Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method
2. The Adjoint-Based Nonlinear/Linear Method (Adj-Nonlinear/Linear)
- Obtain the flow solution and the adjoint solution for a specific performance functional for the baseline geometry;
- Calculate sensitivity for all geometric variables. The sensitivity calculation involves mesh perturbation and the objective function J & residual evaluation, based on the perturbed mesh and the baseline flow solution . The finite difference method is also involved here to determine the sensitivities, see the green box in Figure 1. Therefore, the perturbation size of each design variable is quite important, as too big or too small a value introduces big truncation or rounding errors. The operation has to be performed for each geometric variable (for 1:M, M represents the amount of the geometric variables). In this investigation, there were, in total, 398 geometric variables. This meant that the number of mesh perturbations and residual evaluations was 398;
- Calculate the performance metrics of all samples. The geometry perturbations of all samples() (for i = 1:N, N represents the amount of the samples) are reflected in changes to the baseline geometry variable vector . Then, Equation (9) is used to calculate the performance metric J.
3. Uncertainty Quantification
3.1. Test Case
3.2. Adjoint Solution Verification
3.3. Verification of the MC−Adj−Nonlinear Method at Design and Near-Stall Conditions
3.4. Full Map UQ of Aerodynamic Performance at Four Speeds
- At 100% speed, compared with the MC−adj−linear method, the UQ results predicted by the MC−adj−nonlinear method were more accurate, especially for the near-stall condition, where the nonlinear dependence of performance functionals on geometric variables was stronger. At 50% speed, the differences in the UQ results predicted by the two adjoint-based methods were much smaller, due to the weaker nonlinearity of the flow. The MC−adj−nonlinear method required nearly 30 times less time than the MC−CFD method. Hence, the MC−adj−nonlinear approach provides a satisfactory balance between precision and time cost for UQ.
- Aerodynamic performance is more sensitive to geometric deviations at high speeds than at low speeds. For this particular case, the geometric deviations produced an increased mean of mass flow rate and pressure at low speeds, while incurring a reduced mean at high speeds. The geometric deviations were generally detrimental to the mean efficiency over the four speeds. The reduction of the mean of mass flow rate, pressure ratio, and efficiency became more with increase in shaft speed.
- The standard deviation of performance generally increased with increase in shaft speed. Along a speedline, the standard deviation also increased with increase in pressure ratio. The difference in standard deviation between a near choke point and a near stall point along a speedline was much larger at high speeds than at low speeds.
Data Availability Statement
Conflicts of Interest
- Goodhand, M.N.; Miller, R.J. The Impact of Real Geometries on Three-Dimensional Separations in Compressors. J. Turbomach. 2011, 134, 021007. [Google Scholar] [CrossRef]
- Montomoli, F.; Massini, M.; Salvadori, S. Geometrical uncertainty in turbomachinery: Tip gap and fillet radius. Comput. Fluids 2011, 46, 362–368. [Google Scholar] [CrossRef]
- Wheeler, A.P.S.; Sofia, A.; Miller, R.J. The Effect of Leading-Edge Geometry on Wake Interactions in Compressors. J. Turbomach. 2009, 131, 041013. [Google Scholar] [CrossRef]
- Reid, L.; Urasek, D.C. Experimental Evaluation of the Effects of a Blunt Leading Edge on the Performance of a Transonic Rotor. J. Eng. Power 1973, 95, 199–204. [Google Scholar] [CrossRef]
- Wunsch, D.; Hirsch, C.; Nigro, R.; Coussement, G. Quantification of Combined Operational and Geometrical Uncertainties in Turbo-Machinery Design. In Turbo Expo: Power for Land, Sea, and Air, Volume 2C: Turbomachinery; ASME: Montreal, QC, Canada, 2015. [Google Scholar] [CrossRef]
- Ju, Y.; Zhang, C. Robust design optimization method for centrifugal impellers under surface roughness uncertainties due to blade fouling. Chin. J. Mech. Eng. 2016, 29, 301–314. [Google Scholar] [CrossRef]
- Kumar, A.; Keane, A.; Nair, P.; Shahpar, S. Robust Design of Compressor Fan Blades Against Erosion. J. Mech. Des. 2006, 128, 864–873. [Google Scholar] [CrossRef][Green Version]
- Eldred, M.; Waanders, B.; Wojtkiewicz, S.; Kolda, T.; Adams, B.; Swiler, L.; Williams, P.; Hough, P.; Gay, D.; Dunlavy, D.; et al. DAKOTA, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis Version 6.16 Theory Manual; Sandia National Lab. (SNL-NM): Albuquerque, NM, USA, 2022. [Google Scholar] [CrossRef]
