Efficient uncertainty quantification of CFD problems by combination of proper orthogonal decomposition and compressed sensing. (English) Zbl 1481.76166
Summary: In the current paper, an efficient surrogate model based on combination of Proper Orthogonal Decomposition (POD) and compressed sensing is developed for affordable representation of high dimensional stochastic fields. In the developed method, instead of the full (or classical) Polynomial Chaos Expansion (PCE), the \(\ell_1\)-minimization approach is utilized to reduce the computational work-load of the low-fidelity calculations. To assess the model capability in the real engineering problems, two challenging high-dimensional CFD test cases namely; i) turbulent transonic flow around RAE2822 airfoil with 18 geometrical uncertainties and ii) turbulent transonic flow around NASA Rotor 37 with 3 operational and 21 geometrical uncertainties are considered. Results of Uncertainty Quantification (UQ) analysis in both test cases showed that the proposed multi-fidelity approach is able to reproduce the statistics of quantities of interest with much lower computational cost than the classical regression-based PCE method. It is shown that the combination of the POD with the compressed sensing in RAE2822 and Rotor 37 test cases gives respectively computational gains between 1.26–7.72 and 1.79–9.05 times greater than the combination of the POD with the full PCE.
MSC:
| 76M35 | Stochastic analysis applied to problems in fluid mechanics |
| 65C20 | Probabilistic models, generic numerical methods in probability and statistics |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
Keywords:
uncertainty quantification; proper orthogonal decomposition; multifidelity; compressed sensing; polynomial chaos expansionSoftware:
TOMS659References:
| [1] | Fishman, G., Monte Carlo: Concepts, Algorithms, and Applications (1996), Springer Science & Business Media · Zbl 0859.65001 |
| [2] | Cunha, A.; Nasser, R.; Sampaio, R.; Lopes, H.; Breitman, K., Uncertainty quantification through the monte carlo method in a cloud computing setting, Comput. Phys. Commun., 185, 5, 1355-1363 (2014) |
| [3] | Debusschere, B. J.; Najm, H. N.; Pébay, P. P.; Knio, O. M.; Ghanem, R. G.; Le Maıtre, O. P., Numerical challenges in the use of polynomial chaos representations for stochastic processes, SIAM J. Sci. Comput., 26, 2, 698-719 (2004) · Zbl 1072.60042 |
| [4] | Mathelin, L.; Hussaini, M. Y.; Zang, T. A., Stochastic approaches to uncertainty quantification in CFD simulations, Numer. Algorithms, 38, 1-3, 209-236 (2005) · Zbl 1130.76062 |
| [5] | Xiu, D.; Karniadakis, G. E., Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187, 1, 137-167 (2003) · Zbl 1047.76111 |
| [6] | Hosder, S.; Walters, R.; Perez, R., A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations, AIAA Paper, 891, 2006 (2006) |
| [7] | Ghanem, R. G.; Spanos, P. D., Stochastic Finite Elements: a Spectral Approach (2003), Courier Corporation |
| [8] | Dinescu, C.; Smirnov, S.; Hirsch, C.; Lacor, C., Assessment of intrusive and non-intrusive non-deterministic CFD methodologies based on polynomial chaos expansions, Int. J. Eng. Syst.Model. Simul., 2, 1-2, 87-98 (2010) |
| [9] | Lacor, C.; Dinescu, C.; Hirsch, C.; Smirnov, S., Implementation of intrusive polynomial chaos in CFD codes and application to 3D Navier-Stokes, Uncertainty Quantification in Computational Fluid Dynamics, 193-223 (2013), Springer · Zbl 1276.76065 |
| [10] | Ma, X.; Zabaras, N., An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys., 228, 8, 3084-3113 (2009) · Zbl 1161.65006 |
| [11] | Resmini, A.; Peter, J.; Lucor, D., Sparse grids-based stochastic approximations with applications to aerodynamics sensitivity analysis, Int. J. Numer. Methods Eng., 106, 1, 32-57 (2016) · Zbl 1352.76093 |
| [12] | Blatman, G.; Sudret, B., An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis, Probab. Eng. Mech., 25, 2, 183-197 (2010) |
| [13] | Blatman, G.; Sudret, B., Adaptive sparse polynomial chaos expansion based on least angle regression, J. Comput. Phys., 230, 6, 2345-2367 (2011) · Zbl 1210.65019 |
