## 7 Conclusions  
Grounded in dynamic response theory, this paper establishes a spectral element computational model considering humidity for isotropic semi-rigid base asphalt pavements under dynamic loading. Focusing on typical road structure types in China, the paper compares the proposed method with classical finite element results and applies it in actual engineering projects under the influence of humidity. It analyzes the variation patterns of dynamic deflection of road surfaces and confirms the reliability and applicability of the spectral element method under the influence of groundwater. The main conclusions are as follows:  
(1) Based on wave propagation theory, the chain rule, and the method of separation of variables, this paper calculates the general solution of the potential function in the frequency domain. For finite layers and semi-infinite layers, potential functions were used to describe the principles of wave propagation, which resulted in stiffness matrices for two types of spectral elements.  
(2) The paper derives related formulas for the overall stiffness matrix, displacement matrix, and nodal force matrix of the road structure. The overall displacement in the time domain was obtained through the Fourier inverse transform. A spectral element mechanical model of the semi-rigid base pavement structure under FWD loading was established, along with a computational program for asphalt pavement dynamic response.  
(3) The maximum absolute error in relative terms between the spectral element method and the ABAQUS finite element method was only 1.85%, and the average absolute error was only 0.70%; the finite element method took about 7 min to compute, while the spectral element method took approximately 12 s.  
(4) When considering humidity changes in engineering applications, the correlation coefficient between the computed deflection curve and the measured one was greater than 0.9, with a peak deflection error of less than 6.5%. Compared to traditional finite element methods, the spectral element method model for calculating dynamic deflection of road surfaces offers higher computational speed and accuracy under the influence of groundwater.  
Conceptualization, B.Z. and Y.Z.; methodology, X.L.; software, H.C.; validation, T.W. and D.W.; formal analysis, T.W.; resources, B.Z.; data curation, H.C.; writing--original draft preparation, D.W.; writing--review and editing, D.W., B.Z. and Y.Z.; visualization, T.W.; supervision,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measurement Point Locations** & **D1** & **D2** & **D3** & **D4** & **D5** & **D6** & **D7** \\ \hline Observed data (\(\mu\)m) & 100.58 & 79.87 & 63.94 & 51.24 & 43.11 & 37.53 & 33.72 \\ Proposed Algorithm (\(\mu\)m) & 103.34 & 82.53 & 67.50 & 54.35 & 45.37 & 39.53 & 35.83 \\ Error (\%) & 2.74 & 3.22 & 5.57 & 6.07 & 5.24 & 5.33 & 6.27 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of peak deflection values for each sensor.  
B.Z.; project administration, B.Z. and Y.Z. All authors have read and agreed to the published version of the manuscript.  
The research presented in this work was supported by the Central Plains Technology Innovation Leading Talent Project (Grant No. 234200510014), National Natural Science Foundation of China (Grant No. 51878624), and the Funding Program for Key Scientific Research Projects of Higher Education Institutions in Henan Province (Grant No. 22A580004).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
Author [PERSON] was employed by the company Henan Transportation Science and Technology Research Institute Co., Ltd. The remaining authors declare that the re-search was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.