ZHAO H, GAO Z H, XU F, et al . Review of robust aerodynamic design optimization for air vehicles[J ] . Archives of Computational Methods in Engineering, 2019 , 26 ( 3 ): 685 − 732 .

YAO W, CHEN X Q, LUO W C, et al . Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles[J ] . Progress in Aerospace Sciences, 2011 , 47 ( 6 ): 450 − 479 .

ZANG T A, HEMSCH M J, HIBURGER M W, et al. Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles: NASA/TM-2002-211462[R]. NASA Langley Research Center, 2002.

CHAUDHURI A, WAYCASTER G, PRICE N, et al . NASA uncertainty quantification challenge: An optimization-based methodology and validation[J ] . Journal of Aerospace Information Systems, 2015 , 12 ( 1 ): 10 − 34 .

CRESPO L G, KENNY S P, GIESY D P. The NASA Langley multidisciplinary uncertainty quantification challenge [C]//16th AIAA Non-Deterministic Approaches Conference, National Harbor, Maryland, 2014.

EISFELD B. The German national joint project MUNA: Management and minimization of uncertainties and errors in numerical aerodynamics [C]//5th European Conference on Computational Fluid Dynamics, ECCOMAS CFD, Lisbon, Portugal, 2010.

TEKASLAN H E, YILDIZ S, DEMIROGLU Y, et al. Implementation of multidisciplinary multi-fidelity uncertainty quantification methods in sonic boom prediction[C]// AIAA Aviation 2021 forum, virtual event, Reston, Virginia, 2021.

ROGERS M M. Progress towards the CFD vision 2030 [C]// United Technologies Research Center “Digital Thread: Turbomachinery Design” Workshop, East Hartford, CT, 2018.

ZHOU Z, HUANG J T, HUANG Y, et al. CFD technology in aeronautic engineering field: Applications, challenges and development[J]. Acta Aeronautica et Astronautica Sinica, 2017, 38(3): 20891−020891(in Chinese).

PARK G J, LEE T H, LEE K H, et al . Robust design: An overview[J ] . AIAA Journal, 2006 , 44 ( 1 ): 181 − 191 .

HUYSE L. Free-form airfoil shape optimization under uncertainty using maximum expected value and second-order second-moment strategies[M]. Hampton, Va. : Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 2001.

PADULA S L, GUMBERT C R, LI W . Aerospace applications of optimization under uncertainty[J ] . Optimization and Engineering, 2006 , 7 ( 3 ): 317 − 328 .

LEDOUX S T, VASSBERG J C, YOUNG D P, et al . Study based on the AIAA aerodynamic design optimization discussion group test cases[J ] . AIAA Journal, 2015 , 53 ( 7 ): 1910 − 1935 .

LI W, HUYSE L, PADULA S L . Robust airfoil optimization to achieve drag reduction over a range of Mach numbers[J ] . Structural and Multidisciplinary Optimization, 2002 , 24 ( 1 ): 38 − 50 .

ZINGG D W, ELIAS S . Aerodynamic optimization under a range of operating conditions[J ] . AIAA Journal, 2006 , 44 ( 11 ): 2787 − 2792 .

CHEN Y C, ZHANG M H, ZHANG M, et al . Review of large civil aircraft aerodynamic design[J ] . Acta Aeronautica et Astronautica Sinica, 2019 , 40 ( 1 ): 522759 (in Chinese).

ZHAO H, GAO Z H, XIA L . Efficient robust aerodynamic design optimization method for high-speed NLF airfoil[J ] . Acta Aeronautica et Astronautica Sinica, 2022 , 43 ( 1 ): 124894 (in Chinese).

ZHAO K, GAO Z H, HUANG J T . Robust design of natural laminar flow supercritical airfoil by multi-objective evolution method[J ] . Applied Mathematics and Mechanics, 2014 , 35 ( 2 ): 191 − 202 .

ZHAO H, GAO Z H, WANG C, et al. Robust design of high speed natural-laminar-flow airfoil for high lift[C]// 55th AIAA Aerospace Sciences Meeting, Grapevine, Texas. Reston, Virginia, 2017.

NAJM H N . Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics[J ] . Annual Review of Fluid Mechanics, 2009 , 41 ( 1 ): 35 − 52 .

XIAO H, CINNELLA P . Quantification of model uncertainty in RANS simulations: A review[J ] . Progress in Aerospace Sciences, 2019 , 108 : 1 − 31 .

BERAN P, STANFORD B, SCHROCK C . Uncertainty quantification in aeroelasticity[J ] . Annual Review of Fluid Mechanics, 2017 , 49 : 361 − 386 .

TAGUCHI G, CHOWDHURY S, TAGUCHI S. Robust engineering[M]. New York: McGraw-Hill, 2000.

LI W, HYUSE L, PADULA S L. Robust airfoil optimization to achieve consistent drag reduction over a Mach range: NASA/CR-2001-211042[R]. NASA Langley Research Center, 2001.

PADULA S L, LI W. Options for robust airfoil optimization under uncertainty[C]// 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, Georgia. Reston, Virginia, 2002.

KEANE A J . Cokriging for robust design optimization[J ] . AIAA Journal, 2012 , 50 ( 11 ): 2351 − 2364 .

ZHAO H, GAO Z H, GAO Y, et al . Effective robust design of high lift NLF airfoil under multi-parameter uncertainty[J ] . Aerospace Science and Technology, 2017 , 68 : 530 − 542 .

ZHAO H, GAO Z H . Uncertainty-based design optimization of NLF airfoil for high altitude long endurance unmanned air vehicles[J ] . Engineering Computations, 2019 , 36 ( 3 ): 971 − 996 .

