A method based on non-steady heat diffusion problems for detecting the location of inclusions

Abstract

This work is concerned with the resolution of inverse problems for the detection of defects inside a homogeneous medium using non-steady heat diffusion problem, under the assumption of small contrast on the value of the conductivity coefficient between the matrix material and that of the defect. This is the so-called small amplitude, small contrast or small aspect ratio assumption. Following the idea developed by Allaire and Gutiérrez for optimal design problems, we develop a second-order asymptotic expansion with respect to the aspect ratio, which allows us to simplify the inverse problem, considering it as an optimization problem. According to this, we can develop a gradient-type algorithm, that reduces, in the time interval being considered, the difference between the boundary values obtained from a problem that is numerically solved with full knowledge of the defect distribution and boundary values obtained from solving another problem based on an assumption on the distribution of the defect. In general, by the use of non-steady problems, we can obtain substantially better information of the defects location compared to using steady problems.