# FEA-Based Ultrasonic Focusing Method in Anisotropic Media for Phased Array Systems
## 4 Construction of Wave Propagation Time Database  
It is impossible to determine the time delays in the austenitic welds using the basic TFM algorithm because the ultrasonic waves in the welds are distorted. This section describes the methodology used to construct the wave propagation time database, \(S\), which consists of three parameters: coordinates of elements within aperture (\(x_{i}\), \(z_{i}\)) or (\(x_{j}\), \(z_{j}\)), coordinates of any target scanning points (\(x,z\)), and the wave propagation time from the elements to scanning points, \(t_{i}\) or \(t_{j}\). The wave propagation time can be calculated through FEA for a sufficiently rich set of phased array element positions on the surface of the DMW specimen, after which the wave propagation time can be extracted at the scanning points. The elements in the database are the coordinates of the phased array elements, scanning points, and the corresponding wave propagation time. The database can be expressed as follows:  
\[S=\left\{\left(x_{i},\,z_{i}\right)\,or\,\left(x_{j},\,z_{j}\right),\,\left(x, \,z\right),\,t_{i}\,or\,t_{j}\,x_{i},x_{j},x\in X,z_{i},z_{j},\,\,z\in Z\,\right\} \tag{4}\]  
The wave propagation behavior in the austenitic welds was calculated using elastic FEA. The FEA was performed using the implicit solver in ABAQUS [22]. Figure 3a depicts the schematic of the DMW specimen used in the simulation. Within the \(x\)-\(y\)-plane, the medium appears isotropic; however, in other directions outside the plane, the medium exhibits anisotropic features. The FE model was assumed to have two dimensions, meaning that energy only propagated in the \(x\)-\(z\) plane. The mesh (Figure 9) was constructed using eight-node reduced integration plane strain elements, CPE8R in ABAQUS. The finite element size was 0.25 mm, to ensure that there would be at least seven nodes per wavelength in the spatial domain [13]. The simulation method using FEA was used to verify the longitudinal velocity in the transversely isotropic medium without changing the grain orientation. The longitudinal wave velocity in the \(x\)- and \(z\)-directions was calculated as \(V_{xx}=\sqrt{C_{11}/\rho}\) and \(V_{zz}=\sqrt{C_{33}/\rho}\), obtained from the Christoffel equation for transversely isotropic media. In a previous study [5], the velocities calculated by FEA were well matched with the theoretical solution within 1%. The measured grain orientation distribution described in Section 3 was used as the simulation input describing the grain structures in the austenitic welds, as shown in Figure 9.  
Figure 9: FE mesh for calculating wave propagation time in austenitic welds and schematic representation for phased array element position.  
The array parameters with a linear array transducer with equispaced elements are presented in Table 1. The transmitted load from the phased array elements was simulated as the pressure at which the signal was modeled in the form of a sine function, \(A_{0}/2\cdot(\sin(2\pi\overline{\psi}(t-\lambda/4))+1)\) for \((0\leq\mathrm{t}\leq\lambda)\), as shown in Figure 10. \(A_{0}\) is the amplitude of the input load, \(f\) is the center frequency, and \(\lambda\) is the wavelength in the time domain. The acceleration data were stored at a sampling rate of 200 MHz.  
Figure 11 shows the change in the displacement field with time after the first phased array element located at \(x=-5.75\), \(y=0.0\) was excited. Instead of pure longitudinal and shear waves, quasi-longitudinal and quasi-shear waves were generated in the anisotropic medium, as shown in Figure 11. Figure 12 shows the displacement as time increased at the points presented in Figure 11a (6.0, 16.0), and the method used to identify the wave propagation time. In the FE model, all of the nodes can be the potential scanning point; therefore, the database \(S\) contained the wave propagation time for all of the potential nodes per excited loading point (phased array element position).