Sensitivity analysis of transient non-linear heat conduction

This paper presents a general formulation of the continuous sensitivities for transient non-linear heat conduction problems. The development of general differential equations for the sensitivities is presented for the fully non-linear heat equation. Boundary conditions are developed for both shape and value parameters. For shape parameters, this requires the extraction of first and second derivatives of the temperature along solid surfaces, a very challenging problem. The temperature and sensitivity equations are solved by a finite element method. Grid refinement studies are used to control accuracy. The proposed methodology is verified on a problem with a closed form solution using the method of manufactured solutions. The verified code is then applied to identify physical properties from a single point measurement. We demonstrate the use of sensitivities for fast computation of nearby solutions. We also use sensitivity information for cascading input data uncertainty through the finite element solver to produce uncertainty estimate of the thermal response of the system. (author)