## 5 Finite Element Simulation of Void Structure
### Analysis of the Simulation Results
In order to analyze the errors between the test and simulation results, maximum errors and average errors were calculated using Equations (26)-(28):  
\[\delta_{i}=\left|\frac{\alpha_{1}-\alpha_{2}}{\alpha_{1}}\right| \tag{26}\]  
\[\overline{\delta}=\frac{\sum_{i=1}^{n}\delta_{i}}{n} \tag{27}\]  
\[\delta_{max}=max\{\delta_{i}\} \tag{28}\]where \(\delta_{i}\) is the absolute value of the error between test result and simulation result in a single frequency; \(\alpha_{1}\) is the test SAC of PAC; \(\alpha_{2}\) is the simulation SAC of PAC; \(\overline{\delta}\) is the average error of all frequencies; \(n\) is the total number of each frequency, the taken values space is 5 Hz in this paper and \(\delta_{max}\) is the maximum error at all frequencies. Figure 12: Sound pressure level distribution of PAC with different void contents at different frequencies: (**a**) sample A1; (**b**) sample A2; (**c**) sample A3 and (**d**) sample A4. The calculated results of errors were listed in Figure 13. The maximum errors of A1\(\sim\)A4 samples are 22.00%, 19.41%, 36.93% and 26.82%. The average errors are 9.34%, 9.28%, 15.17% and 14.48%, respectively. For porous asphalt concrete, the sample with the lower void content has the more accurate simulation result. The average errors of A1 and A2 are both less than 10%. Figure 13 shows the simulation results are more precise at low frequencies (about 500-1000 Hz) and less at high frequencies (1000-1500 Hz). Because the frequencies corresponding to the peak SAC of low void content samples is smaller than the high void content samples, the FEM models are more suitable for PAC with small void content. Figure 13: Comparison of SAC spectra between samples A1\(\sim\)A4: (**a**) sample A1; (**b**) sample A2; (**c**) sample A3 and (**d**) sample A4. Furthermore, the measured results were compared to the mean values of the SAC obtained from the simulations. The measured and simulated SACs of A1 to A4 are shown in Figure 14. Both the tested and simulated results have a strong linear relationship with the void content, with correlation coefficients of 0.9368 and 0.9074, respectively. The simulated values are smaller than the measured values, with the simulated mean values of the absorption coefficients of samples A1 to A4 being 74.36%, 74.47%, 76.77% and 75.01% of the test values, respectively. Combined with Figures 12 and 13, the simulation results are more similar to the experimental results at low frequencies but less so at high frequencies. One reason is that the effect of inner roughness on the loss of acoustic waves grows as the frequency increases. The acquired void structure in this study after the CT scanning, image recognition and vectorization conversion causes the recognition accuracy to be limited, resulting in smaller simulation results than test results at high frequencies. Another reason is that the scanning interval between each cross-sectional image will truncate the surface-connected pores, which are cut into some surface-connected voids and inner voids in each image. The damping qualities of the material and the structure of the voids are primarily responsible for the noise reduction effect of PAC. The material properties can be simulated fully during the simulation process. However, the simulation of voids is limited, which reduces the simulation accuracy. The goal of follow-up research is to determine more about 3D void modeling. The SAC and the structural characteristic frequencies of the test results are similar to those of the simulated results. All the simulated average values of the SAC account for approximately 75% of the measured values. By the comparison between test and simulated results, the reasonableness of the model is proved, which shows that the PAC acoustic model that was created using the CT scanning methodology and the finite element approach is practical.