# A Numerical Study on Computational Time Reversal for Structural Health Monitoring
## 5 Conclusions  
The time reversal method has been used for the study of structural health monitoring problems with beam and frame structures. The method is based on solid theoretical investigations in the area of wave propagation in continuous media and must be specialized for classes of structures. Beams and two-dimensional frames are studied here, since these are the most common elements in structural analysis. Future investigations may consider three-dimensional frame structures and plates or shells.  
TR is based on the time reversibility of the wave equation. In theory, if all output signals can be measured and fed, after reversion, onto the structure, the resulting wave will refocus on loading sources or damages and other defects. In practice, the measurement and utilization of all output signals is impossible due to technological and resource limitations. Therefore, the practical question arises where to put measurement devices, how many of them and which DOFs one should capture. These questions must be replied for every specific class of structures, like trusses, frames, plates etc. This justifies the numerical investigation reported in this paper. On top of that, the structure can not be isolated from the environment and parasitic reflections from boundaries or structural elements away from the area of investigation appear. These parasitic parts of data are characterized as noise. The influence of the signal to noise ratio on the effectiveness of the method was the second task of our investigation.  
The numerical experiments reported here have demonstrated that the source and damage identification is possible based on TR technique. The investigation has been based on unit values of elasticity and cross-section constants. The application of real data is straightforward.  
Figure 13: Imaging performed using Equation (21) for several source locations (red vertical line), each time using the same number of receivers (green vertical lines).  
The applicability of the method to a specific structure and task requires the careful consideration of several parameters indicated in the present study, such as the number and placement of the sensors and the quantities to be measured and used. Furthermore, a BEM time reversal implementation utilizing time reversed boundary conditions, which may be of interest for certain applications, is under consideration. For practical applications, signal and measurement errors as well as the existence of stochastically distributed material parameters, must be investigated. Furthermore, an experimental verification is planned. All these aspects are left for future research and development work.  
Methodology, C.G.P.; software, C.G.P.; validation, C.G.P.; formal analysis, C.G.P.; investigation, C.G.P.; writing--original draft preparation, C.G.P.; writing--review and editing, C.G.P. and G.E.S. All authors have read and agreed to the published version of the manuscript.  
C.G.P. acknowledges the financial support of the Stavros Niarchos Foundation within the framework of the project ARCHERS (\"Advancing Young Researchers' Human Capital in Cutting Edge Technologies in the Preservation of Cultural Heritage and the Tackling of Societal Challenges\"). Open access has been covered by research funds of the School of Production Engineering and Management, Technical University of Crete.  
Not applicable.  
Authors would like to acknowledge the assistance and contribution of graduate student [PERSON] as it is also captured in [41]. Most of the work has been completed during the affiliation of C.G.P. to the Institute of Applied & Computational Mathematics of the Foundation for Research and Technology--Hellas.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} SHM & Structural Health Monitoring \\ TR & Tine Reversal \\ DOF & Degree of Freedom \\ SNR & Signal to Noise Ratio \\ FEM & Finite Element Method \\ BEM & Boundary Element Method \\ \end{tabular}