Comprehensive investigation of an inverse geometry problem in heat conduction via adjoint-based optimization method

Abstract

An inverse geometry problem in heat conduction is solved using different versions of an iterative regularization method. The algorithm consists of direct and inverse problems, which aims to modification of geometry. The direct problem is solved using a finite-element method. The employed iterative regularization method is constructed using the adjoint and sensitivity equations that are used to calculate the gradient of the objective function and the optimal step size, respectively. Results show that the Powel-Beale version has the best convergence rate compared to the Fletcher-Reeves and Polak-Ribiere versions of the conjugate gradient method. Effects of geometric parameters, location and number of sensors, heat flux value, error of sensors, and size of meshes are studied. Results show that as the sensors get closer to the unknown boundary, both accuracy and the convergence rate of the algorithm improve. Increasing the number of sensors has a positive effect on accuracy and the convergence rate, only when it is smaller than a certain number. The presence of a measurement error leads to inaccurate estimation of the geometry shape. A proper size of mesh has the best convergence and accuracy in the shape identification problem.