# Application of Finite Layer Method in Pavement Structural Analysis
## 3 EEP Super-Convergence Strategy for EasyFEM
\begin{array}{c}\sum_{m=1}^{M}\sum_{n=1}^{N}\int_{V_{e}}\Big{(}E_{13}^{mn} \widetilde{d}^{mn}+E_{2}^{mn}\frac{\partial\widetilde{d}^{mn}}{\partial y} \Big{)}^{T}[D]\Big{(}E_{13}^{mn}e_{h}^{mn}+E_{2}^{mn}\frac{\partial e_{h}^{mn }}{\partial y}\Big{)}dV=\\ \sum_{m=1}^{M}\sum_{n=1}^{N}\int_{y_{1}}^{y_{2}}\bigg{[}\bigg{(}\frac{ \partial\widetilde{d}^{mn}}{\partial y}\Big{)}^{T}A^{mn}\frac{\partial e_{h}^ {mn}}{\partial y}+\bigg{(}\frac{\partial\widetilde{d}^{mn}}{\partial y}\bigg{)} ^{T}G^{mn}e_{h}^{mn}+\bigg{(}\widetilde{d}^{mn}\bigg{)}^{T}(G^{mn})^{T}\frac{ \partial e_{h}^{mn}}{\partial y}+\\ \bigg{(}\widetilde{d}^{mn}\bigg{)}^{T}H^{mn}e_{h}^{mn}\bigg{]}dy\approx 0, \end{array}  \tag{22}\]  
where  
\[ \begin{array}{c}A^{mn}=\int_{0}^{a}\int_{0}^{c}(E_{2}^{mn})^{T}[D]E_{2}^{mn} dxdz,\\ G^{mn}=\int_{0}^{a}\int_{0}^{c}(E_{2}^{mn})^{T}[D]E_{13}^{mn}dxdz,\\ H^{mn}=\int_{0}^{a}\int_{0}^{c}\int_{0}^{c}(E_{13}^{mn})^{T}[D]E_{13}^{mn}dxdz, \end{array} \]  
Figure 3: A typical EasyFEM element e in the pavement model. with integration by parts along the \(y\)-direction, the EEP can be equivalently converted into the following expression:  
\[ \begin{array}{c}\sum\limits_{m=1}^{M}\sum\limits_{n=1}^{N}\int_{y_{1}}^{y_{2}} \left(-\widetilde{d}^{mn^{T}}A^{mn}\frac{\partial^{2}e_{h}^{mn}}{\partial y^{2} }+\widetilde{d}^{mn^{T}}((G^{mn})^{T}-G^{mn})\frac{\partial e_{h}^{mn}}{ \partial y}+\left(\widetilde{d}^{mn}\right)^{T}H^{mn}e_{h}^{mn}\right)dy\\ +\left(\widetilde{d}^{mn^{T}}(A^{mn}\frac{\partial e_{h}^{mn}}{\partial y}+G^{ mn}e_{h}^{mn})\right)_{y_{1}}^{y_{2}}\approx 0.\end{array}  \tag{23}\]  
Since the test function \(\widetilde{d}^{mn}\) can be arbitrarily selected for any \(m\) and \(n\), the left side of Equation (23) is decoupled into  
\[\int_{y_{1}}^{y_{2}}\widetilde{d}^{mn^{T}}\left(-A^{mn}\frac{\partial^{2}e_{h }^{mn}}{\partial y^{2}}+\left(G^{mn^{T}}-G^{mn}\right)\frac{\partial e_{h}^{ mn}}{\partial y}+H^{mn}e_{h}^{mn}\right)dy+\left(\widetilde{d}^{mn^{T}}(A^{mn} \frac{\partial e_{h}^{mn}}{\partial y}+G^{mn}e_{h}^{mn})\right)_{y_{1}}^{y_{2 }}\approx 0. \tag{24}\]  
Taking two linear polynomials:  
\[\overline{N}_{1}=\frac{y_{2}-y}{y_{2}-y_{1}};\ \overline{N}_{2}=\frac{y-y_{1}}{y_{ 2}-y_{1}} \tag{25}\]  
as two test functions \(\widetilde{d}^{mn}\), respectively, and considering that:  
\[d^{mn}(y_{1})=d_{h}^{mn}(y_{1})d^{mn}(y_{2})=d_{h}^{mn}(y_{2}), \tag{26}\]  
since the nodal displacements are super-convergent based on mathematical theory, one can obtain the following equation to calculate a recovered derivative solution at either of the two end-nodes of element e [39]:  
\[ \begin{array}{ccl}\frac{\partial d_{f}^{mn}(y_{1})}{\partial y}&=&(A^{mn})^ {-1}\big{[}(b,\overline{N}_{1})-a^{e}\big{(}d_{h}^{mn},\overline{N}_{1}\big{)} -G_{1}d_{h}^{mn}(y_{1})\big{]},\\ \frac{\partial d_{f}^{mn}(y_{2})}{\partial y}&=&-(A^{mn})^{-1}\big{[}(b, \overline{N}_{2})-a^{e}\big{(}d_{h}^{mn},\overline{N}_{2}\big{)}+G_{2}d_{h}^{ mn}(y_{2})\big{]},\end{array}  \tag{27}\]  
where:  
\[i=1,\ \ 2,\ \ \ m=1,\cdots,M,\ \ \ n=1,\cdots,N. \tag{28}\]  
Substituting the derivative results calculated from Equation (28) into Equation (4), recovered strains and the corresponding stresses are obtained for the pavement model. This is the EEP super-convergence technique for FLM. When it is implemented in EasyFEM, stresses are obtained with much fewer elements and significantly less computational effort.