A review of life cycle uncertainty factors and modeling methods for aero engine high-pressure compressor

References

1. Xiaoli, LI, Wuli, CHU. Analysis on the influence of variable installation angle on performance of multi-stage axial-flow compressor. Chinese J Turbomach 2008;5:27–9.Search in Google Scholar

2. Zhang, G. Research on the influence mechanism of abnormal installation angle and variable camber blade on compressor performance [Unpublished doctoral dissertation]. Xi’an:Northwestern Polytechnical University; 2016.Search in Google Scholar

3. Egorov, I, Marchukov, E, Popov, G, Baturin, O, Goriachkin, E, Novikova, Y. Optimization of blades stagger angles of the three-spool axial compressor to improve of efficiency of the gas turbine engine. In: 53rd AIAA/SAE/ASEE joint propulsion conference; 2017:4733 p.10.2514/6.2017-4733Search in Google Scholar

4. Zheng, S, Teng, J, Fan, L, Qiang, X. The impact of uncertain stagger angle variation on high-pressure compressor rotor performance. In Proceedings of the Global Power and Propulsion Forum (GPPS Shanghai 2017), Shanghai, China, October 30 – November 1, 2017, Paper No. GPPS-2017-0045. https://www.gpps.global.Search in Google Scholar

5. Zheng, S, Teng, J, Wu, Y, Guo, F, Lu, S, Qiang, X. Impact of nonuniform stagger angle distribution on high-pressure compressor rotor performance. In: Proceedings of the ASME turbo expo 2018: turbomachinery technical conference and exposition. Volume 2A: turbomachinery. Oslo, Norway: ASME; 2018. V02AT39A023.10.1115/GT2018-76067Search in Google Scholar

6. Kun, W, Fu, C, Jianyang, Y, Yanping, S. Nested sparse-grid stochastic collocation method for uncertainty quantification of blade stagger angle. Energy 2020;201:117583. https://doi.org/10.1016/j.energy.2020.117583.Search in Google Scholar

7. Dalbanjan, MS, Sarangi, N. Sensitivity study of stagger angle on the aerodynamic performance of transonic axial flow compressors. In: Sivaramakrishna, G, Kishore Kumar, S, Raghunandan, BN, editors. Proceedings of the national aerospace propulsion conference. Lecture notes in mechanical engineering. Singapore: Springer; 2023.10.1007/978-981-19-2378-4_1Search in Google Scholar

8. Lange, A, Voigt, M, Vogeler, K, Schrapp, H, Johann, E, Gu Mmer, V. Probabilistic CFD simulation of a high-pressure compressor stage taking manufacturing variability into account. Turbo expo: power for land, sea, and air 2010;vol 44014:617–28.Search in Google Scholar

9. Elmstrom, ME, Millsaps, KT, Hobson, GV, Patterson, JS. Impact of nonuniform leading edge coatings on the aerodynamic performance of compressor airfoils. ASME. J Turbomach 2011;133:041004. https://doi.org/10.1115/1.3213550.Search in Google Scholar

10. Li-min, GAO, Yu-tong, CAI, Rui-hui, ZENG, Lin-chuan, TIAN. Effects of blade machining error on compressor cascade aerodynamic performance. J Propuls Technol 2017;38:525.Search in Google Scholar

11. Li, Z, Liu, Y, Agarwal, RK. Uncertainty quantification of geometric and flow variables affecting the performance of a transonic axial compressor. In: 2018 AIAA aerospace sciences meeting; 2018: 0068 p.10.2514/6.2018-0068Search in Google Scholar

12. Cheng, C, Wu, BH, Zheng, H, Gao, LM. Effect of blade machining errors on compressor performance. Acta Aeronautica Astronautica Sinica 2020;41:623237.Search in Google Scholar

13. Liu, J, Yu, X, Meng, D, Shi, W, Liu, B. State and effect of manufacture deviations of compressor blade in high-pressure compressor outlet stage. Acta Aeronautica Astronautica Sinica 2021;42:423796.Search in Google Scholar

14. Domercq, O, Escuret, J-F. Tip clearance effect on high-pressure compressor stage matching. Proc Inst Mech Eng A J Power Energy, 2007;221:759–67. https://doi.org/10.1243/09576509jpe468.Search in Google Scholar

