## 2 Enrichment Scheme for the Edge-Based Smoothed 2D Solid Finite Elements
### Ad-Hoc Problem  
Here, the ad-hoc problem shown in Figure 9 is considered. For the given in-plane displacements  
\[u=(1-x^{2})^{2}(1-y^{2})^{2}e^{my}\cos mx, \tag{30}\]  
\[v=(1-x^{2})^{2}(1-y^{2})^{2}e^{my}\sin mx\text{ with }m=5, \tag{31}\]  
we can obtain the body forces satisfying the equilibrium equations  
\[f_{x}^{B}=-\bigg{(}\frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{ xy}}{\partial y}\bigg{)},\,f_{y}^{B}=-\bigg{(}\frac{\partial\tau_{yy}}{ \partial y}+\frac{\partial\tau_{xy}}{\partial x}\bigg{)}. \tag{32}\]  
Figure 8: Transient response (displacement in \(y\)-direction) at the point A in the [PERSON]’s skew beam problem. The edge-based smoothed four-node element (EN-Q4), and adaptive enrichment scheme for the edge-based smoothed element (adaptive) are used with Mesh I. The body forces are applied to the whole problem domain, and the fixed boundary condition is imposed on the bottom line (\(y=-1\)). The plane stress condition is employed with \(Y=2\times 10^{5}\,\mathrm{N}/\mathrm{m}^{2}\) and \(p=0.3\), and the solution is obtained using the three triangular meshes shown in Figure 9 with the edge-based three-node triangular elements (ES-T3) and the suggested adaptive enrichment scheme. The reference solutions are obtained using a fine mesh of the standard nine-node quadrilateral elements. Figure 10 provides the von Mises stress distributions for the three triangular meshes shown in Figure 9 and the degrees of cover functions applied based on the adaptive enrichment scheme. The relative errors in the energy norm in Equation (27) and maximum von Mises stress that are averaged stress values at each node in Equation (26) are summarized in Table 3. The von Mises stress fields obtained by applying the adaptive enrichment scheme almost converge to the reference solution for all the meshes (Mesh I, II, and III) considered. Figure 11: Automotive wheel problem: (**a**) Problem description. (**b**) Sequence of meshes (Mesh I, II, and III) used. Figure 10: von Mises (\(\tau\)) predictions obtained by the edge-based three-node triangular elements (ES-T3) and the proposed adaptive enrichment scheme for the ad hoc problem shown in Figure 9.  
Numerical results obtained employing the edge-based smoothed finite elements (ES-FE) are compared with the results of the adaptive enrichment scheme. Table 4 gives the relative errors in the energy norm in Equation (27) and maximum von Mises stress that are averaged stress values at each node in Equation (26). The von Mises stress distributions are given in Figure 12 and the reference solutions are obtained employing a mesh of 9058 nine-node quadrilateral elements. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{Mesh} & \multicolumn{2}{c}{\(e_{E}\) [\%]} & \multicolumn{2}{c}{\(e_{\tau}\) [\%]} & \multicolumn{2}{c}{DOFs} \\ \cline{2-9}  & \multicolumn{2}{c}{Linear} & \multicolumn{2}{c}{Adaptive} & \multicolumn{2}{c}{Linear} & \multicolumn{2}{c}{Adaptive} & \multicolumn{2}{c}{Linear} & \multicolumn{2}{c}{Adaptive} \\  & \multicolumn{2}{c}{Elements} & \multicolumn{2}{c}{Covers} & \multicolumn{2}{c}{Elements} & \multicolumn{2}{c}{Covers} & \multicolumn{2}{c}{Elements} & \multicolumn{2}{c}{Covers} \\ \hline ES-Q4\&T3 & I & 6.522 & 2.066 & 36.424 & 1.5739 & 1532 & 4144 \\  & II & 1.690 & 1.289 & 13.230 & 0.429 & 5268 & 8614 \\ \hline \multicolumn{9}{c}{Reference solutions: \(E_{ref}=3.997\times 10^{-7}\) N \(\cdot\) mm, \(\tau_{ref}=8.367\times 10^{-7}\) N \(\cdot\) mm\({}^{2}\) (DOFs = 74,414)} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relative errors (%) in the energy norm (\(E\)) and the maximum von Mises stress (\(\tau\)) at point \(A\) in the wheel problem. Figure 12: von Mises (\(\tau\)) predictions obtained by the edge-based smoothed elements (ES-Q4, ES-T3) and the proposed adaptive enrichment scheme for the wheel problem shown in Figure 11.  
The proposed scheme applies the cover functions to the area where the solution accuracy needs to be improved and there is high stress gradient.