# Non-Invasive Testing of Physical Systems Using Topological Sensitivity
## 3 Application to SHM Using Guided Lamb Waves  
This section deals with the use of the topological sensitivity method to post-process data obtained from ultrasonic transducers (also used for exciting) or laser vibrometer sensors that may (and often) coincide in position with emitters for detecting the presence of defects in solid metals, which may consist of either holes/grooves or inclusions with changes in density or elastic properties [43; 44]. This application is particularly challenging because reflections of the guided waves in the boundaries or diffractions at defects make it very hard to use classical methods relying on the time/direction of flight of echo waves as the measurement principle. The applications below focused on thin plates so that the Lamb wave limit will apply and the system will be modeled as a two-dimensional elastodynamic problem (including variable thickness). However, the extension to the full three-dimensional problem (for items more complex than a thin plate) is not expected to present any new difficulties. In addition, the method can be straightforwardly extended to detect delaminations, kiss bonding phenomena, and defects in composite structures that would be otherwise very hard to detect with classical post-processing methods of NDT data. In this case, the sizes of the degradation vector, \(\tilde{\mathbf{\kappa}}\), and the state vector, \(\mathbf{u}\), are both extremely large, as they have a size proportional to the number of points used to describe the physical properties in both the healthy and damaged item (i.e., the desired resolution of the method). Thus, the classical methods are no longer viable to solve the inverse problem (to find the item physical properties that will produce the measured response in a reduced number of sensors); therefore, the adjoint formulation is needed. In this application, the physical properties that are degraded by defects are the density and Lame coefficients of the material (that are far different from that of the material if an air void appear f.i.). Two relevant and demanding configurations for metallic (thus, with isotropic properties items) thin plates are considered:  
* Rectangular plates with constant thickness, but exhibiting either (i) an _elongate through slit_ or (ii) an _elongated inclusion of a different material_, such as Titanium. Note that, in case (i), the through slit does not permit any transmission of the waves through it. Thus, the elongated slit cannot go from side to side transversally to the plate but should leave free portions of the plate at both sides of the slit. In case (ii), instead, the elongated inclusion permits partial transmission of the waves; thus, it can completely cover a section in the plate, from side to side. The elongated inclusion somewhat mimics the effect of _stringers_ in the outer surface of aircraft wings. This application of topological sensitivity was performed in Reference [43], where it was seen that the method identifies defects, even in cases in which all emitters and receivers are at one side of the elongated artifact, while the defects are located at the other side. * Non-rectangular plates, exhibiting a _complex platform_ or _variable thickness_. Difficulties for classical methods arise here from wave reflections in the boundaries and variable wave propagation speed (for guided waves) due to the variable thickness. This application was considered in Reference [44], where it was seen that, again, the method performs quite well. These complex configurations cannot be efficiently analyzed using flight time-based classical methods. This is because, in the above mentioned cases, the through slit, the inclusion, or the limits of the sample induce a sudden (even infinite) jump in the acoustic impedance, producing a partial or total reflection of the wave that masks the reflections coming from the defects behind. The two above mentioned configurations are addressed below, one after another.