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Bayesian uncertainty quantification of turbulence models based on high-order adjoint. (English) Zbl 1390.76113

Summary: The uncertainties in the parameters of turbulence models employed in computational fluid dynamics simulations are quantified using the Bayesian inference framework and analytical approximations. The posterior distribution of the parameters is approximated by a Gaussian distribution with the most probable value obtained by minimizing the objective function defined by the minus of the logarithm of the posterior distribution. The gradient and the Hessian of the objective function with respect to the parameters are computed using the direct differentiation and the adjoint approach to the flow equations including the turbulence model ones. The Hessian matrix is used both to compute the covariance matrix of the posterior distribution and to initialize the quasi-Newton optimization algorithm used to minimize the objective function. The propagation of uncertainties in output quantities of interest is also presented based on Laplace asymptotic approximations and the adjoint formulation. The proposed method is demonstrated using the Spalart-Allmaras turbulence model parameters in the case of the flat plate flow using DNS data for velocities and the flow through a backward facing step using experimental data for velocities and Reynolds stresses.

MSC:

76F55 Statistical turbulence modeling
62F15 Bayesian inference
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[1] Beck, J. L., Bayesian system identification based on probability logic, Struct Control Health Monit, 17, 7, 825-847, (2010)
[2] Yuen, K. V., Bayesian methods for structural dynamics and civil engineering, (2010), John Wiley & Sons
[3] Papadimitriou, C.; Katafygiotis, L. S., A Bayesian methodology for structural integrity and reliability assessment, Int J Adv Manufact Syst, 4, 1, 93-100, (2001)
[4] Soize, C., A computational inverse method for identification of non-gaussian random fields using the Bayesian approach in very high dimension, Comput Methods Appl Mech Eng, 200, 45-46, 3083-3099, (2011) · Zbl 1230.74241
[5] Beck, J. L.; Katafygiotis, L. S., Updating models and their uncertainties. I: Bayesian statistical framework, J Eng Mech, ASCE, 124, 4, 455-461, (1998)
[6] Angelikopoulos, P.; Papadimitriou, C.; Koumoutsakos, P., Bayesian uncertainty quantification and propagation in molecular dynamics simulations: a high performance computing framework, J Chem Phys, 137, 14, 103-144, (2012)
[7] Wang, J.; Zabaras, N., A Bayesian inference approach to the inverse heat conduction problem, Int J Heat Mass Transf, 47, 3927-3941, (2004) · Zbl 1070.80002
[8] Constantine, P. G.; Doostan, A.; Wang, Q.; Iaccarino, G., A surrogate accelerated Bayesian inverse analysis of the hyshot II flight data, AIAA paper 2011-2037, (2011)
[9] Jategaonkar, R.; Fischenberg, D.; Gruenhagen, W., Aerodynamic modeling and system identification from flight data-recent applications at DLR, J Aircraft, 41, 4, (2004)
[10] Oden, J. T.; Hawkins, A.; Prudhomme, S., General diffuse interface theories and an approach to predictive tumor growth modeling, Math Models Methods Appl Sci., 20, 3, 477-517, (2010) · Zbl 1186.92024
[11] Cheung, S. H.; Oliver, T. A.; Prudencio, E. E.; Prudhomme, S.; Moser, R. D., Bayesian uncertainty analysis with applications to turbulence modeling, Reliab Eng Syst Saf, 96, 1137-1149, (2011)
