A flexible symmetry-preserving Galerkin/POD reduced order model applied to a convective instability problem

Abstract A flexible Galerkin method based on proper orthogonal decomposition (POD) is described to construct the bifurcation diagram, as the Rayleigh number R is varied, in the Rayleigh–Benard convection in a rectangular box for large Prandtl number. The bifurcation diagram is approximated using the POD modes resulting from unconverged snapshots for just one specific value of R, calculated in either Newton iterations or time-dependent runs converging to steady states. Moreover, the selection of the specific value of R is quite flexible. In addition, a horizontal reflection symmetry is taken into account to construct a symmetry-preserving Galerkin system. The resulting un-symmetric and symmetric low-dimensional systems are combined with a basic continuation method, which provide the bifurcation diagram at a quite low computational cost.

[1]  Jeffrey P. Thomas,et al.  Using Automatic Differentiation to Create a Nonlinear Reduced-Order-Model Aerodynamic Solver , 2010 .

[2]  Dan S. Henningson,et al.  Model Reduction of the Nonlinear Complex Ginzburg-Landau Equation , 2010, SIAM J. Appl. Dyn. Syst..

[4]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

[5]  E. Knobloch,et al.  Symmetry and Symmetry-Breaking Bifurcations in Fluid Dynamics , 1991 .

[6]  Roger Temam,et al.  Inertial Manifolds for the Kuramoto-Sivashinsky Equation and an Estimate of their Lowest Dimension , 1986 .

[7]  F. Richter,et al.  Heat transfer and horizontally averaged temperature of convection with large viscosity variations , 1983, Journal of Fluid Mechanics.

[8]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[9]  Juan J. Alonso,et al.  Investigation of non-linear projection for POD based reduced order models for Aerodynamics , 2001 .

[10]  P. Sagaut,et al.  Towards an adaptive POD/SVD surrogate model for aeronautic design , 2011 .

[11]  P. Beran,et al.  Reduced-order modeling: new approaches for computational physics , 2004 .

[12]  José M. Vega,et al.  Construction of Bifurcation Diagrams Using POD on the Fly , 2014, SIAM J. Appl. Dyn. Syst..

[13]  Martin Golubitsky,et al.  Symmetries and pattern selection in Rayleigh-Bénard convection , 1984 .

[14]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[15]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[16]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[17]  José M. Vega,et al.  Reduced order models based on local POD plus Galerkin projection , 2010, J. Comput. Phys..

[18]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[19]  Roger Temam,et al.  Localization and approximation of attractors for the Ginzburg-Landau equation , 1991 .

[20]  C. Bernardi,et al.  Approximations spectrales de problèmes aux limites elliptiques , 2003 .

[21]  Yvon Maday,et al.  RB (Reduced basis) for RB (Rayleigh–Bénard) , 2013 .

[22]  Earl H. Dowell,et al.  Modeling of Fluid-Structure Interaction , 2001 .

[23]  D. Rovas,et al.  Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods , 2002 .

[24]  Kelly Cohen,et al.  Low-dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition , 2008, Journal of Fluid Mechanics.

[25]  Gianluigi Rozza,et al.  Reduced Order Methods for Modeling and Computational Reduction , 2013 .

[26]  P. Sagaut,et al.  Calibrated reduced-order POD-Galerkin system for fluid flow modelling , 2005 .

[27]  O. Zienkiewicz.  The Finite Element Method In Engineering Science , 1971 .

[28]  Ulrich R. Christensen,et al.  Dynamic Earth: Plates, Plumes and Mantle Convection , 2000 .

[29]  Stefan Volkwein,et al.  Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics , 2002, SIAM J. Numer. Anal..

[30]  Gianluigi Rozza,et al.  Model Order Reduction in Fluid Dynamics: Challenges and Perspectives , 2014 .

[31]  J. Vega,et al.  Reduced-Order Modeling of Three-Dimensional External Aerodynamic Flows , 2012 .

[32]  G. Sell,et al.  Inertial manifolds for nonlinear evolutionary equations , 1988 .

[33]  Henri Bénard,et al.  Les tourbillons cellulaires dans une nappe liquide. - Méthodes optiques d'observation et d'enregistrement , 1901 .

[34]  Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio , 2008, 0810.3799.

[35]  I. Kevrekidis,et al.  Equation-free/Galerkin-free POD-assisted computation of incompressible flows , 2005 .

[36]  Andrea Manzoni,et al.  An efficient computational framework for reduced basis approximation and a posteriori error estimation of parametrized Navier–Stokes flows , 2014 .

[37]  José M. Vega,et al.  On the use of POD-based ROMs to analyze bifurcations in some dissipative systems , 2012 .

[38]  M. Jolly.  Bifurcation computations on an approximate inertial manifold for the 2D Navier-Stokes equations , 1993 .

[39]  I. Aranson,et al.  The world of the complex Ginzburg-Landau equation , 2001, cond-mat/0106115.

[40]  David Ryckelynck.  Hyper‐reduction of mechanical models involving internal variables , 2009 .

[41]  M. Haragus,et al.  Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems , 2010 .

[42]  James C. Robinson.  Computing inertial manifolds , 2002 .

[43]  Nadine Aubry,et al.  Preserving Symmetries in the Proper Orthogonal Decomposition , 1993, SIAM J. Sci. Comput..

[44]  Danny C. Sorensen,et al.  A Symmetry Preserving Singular Value Decomposition , 2006, SIAM J. Matrix Anal. Appl..

[45]  Danny C. Sorensen,et al.  Best Nonspherical Symmetric Low Rank Approximation , 2009, SIAM J. Matrix Anal. Appl..

[46]  Gianluigi Rozza,et al.  Reduced basis method for multi-parameter-dependent steady Navier-Stokes equations: Applications to natural convection in a cavity , 2009, J. Comput. Phys..

[47]  D. Rempfer.  LOW-DIMENSIONAL MODELING AND NUMERICAL SIMULATION OF TRANSITION IN SIMPLE SHEAR FLOWS , 2003 .

[48]  Charbel Farhat,et al.  Reduced-order fluid/structure modeling of a complete aircraft configuration , 2006 .

[49]  A. Velazquez,et al.  Reduced-Order Model for Viscous Aerodynamic Flow Past an Airfoil , 2010 .

[50]  Eusebio Valero,et al.  Local POD Plus Galerkin Projection in the Unsteady Lid-Driven Cavity Problem , 2011, SIAM J. Sci. Comput..

[51]  A. Velazquez,et al.  A method to generate computationally efficient reduced order models , 2009 .

[52]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[53]  G. Karniadakis,et al.  Stability and accuracy of periodic flow solutions obtained by a POD-penalty method , 2005 .