15. Ciorciari, R, Lesser, A, Blaim, F, Niehuis, R. Numerical investigation of tip clearance effects in an axial transonic compressor. J Therm Sci 2012;21:109–19. https://doi.org/10.1007/s11630-012-0525-6.Search in Google Scholar

16. Danish, SN, Qureshi, SR, Imran, MM, Khan, SUD, Sarfraz, MM, El-Leathy, A, et al.. Effect of tip clearance and rotor–stator axial gap on the efficiency of a multistage compressor. Appl Therm Eng 2016;99:988–95. https://doi.org/10.1016/j.applthermaleng.2016.01.132.Search in Google Scholar

17. Dalbanjan, MS, Niranjan, S. An effect of tip clearance on aero performance in axial flow compressors for aero gas turbine engines. Int J Mech Prod Eng Res Dev 2019;9:769–76.10.24247/ijmperdaug201977Search in Google Scholar

18. Cheng, H, Lu, X, Zhao, S, Huang, S, Zhu, J. Effect of tip clearance variation in the transonic axial compressor of a miniature gas turbine at different Reynolds numbers. Aero Sci Technol 2022;128:107793. https://doi.org/10.1016/j.ast.2022.107793.Search in Google Scholar

19. Dalbanjan, MS, Niranjan, S. Effect of rotor tip and stator hub clearance on aero performance in axial flow compressors for aero gas turbine engines. In: International conference on intelligent manufacturing and energy sustainability. Singapore: Springer Nature Singapore; 2023.10.1007/978-981-99-6774-2_24Search in Google Scholar

20. Gao, G, Zeng, R, Chen, M. Change law of tip clearance and its influence on aerodynamic performance of compressor. Sci Technol Eng 2024;24:2573–80.Search in Google Scholar

21. Giebmanns, A, Schnell, R, Werner-Spatz, C. A method for efficient performance prediction for fan and compressor stages with degraded blades. In: Proceedings of the ASME turbo expo 2015: turbine technical conference and exposition. Montreal, QC, Canada; 2015.Search in Google Scholar

22. Roberts, WB. Axial compressor performance restoration by blade profile control. Amsterdam, The Netherlands: Proceedings of the ASME 1984 International Gas Turbine Conference and Exhibit; 1984, Volume 1: Turbomachinery.10.1115/84-GT-232Search in Google Scholar

23. Tabakoff, W. Review—turbomachinery performance deterioration exposed to solid particulates environment. ASME. J. Fluids Eng 1984;106:125–34. https://doi.org/10.1115/1.3243088.Search in Google Scholar

24. Suzuki, M, Inaba, K, Yamamoto, M. Numerical simulation of sand erosion phenomena in rotor/stator interaction of compressor. J Therm Sci 2008;17:125–33. https://doi.org/10.1007/s11630-008-0125-7.Search in Google Scholar

25. Walton, K, Blunt, L, Fleming, L, Goodhand, M, Lung, H. Areal parametric characterisation of ex-service compressor blade leading edges. Wear 2014;321:79–86. https://doi.org/10.1016/j.wear.2014.10.007.Search in Google Scholar

26. Brandes, T, Koch, C, Staudacher, S. Estimation of aircraft engine flight mission severity caused by erosion. ASME. J. Turbomach 2021;143:111001. https://doi.org/10.1115/1.4051000.Search in Google Scholar

27. Shi, L, Guo, S, Yu, P, Zhang, X, Xiong, J. A review on leading-edge erosion morphology and performance degradation of aero-engine fan and compressor blades. Energies 2023;16:3068. https://doi.org/10.3390/en16073068.Search in Google Scholar

28. Li, Q, Song, W, Du, Q, Xing, Y. Numerical simulation of high pressure axial compressor with tip clearance in rotor. Comput Simulat 2008;44–7+83.Search in Google Scholar

29. Wang, Z, Wang, Y, Huang, G. Influence of compressor rotor tip clearance on compressor performance. Gas Turbine Exp Res 2012;25:31–5+40.Search in Google Scholar

30. Giebmanns, A, Schnell, R, Werner-Spatz, C. A method for efficient performance prediction for fan and compressor stages with degraded blades. In: Turbomachinery. Montreal, Quebec, Canada: American Society of Mechanical Engineers; 2015, vol 2A. V02AT37A006.10.1115/GT2015-42108Search in Google Scholar

