An application of conjugate gradient technique for determination of thermal conductivity as an inverse engineering problem

Abstract

Inverse problems frequently occur when boundary conditions/thermal properties of a material are unknown, and are determined from known internal temperature measurements. A direct problem is the opposite of an inverse problem. For example, it would be the task of determining the temperature field within a material body, given boundary conditions and thermal properties. In an ill-posed, inverse problem, small variations in internal temperature measurement cause large variations in the estimated quantity such as boundary heat flux. Measured temperature variations may be due to noise, measurement error and instrumentation error, among other factors. There are various methods for solving inverse problems in materials engineering. In the paper, an inverse problem of estimating boundary heat flux for one-dimensional heat conduction is solved using the classical method of Beck and temperature dependent thermal conductivity is estimated using the more recent and advanced Conjugate Gradient method (CGM). The temperature values from the finite difference application to a direct heat transfer problem are used along with noise added to the data, in order to simulate real world, experimental conditions. The data is applied to the inverse problem of finding boundary heat flux and thermal conductivity. It is seen that Conjugate Gradient Method (CGM) along with sequential function specification method recovers the boundary heat flux and thermal conductivity quite accurately for noise levels of 1% and 2%. It is further observed that addition of future time steps stabilize the algorithm and increase accuracy of the desired results. It is also observed that CGM can recover the temperature dependent thermal conductivity quickly within few iterations, for different values of thermal conductivity function coefficients, k0 and k1. CGM is found to be fast and robust when applied to one-dimensional case and can be extended to complex geometries.