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Nonlinear model reduction for the Navier-Stokes equations using residual DEIM method. (English) Zbl 1349.76288

Summary: This article presents a new reduced order model based upon proper orthogonal decomposition (POD) for solving the Navier-Stokes equations. The novelty of the method lies in its treatment of the equation’s non-linear operator, for which a new method is proposed that provides accurate simulations within an efficient framework. The method itself is a hybrid of two existing approaches, namely the quadratic expansion method and the Discrete Empirical Interpolation Method (DEIM), that have already been developed to treat non-linear operators within reduced order models. The method proposed applies the quadratic expansion to provide a first approximation of the non-linear operator, and DEIM is then used as a corrector to improve its representation. In addition to the treatment of the non-linear operator the POD model is stabilized using a Petrov-Galerkin method. This adds artificial dissipation to the solution of the reduced order model which is necessary to avoid spurious oscillations and unstable solutions. A demonstration of the capabilities of this new approach is provided by solving the incompressible Navier-Stokes equations for simulating a flow past a cylinder and gyre problems.comparisons are made with other treatments of non-linear operators, and these show the new method to provide significant improvements in the solution’s accuracy.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
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[1] Fukunaga, K., Introduction to Statistical Recognition, Computer Science and Scientific Computing Series, 5-33 (1990), Academic Press: Academic Press Boston, MA
[2] Pearson, K., On lines and planes of closest fit to systems of points in space, Philos. Mag., 2, 559-572 (1901) · JFM 32.0246.07
[3] Crommelin, D. T.; Majda, A. J., Strategies for model reduction: Comparing different optimal bases, J. Atmos. Sci., 61, 2206-2217 (2004)
[4] Jolliffe, I. T., Principal Component Analysis, 559-572 (2002), Springer · Zbl 1011.62064
[5] Cao, Y.; Zhu, J.; Navon, I. M.; Luo, Z., A reduced order approach to four dimensional variational data assimilation using proper orthogonal decomposition, Int. J. Numer. Methods Fluids, 53, 1571-1583 (2007) · Zbl 1370.86002
[6] Vermeulen, P. T.M.; Heemink, A. W., Model-reduced variational data assimilation, Mon. Weather Rev., 134, 10, 2888-2899 (2006)
[7] Daescu, D. N.; Navon, I. M., A dual-weighted approach to order reduction in 4d-var data assimilation, Mon. Weather Rev., 136, 3, 1026-1041 (2008)
[8] Chen, X.; Fang, F.; Navon, I. M., A dual weighted trust-region adaptive POD 4D-Var applied to a finite-volume shallow-water equations model, Int. J. Numer. Methods Fluids, 65, 520-541 (2011) · Zbl 1428.76145
[9] Chen, X.; Akella, S.; Navon, I. M., A dual weighted trust-region adaptive POD 4D-Var applied to a finite-volume shallow-water equations model on the sphere, Int. J. Numer. Methods Fluids, 68, 377-402 (2012) · Zbl 1426.76347
[10] Altaf, M. U., Model reduced variational data assimilation for shallow water flow models (2011), Delft University of Technology, PhD thesis
[11] Du, J.; Fang, F.; Pain, C. C.; Navon, I. M.; Zhu, J.; Ham, D. A., POD reduced-order unstructured mesh modeling applied to 2D and 3D fluid flow, Comput. Math. Appl., 65, 362-379 (2013) · Zbl 1319.76026
[12] Fang, F.; Pain, C. C.; Navon, I. M.; ElSheikh, A. H.; Du, J.; Xiao, D., Non-linear Petrov-Galerkin methods for reduced order hyperbolic equations and discontinuous finite element methods, J. Comput. Phys., 234, 540-559 (2013) · Zbl 1284.65132