31. Jiangzheng, ZHAN, Guang, ZHANG, Wenyan, SONG. Theoretical analysis and numerical simulation of tip clearance of 12-stage high-pressure compressor rotor. Mech Sci Technol Aero Eng 2019;38:1632–40.Search in Google Scholar

32. Liu, Z, Wang, J, Cao, P, Wang, K. Study of the effect of clearance variation on flow field at the top of blade of an axial compressor rotor under near-stall condition. J Eng Thermal Energy Power 2024;39:68–78. https://doi.org/10.16146/j.cnki.rndlgc.2024.01.008.Search in Google Scholar

33. Bammert, K, Woelk, GU. The influence of the blading surface roughness on the aerodynamic behavior and characteristic of an axial compressor. ASME Eng Power 1980;102:283–7. https://doi.org/10.1115/1.3230249.Search in Google Scholar

34. Dong, LI, Zhaoyuan, FAN, Juan, Z. Influence of blade roughness on compressor performance deterioration. Aeroengine 2009;35:32–5.Search in Google Scholar

35. Wang, L, Sun, W, Sun, T, Sun, H. Research on the effect of roughness on the performance of compressor Cascade. J Eng Therm Energy Power 2022;37:20–8. https://doi.org/10.16146/j.cnki.rndlgc.2022.12.003.Search in Google Scholar

36. Sun, W. Effect of Roughness on Performance Degradation of Compressor Cascade [Unpublished master’s thesis]. Harbin Engineering University, Harbin, China 2022. https://doi.org/10.27060/d.cnki.ghbcu.2022.000245.Search in Google Scholar

37. Yu, H, Chen, W, Cheng, L, Zhang, W. Fouling behavior of aero-engine compressor blades surface. China Surf Eng 2022;34:188–97.Search in Google Scholar

38. Morini, M, Pinelli, M, Spina, PR, Venturini, M. Computational fluid dynamics simulation of fouling on axial compressor stages. Gas Turbines Power 2010;132:072401.10.1115/1.4000128Search in Google Scholar

39. Li, Z, Li, B, Wang, D, Li, L. Analysis of sensitivity of compressor performance parameters to fouling. Aeronaut Comput Tech 2011;41:41–4.Search in Google Scholar

40. Mishra, RK. Fouling and corrosion in an aero gas turbine compressor. J Fail Anal Prev 2015;15:837–45. https://doi.org/10.1007/s11668-015-0023-8.Search in Google Scholar

41. Döring, F, Staudacher, S, Koch, C. Predicting the temporal progression of aircraft engine compressor performance deterioration due to particle deposition. In: Turbo expo: power for land, sea, and air. American Society of Mechanical Engineers; 2017, vol 50817. V02DT48A007.10.1115/GT2017-63544Search in Google Scholar

42. Liu, H. Research on the influence of blade fouling on axial compressor performance [Unpublished master’s thesis]. Civil Aviation University of China,Tianjin, China; 2017.Search in Google Scholar

43. Lange, A, Vogeler, K, Gümmer, V, Schrapp, H, Clemen, C. Introduction of a parameter based compressor blade model for considering measured geometry uncertainties in numerical simulation. In: Proceedings of the ASME turbo expo 2009: power for land, sSea, and air. Volume 6: Structures and Dynamics, Parts A and B. Orlando, Florida, USA: ASME; 2009: 1113–23 pp.10.1115/GT2009-59937Search in Google Scholar

44. Lange, A, Voigt, M, Vogeler, K, Schrapp, H, Johann, E, Gümmer, V. Impact of manufacturing variability on multistage high-pressure compressor performance. ASME J Eng Gas Turbines Power 2012;134:112601. https://doi.org/10.1115/1.4007167.Search in Google Scholar

45. Wang, S, Wang, GH, Han, Q. Compressor performance deterioration caused by blade fouling. J Harbin Eng Univ 2014;35:1524–8.Search in Google Scholar

46. Tang, P, Sun, J, Nian, J, Lu, J, Liu, Q. Uncertainty quantification of the impact of high-pressure compressor blade geometric deviations on aero engine performance. Aerospace 2025;12:767. https://doi.org/10.3390/aerospace12090767.Search in Google Scholar

47. Nikuradse, J. Laws of flow in rough pipes. 1950.Search in Google Scholar

48. Schlichting, H. Experimental investigation of the problem of surface roughness. Washington, D.C.: National Advisory Commitee for Aeronautics; 1937.Search in Google Scholar

