RB (Reduced basis) for RB (Rayleigh–Bénard)

Abstract The reduced basis approximation is a discretization method that can be implemented for solving parameter-dependent problems in cases of many queries. In this work it is applied to a two dimensional Rayleigh-Benard problem that depends on the Rayleigh number, which measures buoyancy. For each fixed aspect ratio, multiple steady solutions can be found for different Rayleigh numbers and stable solutions coexist at the same values of external physical parameters. The reduced basis method permits to speed up the computations of these solutions at any value of the Rayleigh number chosen in a fixed interval associated with a single bifurcation branch while maintaining accuracy.

[1]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[2]  J. Hesthaven,et al.  Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations , 2007 .

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

[4]  O. Lafitte,et al.  Theoretical and numerical study of a thermal convection problem with temperature-dependent viscosity in an infinite layer , 2010 .

[5]  Ahmed K. Noor,et al.  Bifurcation and post-buckling analysis of laminated composite plates via reduced basis technique , 1981 .

[6]  A. Patera,et al.  Certified real‐time solution of the parametrized steady incompressible Navier–Stokes equations: rigorous reduced‐basis a posteriori error bounds , 2005 .

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

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

[9]  M. C. Navarro,et al.  Building a reduced model for nonlinear dynamics in Rayleigh-Bénard convection with counter-rotating disks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[10]  Yvon Maday,et al.  A two-grid finite-element/reduced basis scheme for the approximation of the solution of parameter dependent PDE , 2009 .

[11]  Wolfgang Dahmen,et al.  Convergence Rates for Greedy Algorithms in Reduced Basis Methods , 2010, SIAM J. Math. Anal..

[12]  Ahmed K. Noor,et al.  Multiple‐parameter reduced basis technique for bifurcation and post‐buckling analyses of composite plates , 1983 .

[13]  D. Rovas,et al.  A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations , 2003 .

[14]  T. A. Porsching,et al.  The reduced basis method for initial value problems , 1987 .

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

[16]  Anthony T. Patera,et al.  Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations , 2002 .

[17]  H. Herrero A. M. Mancho.  On pressure boundary conditions for thermoconvective problems , 2001 .

[18]  R. A. Wentzell,et al.  Hydrodynamic and Hydromagnetic Stability. By S. CHANDRASEKHAR. Clarendon Press: Oxford University Press, 1961. 652 pp. £5. 5s. , 1962, Journal of Fluid Mechanics.

[19]  G. Reddien,et al.  On the reduced basis method , 1995 .

[20]  Ahmed K. Noor,et al.  Reduction methods for nonlinear steady‐state thermal analysis , 1984 .

[21]  Werner C. Rheinboldt,et al.  On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element Approximations , 1983 .

[22]  A. Patera,et al.  A PRIORI CONVERGENCE OF THE GREEDY ALGORITHM FOR THE PARAMETRIZED REDUCED BASIS METHOD , 2012 .

[23]  T. A. Porsching,et al.  Estimation of the error in the reduced basis method solution of nonlinear equations , 1985 .

[24]  Anthony T. Patera,et al.  A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations , 2002, J. Sci. Comput..

[25]  P. Stern,et al.  Automatic choice of global shape functions in structural analysis , 1978 .

[26]  Y. Maday,et al.  Une méthode combinée d'éléments finis à deux grilles/bases réduites pour l'approximation des solutions d'une E.D.P. paramétrique , 2009 .