A method based on small amplitude homogenization for detecting defects using elastic waves

This work is concerned with the resolution of inverse problems for the detection of defects inside a medium using the propagation of elastic waves, under the assumption of small contrast on the value of the stiffness 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 framework developed for optimal design problems, we consider a formal 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 and through solving the wave equation in the time domain, 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 with certain defect distribution that is numerically solved and those values obtained from an assumption on the distribution of the defect. An adaptive procedure is presented for locating the wave source in order to improve the results.