REDUCED BASIS METHODS AND A POSTERIORI ERROR ESTIMATORS FOR HEAT TRANSFER PROBLEMS

This paper focuses on the parametric study of steady and unsteady forced and natural convection problems by the certified reduced basis method. These problems are characterized by an input-output relationship in which given an input parameter vector — material properties, boundary conditions and sources, and geometry — we would like to compute certain outputs of engineering interest — heat fluxes and average temperatures. The certified reduced basis method provides both (i) a very inexpensive yet accurate output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization. The

[1]  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..

[2]  A. Quarteroni,et al.  Numerical solution of parametrized Navier–Stokes equations by reduced basis methods , 2007 .

[3]  Simone Deparis,et al.  Reduced Basis Error Bound Computation of Parameter-Dependent Navier-Stokes Equations by the Natural Norm Approach , 2008, SIAM J. Numer. Anal..

[4]  Nguyen Ngoc Cuong,et al.  Certified Real-Time Solution of Parametrized Partial Differential Equations , 2005 .

[5]  I. Kevrekidis,et al.  Low‐dimensional models for complex geometry flows: Application to grooved channels and circular cylinders , 1991 .

[6]  Pinhas Z. Bar-Yoseph,et al.  Stability of multiple steady states of convection in laterally heated cavities , 1999, Journal of Fluid Mechanics.

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

[8]  Janet S. Peterson,et al.  The Reduced Basis Method for Incompressible Viscous Flow Calculations , 1989 .

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

[10]  Vedat S. Arpaci,et al.  Conduction Heat Transfer , 2002 .

[11]  Einar M. Rønquist,et al.  Reduced-basis modeling of turbulent plane channel flow , 2006 .

[12]  G. Rozza,et al.  On the stability of the reduced basis method for Stokes equations in parametrized domains , 2007 .

[13]  M. Gunzburger,et al.  Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data , 2007 .

[14]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[15]  K. ITOy.  REDUCED BASIS METHOD FOR OPTIMAL CONTROL OF UNSTEADY VISCOUS FLOWS , 2006 .

[16]  W. E. Schiesser,et al.  Computational Transport Phenomena: Numerical Methods for the Solution of Transport Problems , 1997 .

[17]  Gianluigi Rozza,et al.  Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation , 2010 .

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

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

[20]  Bernard Haasdonk,et al.  Reduced Basis Method for Finite Volume Approximations of Parametrized Evolution Equations , 2006 .

[21]  B. Roux.  Numerical simulation of oscillatory convection in low-Pr fluids : a GAMM Workshop , 1990 .

[22]  N. Nguyen,et al.  EFFICIENT REDUCED-BASIS TREATMENT OF NONAFFINE AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 2007 .

[23]  Anthony T. Patera,et al.  "Natural norm" a posteriori error estimators for reduced basis approximations , 2006, J. Comput. Phys..

[24]  Sidney Yip,et al.  Handbook of Materials Modeling , 2005 .

[25]  K. N. Seetharamu,et al.  Convection Heat Transfer , 2005 .

[26]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

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

[28]  Gianluigi Rozza,et al.  Real-Time Reliable Simulation of Heat Transfer Phenomena , 2009 .

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

[30]  N. Nguyen,et al.  REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS , 2011 .

[31]  A. Patera,et al.  A Successive Constraint Linear Optimization Method for Lower Bounds of Parametric Coercivity and Inf-Sup Stability Constants , 2007 .

[32]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[33]  Gianluigi Rozza,et al.  Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation , 2009 .

[34]  A. Patera,et al.  A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations , 2005 .

[35]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[36]  B. Haasdonk,et al.  REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATIONS OF PARAMETRIZED LINEAR EVOLUTION EQUATIONS , 2008 .

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