49. Koch, CC, SmithJrLH. Loss sources and magnitudes in axial-flow compressors; 1976.10.1115/1.3446202Search in Google Scholar

50. Bons, JP. A review of surface roughness effects in gas turbines; 2010.10.1115/1.3066315Search in Google Scholar

51. Schlichting, H, Kestin, J. Boundary layer theory. New York: McGraw-Hill; 1961.Search in Google Scholar

52. Montomoli, F, Carnevale, M, D’Ammaro, A, Massini, M, Salvadori, S. Uncertainty quantification in computational fluid dynamics and aircraft engines. London, UK: Springer; 2015:1–90 pp.10.1007/978-3-319-14681-2_1Search in Google Scholar

53. Kraft, J, Sethi, V, Singh, R. Optimization of aero gas turbine maintenance using advanced simulation and diagnostic methods. J Eng Gas Turbines Power 2014;136:111601. https://doi.org/10.1115/1.4027356.Search in Google Scholar

54. Daxiang, LIU, Jie, JIN, Denghuan, LIU. Position and function of numerical simulation technology in aero-engine development. 航空动力学报 2022;37:2017–24.Search in Google Scholar

55. Ivanov, MJ. Perspective problems of gas turbine engines simulation. Neuilly-sur-Seine, France: AGARD-LS-198; 1994.Search in Google Scholar

56. Zhang, X, Wang, Z. Numerical simulation and optimization of aero-engine overall performance. Beijing: Science Press; 2023.Search in Google Scholar

57. McKinney, JS. Simulation of turbofan engine, part I. Description of method and balancing technique. Air Force Aero-Propulsion Laboratory 1967. AFAPC-TR-67-125-PT.-1.Search in Google Scholar

58. Kluiters, SCA, Visser, WPJ, Rademaker, ER. A new combustor and emission model for the gas turbine simulation program GSP. Technical Report 1998.Search in Google Scholar

59. Reed, JA, Afjeh, AA. An object-oriented framework for distributed computational simulation of aerospace propulsion systems. In: COOTS. 1998:149–64.10.2514/6.1998-3565Search in Google Scholar

60. Lytle, JK. The numerical propulsion system simulation: an overview. Comput Aerosciences 2000. (E-12152).Search in Google Scholar

61. Antonio, AGP. Modelling of a gas turbine with modelica. Lund: Lund University; 2001.Search in Google Scholar

62. Mattingly, JD. Aircraft engine design. Reston, Virginia, USA: Aiaa; 2002.10.2514/4.861444Search in Google Scholar

63. Ivanov, M, Nigmatullin, R. Interconnected multi-level design of gas turbine elements. In: 41st aerospace sciences meeting and exhibit. 2003: 1215 p.10.2514/6.2003-1215Search in Google Scholar

64. Kurzke, J. GasTurb 10 user’s manual. Aachen: GasTurb Gmbh; 2005.Search in Google Scholar

65. Bala, A, Sethi, V, Gatto, EL, Pachidis, V, Pilidis, P. PROOSIS—a collaborative venture for gas turbine performance simulation using an object oriented programming schema. In: 18th ISABE Conference, ISABE-2007-1357, Beijing, China; 2007.Search in Google Scholar

66. Hendricks, ES, Gray, JS. pyCycle: a tool for efficient optimization of gas turbine engine cycles. Aerospace 2019;6:87. https://doi.org/10.3390/aerospace6080087.Search in Google Scholar

67. Wei, F. Research on variable specific heat thermodynamic calculation and system-level simulation of aero-engines [Unpublished doctoral dissertation]. Shanghai: Shanghai Jiao Tong University; 2006.Search in Google Scholar

68. Cui, K, Gou, LF. Design of Simulation Platform for Aero-Engine Control System. Computer Measurement & Control 2016; 24, 101–103, 107. https://doi.org/10.16526/j.cnki.11-4762/tp.2016.10.029 Search in Google Scholar

69. Chuankai, L, Hongchao, JIANG, Yanru, L, Yuanyuan, L, Shuiting, D. Coupled simulation model of aero-engine performance and secondary air system. J Aero Power 2017;32:1623–30.Search in Google Scholar

