zbMATH — the first resource for mathematics

A zonal Galerkin-free POD model for incompressible flows. (English) Zbl 1380.65239
Summary: A domain decomposition method which couples a high and a low-fidelity model is proposed to reduce the computational cost of a flow simulation. This approach requires to solve the high-fidelity model in a small portion of the computational domain while the external field is described by a Galerkin-free proper orthogonal decomposition (POD) model. We propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models. This zonal approach can be used to study multi-body configurations or to perform detailed local analyses in the framework of shape optimisation problems. The efficiency of the method to perform predictive low-cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem.

65M55 Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Poinsot, T. J.; Lelef, S., Boundary conditions for direct simulations of compressible viscous flows, J. Comput. Phys., 101, 1, 104-129, (1992) · Zbl 0766.76084
[2] Jin, G.; Braza, M., A nonreflecting outlet boundary condition for incompressible unsteady Navier-Stokes calculations, J. Comput. Phys., 107, 2, 239-253, (1993) · Zbl 0777.76072
[3] Blazek, J., Computational fluid dynamics: principles and applications, (2015), Butterworth-Heinemann · Zbl 1308.76001
[4] Thomas, J. L.; Salas, M., Far-field boundary conditions for transonic lifting solutions to the Euler equations, AIAA J., 24, 7, 1074-1080, (1986)
[5] Gallizio, F.; Iollo, A.; Protas, B.; Zannetti, L., On continuation of inviscid vortex patches, Phys. D: Nonlinear Phenom., 239, 3, 190-201, (2010) · Zbl 1263.76029
[6] Coloşi, T.; Abrudean, M.-I.; Unguresan, M.-L.; Muresan, V., Numerical simulation of distributed parameter processes, (2013), Springer · Zbl 1284.65003
[7] Turner, M. G.; Reed, J. A.; Ryder, R.; Veres, J. P., Multi-fidelity simulation of a turbofan engine with results zoomed into mini-maps for a zero-D cycle simulation, (ASME Turbo Expo 2004: Power for Land, Sea, and Air, (2004), American Society of Mechanical Engineers), 219-230
[8] Quarteroni, A., Modeling the heart and the circulatory system, vol. 14, (2015), Springer
[9] Sirovich, L., Turbulence and the dynamics of coherent structures part I: coherent structures, Q. Appl. Math., 45, 3, 561-571, (1987) · Zbl 0676.76047
[10] Willcox, K.; Peraire, J., Balanced model reduction via the proper orthogonal decomposition, AIAA J., 40, 11, 2323-2330, (2002)
[11] Rowley, C. W.; Colonius, T.; Murray, R. M., Model reduction for compressible flows using pod and Galerkin projection, Phys. D: Nonlinear Phenom., 189, 1, 115-129, (2004) · Zbl 1098.76602
[12] Noack, B. R.; Papas, P.; Monkewitz, P. A., The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows, J. Fluid Mech., 523, 339-365, (2005) · Zbl 1065.76102
[13] Rowley, C. W., Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurc. Chaos, 15, 03, 997-1013, (2005) · Zbl 1140.76443
[14] Sirisup, S.; Karniadakis, G. E.; Xiu, D.; Kevrekidis, I. G., Equation-free/Galerkin-free pod-assisted computation of incompressible flows, J. Comput. Phys., 207, 2, 568-587, (2005) · Zbl 1213.76146
[15] Ilak, M.; Rowley, C. W., Modeling of transitional channel flow using balanced proper orthogonal decomposition, Phys. Fluids, 20, 3, (2008) · Zbl 1182.76341
[16] Buffoni, M.; Willcox, K., Projection-based model reduction for reacting flows, (40th Fluid Dynamics Conference and Exhibit, (2010)), 5008
[17] Caiazzo, A.; Iliescu, T.; John, V.; Schyschlowa, S., A numerical investigation of velocity-pressure reduced order models for incompressible flows, J. Comput. Phys., 259, 598-616, (2014) · Zbl 1349.76050