- Moeckel, C.W.; Darmofal, D.L.; Kingston, T.R.; Norton, R.J.G. Toleranced Designs of Cooled Turbine Blades Through Probabilistic Thermal Analysis of Manufacturing Variability. In Turbo Expo: Power for Land, Sea, and Air, Volume 5: Turbo Expo 2007; ASME: Montreal, QC, Canada, 2007; pp. 1179–1191. [Google Scholar] [CrossRef][Green Version]
- Caflisch, R.E. Monte Carlo and quasi-Monte Carlo methods. Acta Numer. 1998, 7, 1–49. [Google Scholar] [CrossRef][Green Version]
- Garzon, V.E.; Darmofal, D.L. Impact of Geometric Variability on Axial Compressor Performance. J. Turbomach. 2003, 125, 1199–1213. [Google Scholar] [CrossRef][Green Version]
- Wang, J.; Zheng, X. Review of Geometric Uncertainty Quantification in Gas Turbines. J. Eng. Gas Turbines Power 2020, 142, 070801. [Google Scholar] [CrossRef]
- Lange, A.; Voigt, M.; Vogeler, K.; Schrapp, H.; Johann, E.; Guemmer, V. Impact of Manufacturing Variability on Multistage High-Pressure Compressor Performance. J. Eng. Gas Turbines Power 2012, 134, 417–426. [Google Scholar] [CrossRef]
- Najm, H. Uncertainty Quantification and Polynomial Chaos Techniques in Computational Fluid Dynamics. Annu. Rev. Fluid Mech. 2008, 41, 35–52. [Google Scholar] [CrossRef]
- Allaire, G. A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes. Ing. Automob. 2015, 836, 33–36. [Google Scholar]
- Marta, A.; Shankaran, S. Assessing Turbomachinery Performance Sensitivity to Boundary Conditions Using Control Theory. J. Propuls. Power 2014, 30, 1–14. [Google Scholar] [CrossRef]
- Rodrigues, S.; Marta, A. Adjoint formulation of a steady multistage turbomachinery interface using automatic differentiation. Comput. Fluids 2018, 176, 182–192. [Google Scholar] [CrossRef]
- Rodrigues, S.S.; Marta, A.C. Adjoint-Based Shape Sensitivity of Multi-Row Turbomachinery. Struct. Multidiscip. Optim. 2020, 61, 837–853. [Google Scholar] [CrossRef]
- Giebmanns, A.; Backhaus, J.; Frey, C.; Schnell, R. Compressor Leading Edge Sensitivities and Analysis with an Adjoint Flow Solver. In Proceedings of the ASME Turbo Expo 2013: Turbine Technical Conference and Exposition, San Antonio, TX, USA, 3–7 June 2013. [Google Scholar] [CrossRef]
- Luo, J.; Liu, F. Statistical Evaluation of Performance Impact of Manufacturing Variability by an Adjoint Method. Aerosp. Sci. Technol. 2018, 77, 471–484. [Google Scholar] [CrossRef]
- Ghate, D.; Giles, M.B. Inexpensive Monte Carlo Uncertainty Analysis. In Recent Trends in Aerospace Design and Optimization: Symposium on Applied Aerodynamics and Design of Aerospace Vehicles, SAROD-2005; Tata Mc Graw-Hill: Hyderabad, India, 2005. [Google Scholar]
- Zhang, Q.; Xu, S.; Yu, X.; Liu, J.; Wang, D.; Huang, X. Nonlinear Uncertainty Quantification of the Impact of Geometric Variability on Compressor Performance Using an Adjoint Method. Chin. J. Aeronaut. 2022, 35, 5. [Google Scholar] [CrossRef]
- Zhang, Q.; Xu, S.; Yu, X.; Liu, J.; Wang, D.; Huang, X. Quantification of Compressor Aerodynamic Performance Deviation due to Manufacturing Uncertainty Using the Adjoint Method. In Proceedings of the GPPS Xi’an21, Xi’an, China, 18–20 October 2022. [Google Scholar] [CrossRef]
- Xu, S.; Timme, S. Robust and Efficient Adjoint Solver for Complex Flow Conditions. Comput. Fluids 2017, 148, 26–38. [Google Scholar] [CrossRef][Green Version]
- Xu, S.; Mohanamuraly, P.; Wang, D.; Müller, J.D. Newton–Krylov Solver for Robust Turbomachinery Aerodynamic Analysis. AIAA J. 2020, 58, 1320–1336. [Google Scholar] [CrossRef]
- Xu, S.; Li, Y.; Huang, X.; Wang, D. Robust Newton–Krylov Adjoint Solver for the Sensitivity Analysis of Turbomachinery Aerodynamics. AIAA J. 2021, 59, 4014–4030. [Google Scholar] [CrossRef]
|Operating Point||Numerical Method||Mass Flow Rate (kg/s)||Pressure Ratio||Efficiency|
|the design condition||MC−CFD||0.023428||1.420063||0.910661|
|the near-stall condition||MC−CFD||0.021610||1.476226||0.882060|
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Xu, S.; Zhang, Q.; Wang, D.; Huang, X. Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method. Aerospace 2023, 10, 280. https://doi.org/10.3390/aerospace10030280
Xu S, Zhang Q, Wang D, Huang X. Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method. Aerospace. 2023; 10(3):280. https://doi.org/10.3390/aerospace10030280Chicago/Turabian Style
Xu, Shenren, Qian Zhang, Dingxi Wang, and Xiuquan Huang. 2023. "Uncertainty Quantification of Compressor Map Using the Monte Carlo Approach Accelerated by an Adjoint-Based Nonlinear Method" Aerospace 10, no. 3: 280. https://doi.org/10.3390/aerospace10030280