# Non-Invasive Testing of Physical Systems Using Topological Sensitivity
## 4 Application to SHM Using Infrared Thermography  
Let us now consider the topological sensitivity approach for post-processing SHM data obtained via infrared (IR) active thermographic inspection. A topological sensitivity-based analysis using steady thermography was presented by us in Reference [81] (see Reference [82] for a preliminary theoretical analysis using steady thermograms, and also Reference [83] for an analysis using oscillatory, multifrequency, and thermograms) and had been already considered by other authors in References [84; 85]. Steady thermograms produced in three-dimensional media obviously provide less information than dynamic thermograms. However, steady thermography can be more convenient, relevant, and beneficial in certain applications, such as medical tests for tumor detection [86] and in other areas of medical diagnosis [87]. As a general description of the technique, active thermographic inspection of a structure aims at recovering degradations in the structure properties (i.e., localized degradations of the density and thermal conductivity) from thermograms giving the temperature distribution in a part of the structure boundary, as resulting from the illumination using one or various lamps. In comparison with guided waves testing, IR testing has the advantage that it is more non-contact and non-invasive. However, heat transport is short range, and the associated signal-to-noise ratio is less favorable than its counterpart in elastic/acoustic wave propagation. Concentrating, for simplicity in the exposition, on _steady thermographic inspection_ of a rectangular plate, the governing Equation (9) is the conveniently discretized _steady heat equation_ with appropriate boundary conditions. In this problem, the degradation vector \(\tilde{\mathbf{\alpha}}\) is the thermal conductivity function (depending on spatial variables) corresponding to the inhomogeneities of the metallic plate; the state vector \(\mathbf{u}\) is the temperature field at the illuminated face of the plate. Thus, the topological sensitivity of the cost function defined in (5) measures the sensitivity to perturbations of the thermal conductivity in the bulk of a homogeneous aluminum plate, which is otherwise assumed constant in the whole domain. This direct problem does not depend on the density and, thus, only permits localizing degradations in the thermal conductivity, which decreases the performance of the method. Results will be given on steady IR thermographic inspection using twelve lamps contained in a \(y-z\) plane, parallel to the cross section of the plate, in a Cartesian coordinate system (see Figure 3). These twelve lamps steadily illuminate, one after another (waiting for the plate reaching a steady state each time), on one side of the plate. Twelve thermograms capture the temperature distribution at the same side of the plate, after the steady state is reached each time. Thus, the number of available data is increased in a way similar to that developed in Reference [79]. The cross section of the plate is a \(35\times 15\) cm\({}^{2}\) rectangle, and its thickness equals 0.5 cm. A sketch of both the lamp positions and the plate with six small defects (representing no more than 0.1% of the total volume) is given in Figure 3.  
As can be seen, the lamps are located equispacedly in a plane parallel to the plate planform, and the defects are much smaller than the transversal section of the plate. In order to somehow mimic real experimental measurements, some random noise of size 0.05 K (which is larger than the expected error in experimental thermograms) has been added to the synthetic thermograms that are used here. The effect of the defects on the obtained thermograms is given in Figure 4.  
This figure shows that the raw thermograms do not capture at all the presence of the defects.