LI J. Research on aerodynamic configuration design methods for high performance aircraft[D]. Xi’an: Northwestern Polytechnical University, 2014(in Chinese).

ZHAO K. Complex aerodynamic optimization and robust design method based on computational fluid dynamics[D]. Xi’an: Northwestern Polytechnical University, 2015(in Chinese).

LI J, GAO Z H, HUANG J T, et al . Robust design of NLF airfoils[J ] . Chinese Journal of Aeronautics, 2013 , 26 ( 2 ): 309 − 318 .

LI J Z, GAO Z H . The application of multi-objective evolutionary algorithm and surrogate model to aerodynamic robust optimization design[J ] . Acta Aerodynamica Sinica, 2012 , 30 ( 1 ): 46 − 51 (in Chinese).

ZHAO H, GAO Z H, XU F, et al . An efficient adaptive forward–backward selection method for sparse polynomial chaos expansion[J ] . Computer Methods in Applied Mechanics and Engineering, 2019 , 355 : 456 − 491 .

ZHAO K, GAO Z H, HUANG J T, et al . Uncertainty quantification and robust design of airfoil based on polynomial chaos technique[J ] . Chinese Journal of Theoretical and Applied Mechanics, 2014 , 46 ( 1 ): 10 − 19 .

LI J Z, GAO Z H . Multi objective optimization methodology for airfoil robust design under geometry uncertainty[J ] . Chinese Journal of Theoretical and Applied Mechanics, 2011 , 43 ( 3 ): 611 − 615 (in Chinese).

HUANG J T, GAO Z H, BAI J Q, et al . Study of robust winglet design based on arbitrary space shape FFD technique[J ] . Acta Aeronautica et Astronautica Sinica, 2013 , 34 ( 1 ): 37 − 45 (in Chinese).

ZANG T A, HEMSCH M J, HILBURGER M W, et al.Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles: L-18174[R]. NASA, 2002.

ZHAO H, GAO Z H, GAO Y. Design optimization of natural-laminar-flow airfoil for complicated flight conditions[C]// 35th AIAA Applied Aerodynamics Conference, Denver, Colorado. Reston, Virginia, 2017.

HUYSE L, PADULA S L, LEWIS R M, et al . Probabilistic approach to free-form airfoil shape optimization under uncertainty[J ] . AIAA Journal, 2002 , 40 : 1764 − 1772 .

HUYSE L, BUSHNELL D M. Free-form airfoil shape optimization under uncertainty using maximum expected value and second-order second-moment strategies: NASA/CR-2001-211020[R]. NASA Langley Research Center, 2001.

CROICU A M . Robust airfoil optimization using a conditional expected maximum value approach[J ] . AIAA Journal, 2022 , 60 ( 11 ): 6473 − 6477 .

CROICU A M, HUSSAINI M Y, JAMESON A, et al . Robust airfoil optimization using maximum expected value and expected maximum value approaches[J ] . AIAA Journal, 2012 , 50 ( 9 ): 1905 − 1919 .

CROICU A M, HUSSAINI M Y . On the expected optimal value and the optimal expected value[J ] . Applied Mathematics and Computation, 2006 , 180 ( 1 ): 330 − 341 .

KUMAR A, KEANE A J, NAIR P B, et al. Robust design of compressor blades against manufacturing variations[C]// ASME 2006 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Philadelphia, Pennsylvania, USA, 2008.

KEANE A J . Comparison of several optimization strategies for robust turbine blade design[J ] . Journal of Propulsion and Power, 2009 , 25 ( 5 ): 1092 − 1099 .

KAMENIK J, VOUTCHKOV I, TOAL D J J, et al . Robust turbine blade optimization in the face of real geometric variations[J ] . Journal of Propulsion and Power, 2018 , 34 ( 6 ): 1479 − 1493 .

HOSDER S, WALTERS R, PEREZ R. A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations[C]// 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. Reston, Virigina, 2006.

ZHANG Y, HOSDER S, LEIFSSON L, et al. Robust airfoil optimization under inherent and model-form uncertainties using stochastic expansions[C]// 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Nashville, Tennessee. Reston, Virigina, 2012.

SHAH H, HOSDER S, KOZIEL S, et al . Multi-fidelity robust aerodynamic design optimization under mixed uncertainty[J ] . Aerospace Science and Technology, 2015 , 45 : 17 − 29 .

SHAH H, HOSDER S, WINTER T . Quantification of margins and mixed uncertainties using evidence theory and stochastic expansions[J ] . Reliability Engineering System Safety, 2015 , 138 : 59 − 72 .

PADRON A S, ALONSO J J, ELDRED M S. Multi-fidelity methods in aerodynamic robust optimization[C]// 18th AIAA Non-Deterministic Approaches Conference, San Diego, California, USA. Reston, Virginia, 2016.

ELDRED M S, BURKARDT J. Comparison of non-intrusive polynomial chaos and stochastic collocation methods for uncertainty quantification[C]// 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando, Florida. Reston, Virginia, 2009.

ELDRED M S, WEBSTER C, CONSTANTINE P. Design under uncertainty employing stochastic expansion methods[C]// 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Victoria, British Columbia, Canada. Reston, Virginia, 2008.

PAPADIMITRIOU D I, PAPADIMITRIOU C . Aerodynamic shape optimization for minimum robust drag and lift reliability constraint[J ] . Aerospace Science and Technology, 2016 , 55 : 24 − 33 .

