Real-Time Reliable Simulation of Heat Transfer Phenomena

In this paper we discuss the application of the certified reduced basis method and the associated software package rbMIT c to “worked problems” in steady and unsteady conduction. Each worked problem is characterized by an input parameter vector — material properties, boundary conditions and sources, and geometry — and desired outputs — selected fluxes and temperatures. The methodology and associated rbMIT c software, as well as the educational worked problem framework, consists of two distinct stages: an Offline (or “Instructor”) stage in which a new heat transfer worked problem is first created; and an Online (or “Lecturer”/“Student”) stage in which the worked problem is subsequently invoked in (say) various in‐ class, project, or homework settings. In the very inexpensive Online stage, given an input parameter value, the software returns both (i) an accurate reduced basis output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element dis

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

[2]  Ganesh Rajagopalan,et al.  Development of Hands-On CFD Educational Inter face for Under gr aduate Engineer ing Cour ses and Labor ator ies , 2004 .

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

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

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

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

[7]  Robert J. Ribando,et al.  Using Software With Visualization To Teach Heat Transfer Concepts , 2002 .

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

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

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

[11]  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..

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

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

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

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

[17]  A. M. Khounsary.  Heat transfer education : Keeping it relevant and vibrant. , 1998 .

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

[19]  M. A. Starnes.  Development of technical bases for using infrared thermography for nondestructive evaluation of fiber reinforced polymer composites bonded to concrete , 2002 .

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

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

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

[23]  Robert J. Ribando,et al.  A “Hands-On” Approach to Teaching Undergraduate Heat Transfer , 2004 .