## 2 Fast Finite Element Simulation Method for PAUT  
### Fast Finite Element Simulation Method  
In ultrasonic testing, the probe generates an impulse force on the surface of the specimen under the excitation of pulse voltage, so as to generate an ultrasonic wave in the specimen. According to the basic theory of elasticity, the basic control equation of an ultrasonic wave in solid materials can be obtained as follows [24]:  
\[\mu\
abla^{2}\mathbf{u}+(\lambda+\mu)\
abla(\
abla\cdot\mathbf{u})+\mathbf{f} =\rho\ddot{\mathbf{u}}+\gamma\dot{\mathbf{u}} \tag{1}\]  
where \(\lambda\) and \(\mu\) are the Lame elastic constant of the material, \(\rho\) is the density of the material, \(\gamma\) is the ultrasonic damping coefficient, \(\mathbf{u}\) is the particle displacement vector, and \(\mathbf{f}\) is the excitation force vector. According to the basic principle of the finite element method, the above differential control equation can be transformed into the following discrete finite element control equation:  
\[[M]\Big{\{}\ddot{U}\Big{\}}+[D]\Big{\{}\dot{U}\Big{\}}+[K]\{U\}=\{F\} \tag{2}\]  
where \([M]\) is the mass matrix, \([D]\) is the damping matrix, \([K]\) is the stiffness matrix, \(\{F\}\) is the node load vector, and \(\{U\}\) is the displacement vector of the node to be solved. In the calculation of the wave problem, the consistent (coordinated) mass matrix of an element can be transformed into a diagonally agglomerated mass matrix to obtain more reliable calculation results. It is assumed that the mass of the element is agglomerated at the node of the element; that is, the element mass matrix becomes a diagonal matrix [25]:  
\[M^{\mathrm{e}}_{ij}=\left\{ \begin{array}{ll}\sum\limits_{k=1}^{8}M^{ \mathrm{e}}_{ik}&(i=j)\\ 0&(i\
eq j)\end{array} \right.\;(i,j=1,\ldots,8) \tag{3}\]  
Equation (2) is actually a discrete dynamic initial value problem. For the general dynamic initial value problem, the time-domain step-by-step integration algorithm can be used to solve it. The current time-domain integration algorithms are roughly divided into two categories, namely the implicit integration method and explicit integration method [26,27]. The implicit integration method is characterized by a high computational stability and unconditional stability, but it needs to solve the coupled linear equations, which require a large amount of storage and calculations. The characteristic of the explicit integration method is that it does not need to solve the coupled linear equations, and can be combined with one-dimensional compressed storage technology. Its storage and calculation can be greatly reduced, but its calculation is conditionally stable. In order to further reduce the amount of storage and calculations in the process of ultrasonic numerical simulation and to meet the need for more than second-order calculation accuracy in engineering applications, a new and more concise explicit integration improved algorithm based on the central difference method [26] and Newmark average velocity method [27] was established as follows:  
\[\left\{ \begin{array}{l}\left\{\dot{U}\right\}_{t+\Delta t}=\left\{\dot{U} \right\}_{t-\Delta t}-2\Delta t\left[\widetilde{C}\right]\left\{\dot{U}\right\} _{t}-2\Delta t\left[\widetilde{K}\right]\left\{U\right\}_{t}+2\Delta t[M]^{-1} \{F\}_{t}\\ \left\{U\right\}_{t+\Delta t}=\left\{U\right\}_{t}+\frac{\left\{\dot{u}\right\} _{t+\Delta t}+\left\{\dot{u}\right\}_{t}}{2}\Delta t\end{array} \right. \tag{4}\]  
where \(\left[\widetilde{K}\right]=[M]^{-1}[K]\), \(\left[\widetilde{C}\right]=[M]^{-1}[C]\). It can be seen that solving Equation (4) does not need to solve the coupled linear equations, and the amount of calculations are greatly reduced compared with the traditional implicit integration method. Compared with the existing explicit integration method ([PERSON]'s method) [28], the storage and calculation amounts in the calculation process are also reduced by about half.