PAPADIMITRIOU D I, GIANNAKOGLOU K C . Third-order sensitivity analysis for robust aerodynamic design using continuous adjoint[J ] . International Journal for Numerical Methods in Fluids, 2013 , 71 ( 5 ): 652 − 670 .

PAPAGEORGIOU A K, PAPOUTSIS-KIACHAGIAS E M, GIANNAKOGLOU K C . Development and assessment of an intrusive polynomial chaos expansion-based continuous adjoint method for shape optimization under uncertainties[J ] . International Journal for Numerical Methods in Fluids, 2022 , 94 ( 1 ): 59 − 75 .

PAPADIMITRIOU D, PAPADIMITRIOU C. Robust reliability-based aerodynamic shape optimization[J]. Engineering Optimization, 2014, 353-358.

PAPOUTSIS-KIACHAGIAS E M, PAPADIMITRIOU D I, GIANNAKOGLOU K C . Robust design in aerodynamics using third-order sensitivity analysis based on discrete adjoint. Application to quasi-1D flows[J ] . International Journal for Numerical Methods in Fluids, 2012 , 69 ( 3 ): 691 − 709 .

SHIMOYAMA K, OYAMA A, FUJII K. A new efficient and useful robust optimization approach - design for multi-objective six sigma[C]// 2005 IEEE Congress on Evolutionary Computation, Edinburgh, UK, 2005: 950-957.

SHIMOYAMA K, LIM J N, JEONG S, et al . Practical implementation of robust design assisted by response surface approximation and visual data-mining[J ] . Journal of Mechanical Design, 2009 , 131 ( 6 ): 061007 .

SHIMOYAMA K. Robust aerodynamic design of Mars exploratory airplane wing with a new optimization method [M]. School of Engineering, The University of Tokyo, Diss, 2006.

DODSON M, PARKS G T . Robust aerodynamic design optimization using polynomial chaos[J ] . Journal of Aircraft, 2009 , 46 ( 2 ): 635 − 646 .

GHISU T, JARRETT J P, PARKS G T . Robust design optimization of airfoils with respect to ice accretion[J ] . Journal of Aircraft, 2011 , 48 ( 1 ): 287 − 304 .

TIZIANO G. Robust aerodynamic design optimisation of compression systems[D]. Cambridge, East of England, UK: University of Cambridge, 2009.

GHISU T, PARKS G T, JARRETT J P, et al . Robust design optimization of gas turbine compression systems[J ] . Journal of Propulsion and Power, 2011 , 27 ( 2 ): 282 − 295 .

SCHULZ V, SCHILLINGS C. Optimal aerodynamic design under uncertainty[M]// Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamics, Berlin, Heidelberg, 2013.

LITVINENKO A, MATTHIES H. Uncertainty quantification in numerical aerodynamics [J]. Proceedings in Applied Mathematics and Mechanics, 2017, 12.

SCHULZ V, SCHILLINGS C . Problem formulations and treatment of uncertainties in aerodynamic design[J ] . AIAA Journal, 2009 , 47 ( 3 ): 646 − 654 .

SCHILLINGS C, SCHMIDT S, SCHULZ V . Efficient shape optimization for certain and uncertain aerodynamic design[J ] . Computers Fluids, 2011 , 46 ( 1 ): 78 − 87 .

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 .

SCHULZ V, SCHILLINGS C. Optimal aerodynamic design under shape uncertainties [M]. Berlin, Heidelberg: Springer, 2013.

SABATER C, BEKEMEYER P, GÖRTZ S . Robust design of transonic natural laminar flow wings under environmental and operational uncertainties[J ] . AIAA Journal, 2022 , 60 ( 2 ): 767 − 782 .

MARUYAMA D, LIU D S, GÖRTZ S. An efficient aerodynamic shape optimization framework for robust design of airfoils using surrogate models[C]// VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016), Crete Island, Greece, Athens, 2016.

MARUYAMA D, LIU D S, GÖRTZ S. Surrogate model-based approaches to UQ and their range of applicability[M]. Cham: Springer, 2019.

SABATER C, BEKEMEYER P, GÖRTZ S . Efficient bilevel surrogate approach for optimization under uncertainty of shock control bumps[J ] . AIAA Journal, 2020 , 58 ( 12 ): 5228 − 5242 .

MARUYAMA D, GÖRTZ S, LIU D S. Robust design measures for airfoil shape optimization[M]. Cham: Springer, 2019.

WANG Y Y, ZHANG B Q, CHEN Y C . Robust airfoil optimization based on improved particle swarm optimization method[J ] . Applied Mathematics and Mechanics, 2011 , 32 ( 10 ): 1161 − 1168 (in Chinese).

WANG Y. Uncertainty-based optimization method and its application in aircraft design[D]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2010(in Chinese).

ZENG H H. Robust optimization for aerodynamic design and its application to design of a micro air vehicle[D]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2006 (in Chinese).

ZHANG W, WANG Q, LU J C, et al . Robust optimization design under geometric uncertainty based on PCA-HicksHenne method[J ] . Journal of Beijing University of Aeronautics and Astronautics, 2022 , 48 ( 12 ): 2473 − 2481 (in Chinese).

ZHANG W, WANG Q, ZENG F Z, et al . A novel robust aerodynamic optimization technique coupled with adjoint solvers and polynomial chaos expansion[J ] . Chinese Journal of Aeronautics, 2022 , 35 ( 10 ): 35 − 55 .

CHENG H Z, ZHOU C X, LU X G, et al . Robust aerodynamic optimization and design exploration of a wide-chord transonic fan under geometric and operational uncertainties[J ] . Energy, 2023 , 278 : 128011 .