[13] Pain, C. C.; Piggott, M. D.; Goddard, A. J.H., Three-dimensional unstructured mesh ocean modelling, Ocean Model., 10, 5-33 (2005)
[14] Nguyen, N. C.; Peraire, J., An efficient reduced-order modeling approach for non-linear parametrized partial differential equations, Int. J. Numer. Methods Eng., 76, 27-55 (2008) · Zbl 1162.65407
[15] 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. Acad. Sci. Paris, Ser., 339, 667-672 (2004) · Zbl 1061.65118
[16] Stefanescu, R.; Navon, I. M., POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model, J. Comput. Phys., 237, 95-114 (2013) · Zbl 1286.76106
[17] Chaturantabut, S., Dimension reduction for unsteady nonlinear partial differential equations via empirical interpolation methods (2008), Rice University, Master’s thesis
[18] Chaturantabut, S.; Sorensen, D. C., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 2737-2764 (2010) · Zbl 1217.65169
[19] Chaturantabut, S.; Sorensen, D. C., A state space error estimate for POD-DEIM nonlinear model reduction, SIAM J. Numer. Anal., 50, 46-63 (2012) · Zbl 1237.93035
[20] Rozza, G.; Huynh, D. B.P.; Patera, A. T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations - application to transport and continuum mechanics, Arch. Comput. Methods Eng., 15, 3, 229-275 (2008) · Zbl 1304.65251
[21] Nguyen, N. C.; Rozza, G.; Patera, A. T., Reduced basis approximation and a posteriori error estimation for the time-dependent viscous burgers equation, Calcolo, 46, 3, 157-185 (2009) · Zbl 1178.65109
[22] Boyaval, S.; Le Bris, C.; Lelièvre, T.; Maday, Y.; Nguyen, N. C.; Patera, A. T., Reduced basis techniques for stochastic problems, Arch. Comput. Methods Eng., 17, 4, 435-454 (2010) · Zbl 1269.65005
[23] Eftang, J. L.; Knezevic, D. J.; Patera, A. T., An hp certified reduced basis method for parametrized parabolic partial differential equations, Math. Comput. Model. Dyn. Syst., 17, 4, 395-422 (2011) · Zbl 1302.65223
[24] Rozza, Gianluigi; Manzoni, Andrea, Model order reduction by geometrical parametrization for shape optimization in computational fluid dynamics, (Proceedings of ECCOMAS CFD (2010))
[25] Rozza, G.; Manzoni, A.; Quarteroni, A., Shape optimization for viscous flows by reduced basis methods and free-form deformation, Int. J. Numer. Methods Fluids, 70, 5, 646-670 (2012) · Zbl 1412.76031
[26] Negri, F.; Rozza, G.; Manzoni, A.; Quarteroni, A., Reduced basis method for parametrized elliptic optimal control problems, SIAM J. Sci. Comput., 35, 5, A2316-A2340 (2013) · Zbl 1280.49046
[27] Everson, R.; Sirovich, L., The Karhunen-Loeve procedure for gappy data, J. Opt. Soc. Am., 12, 1657-1664 (1995)
[28] Willcox, Karen, Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition, Comput. Fluids, 35, 2, 208-226 (2006) · Zbl 1160.76394
[29] Bou-Mosleh, C.; Carlberg, K.; Farhat, C., Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Int. J. Numer. Methods Eng., 86, 155-181 (2011) · Zbl 1235.74351
[30] Xiao, D.; Fang, F.; Du, J.; Pain, C. C.; Navon, I. M.; Buchan, A. G.; ElSheikh, A. H.; Hu, G., Non-linear Petrov-Galerkin methods for reduced order modelling of the Navier-Stokes equations using a mixed finite element pair, Comput. Methods Appl. Mech. Eng., 255, 147-157 (2013) · Zbl 1297.76107
[31] Baiges, J.; Codina, R.; Idelsohn, S., Explicit reduced-order models for the stabilized finite element approximation of the incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 72, 12, 1219-1243 (2013) · Zbl 1455.76078
[32] Aubry, N.; Holmes, P.; Lumley, J. L., The dynamics of coherent structures in the wall region of a turbulent boundary layer, J. Fluid Mech., 192, 1, 115-173 (1988) · Zbl 0643.76066
[33] Chaturantabut, S., Nonlinear model reduction via discrete empirical interpolation (2011), Rice University, PhD thesis
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