## 2 Enrichment Scheme for the Edge-Based Smoothed 2D Solid Finite Elements
### L-Shape Structure Problem  
The L-shape structure is subjected to pressure load varying along the edges, as shown in Figure 13 and the boundary conditions are imposed as: \(u=0\) along AD and \(v=0\) along CD. The plane stress condition is used with material properties \(Y=200\times 10^{9}\,\mathrm{N}/\mathrm{m}^{2}\) and \(p=0.3\).  
The finite element models are constructed using the standard four-node and three-node elements (ST-Q4, ST-T3) and the edge-based smoothed four-node and four-node elements (ES-Q4, ES-T3) for three different meshes (Mesh I, II, III), as shown in Figure 13. Triangular element meshes are generated from the quadrilateral mesh by dividing each quadrilateral element into two triangular elements. To obtain an improved solution, the enrichment schemes for the ST-FE and ES-FE are employed, and accuracy improvement and computational efficiency in terms of the number of DOFs used are compared. The reference solutions are calculated using a mesh of 3072 nine-node quadrilateral elements.  
The von Mises stress fields and the strain component \(\varepsilon_{xx}\) obtained from the linear elements and adaptive enrichment scheme are presented in Figures 14-16, and the relative errors of strain energy and maximum von Mises stress are summarized in Tables 5 and 6.  
Figure 14: von Mises (\(\tau\)) predictions obtained by the standard and edge-based smoothed four-node quadrilateral finite elements (ST-Q4 and ES-Q4) and the adaptive enrichment scheme for the L-shape structure problem when the Mesh I is used.  
Figure 15: Strain (\(\epsilon_{xx}\)) predictions obtained by the standard and edge-based smoothed four-node triangular finite elements (ST-Q4 and ES-Q4) and the adaptive enrichment scheme for the L-shape structure problem when the Mesh I is used.  
As for the quadrilateral mesh, the ES-Q4 provides greatly improved strain energy prediction compared to the ST-Q4, whereas the ES-Q4 appears to be less accurate in predicting the maximum von Mises stress. By applying the suggested procedure, the solution of the ES-Q4 is significantly improved, especially in stress prediction, and the increase of DOFs used is less than the ST-Q4, as seen in Table 5. For the triangular mesh, the ES-T3 provides more accurate results than the ST-T3 for the same mesh. Therefore,  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & \multicolumn{2}{c}{\(e_{E}\) [\%]} & \multicolumn{2}{c}{\(e_{\tau}\) [\%]} & \multicolumn{2}{c}{DOFs} \\ \cline{3-8} \multicolumn{1}{c}{Mesh} & \multicolumn{1}{c}{Linear} & Adaptive & Linear & Adaptive & Linear & Adaptive \\  & & Elements & Covers & Elements & Covers & Elements & Covers \\ \hline \multirow{3}{*}{ST-T3} & I & 21.206 & 0.120 & 49.037 & 4.598 & 120 & 890 \\  & II & 6.980 & 1.483 & 33.617 & 1.075 & 432 & 1946 \\  & II & 1.977 & 0.884 & 18.077 & 0.174 & 1632 & 4050 \\ \hline \multirow{3}{*}{EN-T3} & I & 3.846 & 0.123 & 38.008 & 4.541 & 120 & 836 \\  & II & 0.733 & 0.976 & 27.116 & 1.660 & 432 & 1716 \\ \cline{1-1}  & III & 0.139 & 0.208 & 17.314 & 0.185 & 1632 & 3826 \\ \hline \multicolumn{8}{c}{Reference solutions: \(E_{nf}=5.120\times 10^{-9}\) N - m, \(\tau_{ref}=1.751\times 10^{9}\) N / m\({}^{2}\)(DOFs = 6336)} \\ \hline \hline \end{tabular}
\end{table}
Table 6: Relative errors (%) in the energy norm (\(E\)) and the maximum von Mises stress (\(\tau\)) in the L-shape structure problem when the meshes of triangular elements (ST-T3, ES-T3) are used.  
Figure 16: Von Mises (\(\tau\)) predictions obtained by the standard and edge-based smoothed four-node triangular finite elements (ST-T3 and ES-T3) and the adaptive enrichment scheme for the L-shape structure problem when the Mesh III is used.  
when the same procedure is applied, the smaller number of DOFs is used for the ES-T3, but sufficiently accurate results are provided (see Table 6).