Estimation of a Line Heat Source Using an Adjoint Free Gradient Based Inverse Analysis

An inverse analysis is presented to estimate line heat source in two-dimensional steady-state and transient heat transfer problems. A constant heat source is considered in the steady-state heat transfer problem (a parameter estimation problem) and a time-varying heat source is considered in the transient heat transfer problem (a function estimation problem). Since a general irregular 2D heat conducting body is considered, a body-fitted grid generation is used to mesh the domain. Then governing equations and associated boundary and initial conditions are transformed from the physical domain to the computational domain and finite difference method is used to solve the governing equations to obtain the temperature distribution in the body. Using an efficient, accurate, and very easy to implement sensitivity analysis incorporated in a gradient based minimization method (here, steepest descent method), the unknown heat source is estimated accurately. In the function estimation part, it is assumed that there is no prior information on the functional form of the heat source and the estimation process can be performed with a reasonable initial guess for the heat source. The main advantage of the proposed inverse analysis is that the sensitivity matrix (and hence, the objective function gradient with respect to the unknown variables) can be computed during the direct heat transfer solution through new yet simple explicit expressions with no need to solve extra equations such as the sensitivity and adjoint problems and impose additional computational costs comparable to the direct problem solution ones. Some test cases are presented to investigate the accuracy, efficiency, and effect of measurement error on the estimated parameter and function for the line heat source.