[12] Oliver, T. A.; Moser, R. D., Bayesian uncertainty quantification applied to RANS turbulence models, J Phys Conf Ser, 318, (2011)
[13] Edeling, W. N.; Cinnella, P.; Dwight, R. P.; Bijl, H., Bayesian estimates of parameter variability in the k-\(\epsilon\) turbulence model, J Comput Phys, 258, 73-94, (2014) · Zbl 1349.76110
[14] Arnst, M.; Ghanem, R.; Soize, C., Identification of Bayesian posteriors for coefficients of chaos expansions, J Comput Phys, 229, 9, 3134-3154, (2010) · Zbl 1184.62034
[15] Papadimitriou, C.; Beck, J. L.; Katafygiotis, L. S., Updating robust reliability using structural test data, Probab Eng Mech, 16, 2, 103-113, (2001)
[16] Beck, J. L.; Yuen, K. V., Model selection using response measurements: Bayesian probabilistic approach, J Eng Mech, ASCE, 130, 2, 192-203, (2004)
[17] Cheung, S. H.; Beck, J. L., Calculation of posterior probabilities for Bayesian model class assessment and averaging from posterior samples based on dynamic system data, J Comput-Aided Civil Infrastruct Eng, 25, 5, 304-321, (2010)
[18] Papadimitriou, C.; Katafygiotis, L. S., Bayesian modeling and updating, (Nikolaidis, N.; Ghiocel, D. M.; Singhal, S., Engineering design reliability handbook, (2004), CRC Press)
[19] Tierney, L.; Kadane, J. B., Accurate approximations for posterior moments and marginal densities, J Am Stat Assoc, 81, 393, 82-86, (1986) · Zbl 0587.62067
[20] Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E., Equation of state calculations by fast computing machines, J Chem Phys, 21, 6, 1087-1092, (1953)
[21] Beck, J. L.; Au, S. K., Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation, ASCE J Eng Mech, 128, 4, 380-391, (2002)
[22] Ching, J.; Chen, Y. C., Transitional Markov chain Monte Carlo method for Bayesian updating, model class selection, and model averaging, J Eng Mech, ASCE, 133, 816-832, (2007)
[23] Haario, H.; Laine, M.; Mira, A.; Saksman, E., DRAM: efficient adaptive MCMC, Stat Comput, 16, 339-354, (2006)
[24] Cheung, S. H.; Beck, J. L., Bayesian model updating using hybrid Monte Carlo simulation with application to structural dynamic models with many uncertain parameters, J Eng Mech, ASCE, 135, 4, 243-255, (2009)
[25] Margheri, L.; Meldi, M.; Salvetti, M. V.; Sagaut, P., Epistemic uncertainties in RANS model free coefficients, Comput Fluids, 102, 315-335, (2014) · Zbl 1391.76189
[26] Angelikopoulos, P.; Papadimitriou, C.; Koumoutsakos, P., X-TMCMC: adaptive Kriging for Bayesian inverse modeling, Comput Methods Appl Mech Eng, 289, 409-428, (2015)
[27] Lions, J. L., Optimal control of systems governed by partial differential equations, (1971), Springer-Verlag New York · Zbl 0203.09001
[28] Pironneau, O., On optimum design in fluid mechanics, J Fluid Mech, 64, 97-110, (1974) · Zbl 0281.76020
[29] Pironneau, O., Optimal shape design for elliptic systems, (1984), Springer-Verlag New York · Zbl 0496.93029
[30] Jameson, A., Aerodynamic design via control theory, J Sci Comput, 3, 233-260, (1988) · Zbl 0676.76055
[31] Jameson, A.; Reuther, J., Control theory based airfoil design using the Euler equations, AIAA paper 94-4272, (1994)
[32] Jameson, A., Optimum aerodynamic design using CFD and control theory, AIAA paper 95-1729, (1995)
[33] Jameson, A.; Pierce, N.; Martinelli, L., Optimum aerodynamic design using the Navier-Stokes equations, Theor Comput Fluid Dyn, 10, 213-237, (1998) · Zbl 0912.76067
