# Application of Finite Layer Method in Pavement Structural Analysis
## 3 EEP Super-Convergence Strategy for EasyFEM  
When Equation (16) is solved for \(U^{mn}\), both \(d^{mn}\) and \(d\) can be obtained from Equation (2). These two approximate solutions are denoted as \(d^{mn}_{h}\) and \(d_{h}\) to distinguish them from the exact solutions. It is clear, that if \(M\) and \(N\) are fixed, the accuracy of \(d_{h}\) mainly depends on the accuracy of \(d^{mn}_{h}\). For conventional FEM models, a rather extensive mesh should be used to attain more accurate \(d^{mn}_{h}\), which leads to a large number of degrees of freedom in the calculation. In this case, the newly-developed super-convergence technique EEP was introduced into EasyFEM as to improve the accuracy of the solutions for strains and stresses over the whole pavement model. The original idea of the EEP technique came from a well-known mathematical theorem for FEM called the projection theorem [37]. Actually, the principle of minimum potential energy with respect to nodal displacements \(\{d\}\) in Equation (8) can be rewritten in the same manner with respect to the continuous 1D displacement \(d(y)\) as follows:  
\[\delta\Pi(d(y))=\int_{V}\delta d(y)^{T}[E]^{T}[D][E]d(y)dV-\int_{V}\delta d(y )^{T}[N(x,z)]^{T}\{b\}dV-\int_{S}\delta d(y)^{T}[N(x,z)]^{T}\{f\}dS\ =0. \tag{17}\]  
By replacing \(\delta d(y)\) in Equation (17) with the test displacement \(\widetilde{d}(y)\), an equivalent expression for the above principle of minimum potential energy is the virtual work equation as follows:  
\[ \begin{array}{c}a(d(y),\widetilde{d}(y))=(f,\widetilde{d}(y)),\forall \widetilde{d}(y)\in S^{h},\\ \text{with }a\left(d(y),\widetilde{d}(y)\right)=\int_{V}\widetilde{d}(y)^{T}[E]^{T }[D][E]d(y)dV,\\ (\text{f},\ \widetilde{d}(y))=\int_{V}\widetilde{d}(y)^{T}[\text{N}(\text{x}, \text{z})]^{T}\{\text{b}\}dV+\int_{S}\widetilde{d}(y)^{T}[\text{N}(\text{x}, \text{z})]^{T}\{\text{f}\}\text{d}S,\end{array}  \tag{18}\]  
where \(a(\cdot,\cdot)\) and \((\cdot,\cdot)\) are defined as the energy product and the linear form in mathematics, respectively, and \(S^{h}\) is the test space of FLM. It can be easily understood that Equation (18) holds true both for the exact solution \(d(y)\) and the approximate solution \(d_{h}(y)\). Always taking the same test displacement for these two cases, the following projection theorem is derived for FLM:  
\[a(d(y)-d_{h}(y),\widetilde{d}(y))=0,\sqrt{\widetilde{d}(y)}\in S^{h}. \tag{19}\]  
This theorem holds true over the entire 3D pavement model. By assuming that it is approximately true over one element of FLM, the EEP equation was obtained:  
\[a^{e}(d(y)-d_{h}(y),\widetilde{d}(y))\approx 0,\quad\forall\widetilde{d}(y)\in S ^{h}. \tag{20}\]  
A typical element \(e\) of the FLM is shown in Figure 3, which is actually a small layer of the pavement with the \(y\)-coordinate ranging from \(y_{1}\) to \(y_{2}\). Denote \(e_{h}=d(y)-d_{h}(y)\). Then:  
\[ \begin{array}{c}a^{e}\Big{(}d(y)-d_{h}(y),\widetilde{d}(y)\Big{)}=\int_{V_{e }}\widetilde{d}(y)^{T}[E]^{T}[D][E]e_{h}dV\\ =\sum_{m=1}^{M}\sum_{n=1}^{N}\int_{V_{e}}\widetilde{d}^{mn^{T}}[E^{mn}]^{T}[D ][E^{mn}]e_{h}^{mn}dV=\sum_{m=1}^{M}\sum_{n=1}^{N}a^{e}(e_{h}^{mn},\widetilde {d}^{mn})\approx 0\end{array}  \tag{21}\]  
Denoting \(E_{13}^{mn}=L_{1}\frac{\partial N^{mn}}{\partial x}+L_{3}\frac{\partial N^{mn }}{\partial z}\), \(E_{2}^{mn}=L_{2}N^{mn}\), and considering the zeros from integration of the Fourier series as shown in Equation (12), Equation (21) can be further rewritten as follows:  
\[