70. Turner, M. Lessons learned from the GE90 3-D full engine simulations. In: 48th AIAA aerospace sciences meeting including the new Horizons forum and aerospace exposition. 2010. 1606 p.10.2514/6.2010-1606Search in Google Scholar

71. Lawrence, C. An overview of three approaches to multidisciplinary aero propulsion simulations. 1997.Search in Google Scholar

72. Liu, W. High precision quasi-3d and 3d variable dimension coupling simulation of aero-engine [Unpublished doctoral dissertation]. Heilongjiang: Harbin Institute of Technology; 2022.Search in Google Scholar

73. Melloni, L, Kotsiopoulos, P, Jackson, A, Pachidis, V, Pilidis, P. Military engine response to compressor inlet stratified pressure distortion by an integrated CFD analysis. Turbo expo: power for land, sea, and air 2006;Vol. 42398:267–78.10.1115/GT2006-90805Search in Google Scholar

74. Reitenbach, S, Schnös, M, Becker, R, Otten, T. “Optimization of Compressor Variable Geometry Settings Using Multi-Fidelity Simulation.” Proceedings of the ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. Volume 2C: Turbomachinery. Montreal, Quebec, Canada. June 15–19, 2015. V02CT45A010. ASME. https://doi.org/10.1115/GT2015-42832 Search in Google Scholar

75. Huang, JH, Feng, GT, Yu, TC, Zhou, C. Numerical simulation of redesigned turbine matching in turbojet engine. Tui** Jishu/J Propuls Technol (China) 2005;26:151–4.Search in Google Scholar

76. Reitz, G, Friedrichs, J. Impact of front-and rear-stage high pressure compressor deterioration on jet engine performance. Int J Turbomach Propuls Power 2018;3:15. https://doi.org/10.3390/ijtpp3020015.Search in Google Scholar

77. Zhan-xue, WANG, Fu, SONG, Li, ZHOU, **ao-bo, ZHANG, Ming-yang, ZHANG. Research progress in numerical zooming technology of aero-engine. J Propuls Technol 2018;39:1441.Search in Google Scholar

78. Follen, G, AuBuchon, M. Numerical zooming between a NPSS engine system simulation and a one-dimensional high compressor analysis code. 2000.Search in Google Scholar

79. Sampath, R, Plybon, R, Meyers, C, Irani, R, Balasubramaniam, M. High fidelity system simulation of aerospace vehicles using NPSS. In: 42nd AIAA aerospace sciences meeting and exhibit. 2004: 371 p.10.2514/6.2004-371Search in Google Scholar

80. Klein, C, Wolters, F, Reitenbach, S, Schönweitz, D. Integration of 3d-cfd component simulation into overall engine performance analysis for engine condition monitoring purposes. Turbo expo: power for land, sea, and air. American Society of Mechanical Engineers 2018; vol 50985. V001T01A013.10.1115/GT2018-75719Search in Google Scholar

81. Song, F, Zhou, L, Wang, Z, Zhang, X, Hao, W. Effects of boundary conditions on multi-level variable cycle engine simulation model. J Propuls Technol 2020;41:974–83. https://doi.org/10.13675/j.cnki.tjjs.190304.Search in Google Scholar

82. Zhu, X, Zhang, W, Feng, W, Li, S, Lu, Y, Li, Z, et al.. Serial collaborative simulation method for aero-engine components based on test data. J Aero Power 2023;38:1648–57.Search in Google Scholar

83. Klein, C, Reitenbach, S, Schoenweitz, D, Wolters, F. A fully coupled approach for the integration of 3D-CFD component simulation in overall engine performance analysis. In: Turbo expo: power for land, sea, and air. American Society of Mechanical Engineers 2017; vol 50770. V001T01A014.10.1115/GT2017-63591Search in Google Scholar

84. Fu, SONG, Li, ZHOU, Zhan-xue, WANG, Ming-yang, ZHANG, **ao-bo, ZHANG. Application of different zooming strategies in aero-engine simulation. J Propuls Technol 2020;41:974.Search in Google Scholar

85. Song, F, Zhou, L, Wang, Z, Zhang, X, Hao, W. Coupling method between three-dimensional component simulation model and aero-engine cycle parameter analysis. J Propuls Technol 2022;43:201000.Search in Google Scholar

86. Paniagua, G, Denos, R, Almeida, S. Effect of the hub endwall cavity flow on the flow-field of a transonic high-pressure turbine. 2004.10.1115/GT2004-53458Search in Google Scholar

