## 2 Enrichment Scheme for the Edge-Based Smoothed 2D Solid Finite Elements
### Cook's Skew Beam Problem  
The first numerical example is the Cook beam problem shown in Figure 5[4]. A distributed force is applied on the right edge, and the left edge is clamped. The plane stress condition is employed with [PERSON]'s modulus \(Y=3\times 10^{7}\) N/m\({}^{2}\), [PERSON]'s ratio \(p=0.3\), and density \(\rho=1\,\mathrm{kg/m^{3}}\). The two quadrilateral meshes are shown in Figure 5, where edge-based smoothed four-node quadrilateral element (ES-Q4) are used, and the proposed scheme is applied for each mesh to obtain solutions with improved accuracy. The relative errors of the vertical displacement in Equation (25) and the energy norm in Equation (27) are given in Table 1. Figure 6 provides the distributions of the strain component \(\varepsilon_{xy}\) when the edge-based four-node quadrilateral elements (ES-Q4) and the proposed scheme are applied to the meshes shown in Figure 6. The distributions of the von Mises stress along line BC is given in Figure 7. The reference solution is calculated with a fine mesh of nine-node quadrilateral elements. The actual computational time for constructing the stiffness matrix and solving the linear system of equations are summarized in Table 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{Mesh} & \multicolumn{2}{c}{\(\varepsilon_{E}\) [\%]} & \multicolumn{2}{c}{\(\varepsilon_{v}\) [\%]} & \multicolumn{2}{c}{DOFs} \\ \cline{2-9}  & \multicolumn{2}{c}{Linear} & \multicolumn{1}{c}{Adaptive} & \multicolumn{1}{c}{Linear} & \multicolumn{1}{c}{Adaptive} & \multicolumn{1}{c}{Linear} & \multicolumn{1}{c}{Adaptive} \\  & \multicolumn{2}{c}{Elements} & \multicolumn{1}{c}{Covers} & \multicolumn{1}{c}{Elements} & \multicolumn{1}{c}{Covers} & \multicolumn{1}{c}{Elements} & \multicolumn{1}{c}{Covers} \\ \hline \multirow{2}{*}{ES-Q4} & I & 12.277 & 0.244 & 13.348 & 0.889 & 40 & 360 \\  & II & 0.183 & 0.112 & 0.653 & 0.116 & 2112 & 3986 \\ \hline \multicolumn{9}{c}{Reference solutions: \(E_{ref}=3.997\times 10^{-7}\) N \(\cdot\) m, \(\varepsilon_{ref}=8.367\times 10^{-7}\) m (DOFs = 18,624)} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relative errors (%) in the energy norm (\(E\)) and the vertical displacement (\(v\)) at the point \(A\) in the [PERSON]’s skew beam problem. Figure 5: [PERSON]’s skew beam problem: (**a**) Problem description. (**b**) Quadrilateral meshes (Mesh I and II) used. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{} & \multirow{2}{*}{Mesh} & \multicolumn{2}{c}{**Solution Times [s]**} \\ \cline{3-6}  & & & \multicolumn{2}{c}{**Stiffness Construction**} & \multicolumn{1}{c}{**Equation Solver**} & \multicolumn{1}{c}{Total} \\ \hline \multirow{4}{*}{ES-Q4} & I & Linear elements & 40 & 0.006 & 0.000 & 0.006 \\  & Adaptive covers & 360 & 0.372 & 0.001 & 0.373 \\ \cline{2-6}  & \multirow{2}{*}{II} & Linear elements & 2112 & 1.102 & 0.001 & 1.103 \\  & Adaptive covers & 3986 & 10.202 & 0.004 & 10.206 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Solution times (in seconds) for constructing the stiffness matrix and solving linear equations for the [PERSON]’s skew beam problem. Figure 6: Strain (\(\epsilon_{xy}\)) predictions obtained by the ES-Q4 and the proposed adaptive enrichment scheme for the [PERSON]’s skew beam problem. Figure 7: von Mises stress (\(\tau\)) distributions along the line BC of the cook’s skew beam problem for the quadrilateral meshes shown in Figure 5.  
It is shown that the calculated results of the strain energy norm, displacement, strain, and von Mises stress with the adaptive enrichment scheme are more accurate than those without cover functions. Furthermore, both the case of the adaptive cover in Mesh 1 and the case of using the linear element in Mesh 2 show sufficiently accurate results; however, the adaptive method consumes approximately three times less computational time. Transient response at point A is also calculated, and Newmark time integration method with \(\Delta t=0.001\,\mathrm{s}\) is used.