[18] Chinesta, F.; Leygue, A.; Bordeu, F.; Aguado, J. V.; Cueto, E.; González, D.; Alfaro, I.; Ammar, A.; Huerta, A., PGD-based computational vademecum for efficient design, optimization and control, Arch. Comput. Methods Eng., 20, 1, 31-59, (2013) · Zbl 1354.65100
[19] Quarteroni, A.; Rozza, G.; Manzoni, A., Certified reduced basis approximation for parametrized partial differential equations and applications, J. Math. Ind., 1, 1, (2011) · Zbl 1273.65148
[20] Quarteroni, A.; Rozza, G., Reduced order methods for modeling and computational reduction, vol. 9, (2014), Springer
[21] Schmid, P. J., Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656, 5-28, (2010) · Zbl 1197.76091
[22] Barrault, M.; Maday, Y.; Nguyen, N. C.; Patera, A. T., An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math., 339, 9, 667-672, (2004) · Zbl 1061.65118
[23] Chaturantabut, S.; Sorensen, D. C., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 5, 2737-2764, (2010) · Zbl 1217.65169
[24] Perotto, S.; Reali, A.; Rusconi, P.; Veneziani, A., Higamod: a hierarchical isogeometric approach for model reduction in curved pipes, Comput. Fluids, 142, 21-29, (2017) · Zbl 1390.76347
[25] Bergmann, M.; Bruneau, C.-H.; Iollo, A., Enablers for robust POD models, J. Comput. Phys., 228, 2, 516-538, (2009) · Zbl 1409.76099
[26] Noack, B. R.; Morzynski, M.; Tadmor, G., Reduced-order modelling for flow control, vol. 528, (2011), Springer Science & Business Media
[27] Weller, J.; Lombardi, E.; Iollo, A., Robust model identification of actuated vortex wakes, Phys. D: Nonlinear Phenom., 238, 4, 416-427, (2009) · Zbl 1157.93012
[28] Weller, J.; Camarri, S.; Iollo, A., Feedback control by low-order modelling of the laminar flow past a bluff body, J. Fluid Mech., 634, 405-418, (2009) · Zbl 1183.76704
[29] Wang, Z.; Akhtar, I.; Borggaard, J.; Iliescu, T., Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison, Comput. Methods Appl. Mech. Eng., 237, 10-26, (2012) · Zbl 1253.76050
[30] Weller, J.; Lombardi, E.; Bergmann, M.; Iollo, A., Numerical methods for low-order modeling of fluid flows based on POD, Int. J. Numer. Methods Fluids, 63, 2, 249-268, (2010) · Zbl 1423.76356
[31] Östh, J.; Noack, B.; Krajnovic, S.; Barros, D.; Borée, J., On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an ahmed body, J. Fluid Mech., 747, 518-544, (2014) · Zbl 1371.76085
[32] Baiges, J.; Codina, R.; Idelsohn, S., Reduced-order subscales for POD models, Comput. Methods Appl. Mech. Eng., 291, 173-196, (2015)
[33] Giere, S.; Iliescu, T.; John, V.; Wells, D., SUPG reduced order models for convection-dominated convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Eng., 289, 454-474, (2015)
[34] Carlberg, K.; Bou-Mosleh, C.; Farhat, C., Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Int. J. Numer. Methods Eng., 86, 2, 155-181, (2011) · Zbl 1235.74351
[35] Amsallem, D.; Zahr, M. J.; Farhat, C., Nonlinear model order reduction based on local reduced-order bases, Int. J. Numer. Methods Eng., 92, 10, 891-916, (2012) · Zbl 1352.65212
[36] Zhan, Z.; Habashi, W. G.; Fossati, M., Local reduced-order modeling and iterative sampling for parametric analyses of aero-icing problems, AIAA J., 53, 8, 2174-2185, (2015)
[37] Buffoni, M.; Telib, H.; Iollo, A., Iterative methods for model reduction by domain decomposition, Comput. Fluids, 38, 6, 1160-1167, (2009) · Zbl 1242.76231
[38] Spalart, P.; Allmaras, S., A one-equation turbulence model for aerodynamic flows, (30th Aerospace Sciences Meeting and Exhibit, (1992)), 439
[39] Jasak, H.; Jemcov, A.; Tukovic, Z., Openfoam: a C++ library for complex physics simulations, (International Workshop on Coupled Methods in Numerical Dynamics, vol. 1000, (2007), IUC Dubrovnik, Croatia), 1-20