87. Montomoli, F, Carnevale, M, D’Ammaro, A, Massini, M, Salvadori, S. Uncertainty quantification in computational fluid dynamics and aircraft engines. Cham: Springer International Publishing; 2015.10.1007/978-3-319-14681-2Search in Google Scholar

88. Zang, T, Hemsch, M, Hilburger, M, Kenny, S, Luckring, J, Maghami, P, et al.. Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles. National Aeronautics and Space Administration, Langley Research Center; 2002.Search in Google Scholar

89. Schmidt, R, Voigt, M, Mailach, R, Hirsch, C, Wunsch, D, Szumbarski, J, et al.. Uncertainty management for robust industrial design in aeronautics. Cham: Springer International Publishing; 2019, vol 140.Search in Google Scholar

90. Alonso, J, Eldred, M, Constantine, P, Duraisamy, K, Farhat, C, Iaccarino, G, et al.. Scalable environment for quantification of uncertainty and optimization in industrial applications (SEQUOIA). In: 19th AIAA Non-Deterministic Approaches Conference; 2017.10.2514/6.2017-1327Search in Google Scholar

91. Wang, J, Zheng, X. Review of geometric uncertainty quantification in gas turbines. J Eng Gas Turbines Power 2020;142:070801. https://doi.org/10.1115/1.4047179.Search in Google Scholar

92. Wasserstein, RL. Monte carlo: concepts, algorithms, and applications. Technometrics 1997.10.2307/1271146Search in Google Scholar

93. Ng, LWT, Willcox, KE. Monte Carlo information-reuse approach to aircraft conceptual design optimization under uncertainty. J Aircraft 2016;53:427–38. https://doi.org/10.2514/1.c033352.Search in Google Scholar

94. Hu, X, Chen, X, Parks, GT, Yao, W. Review of improved Monte Carlo methods in uncertainty-based design optimization for aerospace vehicles. Prog Aero Sci 2016;86:20–7. https://doi.org/10.1016/j.paerosci.2016.07.004.Search in Google Scholar

95. Lange, A, Voigt, M, Vogeler, K, Schrapp, H, Johann, E, Gu Mmer, V. Probabilistic CFD simulation of a high-pressure compressor stage taking manufacturing variability into account. In: Structures and dynamics, parts A and B. Glasgow, UK: ASMEDC; 2010, vol 6:617–28 pp.10.1115/GT2010-22484Search in Google Scholar

96. Huyse, L, Enright, M. Efficient statistical analysis of failure risk in engine rotor disks using importance sampling techniques. 44th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference. American Institute of Aeronautics and Astronautics; 2024.Search in Google Scholar

97. Bucher, CG. Adaptive sampling — an iterative fast Monte Carlo procedure. Struct Saf 1988;5:119–26. https://doi.org/10.1016/0167-4730(88)90020-3.Search in Google Scholar

98. Xu, B, Zang, C, Zhang, G. Robust tolerance design for rotor dynamics based on possibilistic concepts. Arch Appl Mech 2022;92:755–70. https://doi.org/10.1007/s00419-021-02070-5.Search in Google Scholar

99. Gao, LM, Cai, YT, Xu, HL, Deng, W. Uncertainty analysis of machining error influence of compressor blade. J Aero Power 2017;32:2253–9.Search in Google Scholar

100. He, X, Zheng, X. Performance improvement of transonic centrifugal compressors by optimization of complex three-dimensional features. Proc Inst Mech Eng G J Aerosp Eng 2017; 231: 2723–38. https://doi.org/10.1177/0954410016673395.Search in Google Scholar

101. Cai, Y, Gao, L, Ma, C, Zheng, T. Uncertainty quantifcation on compressor blade considering ManufacturingError based on NIPC method. J Eng Thermophys 2017;38:490–7.Search in Google Scholar

102. Wiener, N. Nonlinear problems in random theory. 1966.Search in Google Scholar

103. Xiu, D, Karniadakis, GE. The wiener–askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 2002;24:619–44.10.1137/S1064827501387826Search in Google Scholar

104. Ghanem, RG, Spanos, PD. Stochastic finite elements: a spectral approach. New York Berlin Heidelberg: Springer; 1991.10.1007/978-1-4612-3094-6Search in Google Scholar