Thin plate approximation in active infrared thermography

In this work, we find and test a new approximated formula (based on the thin plate approximation), for recovering small, unknown damages on the inaccessible surface of a thin conducting (aluminium) plate. We solve this inverse problem from a controlled heat flux and a sequence of temperature maps on the accessible front boundary of our sample. We heat the front boundary by means of a sinusoidal flux. In the meanwhile, we take a sequence of temperature maps of the same side by means of an infrared camera. This procedure is called active infrared thermography. The solution of the heat equation on the accessible boundary of the damaged sample simulates the collection of data. We use domain derivative to linearize the boundary value problem for heat equation. Then, Fourier analysis on the periodic component of solutions leads us to an elliptic BVP. Finally, we apply perturbation theory in order to find out an approximation of the damage. Numerical tests obtained with synthetic data are encouraging. The solution of the heat equation on the accessible boundary of the damaged sample simulates the collection of data by means of the infrared camera. We use domain derivative to linearize the BVP for heat equation. Then, Fourier analysis on the periodic component of solutions leads us to an elliptic BVP. Finally, we apply the perturbation theory in (Formula presented.) (a is the thickness of our sample) in order to find out an approximation of the damage. Numerical tests obtained with synthetic data are encouraging.