CHENG H Z, ZHOU C X, LI Z L, et al . Uncertainty quantification and sensitivity analysis on the aerodynamic performance of a micro transonic compressor[J ] . Aerospace Science and Technology, 2023 , 141 : 108569 .

ZHANG J S, LI L, DONG X, et al . A discrete adjoint framework coupled with adaptive PCE for robust aerodynamic optimization of turbomachinery under flow uncertainty[J ] . Aerospace Science and Technology, 2023 , 142 : 108592 .

LUO J Q, XIA Z H, LIU F . Robust design optimization considering inlet flow angle variations of a turbine cascade[J ] . Aerospace Science and Technology, 2021 , 116 : 106893 .

LUO J Q, CHEN Z S, ZHENG Y . A gradient-based method assisted by surrogate model for robust optimization of turbomachinery blades[J ] . Chinese Journal of Aeronautics, 2022 , 35 ( 10 ): 1 − 7 .

LUO J Q, CHEN Z S, ZENG X . Robust aerodynamic design optimization of turbine cascades considering uncertainty of geometric design parameters[J ] . Acta Aeronautica et Astronautica Sinica, 2020 , 41 ( 10 ): 123826 (in Chinese).

XIA Z H, LUO J Q, LIU F . Statistical evaluation of performance impact of flow variations for a transonic compressor rotor blade[J ] . Energy, 2019 , 189 : 116285 .

TANG Z L, PÉRIAUX J . Uncertainty based robust optimization method for drag minimization problems in aerodynamics[J ] . Computer Methods in Applied Mechanics and Engineering, 2012 , 217-220 : 12 − 24 .

TANG Z L, ZHANG L H . A new Nash optimization method based on alternate elitist information exchange for multi-objective aerodynamic shape design[J ] . Applied Mathematical Modelling, 2019 , 68 : 244 − 266 .

TANG Z L . Uncertainty based robust optimization in aerodynamics[J ] . Modern Physics Letters B, 2009 , 23 ( 3 ): 477 − 480 .

CHEN Y F, RAO H Y, XIONG N, et al . Adjoint-based robust optimization design of laminar flow airfoil under flight condition uncertainties[J ] . Aerospace Science and Technology, 2023 , 140 : 108465 .

CHEN Y F, RAO H Y, DENG Y J, et al . Adjoint-based robust optimization design of laminar flow wi ng under flight condition uncertainties[J ] . Chinese Journal of Aeronautics, 2023 , 36 ( 6 ): 19 − 34 .

COOK L W, JARRETT J P . Robust airfoil optimization and the importance of appropriately representing uncertainty[J ] . AIAA Journal, 2017 , 55 ( 11 ): 3925 − 3939 .

CHATTERJEE T, CHAKRABORTY S, CHOWDHURY R . A critical review of surrogate assisted robust design optimization[J ] . Archives of Computational Methods in Engineering, 2019 , 26 ( 1 ): 245 − 274 .

ZHAO H. Research on efficient surrogate-based aerodynamic optimization and robust aerodynamic design methods[D]. Xi’an: Northwestern Polytechnical University, 2020(in Chinese).

MATHELIN L, HUSSAINI M Y, ZANG T A . Stochastic approaches to uncertainty quantification in CFD simulations[J ] . Numerical Algorithms, 2005 , 38 ( 1 ): 209 − 236 .

PUTKO M M, TAYLOR A C III, NEWMAN P A, et al . Approach for input uncertainty propagation and robust design in CFD using sensitivity derivatives[J ] . Journal of Fluids Engineering, 2002 , 124 ( 1 ): 60 − 69 .

HELTON J, JOHNSON J, OBERKAMPF W . An exploration of alternative approaches to the representation of uncertainty in model predictions[J ] . Reliability Engineering and System Safety, 2004 , 85 ( 1 ): 39 − 71 .

AGARWAL H, RENAUD J E, PRESTON E L, et al . Uncertainty quantification using evidence theory in multidisciplinary design optimization[J ] . Reliability Engineering System Safety, 2004 , 85 ( 1-3 ): 281 − 294 .

HELTON J C, JOHNSON J D, OBERKAMPF W L, et al . Sensitivity analysis in conjunction with evidence theory representations of epistemic uncertainty[J ] . Reliability Engineering System Safety, 2006 , 91 ( 10-11 ): 1414 − 1434 .

DU X P . Reliability-based design optimization with dependent interval variables[J ] . International Journal for Numerical Methods in Engineering, 2012 , 91 ( 2 ): 218 − 228 .

DU X S, LEIFSSON L T, KOZIEL S. Robust airfoil design optimization using stochastic expansions and utility theory[C]// 18th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Denver, Colorado. Reston, Virginia, 2017.

TURÁNYI T . Sensitivity analysis of complex kinetic systems. Tools and applications[J ] . Journal of Mathematical Chemistry, 1990 , 5 ( 3 ): 203 − 248 .

BUZZARD G T . Global sensitivity analysis using sparse grid interpolation and polynomial chaos[J ] . Reliability Engineering and System Safety, 2012 , 107 : 82 − 89 .

HELTON J C, JOHNSON J D, SALLABERRY C J, et al . Survey of sampling-based methods for uncertainty and sensitivity analysis[J ] . Reliability Engineering System Safety, 2006 , 91 ( 10-11 ): 1175 − 1209 .

JACQUES J, LAVERGNE C, DEVICTOR N . Sensitivity analysis in presence of model uncertainty and correlated inputs[J ] . Reliab ility Engineering and System Safety, 2005 , 91 ( 10 ): 1126 − 1134 .

