# FEA-Based Ultrasonic Focusing Method in Anisotropic Media for Phased Array Systems
## 5 Implementation of TFM Imaging Algorithm in Anisotropic Material  
Using the wave propagation time database described in the previous section, the time delays can be determined in anisotropic media, after which the TFM image can be formed. This section describes the implementation of the TFM imaging algorithm for visualizing defects in austenitic welds. An FEA-based simulation was performed to calculate the FMC signal matrix, \(\mathbf{S}(t)\), which represents the full matrix of time-domain impulse response signals for all transmitting and receiving combinations. The finite element model described in Section 4 was used, but the model had a notch at \(x=5.25\), \(z=13\) mm with a width of 0.5 mm and a length of 4.5 mm, which corresponded to approximately 1.5 \(\lambda\), as shown in Figure 13. Each transmitting element was excited with the impulse defined in Section 4, and the response signals were calculated in the position of all the receiving elements. Thus, 256 time signals were generated for all combinations of transmitting and receiving elements (16 \(\times\) 16). Figure 14 shows an example of a set of 16 A-scan signals, which were calculated for all the receiving elements (elements 1-16) when element 1 was excited. All reflected signals did not arrive in phase because of the difference in propagation distance. Figure 11: Change of displacement field with increase in time. Figure 12: Definition of wave propagation time form an element of aperture to any scanning point. The TFM imaging algorithm is performed by first discretizing the target region (in the \(x\)-\(z\)-plane within the medium) into the grid. The FMC matrix \(\mathbf{S}(t)\) is delayed to produce a high intensity by aligning the reflected signals, and summed to synthesize a focus on every point in the grid. The intensity of the image \(I(x,z)\) at any target scanning point can be calculated using the propagation time database of the austenitic welds. Figure 15 shows the simulated A-scan signals at elements 1-16 under the excitation of elements 1-16, and the reflected signals from the notch. In addition, Figure 15 shows the delayed reflected signals from the center of the notch using the wave propagation database proposed in this study, using the basic traditional TFM method; the wave propagation time is also represented in the detailed figure. When the wave propagation time database was used to delay the A-scan signals, the signals arrived in phase at the target scanning point (notch center). However, when the wave propagation time calculated using the traditional TFM method was used, the A-scan signals were not aligned at the target point. This resulted in errors in the positioning of the defect. Figure 16 shows the method used to calculate the intensity using Equation (2) at the scanning point of the notch center. In the graph, the line indicates the _function_\(h\) in Equation (2), and \(t_{ij}\) is the wave propagation time from the transmitting element to the receiving element. Figure 17 shows the TFM imaging results for the scanning area (as shown in Figure 13) from the starting point of \(x=-10\), \(z=0\) mm to the ending point of \(x=10\), \(z=20\) mm. The imaging results of the basic TFM and the proposed TFM are displayed in Figure 16, in which the known locations of the defects are marked by red dotted lines. The proposed database-based method accurately estimated the defect position, but the traditional method incorrectly predicted the defect location by an error of approximately  
Figure 14: Simulated A-scan signals at element 1–16 under excitation of element 1. Figure 13: Simulation model with a notch. 3.2 mm. The error was caused by the wrong information of the wave propagation time, which was calculated using the basic TFM algorithm. It can be concluded from the comparisons in Figure 13 that the proposed TFM imaging results have a significant improvement in the performance of defect localization accuracy compared with basic TFM imaging results, and the usefulness of the proposed TFM imaging methodology in this study is validated.