Reconstitution of a non uniform interface thermal resistance by inverse conduction

Abstract

Diffusive heat transfer problem within a non isotropic stratified structure, containing a plane defect of non uniform resistance, is solved analytically by the method of integral transforms. The idea consists in applying Fourier cosine transforms on the space variables and a Laplace transform on the time variable. The thermal quadrupole formalism allows to reduce the mathematical model to a product of matrices in the transformed space. The direct modeling, within the framework of a 2D geometry, has been followed by the construction of an inverse procedure that reconstitutes the variation of the thermal resistance from the measure of the surface temperature. But in the presence of noise, this solution becomes unstable because the problem is ill-posed. To avoid an unstable solution, a method of regularization is necessary. It consists in filtering the data to make our inverse problem well-posed. The inversion can then be undertaken in an explicit way either in the transformed space, or in the real space. The established technique has been completed by a method of constrained optimization, in order to guarantee the positivity of the solution. The developed inverse codes have been validated by noised numerical simulations and by a NDT (non destructive testing) operation by stimulated infrared thermography on a bonding defect between two PVC plates.