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 discretization; as required in the educational context, the response is both rapid \emph{and} reliable. We present illustrative results for two worked problems: a steady thermal fin, and unsteady thermal analysis of a delamination crack.