[40] Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math. Comput., 22, 104, 745-762, (1968) · Zbl 0198.50103
[41] Mittal, R.; Dong, H.; Bozkurttas, M.; Najjar, F.; Vargas, A.; von Loebbecke, A., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J. Comput. Phys., 227, 10, 4825-4852, (2008) · Zbl 1388.76263
[42] Ghias, R.; Mittal, R.; Dong, H., A sharp interface immersed boundary method for compressible viscous flows, J. Comput. Phys., 225, 1, 528-553, (2007) · Zbl 1343.76043
[43] Bergmann, M.; Hovnanian, J.; Iollo, A., An accurate Cartesian method for incompressible flows with moving boundaries, Commun. Comput. Phys., 15, 05, 1266-1290, (2014) · Zbl 1373.76156
[44] Patankar, S. V., Numerical heat transfer and fluid flow, (1980), CRC Press · Zbl 0521.76003
[45] Jasak, H., Error analysis and estimation for the finite volume method with applications to fluid flows, (1996), Department of Mechanical Engineering - Imperial College, Ph.D. thesis
[46] Vasile, M.; Minisci, E.; Quagliarella, D.; Guénot, M.; Lepot, I.; Sainvitu, C.; Goblet, J.; Coelho, R. Filomeno, Adaptive sampling strategies for non-intrusive POD-based surrogates, Eng. Comput., 30, 4, 521-547, (2013)
[47] Braconnier, T.; Ferrier, M.; Jouhaud, J.-C.; Montagnac, M.; Sagaut, P., Towards an adaptive POD/SVD surrogate model for aeronautic design, Comput. Fluids, 40, 1, 195-209, (2011) · Zbl 1245.76063
[48] Carlberg, K.; Farhat, C., A compact proper orthogonal decomposition basis for optimization-oriented reduced-order models, (12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, (2008)), 5964
[49] Bergmann, M.; Colin, T.; Iollo, A.; Lombardi, D.; Saut, O.; Telib, H., Reduced order models at work in aeronautics and medicine, (Reduced Order Methods for Modeling and Computational Reduction, (2014), Springer), 305-332 · Zbl 1395.65113
[50] Noack, B. R.; Eckelmann, H., A global stability analysis of the steady and periodic cylinder wake, J. Fluid Mech., 270, 297-330, (1994) · Zbl 0813.76025
[51] Noack, B. R.; Eckelmann, H., A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder, Phys. Fluids, 6, 1, 124-143, (1994) · Zbl 0826.76071
[52] Barbagallo, A.; Sipp, D.; Schmid, P. J., Closed-loop control of an open cavity flow using reduced-order models, J. Fluid Mech., 641, 1-50, (2009) · Zbl 1183.76701
[53] Bagheri, S.; Schlatter, P.; Schmid, P. J.; Henningson, D. S., Global stability of a jet in crossflow, J. Fluid Mech., 624, 33-44, (2009) · Zbl 1171.76372
[54] Geisser, S., Predictive inference, vol. 55, (1993), CRC Press
[55] Kurtulus, D. F., On the unsteady behavior of the flow around NACA 0012 airfoil with steady external conditions at re = 1000, Int. J Micro Air Veh., 7, 3, 301-326, (2015)
[56] Kelleners, P.; Heinrich, R., Simulation of interaction of aircraft with gust and resolved LES-simulated atmospheric turbulence, (Advances in Simulation of Wing and Nacelle Stall, (2016), Springer), 203-221
[57] Sederberg, T. W.; Parry, S. R., Free-form deformation of solid geometric models, ACM SIGGRAPH Comput. Graph., 20, 4, 151-160, (1986)
[58] Pygem: python geometrical morpher
[59] Jones, D. R.; Schonlau, M.; Welch, W. J., Efficient global optimization of expensive black-box functions, J. Glob. Optim., 13, 4, 455-492, (1998) · Zbl 0917.90270
[60] Adams, B.; Bauman, L.; Bohnhoff, W.; Dalbey, K.; Ebeida, M.; Eddy, J.; Eldred, M.; Hough, P.; Hu, K.; Jakeman, J.; Stephens, J.; Swiler, L.; Vigil, D.; Wildey, T., Dakota, A multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: version 6.0 User’s manual, (2014), Sandia National Laboratories, Tech. Rep. SAND2014-4633
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.