| [14] | Mohammadi-Ahmar, A.; Mohammadi, A.; Raisee, M., Efficient uncertainty quantification of turbine blade leading edge film cooling using bi-fidelity combination of compressed sensing and Kriging, Int. J. Heat Mass Transf., 162, 120360 (2020) |
| [15] | Nair, P. B.; Keane, A. J., Stochastic reduced basis methods, AIAA J., 40, 8, 1653-1664 (2002) |
| [16] | Nme.4900 · Zbl 1352.80004 |
| [17] | Fld.4066 · Zbl 1455.76099 |
| [18] | Chen, W.; Hesthaven, J. S.; Junqiang, B.; Qiu, Y.; Yang, Z.; Tihao, Y., Greedy nonintrusive reduced order model for fluid dynamics, AIAA J., 56, 12, 4927-4943 (2018) |
| [19] | Margheri, L.; Sagaut, P., A hybrid anchored-ANOVA POD/Kriging method for uncertainty quantification in unsteady high-fidelity CFD simulations, J. Comput. Phys., 324, 137-173 (2016) · Zbl 1371.76084 |
| [20] | Mohammadi, A.; Raisee, M., Stochastic field representation using bi-fidelity combination of proper orthogonal decomposition and Kriging, Comput. Methods Appl. Mech. Eng., 357, 112589 (2019) · Zbl 1442.65114 |
| [21] | Guo, M.; Hesthaven, J. S., Reduced order modeling for nonlinear structural analysis using gaussian process regression, Comput. Methods Appl. Mech.Eng., 341, 807-826 (2018) · Zbl 1440.65206 |
| [22] | Zhang, Z.; Guo, M.; Hesthaven, J. S., Model order reduction for large-scale structures with local nonlinearities, Comput. Methods Appl. Mech.Eng., 353, 491-515 (2019) · Zbl 1441.74285 |
| [23] | Wang, Q.; Hesthaven, J. S.; Ray, D., Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem, J. Comput. Phys., 384, 289-307 (2019) · Zbl 1459.76117 |
| [24] | Hu, R.; Fang, F.; Pain, C.; Navon, I., Rapid spatio-temporal flood prediction and uncertainty quantification using a deep learning method, J. Hydrol., 575, 911-920 (2019) |
| [25] | Jensen, H.; Mayorga, F.; Valdebenito, M.; Chen, J., An effective parametric model reduction technique for uncertainty propagation analysis in structural dynamics, Reliab. Eng. Syst. Saf., 195, 106723 (2020) |
| [26] | El Moçayd, N.; Mohamed, M. S.; Ouazar, D.; Seaid, M., Stochastic model reduction for polynomial chaos expansion of acoustic waves using proper orthogonal decomposition, Reliab. Eng. Syst. Saf., 195, 106733 (2020) |
| [27] | Skinner, R. W.; Doostan, A.; Peters, E. L.; Evans, J. A.; Jansen, K. E., Reduced-basis multifidelity approach for efficient parametric study of NACA airfoils, AIAA J., 57, 4, 1481-1491 (2019) |
| [28] | Nagel, J. B.; Rieckermann, J.; Sudret, B., Principal component analysis and sparse polynomial chaos expansions for global sensitivity analysis and model calibration: Application to urban drainage simulation, Reliab. Eng. Syst. Saf., 195, 106737 (2020) |
| [29] | Special Issue Honoring the 80th Birthday of Professor Ivo Babuka · Zbl 1173.74411 |
| [30] | Kumar, D.; Raisee, M.; Lacor, C., An efficient non-intrusive reduced basis model for high dimensional stochastic problems in CFD, Comput. Fluids, 138, 67-82 (2016) · Zbl 1390.76800 |
| [31] | Montgomery, D. C., Design and Analysis of Experiments (2017), John Wiley & Sons |
| [32] | Todor, R. A.; Schwab, C., Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Numer. Anal., 27, 2, 232-261 (2007) · Zbl 1120.65004 |
| [33] | Bieri, M.; Schwab, C., Sparse high order FEM for elliptic sPDEs, Comput. Methods Appl. Mech.Eng., 198, 13-14, 1149-1170 (2009) · Zbl 1157.65481 |
| [34] | Mohammadi, A.; Raisee, M., Efficient uncertainty quantification of stochastic heat transfer problems by combination of proper orthogonal decomposition and sparse polynomial chaos expansion, Int. J. Heat Mass Transf., 128, 581-600 (2019) |
| [35] | Doostan, A.; Owhadi, H., A non-adapted sparse approximation of PDEs with stochastic inputs, J. Comput. Phys., 230, 8, 3015-3034 (2011) · Zbl 1218.65008 |
| [36] | Yang, X.; Karniadakis, G. E., Reweighted \(\ell_1\)-minimization method for stochastic elliptic differential equations, J. Comput. Phys., 248, 87-108 (2013) · Zbl 1349.60113 |
| [37] | Peng, J.; Hampton, J.; Doostan, A., A weighted \(\ell_1\)-minimization approach for sparse polynomial chaos expansions, J. Comput. Phys., 267, 92-111 (2014) · Zbl 1349.65198 |
