# FEA-Based Ultrasonic Focusing Method in Anisotropic Media for Phased Array Systems
## 3 Measurement of Grain Orientation Distribution and Anisotropic Elastic Constants
### Anisotropic Elastic Constants  
In the austenitic weldment, the material properties along the welding pass (\(y\)-axis) were assumed to be isotropic; however, in the other directions, to be anisotropic [9; 10]. The material behavior can be properly simulated well with a transversely isotropic material, for which the elastic constants are expressed as follows:  
\[[C]=\left[\begin{array}{cccccc}C_{11}&C_{12}&C_{13}&&0&0&0\\ C_{12}&C_{11}&C_{13}&&0&0&0\\ C_{13}&C_{13}&C_{33}&&0&0&0\\ 0&0&0&&C_{44}&0&0\\ 0&0&0&&0&&C_{44}&0\\ 0&0&0&&0&&\frac{1}{2}(C_{44}-C_{12})\end{array}\right] \tag{3}\]  
The prediction of the elastic constants can be formulated as an inverse problem in which the objective is to find an optimal set of the elastic constants with which wave propagation behavior in austenitic welds can be simulated with minimal error. In general,  
Figure 3: Grain orientations extracted from macrograph in weld region.  
the behavior is measured using a set of sensors attached on the welds. The inverse problems can be solved by a trial-and-error method: guessing the unknown information, solving the forward problem, and then updating the guessed information in the forward data [19; 20; 21]. A user-defined error function describing the discrepancy between the measured wave velocity and the predicted one is minimized during the process.  
The objective function is defined as the difference between the measured and calculated wave velocities, and the design variable is the five elastic constants in the transversely isotropic materials, \(C_{11}\), \(C_{12}\), \(C_{13}\), \(C_{33}\), and \(C_{44}\). The details of the FEA are described in Section 4.  
Figure 5 shows the schematic for the wave propagation test setup for the measurement of anisotropic elastic constants in the welding and buttering parts. A transmitter was placed on the top surface of the welds, and three receivers were attached on the bottom surface. The input load represented in Figure 6 was excited by a transmitter, of which the center frequency was 2.25 MHz, after which the response was stored by the three receivers at a sampling rate of 100 MHz. Figure 7 represents the stored signal by the receivers.  
Figure 5: Position of a transmitter and three receivers for wave propagation response.  
Figure 6: Excited signal from transmitter.  
The test and simulation signals were filtered to extract the time signals containing an interesting frequency, after which the wave velocity could be well observed. The signals were used to optimize the elastic constants by minimizing the difference between the wave velocities of the test and the simulation. Figure 8, calculated using the optimized elastic constants, shows the normalized filtering signals with a 2 nd order band-pass filter for measured signals and simulated signals, of which the upper and lower limits were 2.0 and 2.5 MHz, respectively. As shown in Figure 8, which demonstrates the response on the basement, weld, and buttering regions, the simulated wave propagation time from the transmitter to the receivers showed a good agreement with the tested one; the differences between simulated wave propagation times from the transmitter to receivers 1, 2, and 3, compared with the tested one, were \(-11\), 1, and 0 usec, respectively. The density of the welding and buttering parts was assumed to be 8190 kg/m\({}^{3}\), and the elastic constants of the welding and buttering parts were optimized as follows:  
\[\left[\begin{array}{cccccc}226.5&131.0&115.6&&&0&0&0\\ 131.0&226.5&115.6&&&0&0&0\\ 115.6&115.6&250.0&&&0&0&0\\ 0&0&0&&&111.0&&&0\\ 0&0&0&&&0&&&47.8\\ \end{array}\right]\]  
\[\left[\begin{array}{cccccc}202.7&119.4&115.6&&&0&0&0\\ 119.4&202.7&115.6&&&0&0&0\\ 115.6&115.6&246.0&&&0&0&0\\ 0&0&0&&&101.8&&&0\\ 0&0&0&&&101.8&&&0\\ 0&0&0&&&41.4\\ \end{array}\right]\]  
Figure 7: Wave propagation response measured on bottom surface.  
Figure 8: Comparison between measured and simulated signals.