# Application of Finite Layer Method in Pavement Structural Analysis
## 4 Numerical Analysis Using EasyFEM and EEP Techniques
### Analytical Verification of EasyFEM
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Points**} & \multirow{2}{*}{**Results**} & \multicolumn{3}{c}{**Winter**} \\ \cline{3-5}  & & **EasyFEM** & **ABAQUS** & **Difference** \\ \hline
1 & Vertical displacement [mm] & \(-1.63\times 10^{-1}\) & \(-1.64\times 10^{-1}\) & \(\begin{array}{c}7.72\times 10^{-4}\\ (-0.47\%)\end{array}\) \\ \hline
2 & Horizontal strain [-] & \(3.54\times 10^{-5}\) & \(3.25\times 10^{-5}\) & \(\begin{array}{c}2.82\times 10^{-6}\\ (6.86\%)\end{array}\) \\ \hline
3 & Vertical stress [MPa] & \(-6.57\times 10^{-2}\) & \(-6.09\times 10^{-2}\) & \(\begin{array}{c}-4.84\times 10^{-3}\\ (7.95\%)\end{array}\) \\ \hline
4 & Vertical stress [MPa] & \(-9.70\times 10^{-3}\) & \(-9.62\times 10^{-3}\) & \(\begin{array}{c}-8.61\times 10^{-5}\\ (0.90\%)\end{array}\) \\ \hline
5 & Vertical stress [MPa] & \(-7.57\times 10^{-3}\) & \(-7.55\times 10^{-3}\) & \(\begin{array}{c}-1.51\times 10^{-5}\\ (0.20\%)\end{array}\) \\ \hline
**Points** & **Results** & \multicolumn{3}{c}{**Summer**} \\ \cline{3-5}  & **EasyFEM** & **ABAQUS** & **Difference** \\ \hline
1 & Vertical displacement [mm] & \(-2.99\times 10^{-1}\) & \(-3.02\times 10^{-1}\) & \(\begin{array}{c}3.24\times 10^{-3}\\ (-1.07\%)\end{array}\) \\ \hline
2 & Horizontal strain [-] & \(1.01\times 10^{-4}\) & \(9.33\times 10^{-5}\) & \(\begin{array}{c}7.39\times 10^{-6}\\ (7.92\%)\end{array}\) \\ \hline
3 & Vertical stress [MPa] & \(-1.75\times 10^{-1}\) & \(-1.62\times 10^{-1}\) & \(\begin{array}{c}-1.27\times 10^{-2}\\ (7.85\%)\end{array}\) \\ \hline
4 & Vertical stress [MPa] & \(-2.06\times 10^{-2}\) & \(-2.04\times 10^{-2}\) & \(\begin{array}{c}-1.75\times 10^{-4}\\ (0.86\%)\end{array}\) \\ \hline
5 & Vertical stress [MPa] & \(-1.43\times 10^{-2}\) & \(-1.43\times 10^{-2}\) & \(\begin{array}{c}-1.01\times 10^{-5}\\ (0.07\%)\end{array}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between ABAQUS and EasyFEM based on the results at critical points below the loading center. Figure 6: Comparison of vertical displacement at the top of the asphalt surface in (**a**) winter, (**b**) summer; comparison of horizontal strain along transverse direction at the bottom of the asphalt base course in (**c**) winter, (**d**) summer; comparison of vertical stress at the top of the road base layer in (**e**) winter, (**f**) summer; comparison of vertical stress at the top of the sub-base layer in (**g**) winter, (**h**) summer; comparison of vertical stress at the top of the sub-grade layer in (**i**) winter, (**j**) summer. It can be stated that the results from both programs have a high correlation except for a slightly larger difference in the horizontal strain at the bottom of the asphalt base course and vertical stresses at the top of the sub-base. Considering the different element types and mesh algorithms in the two programs, the differences are judged to be acceptable. Moreover, the computational time of the EasyFEM is much shorter than that of the ABAQUS. Both analyses were run on a computer with an Intel Core Duo 3.4 GHz, 32 GB RAM (Santa Clara, CA, USA). On average, the computational time required by the ABAQUS is about 420 s, whereas the EasyFEM model requires 120 s. It is worth mentioning that the model in ABAQUS is one-fourth symmetrical, but the one in EasyFEM is full-scale. With code optimization, the computation time of the EasyFEM can be reduced further. In summary, the accuracy and efficiency of the EasyFEM are proved by these comparisons. Comparison of Convergence Rate between Ordinary EasyFEM and EasyFEM with EEP Post-Processing Techniques  
As discussed above, by using 275 3-node quadratic finite elements along the \(y\)-direction, EasyFEM has performed fairly well compared to ABAQUS. If the number of elements can be further reduced without decreasing the computational accuracy, the EasyFEM will be even more suitable for application in pavement design and assessment from an engineering point of view. The EEP technique for recovering derivatives makes this possible.