CRESTAUX T, MAIˆTRE O L, MARTINEZ J M . Polynomial chaos expansion for sensitivity analysis[J ] . Reliability Engineering System Safety, 2009 , 94 ( 7 ): 1161 − 1172 .

SUDRET B . Global sensitivity analysis using polynomial chaos expansions[J ] . Reliability Engineering System Safety, 2008 , 93 ( 7 ): 964 − 979 .

GUO J, DU X P . Sensitivity analysis with mixture of epistemic and aleatory uncertainties[J ] . AIAA Journal, 2007 , 45 ( 9 ): 2337 − 2349 .

BAE H R, GRANDHI R V, CANFIELD R A . Sensitivity analysis of structural response uncertainty propagation using evidence theory[J ] . Structural and Multidisciplinary Optimization, 2006 , 31 ( 4 ): 270 − 279 .

TRIPATHY R K, BILIONIS I . Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification[J ] . Journal of Computational Physics, 2018 , 375 : 565 − 588 .

PEHERSTORFER B, WILLCOX K, GUNZBURGER M . Survey of multifidelity methods in uncertainty propagation, inference, and optimization[J ] . SIAM Review, 2018 , 60 ( 3 ): 550 − 591 .

XIU D B, KARNIADAKIS G E . Modeling uncertainty in flow simulations via generalized polynomial chaos[J ] . Journal of Computational Physics, 2003 , 187 ( 1 ): 137 − 167 .

JANSSEN H . Monte-Carlo based uncertainty analysis: Sampling efficiency and sampling convergence[J ] . Reliability Engineering System Safety, 2013 , 109 : 123 − 132 .

DIAZ P, DOOSTAN A, HAMPTON J . Sparse polynomial chaos expansions via compressed sensing and D-optimal design[J ] . Computer Methods in Applied Mechanics and Engineering, 2018 , 336 : 640 − 666 .

HAMPTON J, DOOSTAN A . Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies[J ] . Journal of Computational Physics, 2015 , 280 : 363 − 386 .

HADIGOL M, DOOSTAN A . Least squares polynomial chaos expansion: A review of sampling strategies[J ] . Computer Methods in Applied Mechanics and Engineering, 2018 , 332 : 382 − 407 .

PEHERSTORFER B, GUNZBURGER M, WILLCOX K . Convergence analysis of multifidelity Monte Carlo estimation[J ] . Numerische Mathematik, 2018 , 139 ( 3 ): 683 − 707 .

HU X Z, CHEN X Q, PARKS G T, et al . Review of improved Monte Carlo methods in uncertainty-based design optimization for aerospace vehicles[J ] . Progress in Aerospace Sciences, 2016 , 86 : 20 − 27 .

LEE S H, CHEN W, KWAK B M . Robust design with arbitrary distributions using Gauss-type quadrature formula[J ] . Structural and Multidisciplinary Optimization, 2009 , 39 ( 3 ): 227 − 243 .

RAHMAN S, XU H . A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics[J ] . Probabilistic Engineering Mechanics, 2004 , 19 ( 4 ): 393 − 408 .

XU H, RAHMAN S . A generalized dimension-reduction method for multidimensional integration in stochastic mechanics[J ] . International Journal for Numerical Methods in Engineering, 2004 , 61 ( 12 ): 1992 − 2019 .

PADULO M, CAMPOBASSO M S, GUENOV M D . Novel uncertainty propagation method for robust aerodynamic design[J ] . AIAA Journal, 2011 , 49 ( 3 ): 530 − 543 .

GERSTNER T, GRIEBEL M . Numerical integration using sparse grids[J ] . Numerical Algorithms, 1998 , 18 ( 3 ): 209 − 232 .

XIU D B, KARNIADAKIS G E . The wiener-askey polynomial chaos for stochastic differential equations[J ] . SIAM Journal on Scientific Computing, 2002 , 24 ( 2 ): 619 − 644 .

CAMERON R H, MARTIN W T . The orthogonal development of non-linear functionals in series of fourier-Hermite functionals[J ] . The Annals of Mathematics, 1947 , 48 ( 2 ): 385 .

BLATMAN G, SUDRET B . Adaptive sparse polynomial chaos expansion based on least angle regression[J ] . Journal of Computational Physics, 2011 , 230 ( 6 ): 2345 − 2367 .

PERKÓ Z, GILLI L, LATHOUWERS D, et al . Grid and basis adaptive polynomial chaos techniques for sensitivity and uncertainty analysis[J ] . Journal of Computational Physics, 2014 , 260 : 54 − 84 .

DONOHO D L . Compressed sensing[J ] . IEEE Transactions on Information Theory, 2006 , 52 ( 4 ): 1289 − 1306 .

RAUHUT H, WARD R . Sparse Legendre expan sions via l 1 -minimization[J ] . Journal of Approximation Theory, 2012 , 164 ( 5 ): 517 − 533 .

TROPP J A, GILBERT A C . Signal recovery from random measurements via orthogonal matching pursuit[J ] . IEEE Transactions on Information Theory, 2007 , 53 ( 12 ): 4655 − 4666 .

PENG J, HAMPTON J, DOOSTAN A . A weighted-minimization approach for sparse polynomial chaos expansions[J ] . Journal of Computational Physics, 2014 , 267 : 92 − 111 .

GUO L, NARAYAN A, ZHOU T . A gradient enhanced ℓ1-minimization for sparse approximation of polynomial chaos expansions[J ] . Journal of Computational Physics, 2018 , 367 : 49 − 64 .

