# Sound Field Modeling Method and Key Imaging Technology of an Ultrasonic Phased Array: A Review
## 2 Modelling Method
### Finite Difference Method  
FDM is a numerical method used to simulate elastic wave propagation [33; 34]. In FDM ultrasonic modeling, the propagation medium is divided into staggered grids and probe excitation is applied to the nodes of the corresponding grids in the form of a displacement or stress load. Then, the differential expression is written by the wave equation, and the transient propagation diagram of the sound field at the required time can be calculated. The difference equation can be regarded as the finite approximation of the original equation in terms of a Taylor series expansion. According to the interception method of the Taylor series continuous approximation, the difference scheme can be divided into three kinds:  
1. The Lax-Friedrichs format, explicit, taking three discrete points in space, stable at \(|a\Delta t/\Delta x|\leq 1\). \[\frac{p_{k}^{n+1}-\left(p_{k+1}^{n}+p_{k-1}^{n}\right)/2}{\Delta t}+\frac{a} {\Delta x}\frac{p_{k+1}^{n}-p_{k-1}^{n}}{2}=0\] (4)
2. The Leap-Frog format, explicit, taking three discrete points in time, stable at \(|a\Delta t/\Delta x|\leq 1\) \[\frac{p_{k}^{n+1}-p_{k}^{n-1}}{2\Delta t}+\frac{a}{\Delta x}\frac{p_{k+1}^{n} -p_{k-1}^{n}}{2}=0\] (5)
3. The Crank-Nicholson format, implicit, taking three points in space and three points in time. \[\frac{p_{k}^{n+1}-p_{k}^{n-1}}{\Delta t}+\frac{a}{2\Delta x}\left(\frac{p_{k +1}^{n+1}-p_{k-1}^{n+1}}{2}+\frac{p_{k+1}^{n}-p_{k-1}^{n}}{2}\right)=0\] (6) where \(\alpha\) is the coefficient of the original equation; \(\Delta t\) is the time step; \(\Delta x\) is the space step size; and \(P\) is the target value. The finite difference method is commonly used to model the propagation of elastic waves and has three forms of expression: forward, backward, and central differences, which need to be chosen wisely to reduce computational errors. It uses a standard quadrilateral grid, so it is very effective in dealing with regular boundaries, but not as effective as the finite element method for irregular boundaries.  
In the 1980s, [PERSON] et al. [35] presented a modification of the FDM that enables elimination of single or poorly conditioned stars which made local condensation of the grid and the boundary conditions easy to discretize in the case of an arbitrary shape of the domain. In 1996, [PERSON] et al. [36] used the finite difference method to simulate the S0 and A1 Lamb wave modes in a flat metal sheet. The time domain information of the field displacement perpendicular to the group of sheet points is obtained and compared with the Lamb mode dispersion wave. In 2001, [PERSON] et al. [37] used an improved nodal calculation method following fundamental consideration of the elastic wave equations. This improvement is expected to solve the calculation of internal nodes and unified treatment of solid boundaries and applies to composite materials and anisotropic materials. In the literature [38], simulation of the ultrasonic array wave propagation in a 10-mm thick mild-steel pipe specimen has been demonstrated using the FDM method. The simulation results were compared with the experiments of phased array ultrasonic wave interaction with the defects. After 40 years of development, FDM technology has achieved fruitful results in temperature field, flow field simulation, and defect prediction microstructure simulation. A large amount of FDM-based casting process numerical simulation software has emerged, including Magmasoft, AnyCasting, AFSolid, Solidia, NOVCAST, FT-Star, InteCast, CAST soft, etc. [39].