |
| 34 | PALAR P S， TSUCHIYA T， PARKS G. Decomposition-based evolutionary aerodynamic robust optimization with multi-fidelity point collocation non-intrusive polynomial chaos［C］∥17th AIAA Non-Deterministic Approaches Conference. Reston： AIAA， 2015. |
| 35 | 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）. | |
| 36 | PADULO M， CAMPOBASSO M S， GUENOV M D. Novel uncertainty propagation method for robust aerodynamic design［J］. AIAA Journal， 2011， 49（3）： 530-543. |
| 37 | SHANKARAN S， JAMESON A. Robust optimal control using polynomial chaos and adjoints for systems with uncertain inputs［C］∥20th AIAA Computational Fluid Dynamics Conference. Reston： AIAA， 2011. |
| 38 | SCHILLINGS C， SCHULZ V. On the influence of robustness measures on shape optimization with stochastic uncertainties［J］. Optimization and Engineering， 2015， 16（2）： 347-386. |
| 39 | RALLABHANDI S K， WEST T K， NIELSEN E J. Uncertainty analysis and robust design of low-boom concepts using atmospheric adjoints［J］. Journal of Aircraft， 2017， 54（3）： 902-917. |
| 40 | BOOPATHY K， KENNEDY G. Semi-intrusive uncertainty propagation and adjoint sensitivity analysis using the stochastic Galerkin method［C］∥AIAA Scitech 2020 Forum. Reston： AIAA， 2020. |
| 41 | SABATER C， G?RTZ S. Gradient-based aerodynamic robust optimization using the adjoint method and Gaussian processes［M］∥Advances in evolutionary and deterministic methods for design， optimization and control in engineering and sciences. Cham： Springer， 2021： 211-226. |
| 42 | MAYEUR J， DUMONT A， DESTARAC D， et al. Reynolds-averaged navier-stokes simulations on NACA0012 and ONERA-M6 wing with the ONERA elsA solver［J］. AIAA Journal， 2016， 54（9）： 2671-2687. |
| 43 | FUJINO M， YOSHIZAKI Y， KAWAMURA Y. Natural-laminar-flow airfoil development for a lightweight business jet［J］. Journal of Aircraft， 2003， 40（4）： 609-615. |
| 44 | SUDRET B. Global sensitivity analysis using polynomial chaos expansions［J］. Reliability Engineering & System Safety， 2008， 93（7）： 964-979. |
| 45 | 赵轲. 基于CFD的复杂气动优化与稳健设计方法研究［D］. 西安： 西北工业大学， 2015. |
| ZHAO K. Complex aerodynamic optimization and robust design method based on computational fluid dynamics［D］. Xi’an： Northwestern Polytechnical University， 2015 （in Chinese）. |
/
| 〈 |
|
〉 |
CJA