# Non-Invasive Testing of Physical Systems Using Topological Sensitivity
## 3 Application to SHM Using Guided Lamb Waves
In both cases, the set of emitters and sensors are either non-contact, optical laser devices [80], or contact piezo-electric, which may act as both emitters and receivers that operate as follows. Each time one of the devices acts as emitter, producing guided waves, the remaining devices act as receivers, sensing the incoming waves. A few dual devices will be used. These devices are located very close to the boundary of the plate planform, which makes the analysis more demanding but will use the possible (future) practical implementation of the method. In fact, locating these devices near the boundary highly worsens the performance of flight time-based classical methods. This is because the relevant outgoing (from emitters) and incoming (at the receivers) wave packets mix with their counterparts reflected at the nearby boundary, which distorts the relevant wave packets shape and, what is even worse, increases their thickness. Note that this decreases the accuracy in the computation of the flight time. The topological sensitivity will not be computed in the whole plate but only in a slightly reduced in size _interrogation window_. This window is centered in the plate and excludes both a vicinity of the lateral boundary of the plate and the dual emitting/sensing devices themselves, to avoid misleading artifacts in the sensitivity that will appear due to the wave reflection at the lateral wall and the large values of the sensitivity near the emitting devices. On the other hand, _synthetic experimental data_ are used (obtained using an FEM solver), instead of actual experimental data. However, these synthetic data are calculated by solving the elastodynamic equations in a mesh that concentrates near the defects, while the mesh used in the computation of the direct and adjoint problems (needed to solve the inverse problem) is more equispaced, since the position and size of the defects are not known in the application of the method. In order to increase the amount of data collected in the sensors, the strategy developed in Reference [79] is used, in which the topological sensitivity-based method uses data obtained by considering a set of monochromatic excitations, with several (of the order of several tens) forcing frequencies of various amplitudes in the emitters. Beginning with the case of a rectangular plate of constant thickness, but containing elongated slits or inclusions of a different material, two representative results, obtained upon application of the topological sensitivity method based on two-dimensional elastodynamics, are displayed in Figure 1, where the elongated artifacts, the emitters/receivers, and the defects are displayed. As can be seen in this figure, the dual emitting/sensing devices are located very close to the boundary of the plate planform, at the upper side of the elongated slit (which leaves a lateral portion of the plate free to allow wave propagation, as anticipated) or inclusion of a different material (which goes from side to side of the plate, as also anticipated), while the defect is located at the lower side. In spite of all these challenges, the method identifies the position of the defect fairly well, using 30 frequencies in the case with an elongated slit and 25 frequencies in the case of an elongated Titanium inclusion. Turning now to the case in which the plate exhibits a complex planform or variable thickness, the outcome of the topological sensitivity method is given in Figure 2, where two representative cases are addressed, one with constant thickness but complex planform (which is addressed using two-dimensional elastodynamics) and the other with rectangular planform but variable thickness (which requires using three-dimensional elastodynamics). The complex planform configuration considered in the left plot contains a through slit to make the identification of the defect more demanding. The variable thickness plate, considered in the right plot, is thicker in the middle and thinner in the upper and lower sides of the plate.