SALEHI S, RAISEE M, CERVANTES M J, et al . An efficient multifidelity ℓ1-minimization method for sparse polynomial chaos[J ] . Computer Methods in Applied Mechanics and Engineering, 2018 , 334 : 183 − 207 .

JUNG D H, LEE B C . Development of a simple and efficient method for robust optimization[J ] . International Journal for Numerical Methods in Engineering, 2002 , 53 ( 9 ): 2201 − 2215 .

GENEVA N, ZABARAS N . Quantifying model form uncertainty in Reynolds-averaged turbulence models with Bayesian deep neural networks[J ] . Journal of Computational Physics, 2019 , 383 : 125 − 147 .

ASAFUDDOULA M, SINGH H K, RAY T . Six-sigma robust design optimization using a many-objective decomposition-based evolutionary algorithm[J ] . IEEE Transactions on Evolutionary Computation, 2015 , 19 ( 4 ): 490 − 507 .

JAMESON A. Optimum aerodynamic design using CFD and control theory[C]// 12th Computational Fluid Dynamics Conference, San Diego, CA, USA. Reston, Virigina, 1995.

QUEIPO N V, HAFTKA R T, SHYY W, et al . Surrogate-based analysis and optimization[J ] . Progress in Aerospace Sciences, 2005 , 41 ( 1 ): 1 − 28 .

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, 2016 , 54 ( 3 ): 902 − 917 .

FRAGKOS K B, PAPOUTSIS-KIACHAGIAS E M, GIANNAKOGLOU K C . pFOSM: An efficient algorithm for aerodynamic robust design based on continuous adjoint and matrix-vector products[J ] . Computers Fluids, 2019 , 181 : 57 − 66 .

BONS N P, HE X L, MADER C A, et al . Multimodality in aerodynamic wing design optimization[J ] . AIAA Journal, 2019 , 57 ( 3 ): 1004 − 1018 .

POOLE D J, ALLEN C B, RENDALL T C S . Global optimization of wing aerodynamic optimization case exhibiting multimodality[J ] . Journal of Aircraft, 2018 , 55 ( 4 ): 1576 − 1591 .

CHERNUKHIN O, ZINGG D W . Multimodality and global optimization in aerodynamic design[J ] . AIAA Journal, 2013 , 51 ( 6 ): 1342 − 1354 .

WU X J, ZHANG W W, SONG S F . Robust aerodynamic shape design based on an adaptive stochastic optimization framework[J ] . Structural and Multidisciplinary Optimization, 2018 , 57 ( 2 ): 639 − 651 .

LEE S W, KWON O J . Robust airfoil shape optimization using design for six sigma[J ] . Journal of Aircraft, 2006 , 43 ( 3 ): 843 − 846 .

HUANG B Q, DU X P . A robust design method using variable transformation and Gauss–Hermite integration[J ] . International Journal for Numerical Methods in Engineering, 2006 , 66 ( 12 ): 1841 − 1858 .

ZHAO H, GAO Z H, XIA L . Novel multi-fidelity surrogate model assisted many-objective optimization method[J ] . Acta Aeronautica et Astronautica Sinica, 2023 , 44 ( 6 ): 126962 (in Chinese).

HU W, LI M, AZARM S, et al . Multi-objective robust optimiz ation under interval uncertainty using online approximation and constraint cuts[J ] . Journal of Mechanical Design, 2011 , 133 ( 6 ): 061002 .

GUNAWAN S, AZARM S . Multi-objective robust optimization using a sensitivity region concept[J ] . Structural and Multidisciplinary Optimization, 2005 , 29 ( 1 ): 50 − 60 .

MESSAC A, ISMAIL-YAHAYA A . Multiobjective robust design using physical programming[J ] . Structural and Multidisciplinary Optimization, 2002 , 23 ( 5 ): 357 − 371 .

DU X P, VENIGELLA P K, LIU D S . Robust mechanism synthesis with random and interval variables[J ] . Mechanism and Machine Theory, 2009 , 44 ( 7 ): 1321 − 1337 .

WU X J, ZHANG W W, SONG S F . Uncertainty quantification and sensitivity analysis of transonic aerodynamics with geometric uncertainty[J ] . International Journal of Aerospace Engineering, 2017 , 2017 : 8107190 .

IULIANO E . Global optimization of benchmark aerodynamic cases using physics-based surrogate models[J ] . Aerospace Science and Technology, 2017 , 67 : 273 − 286 .

LYU Z J, KENWAY G K W, MARTINS J R R A . Aerodynamic shape optimization investigations of the common research model wing benchmark[J ] . AIAA Journal, 2014 , 53 ( 4 ): 968 − 985 .

LIEM R P, MARTINS J R R A, KENWAY G K W . Expected drag minimization for aerodynamic design optimization based on aircraft operational data[J ] . Aerospace Science and Technology, 2017 , 63 : 344 − 362 .

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 .

ZHAO H, GAO Z H. Robust design optimization of benchmark aerodynamic case based on polynomial chaos expansion [C]// 31st Congress of the International Council of the Aeronautical Sciences, Belo Horizonte, Brazil, 2018.

ZHAO H, GAO Z H, XIA L . Efficient aerodynamic analysis and optimization under uncertainty using multi-fidelity polynomial chaos-Kriging surrogate model[J ] . Computers Fluids, 2022 , 246 : 105643 .

DU X P, CHEN W . Towards a better understanding of modeling feasibility robustness in engineering design[J ] . Journal of Mechanical Design, 2000 , 122 ( 4 ): 385 − 394 .

