# Computational Vibro-Acoustic Time Reversal for Source and Novelty Localization
## 4 Computational Vibro-Acoustic Time Reversal  
In computational time reversal, at least one of the two steps (forward or backward), is supposed to be performed numerically by using some appropriate method, e.g., finite differences or a finite element method (FEM), as it is conducted in this work. Here, we use the conventional FEM, which is very common for the solution of structural dynamics problems. We present numerical experiments both for data acquisition during the forward step as well as for refocusing during the backward step. The methodology could be described according to the set-up presented in Figure 1. Consider a coupled vibro-acoustic system, which is here represented by an acoustic field incorporating a number of beams and frame structures. In the forward step, we assume that a source (acoustic or mechanic) acts on some point \(x_{s}\) and that the response \(u_{r}(t)\) is recorded on some point \(x_{r}\) for a time duration \(T\). It is possible to record both acoustic pressure and kinematic motion signals. Then, in the backward step, the time reversed signal response \(u_{r}(T-t)\) is re-emitted as excitation from the respective \(x_{r}\) point. In that backward step, an energy refocusing will appear on the \(x_{s}\) point after the same duration of time \(T\). This main application is called source localization. Furthermore, another application is that of novelty localization, which considers location \(x_{d}\) as the point at which a secondary source exists (novelty). In this case, we use the difference between the original signal and the one recorded after the novelty has been introduced. This can be expressed as:  
\[u_{r}^{n}=u_{r}^{t}-u_{r} \tag{7}\]  
In other words, \(u_{r}^{t}\) contains both primal and secondary source influence on the response, \(u_{r}\) contains only the influence of the original source, while \(u_{r}^{n}\) the influence of the novelty related field. Therefore, by re-emitting (in the backward step) the signal reversed \(u_{r}\) in time, one could locate the original source; by re-emitting \(u_{r}^{n}\), one could locate the secondary source due to novelty; and finally, by \(u_{r}^{t}\) re-emission, one could, possibly, locate both the original and secondary sources. Of crucial importance is the choice of quantity to be monitored in order to observe the refocusing during the backward step. That is more prominent in the case of different physics domain coupling, where variables with different scales may coexist. The refocusing  
Figure 2: Beam fully immersed in the fluid. The beam element has been implemented as two node straight line element. On the beam’s axis, it is assumed that there is continuity of pressure. at an appropriate discrete time [5] or using time averaging techniques [20] will indicate the spatial location of the original source excitation used during the forward step. Regarding the monitoring quantity, here we consider the pressure field, while some more general quantity, e.g., the Euclidean norm of all the components of existing degrees of freedom on each spatial position, could also be appropriate. For the source localization problem, we expect refocusing during the backward step, at the location of original source point at time \(T-t_{0}\), \(t_{0}\) being the peak of the pulse. The refocusing time in the novelty localization will be \(T-t_{0}-t_{1}\), where \(t_{1}\) is the time needed for the wave to propagate from the source point to the novelty location. Since the location of the novelty is the main unknown in our problem, the time \(t_{1}\) is also unknown. A solution to this problem is to use a time average approach [20], or to utilize the evolution of appropriate measures (e.g., Shannon entropy, Bounded Variation) [21]. A simple yet efficient approach for defining the refocusing time could be based on the arrival time of waves from the source to the sensor while interacting with the novelty location [22].