[34] Burgreen, G. W.; Baysal, O., Three-dimensional aerodynamic shape optimization using discrete sensitivity analysis, AIAA J, 34, 9, 1761-1770, (1996) · Zbl 0909.76082
[35] Anderson, W. K.; Venkatakrishnan, V., Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, AIAA paper 97-0643, (1997)
[36] Duta, M. C.; Giles, M. B.; Campobasso, M. S., The harmonic adjoint approach to unsteady turbomachinery design, Int J Numer Methods Fluids, 40, 3-4, 323-332, (2002) · Zbl 1036.76056
[37] Hazra, S.; Schulz, V.; Brezillon, J.; Gauger, N., Aerodynamic shape optimization using simultaneous pseudo-timestepping, J Comput Phys, 204, 1, 46-64, (2005) · Zbl 1143.76564
[38] Papadimitriou, D. I.; Giannakoglou, K. C., A continuous adjoint method with objective function derivatives based on boundary integrals for inviscid and viscous flows, Comput Fluids, 36, 325-341, (2007) · Zbl 1177.76369
[39] Sherman, L. L.; Taylor, A. C.; Green, L. L.; Newman, P. A.; Hou, J. W.; Korivi, V. M., First- and second-order aerodynamic sensitivity derivatives via automatic differentiation with incremental iterative methods, J Comput Phys, 129, 307-331, (1996) · Zbl 0933.76070
[40] Papadimitriou, D. I.; Giannakoglou, K. C., Direct, adjoint and mixed approaches for the computation of Hessian in airfoil design problems, Int J Numer Methods Fluids, 56, 1929-1943, (2008) · Zbl 1141.76058
[41] Zervogiannis, T.; Papadimitriou, D. I.; Giannakoglou, K. C., Total pressure losses minimization in turbomachinery cascades using the exact Hessian, J Comput Methods Appl Mech Eng, 199, 2697-2708, (2010) · Zbl 1231.76247
[42] Papadimitriou, D. I.; Giannakoglou, K. C., Computation of the Hessian matrix in aerodynamic inverse design using continuous adjoint formulations, Comput Fluids, 37, 1029-1039, (2008) · Zbl 1237.76162
[43] Papadimitriou, D. I.; Giannakoglou, K. C., Aerodynamic shape optimization using first and second order adjoint and direct approaches, Archiv Comput Methods Eng, (State Art Rev), 15, 4, 447-488, (2008) · Zbl 1170.76348
[44] Papadimitriou, D. I.; Giannakoglou, K. C., The continuous direct-adjoint approach for second order sensitivities in viscous aerodynamic inverse design problems, Comput Fluids, 38, 1539-1548, (2009) · Zbl 1242.76300
[45] 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, Int J Numer Methods Fluids, 69, 691-709, (2012)
[46] Papadimitriou, D. I.; Giannakoglou, K. C., Third-order sensitivity analysis for robust aerodynamic design using continuous adjoint, Int J Numer Methods Fluids, 71, 5, 652-670, (2013)
[47] Tortorelli, D.; Michaleris, P., Design sensitivity analysis: overview and review, Inverse Probl Eng, 1, 71-105, (1994)
[48] Hou, G. W.; Sheen, J., Numerical methods for second-order shape sensitivity analysis with applications to heat conduction problems, Int J Numer Methods Eng, 36, 417-435, (1993) · Zbl 0770.73086
[49] Dimet, F. X.L.; Navon, I. M.; Daescu, D. N., Second-order information in data assimilation, Mon Weather Rev, 130, 3, 629-648, (2002)
[50] Turgeon, E.; Pelletier, D.; Borggaard, J.; Etienne, S., Application of a sensitivity equation method to the k-\(\epsilon\) model of turbulence, Optimiz Eng, 8, 341-372, (2007) · Zbl 1421.76126
[51] Caro, R.; Hay, A.; Etienne, S.; Pelletier, D., Application of a shape sensitivity equation method to turbulent flow over obstacles, AIAA paper 2007-4207, (2007)
