# A Numerical Study on Computational Time Reversal for Structural Health Monitoring
## 4 Applications with Numerical Examples
### Damage Identification
#### 4.4.2 Two-Dimensional Frame Structure  
Some preliminary results from an ongoing study are presented for the case of some scatter, i.e., damage, in a two-dimensional frame. To be more specific, damage is assumed to be located at some small area ranging 10 nodes (nine finite elements), located from \((15.0,4.5)\) to \((15.0,5.0)\) (see Figure 5). Moreover, the mechanical features of this area is  
Figure 10: Refocusing for damage detection using the displacements (m), the absolute value of axial response in the left column and the norm for the transverse and rotational DOFs in the right column of the scattered field. The green line is the location of the receiver while the red line is that of the source and the grey line indicates the damage.  
the material's modulus of elasticity \(E_{sc}=\frac{E}{5}=0.2\) and the cross-section \(A_{sc}=\frac{A}{2}=0.5\). These values result an important reduction in wave velocities within the damaged area. The time for which the wave reaches the damaged area is estimated to be between 655 and 670 time steps. As a result, because of time reversibility, the expected discrete refocusing time range turns out from the subtraction of \(N_{t}=2^{11}\) resulting a time step range between 1378 and 1393. In Figure 11 (left) it can be observed that in the discrete imaging time step 1373, there is a bright red colour near the scattered area. In Figure 11 (right), the evolution in time of the maximum value of the monitored variable is plotted, which may serve as an indicator for defining the appropriate time step for the reconstruction of refocusing. It was mentioned here that one should further critically consider both the selection of the most appropriate monitored variable (displacement, velocity, energy based, etc.) as well as the most proper tracker of that imaging variable's norm evolution in time (maximum value of Euclidean norm, Shannon entropy, etc.).  
Figure 12 presents the SNR for 25 random receiver sets. It is worth mentioning that SNR has an almost linear attitude both for displacements or velocities as monitored variables. However, there is a considerable focused signal in the damage area when more than eight or nine receivers are used. As a result, here the ratio is always calculated using the value of the monitored variable in the damage area divided by the maximal value outside this area. Figure 12 (right) indicates that nine receivers are required in order to identify the defect. For this reason, if less than nine receivers are used, the SNR is lower than 1.  
Figure 11: Imaging (**a**) using the total number of possible receivers and choosing as the imaging variable the norm of displacements (**m**); the intersection of the two horizontal and one vertical grey line show the location of scattered field. (**b**) plot, evolution in time (s) of the maximum value of the norm of displacements (**m**).  
Figure 12: SNR for various sets of receivers using the norm of displacements (**a**) and velocities (**b**). On the horizontal axis, the number of receivers is indicated.