# Non-Invasive Testing of Physical Systems Using Topological Sensitivity
## 1 Introduction
In addition, in thin plates, the wave velocity in the case of guided waves depends on the plate thickness, which is an additional difficulty when using these methods for the inspection of plates with variable thickness. A different approach relies on the concept of _topological sensitivity_ that, as presented in this paper, is closely related to the so called _topological derivative_[15; 16], which has been extensively used to solve inverse problems in a variety of fields, including acoustics [4; 5; 17; 18; 19; 20; 21], elasticity [22; 23; 24; 25; 26; 27], photothermal problems [28; 29], electrical impedance tomography [30; 31; 32; 33; 34], electromagnetism [6; 7; 35; 36; 37; 38; 39], and holography [40; 41]; also see Reference [42] for additional related applications. In the context of SHM [43; 44], this method identifies defects as peaks of the distributed _sensitivity_ (computed using the full set of _elastodynamic equations_) of the system properties to possible degradations. In this sense, classical methods only use the (easy to compute in plates with constant thickness) propagation velocity of the waves, while the topological sensitivity approach uses the full physics and, in principle, should produce better results. The drawback of the latter approach, in connection with its practical implementation, is that the elastodynamic equations should be calibrated to robustly compute the response of the system. On the other hand, in the applications considered below, the topological sensitivity computation will rely on (numerically obtained) _synthetic data_, instead of on actual experimental data. This is a reasonable first step to test (and improve) the performance of the method in the considered fields. The performance of topological sensitivity based methods has been proven in the past initially against synthetic experiments, although also some works deal with real experimental data, showing very promising results too (see References [45; 46; 47; 48; 49; 50]). [PERSON] et al. [51] succeeded in finding the position of several defects in a thin plate with a single emitter/receiver with the line-of-view to the defects obstructed by a saw line. They used an approach equivalent to the one here described, but, instead of computing direct and adjoint problems from numerical models, they computed them after the measurement of impulse response in the healthy plate. Using actual experimental data in the approach here described is obviously the next step, which has been already performed via the topological derivative method in some, scientifically-oriented applications, and will be addressed in the near future for the more industrially-oriented cases considered in this review. Recently, an application of NDT ultrasound techniques [52] analysis has been developed for analysis of Shielded Metal Arc weldings. Other methods for solving inverse problems in NDT and SHM, which also take advantage of the full physics, can be classified into two groups: (a) imaging-based direct methods, like the linear sampling method, the factorization method, the MUSIC algorithm (see References [53; 54; 55; 56; 57; 58; 59; 60]), and (b) non-linear optimization iterative methods, like level sets or Gauss-Newton methods, see References [61; 62; 63; 64; 65; 66; 67]. Compared to these methods, the topological sensitivity based approach requires a rather smaller computational effort. On the other hand, besides full physics-based methods, pure data-driven strategies, specially those relying on deep learning tools, have recently attracted attention in the SHM community. Applications range from damage detection in aluminum beams [68], monitoring of aeronautical composite plates [69], impedance based damage detection [70], concrete crack detection [71], and pavement crack detection [72], to mention a few. For a very detailed state-of-the art of this kind of methods, we refer to the recent reviews [73; 74]. Deep learning tools, however, rely on the acquisition of large databases to train the models, which, in practice, could not be easily available.