## 3 Simulation  
In order to analyze the propagation mechanism of an ultrasonic wave in a complex chip and the energy distribution of an ultrasonic wave acting on a chip micro defect more intuitively, the ultrasonic detection of flip chip is simulated by the \"Acoustic-Solid Interaction, Transient\" in COMSOL Multiphysics. The ultrasonic echo signal is obtained by simulation. The one-dimensional and cross-section sparse reconstruction of ultrasonic echo are carried out to improve the efficiency and accuracy of testing for micro defects. In order to reduce the amount of calculation in the simulation process, the Sonoscan 230 SP probe(D9500, Sonoscan, Elk Grove Village, IL, USA) is scaled down according to the principle of equal scale reduction. The size of the probe model is shown in Figure 2a. According to the principle of COMSOL modeling and the reality of the flip chip, three typical defects of flip chip solder balls and a normal one are simulated. The parameters in the model are taken from the material database of COMSOL software as shown in Table 1, and the models are shown in Figure 2.  
Figure 2: Models of typical defects of solder balls. (**a**) Dimension of virtual probe; (**b**) Model of single solder ball without defect; (**c**) Ball missing; (**d**) Crack; (**e**) Voids. According to the principle of modeling, the \"Pressure Acoustics, Transient\" module and \"Solid Mechanics\" module are introduced into COMSOL. For the pressure acoustic module, an inward normal displacement (Gabor function, length of 10 ns) is loaded on the incident arc to simulate ultrasound. The Gabor function can be described as:  
\[f(t)=A\cdot\exp(-\frac{\pi(t-2T_{0})^{2}}{\left(T_{0}/2\right)^{2}})\cdot\sin(2 \pi f_{0}t), \tag{4}\]  
At the same time, for the solid mechanics module, the solid boundary is set as a \"low reflection boundary\" to simulate the linear elastic material characteristics used in the simulation. The interfaces of micro defects are set as the acoustic-solid coupling boundary. According to the operation speed of the personal computer, the free triangular mesh is used for gridding operation in the model, and a wavelength of 6 grids is set. The simulation is carried out in two steps and the step size is set to \(T_{0}\)/20. Firstly, the normal displacement is loaded on the probe for 20 ns, which is used as the condition of the Study 1 structure of COMSOL simulation. After 20 ns, the excitation signal is removed, and the boundary condition of the normal displacement is modified to the scattering boundary. This process lasted for 110 ns as a condition for Study 2. The sound pressure measured in the region where the probe center frequency located is integrated and processed as the received signal. The time-domain signal graph of an ultrasound, namely the A-scan, is obtained by simulating the void defect detection of solder balls. The A-scan graph is shown in Figure 3. The first signal peak in Figure 3 is the echo signal of the upper surface (the interface of water and silicon). The second signal peak is the echo signal of silicon and the tin-silver-copper interface. The third signal peak is the echo signal of tin-silver-copper and the void interface. The energy attenuation caused by transmission and refraction occurs at the interface of acoustic impedance, so the signal peak behind is smaller. The reason why the amplitude of the second signal peak is smaller than that of the third one is that the acoustic impedance between silicon and tin-silver-copper is smaller, so the ultrasonic reflection at the second interface is weaker and the signal peak reflected in the figure is smaller. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Ultrason Velocity (m/s)** & **Density (Kg/m\({}^{2}\))** & **Poisson Ratio** & **Young Modulus (GPa)** \\ \hline Water & 1484 & 1000 & / & / \\ Silicon & 9370 & 2329 & 0.28 & 170 \\ Sn-3.5 Ag-0.5 Cu & 4660 & / & / & / \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relevant parameters of simulation materials.