AHN J, KIM S, KWON J H . Reliability-based wing design optimization using trust region-sequential quadratic programming framework[J ] . Journal of Aircraft, 2005 , 42 ( 5 ): 1331 − 1336 .

PAIVA R M, CRAWFORD C, SULEMAN A . Robust and reliability-based design optimization framework for wing design[J ] . AIAA Journal, 2014 , 52 ( 4 ): 711 − 724 .

GAO Z H. Advanced laminar flow aerodynamic configuration optimization design for green aviation[C]// 32nd AIAA Applied Aerodynamics Conference, Atlanta, GA. Reston, Virginia, 2014.

KRISHNAN K S G, BERTRAM O, SEIBEL O . Review of hybrid laminar flow control systems[J ] . Progress in Aerospace Sciences, 2017 , 93 : 24 − 52 .

SABATER C , BEKEMEYER P , GÖRTZ S . Robust design of transonic natural laminar flow wings under environmental and operational uncertainties[J ] . AIAA Journal, 2021 , 60 ( 2 ): 1 − 16 .

XIONG N, TAO Y, LIU Z Y, et al . Uncertainty quantification-based robust aerodynamic optimization of laminar flow nacelle[J ] . Modern Physics Letters B, 2018 , 32 : 1840048 .

YAO Y, MA D L, YANG M Q, et al . Adaptive-surrogate-based robust optimization of transonic natural laminar flow nacelle[J ] . Chinese Journal of Aeronautics, 2021 , 34 ( 10 ): 36 − 52 .

TANG Z L, ZHANG M F, HU X . Optimal shape design and transition uncertainty analysis of transonic axisymmetric natural lami nar flow nacelle at high Reynolds number[J ] . Aerospace Science and Technology, 2022 , 121 : 107345 .

YANG T H, CHEN Y F, SHI Y Y, et al . Stochastic investigation on the robustness of laminar-flow wings for flight tests[J ] . AIAA Journal, 2022 , 60 ( 4 ): 2266 − 2286 .

SHI Y Y, LAN Q S, LAN X Y, et al . Robust optimization design of a flying wing using adjoint and uncertainty-based aerodynamic optimization approach[J ] . Structural and Multidisciplinary Optimization, 2023 , 66 ( 5 ): 110 .

QUAGLIARELLA D, IULIANO E . Robust design of a supersonic natural laminar flow wing-body[J ] . IEEE Computational Intelligence Magazine, 2017 , 12 ( 4 ): 14 − 27 .

QUAGLIARELLA D, IULIANO E. UQ sensitivity analysis and robust design optimization of a supersonic natural laminar flow wing-body[M]. Cham: Springer International Publishing, 2018.

ZHAO H, GAO Z H, WANG C, et al . Resear ch on computational grid suitable for high-speed laminar airfoil[J ] . Chinese Journal of Applied Mechanics, 2018 , 35 ( 2 ): 351 − 357 (in Chinese).

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 .

KUMAR A, KEANE A J, NAIR P B, et al . Robust design of compressor fan blades against erosion[J ] . Journal of Mechanical Design, 2006 , 128 ( 4 ): 864 − 873 .

TANG X Z, WANG Z, XIAO P, et al . Uncertainty quantification based optimization of centrifugal compressor impeller for aerodynamic robustness under stochastic operational conditions[J ] . Energy, 2020 , 195 : 116930 .

KESHAVARZZADEH V, GHANEM R G, TORTORELLI D A . Shape optimization under uncertainty for rotor blades of horizontal axis wind turbines[J ] . Computer Methods in Applied Mechanics and Engineering, 2019 , 354 : 271 − 306 .

JUN S, YEE K, LEE J, et al . Robust design optimization of unmanned aerial vehicle coaxial rotor considering operational uncertainty[J ] . Journal of Aircraft, 2011 , 48 ( 2 ): 353 − 367 .

KWON H, CHOI S, KWON J H, et al . Surrogate-based robust optimization and design to unsteady low-noise open rotors[J ] . Journal of Aircraft, 2016 , 53 ( 5 ): 1448 − 1467 .

FUSI F, GUARDONE A, QUARANTA G, et al . Multifidelity physics-based method for robust optimization applied to a hovering rotor airfoil[J ] . AIAA Journal, 2015 , 53 ( 11 ): 3448 − 3465 .

JAEGER L, GOGU C, SEGONDS S, et al . Aircraft multidisciplinary design optimization under both model and design variables uncertainty[J ] . Journal of Aircraft, 2013 , 50 ( 2 ): 528 − 538 .

SAPORITO M, DA RONCH A, BARTOLI N, et al . Robust multidisciplinary analysis and optimization for con ceptual design of flexible aircraft under dynamic aeroelastic constraints[J ] . Aerospace Science and Technology, 2023 , 138 : 108349 .

LIU G, HUANG J T, ZHOU Z, et al . Investigation of supersonic low sonic boom aerodynamic configuration design[J ] . Acta Aerodynamica Sinica, 2020 , 38 ( 5 ): 858 − 865 (in Chinese).

QIAN Z S, HAN Z H . Progress and challenges of sonic boom research[J ] . Acta Aerodynamica Sinica, 2019 , 37 ( 4 ): 601 − 619 (in Chinese).

COLONNO M, ALONSO J. Sonic boom minimization revisited: the robustness of optimal low-boom designs[C]// 13th AIAA/ISSMO Multidisciplinary Analysis Optimization Conference, Fort Worth, Texas. Reston, Virginia, 2010.

SCHAEFER J A, LAZZARA D S, LEDOUX S T. Robust Design of Sonic Boom Performance using Spatially Accurate Polynomial Chaos[C]// AIAA SCITECH 2023 Forum, National Harbor, MD Online. Reston, Virginia, 2023.

