## Theory and Model  
### Finite Element Method  
Due to the different physical properties of medium, the modes of ultrasonic wave in different medium are also different. For example, when ultrasonic wave propagates in air and liquid, only longitudinal wave exist because viscosity is negligible; when ultrasonic wave propagates in solid,due to the effect of shear force, there are both longitudinal wave and transverse wave. However, when there are both solid and air mediums in the process of ultrasonic propagation, the situation will become more complicated. For honeycomb sandwich composite, the ultrasonic propagation in air can be attributed to pressure acoustics problem in fluid field. The ultrasonic propagation in aluminum, glass fiber reinforce polymers (GFRP), epoxy resin adhesive layer can be attributed to stress-strain problem in solid field. The ultrasonic propagation at the fluid-solid coupling interface can be attributed to fluid-solid coupling stress-strain problem under certain continuous conditions. Therefore, it is necessary to analyze and discuss wave equation in fluid mechanics, structural mechanics equation, fluid-solid dynamic coupling equation, and boundary conditions to provide an accurate and reliable theoretical basis for finite element analysis.  
The propagation of sound wave in air can be expressed as [22]:  
\[[\mathrm{M}_{\mathrm{f}}]\{\ddot{\mathrm{P}}_{\mathrm{f}}\}+[\mathrm{K}_{ \mathrm{f}}]\{\mathrm{P}_{\mathrm{f}}\}=\{\mathrm{F}_{\mathrm{f}}\} \tag{1}\]  
where \([\mathrm{M}_{\mathrm{f}}]\) is mass matrix of air medium; \([\mathrm{K}_{\mathrm{f}}]\) is stiffness matrix of air medium; \(\{\mathrm{P}_{\mathrm{f}}\}\) is acoustic pressure vector of unit node in air medium, \(\{\mathrm{F}_{\mathrm{f}}\}\) is the external force vector.  
The propagation of sound wave in aluminum, glass fiber reinforce polymers (GFRP), epoxy resin adhesive layer can be written as [22]:  
\[[\mathrm{M}_{\mathrm{s}}]\{\ddot{\mathrm{U}}\}+[\mathrm{K}_{\mathrm{s}}]\{ \mathrm{U}\}=\{\mathrm{F}_{\mathrm{s}}\} \tag{2}\]  
where \([\mathrm{M}_{\mathrm{s}}]\) is mass matrix of old medium, \(\{\mathrm{F}_{\mathrm{s}}\}\) is pressure load of solid medium,\([\mathrm{K}_{\mathrm{s}}]\) is stiffness matrix of solid medium, \(\{\mathrm{U}\}\) is displacement vector of solid medium.  
However, when sound wave reaches the fluid-solid coupling interface, the dynamic interaction between the fluid medium at fluid interface and the solid medium at solid interface need to be considered. Therefore, the discrete form of sound wave control equation of fluid-solid coupling interface [22] can be expressed as:  
\[\left[\begin{array}{c}\rho_{0}\mathrm{R}_{\mathrm{f}}^{\mathrm{T}}[\mathrm{M }_{\mathrm{f}}]\\ [\mathrm{M}_{\mathrm{s}}]0\end{array}\right]\left\{\begin{array}{c}\{\ddot{ \mathrm{U}}\}\\ \{\ddot{\mathrm{P}}_{\mathrm{f}}\}\end{array}\right\}+\left[\begin{array}{c}0 [\mathrm{K}_{\mathrm{f}}]\\ [\mathrm{K}_{\mathrm{s}}]-\mathrm{R}]\end{array}\right]\left\{\begin{array}{c} \{\mathrm{U}\}\\ \{\mathrm{P}_{\mathrm{f}}\}\end{array}\right\}=\left\{\begin{array}{c}\{ \mathrm{F}_{\mathrm{f}}\}\\ \{\mathrm{F}_{\mathrm{s}}\}\end{array}\right\} \tag{3}\]  
where \([\mathrm{R}]\) is the coupling matrix at the fluid-solid coupling interface. This equations mean that the nodes at fluid-solid interface are continuous and the nodes both have displacement component and pressure component.