| [38] | Diaz, P.; Doostan, A.; Hampton, J., Sparse polynomial chaos expansions via compressed sensing and d-optimal design, Comput. Methods Appl. Mech.Eng., 336, 640-666 (2018) · Zbl 1441.65005 |
| [39] | Abraham, S.; Raisee, M.; Ghorbaniasl, G.; Contino, F.; Lacor, C., A robust and efficient stepwise regression method for building sparse polynomial chaos expansions, J. Comput. Phys., 332, 461-474 (2017) · Zbl 1384.62216 |
| [40] | Salehi, S.; Raisee, M.; Cervantes, M. J.; Nourbakhsh, A., An efficient multifidelity \(\ell_1\)-minimization method for sparse polynomial chaos, Comput. Methods Appl. Mech.Eng., 334, 183-207 (2018) · Zbl 1440.94017 |
| [41] | Cheng, K.; Lu, Z.; Zhen, Y., Multi-level multi-fidelity sparse polynomial chaos expansion based on gaussian process regression, Comput. Methods Appl. Mech.Eng., 349, 360-377 (2019) · Zbl 1441.60003 |
| [42] | Abraham, S.; Tsirikoglou, P.; Miranda, J.; Lacor, C.; Contino, F.; Ghorbaniasl, G., Spectral representation of stochastic field data using sparse polynomial chaos expansions, J. Comput. Phys., 367, 109-120 (2018) · Zbl 1415.65025 |
| [43] | Berveiller, M.; Sudret, B.; Lemaire, M., Stochastic finite element: a non intrusive approach by regression, Eur. J. Comput. Mech., 15, 1-3, 81-92 (2006) · Zbl 1325.74171 |
| [44] | Hosder, S.; Walters, R. W.; Balch, M., Efficient sampling for non-intrusive polynomial chaos applications with multiple uncertain input variables, AIAA Paper, 1939, 2007 (2007) |
| [45] | Bratley, P.; Fox, B. L., Algorithm 659: Implementing sobol’s quasirandom sequence generator, ACM Trans. Math. Softw., 14, 1, 88-100 (1988) · Zbl 0642.65003 |
| [46] | Blatman, G., Adaptive Stochastic Finite Element Methods and Metamodels for Uncertainty Propagation (2007), PhD thesis, Université Blaise Pascal, Clermont-Ferrand, 2007. in preparation, Ph.D. thesis |
| [47] | Molinaro, A. M.; Simon, R.; Pfeiffer, R. M., Prediction error estimation: a comparison of resampling methods, Bioinformatics, 21, 15, 3301-3307 (2005) |
| [48] | Muthukrishnan, S., Data streams: algorithms and applications, Found. Trends® Theor. Comput. Sci., 1, 2, 117-236 (2005) · Zbl 1128.68025 |
| [49] | Eldar, Y. C.; Kutyniok, G., Compressed Sensing: Theory and Applications (2012), Cambridge university press |
| [50] | ICCFD8 · Zbl 1390.76801 |
| [51] | Witteveen, J.; Doostan, A.; Chantrasmi, T.; Pecnik, R.; Iaccarino, G., Comparison of stochastic collocation methods for uncertainty quantification of the transonic RAE 2822 airfoil, Proceedings of Workshop on Quantification of CFD Uncertainties (2009) |
| [52] | Wang, Q.; Chen, H.; Hu, R.; Constantine, P., Conditional sampling and experiment design for quantifying manufacturing error of transonic airfoil, 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 658 (2011) |
| [53] | Cook, P.; Firmin, M.; McDonald, M., Aerofoil RAE 2822: Pressure Distributions, and Boundary Layer and Wake Measurements (1977), RAE |
| [54] | L. Reid, R.D. Moore, Performance of single-stage axial-flow transonic compressor with rotor and stator aspect ratios of 1.19 and 1.26, respectively, and with design pressure ratio of 1.82(1978). |
| [55] | Simoes, M. R.; Montojos, B. G.; Moura, N. R.; Su, J., Validation of turbulence models for simulation of axial flow compressor, 20th International Congress of Mechanical Engineering (2009) |
| [56] | Dunham, J., CFD Validation for Propulsion System Components (la validation CFD des organes des propulseurs), Technical Report (1998), ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT NEUILLY-SUR-SEINE (FRANCE) |
| [57] | Wang, X.; Hirsch, C.; Liu, Z.; Kang, S.; Lacor, C., Uncertainty-based robust aerodynamic optimization of rotor blades, Int. J. Numer. Methods Eng., 94, 2, 111-127 (2013) · Zbl 1352.74232 |
| [58] | Loeven, A.; Bijl, H., The application of the probabilistic collocation method to a transonic axial flow compressor, 51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 18th AIAA/ASME/AHS Adaptive Structures Conference 12th, 2923 (2010) |
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