Development and Implementation of Sensitivity Coefficient Equations for Heat Conduction Problems

Methods are discussed for computing the sensitivity of the temperature field to changes in material properties and initial boundary condition parameters for heat conduction problems. The most general method is to derive sensitivity equations by differentiating the energy equation with respect to the parameter of interest and solving the resulting sensitivity equations numerically. An example problem in which there are 12 parameters of interest is presented and the resulting sensitivity equations and associated boundary initial conditions are derived. The sensitivity equations are implemented in a general-purpose unstructured-grid control-volume finite-element code. Numerical results are presented for thermal conductivity and volumetric heat capacity sensitivity coefficients for heat conduction in a 2-D orthotropic body. The numerical results are compared with the analytical solution to demonstrate that the numerical sensitivity method is second-order accurate as the mesh is refined spatially.