# Non-Invasive Testing of Physical Systems Using Topological Sensitivity
## 4 Application to SHM Using Infrared Thermography
The application of the topological sensitivity method, instead, produces topological sensitivities that do localize the position of the defects, as seen in Figure 5.  
Figure 4: The thermograms obtained for two positions of the lamp (left and middle plots), as obtained in the clean plate (without defects) and in the plate with internal defects. The difference between both thermograms is also given in the right plots. In each plot, the active lamp is represented by an orange circle. Figure 3: Sketch showing the lamp positions and the plate (in which scale along the \(x\) axis in the plate is enlarged to facilitate visualization, since the plate is really thin), with six small internal defects (in green) included. The volume of these defects represent no more than 0.1% of the total volume of the plate. Where the sections of the three-dimensional topological sensitivity with three representative planes are given, note that the method identifies the transversal position of the defects quite well, in contrast to what happened with the thermograms plotted in Figure 4. It is important to note that the topological sensitivity is computed from just the solution of the unperturbed direct and adjoint problems. The information about the defects characteristics is only introduced through the measured thermograms. Due to the weaker influence of depth in the topological sensitivity as compared with that of position or size the depth of the defects, the depth of the defects is not well identified by the topological sensitivity. In addition, treating steady thermograms exhibits two drawbacks in connection with its practical use:  
* Precise modeling of the process, which is needed to solve the direct and adjoint problems, is problematic, specially in connection with modeling the lamps energy deposition. * Experimental steady thermograms are difficult to obtain, specially due to the large thermal relaxation time needed to reach each steady state. It is because of the second difficulty that thermographic inspection in SHM is usually performed experimentally in highly unsteady conditions. Specifically, unsteady thermograms are recorded after illuminating one side of the plate by a quite short (of the order of ms) flash from the lamp. In this case, the thermal relaxation process towards the ambient temperature is recorded by the thermographic camera. In other words, this relaxation process can be modeled by the _unsteady, unforced, homogeneous heat equation_ with initial conditions corresponding to the thermal state of the plate just after the flash. Moreover, in such thermal relaxation process, the instantaneous temperature in the unsteady termograms (computed at the plane illuminated by the lamp flash) can be written as  
\[\theta(y,z,t)=\sum_{n=1}^{N}\Theta_{n}(y,z)\,\mathrm{e}^{-\delta_{n}t}, \tag{22}\]  
where \(\Theta_{n}(y,z)\) correspond to _decaying natural modes_ of the homogeneous heat equation, and \(\delta_{n}\) are the associated _damping rates_. Moreover, the modes and damping rates can be identified using a convenient data processing tool to the decaying thermograms. A good candidate tool to process these data is the recently developed _higher order dynamic mode decomposition_[88] (also see Reference [89] for a reader-friendly presentation of the method and its multiple applications), which is a very robust extension of standard dynamic mode decomposition [90, 91]. Once the modes and damping rates have been identified, these permit applying a topological sensitivity approach in which the direct problem is robustly modeled by a Helmholtz-like equation associated with the homogeneous heat equation that applies here; the adjoint problem is modeled by the same Helmholtz-like equation, forced by the obtained modes \(\Theta_{n}\), displayed in (22). This approach for solving the problem is in progress; thus, it is well beyond the scope of this review. Figure 5: Color maps for the sections of the topological sensitivity with the planes \(x=0\) (**left**), \(x=0.25\) (**middle**), and \(x=0.5\) (**right**).