## 5 Numerical Simulation  
To further understand the principle of the proposed multi-sensing EMI method, an FEM of grouting compactness was established by using COMSOL Multiphysics. As shown in Figure 10, a two-dimensional finite element model, which consists of a duct, grouting, and three PZT patches, was established. The outer diameter of the tendon duct is 100 mm, and its thickness is 2 mm. Three PZT patches with the lengths of 20 mm, 16 mm, and 12 mm and the thickness of 0.3 mm are labeled as PZT1, PZT2, and PZT3. The physical properties of the material in the model are shown in Table 3, and the physical properties of the PZT patches are shown in Table 2.  
Figure 8: The measured impedance signature of PZTs at different grouting compactness levels. Figure 9: 3-dB mean absolute percentage deviation (MAPD) for different grouting compactness levels. In the numerical simulation model, three PZT patches, PZT1, PZT2, and PZT3 were mounted onto the bottom, middle, and top outside of the duct using a thin layer of epoxy. The numerical model ignores the thin layer of epoxy and connects PZT patches with duct by using the Form Union. In order to have sufficient contact with the PZT patches, the local area of the duct surface is ground to a flat surface, which is shown in Figure 10. a terminal voltage with the amplitude of 1 V was applied to the upper surface of the top PZT patch, and the lower surface of the bottom PZT patch is grounded. Suspension potential was used to simulate the serial electrical connectivity of PZT patches. Roll supports were placed on the side of each PZT patch away from the duct to limit its normal displacement. In order to simulate the piezoelectric effect, a multi-physical coupling of the piezoelectric effect was established. The corresponding current of the series PZT patches was acquired after a frequency analysis over a certain frequency range from 20 to 200 kHz; then, the electrical impedance signatures of the series PZT patches can be calculated by dividing the terminal voltage by current. Five routing levels--empty, 25%, 50%, 75% and 100%--were established to simulate the grouting process. For the accuracy of the simulation results, the maximum size of the mesh partition is 1/10 of the wavelength [73]. Figure 11 shows the simulation impedance signatures of at different grouting compactness levels. The three frequency peaks, which are labeled P1, P2 and P3, are corresponding to PZT1, PZT2, and PZT3 respectively. It can be seen that when the grouting level reaches 25%, 50% and 100%, the impedance signature values of P1, P2, and P3 successively change significantly, which is consistent with the experimental results. However, there are two significant changes in the impedance of P2 when the grouting level reached 50% and 75%, while the experimental results showed only one significant change in the impedance of P2 when the grouting compactness reached 50%. This is because it is difficult to ensure that the grouting compactness is exactly 50% during the experiment. During the experiment, the grouting level was slightly above 50%, so the stiffness of the structure near PZT2 did not change significantly during the subsequent grouting. This also indicates that PZT2 is more sensitive to changes in physical properties of local regions. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Parameters** & **Values** \\ \hline Grouting & Density & 2300 kg\(\cdot\)m\({}^{-3}\) \\  & Young’s modulus & 25 GPa \\  & Poisson ratio & 0.2 \\ Duct & Density & 7850 kg\(\cdot\)m\({}^{-3}\) \\  & Young’s modulus & 200 GPa \\  & Poisson ratio & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Material properties. Figure 10: Finite element model of grouting compactness.