[52] Colin, E.; Etienne, S.; Pelletier, D.; Borggaard, J., Application of a sensitivity equation method to turbulent flows with heat transfer, Int J Thermal Sci, 44, 1024-1038, (2005)
[53] Lee, B. J.; Kim, C., Automated design methodology of turbulent internal flow using discrete adjoint formulation, Aerospace Sci Technol, 11, 163-173, (2007) · Zbl 1195.76348
[54] Zymaris, A. S.; Papadimitriou, D. I.; Giannakoglou, K. C.; Othmer, C., Continuous adjoint approach to the Spalart\(-\)Allmaras turbulence model for incompressible flows, Comput Fluids, 38, 1528-1538, (2009) · Zbl 1242.76064
[55] Zymaris, A. S.; Papadimitriou, D. I.; Giannakoglou, K. C.; Othmer, C., Adjoint wall functions: A new concept for use in aerodynamic shape optimization, J Comput Phys, 229, 5228-5245, (2010) · Zbl 1346.76059
[56] Emory, M.; Pecnik, R.; Iaccarino, G., Modeling structural uncertainties in Reynolds-averaged computations of shock/boundary layer interactions, AIAA paper 2011-479, (2011)
[57] Dow, E.; Wang, Q., Quantification of structural uncertainties in the k-omega turbulence model, AIAA paper 2011-1762, (2011)
[58] Kennedy, M. C.; O’Hagan, A., Bayesian calibration of computer models, J R Stat Soc Ser B (Stat Methodol), 63, 3, 425-464, (2001) · Zbl 1007.62021
[59] Peter, J. E.V.; Dwight, R. P., Numerical sensitivity analysis for aerodynamic optimization: A survey of approaches and applications, Comput Fluids, 39, 3, 373-391, (2010) · Zbl 1242.76301
[60] Bui-Thanh, T.; Ghattas, O.; Higdon, D., Adaptive hessian-based non-stationary gaussian process response surface method for probability density approximation with application to Bayesian solution of large-scale inverse problems, ICES REPORT 11-32, (2011)
[61] Merle, X.; Cinnella, P., Bayesian quantification of thermodynamic uncertainties in dense gas flows, Reliab Eng Syst Saf, 134, 305-323, (2015)
[62] Papadimitriou, C.; Beck, J. L.; Katafygiotis, L. S., Asymptotic expansions for reliability and moments of uncertain systems, Journal of Engineering Mechanics, ASCE, 123, 12, 1219-1229, (1997)
[63] Spalart, P.; Allmaras, S., A one-equation turbulence model for aerodynamic flows, AIAA paper, 92-0439, (1992)
[64] Ghate, D. P.; Giles, M. B., Efficient Hessian calculation using automatic differentiation, 25th AIAA applied aerodynamics conference, (2007)
[65] Pini, M.; Cinnella, P., Hybrid adjoint-based robust optimization approach for fluid-dynamics problems, AIAA paper 2013-1814, 15th non-deterministic approaches conference, (2013)
[66] Edeling, W. N.; Cinnella, P.; Dwight, R. P., Predictive RANS simulations via Bayesian model-scenario averaging, J Comput Phys, 275, 65-91, (2014) · Zbl 1349.76106
[67] Simens, M. P.; Jimenez, J.; Hoyas, S.; Mizuno, Y., A high-resolution code for turbulent boundary layers, J Comput Phys, 228, 11, 4218-4231, (2009) · Zbl 1273.76009
[68] Borrell, G.; Sillero, J. A.; Jimenez, J., A code for direct numerical simulation of turbulent boundary layers at high Reynolds numbers in BG/P supercomputers, Comput Fluids, 80, 37-43, (2013) · Zbl 1284.76007
[69] Driver, D. M.; Seegmiller, H. L., Features of reattaching turbulent shear layer in divergent channel flow, AIAA J, 23, 2, 163-171, (1985)
[70] Papadimitriou, C.; Lombaert, G., The effect of prediction error correlation on optimal sensor placement in structural dynamics, Mech Syst Signal Process, 28, 105-127, (2012)
[71] Bertsekas, D. P., Nonlinear programming, (1999), Athena Scientific · Zbl 1015.90077
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