Shimoyama K, Inoue A . Uncertainty quantification by the nonintrusive polynomial chao s expansion with an adjustment strategy[J ] . AIAA Journal, 2016 , 54 ( 10 ): 3107 − 3116 .

George A, Seebass R . Sonic boom minimization including both front and rear shocks[J ] . AIAA Journal, 1971 , 9 ( 10 ): 2091 − 2093 .

TEKASLAN E H, YILDIZ S, DEMIROGLU Y, et al . Implementation of multidisciplinary and multifidelity uncertainty quantification methods for sonic boom prediction[J ] . Journal of Aircraft, 2022 , 60 ( 2 ): 410 − 422 .

ZHAO H, HUANG Y J, XING H N . Adaptive sparse polynomial chaos-based flow field/sonic boom uncertainty quantification under multi-parameter uncertainties[J ] . Chinese Journal of Theoretical and Applied Mechanics, 2023 , 55 ( 9 ): 2027 − 2042 (in Chinese).

PAN X Y, WANG X J, WANG R X, et al . Infrared radiation and stealth characteristics prediction for supersonic aircraft with uncertainty[J ] . Infrared Physics Technology, 2015 , 73 : 238 − 250 .

PAGANI P, MINVIELLE-LARROUSSE P, SESQUES M. Uncertainty quantification for the RCS of a coated target using an IBC-based metamodel[C]// 2021 51st European Microwave Conference (EuMC), London, United Kingdom, 2022.

WANG K C, HE Z, DING D Z, et al . Uncertainty scattering analysis of 3-D objects with varying shape based on method of moments[J ] . IEEE Transactions on Antennas and Propagation, 2019 , 67 ( 4 ): 2835 − 2840 .

NIE Y, YU X Q. Research on robust optimization design of electromagnetic scattering characteristics of wing stealth structure[J]. Acta Aeronautica et Astronautica Sinica, 2007, 28(增): 104-108(in Chinese).

赵欢, 高正红. 基于新型多可信度代理模型的高效气动稳健设计方法[C], 第二届中国空气动力学大会, 天津, 2022.

ALLEN M, MAUTE K . Reliability-based design optimization of aeroelastic structures[J ] . Structural and Multidisciplinary Optimization, 2004 , 27 ( 4 ): 228 − 242 .

ALLEN M, MAUTE K . Reliability-based sha pe optimization of structures undergoing fluid–structure interaction phenomena[J ] . Computer Methods in Applied Mechanics and Engineering, 2005 , 194 ( 30-33 ): 3472 − 3495 .

STANFORD B, BERAN P . Direct flutter and limit cycle computations of highly flexible wings for efficient analysis and optimization[J ] . Journal of Fluids and Structures, 2013 , 36 : 111 − 123 .

MANI K, MAVRIPLIS D J . Unsteady discrete adjoint formulation for two-dimensional flow problems with deforming meshes[J ] . AIAA Journal, 2008 , 46 ( 6 ): 1351 − 1364 .

NIKBAY M, KURU M N . Reliability based multidisciplinary optimization of aeroelastic systems with structural and aerodynamic uncertainties[J ] . Journal of Aircraft, 2013 , 50 ( 3 ): 708 − 715 .

VERHOOSEL C V, SCHOLCZ T P, HULSHOFF S J, et al . Uncertainty and reliability analysis of fluid-structure stability boundaries[J ] . AIAA Journal, 2009 , 47 ( 1 ): 91 − 104 .

MANAN A, COOPER J . Design of composite wings including uncertainties: A probabilistic approach[J ] . Journal of Aircraft, 2009 , 46 ( 2 ): 601 − 607 .

SCARTH C, COOPER J E, WEAVER P M, et al . Uncertainty quantification of aeroelastic stability of composite plate wings using lamination parameters[J ] . Composite Structures, 2014 , 116 : 84 − 93 .

HOSDER S, WALTERS R, BALCH M. Efficient uncertainty quantification applied to the aeroelastic analysis of a transonic wing[C]// 46th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. Reston, Virginia, 2008.

MISSOUM S, DRIBUSCH C, BERAN P . Reliability-based design optimization of nonlinear aeroelasticity problems[J ] . Journal of Aircraft, 2010 , 47 ( 3 ): 992 − 998 .

SCARTH C, SARTOR P, COOPER J E, et al. Robust aeroelastic design of a composite wing-box[C]// 17th AIAA Non-Deterministic Approaches Conference, Kissimmee, Florida. Reston, Virginia, 2015.

SCARTH C, SARTOR P N, COOPER J E, et al . Robust and reliability-based aeroelastic design of composite plate wings[J ] . AIAA Journal, 2017 , 55 ( 10 ): 3539 − 3552 .

DA RONCH A, DROFELNIK J, VAN ROOIJ M P C, et al . Aerodynamic and aeroelastic uncertainty quantification of nato sto avt-251 unmanned combat aerial vehicle[J ] . Aerospace Science and Technology, 2019 , 91 : 627 − 639 .

PHILLIPS B D, WEST T K. Aeroelastic uncertainty quantification of a low-boom aircraft configuration[C]// 2018 AIAA Aerospace Sciences Meeting, Kissimmee, Florida. Reston, Virginia, 2018.

ELDRED M S, SWILER L P, TANG G . Mixed aleatory-epistemic uncertainty quantification with stochastic expansions and optimization-based interval estimation[J ] . Reliability Engineering System Safety, 2011 , 96 ( 9 ): 1092 − 1113 .