## 2 Enrichment Scheme for the Edge-Based Smoothed 2D Solid Finite Elements
### Entriched 2D Finite Elements
For the linear cover function used (\(d=1\)), the following components of the interpolation matrix and the enriched DOFs vector are employed  
\[\widetilde{\mathbf{u}}_{i}^{n}(r,s)=h_{i}^{n}(r,s)\big{[}\ \ \xi_{i}\ \ \ \eta_{i}\big{]}\ \,\ \widetilde{\mathbf{u}}_{i}^{u}=\Big{[}\ \ \widetilde{u}_{i}^{\zeta}\ \ \ \widetilde{u}_{i}^{\eta}\ \ \Big{]}^{T},\ \widetilde{\mathbf{u}}_{i}^{v}=\Big{[}\ \ \widetilde{v}_{i}^{\zeta}\ \ \ \widetilde{v}_{i}^{\eta}\ \ \Big{]}^{T}. \tag{14}\]  
When the quadratic cover function is used (\(d=2\)), the components of the interpolation matrix and the enriched DOFs vector are  
\[\widetilde{\mathbf{h}}_{i}^{n}(r,s)=\mathbf{h}_{i}^{n}(r,s)\big{[}\ \xi_{i}\ \ \ \eta_{i}\ \ \ \xi_{i}^{2}\ \ \ \xi_{i}\eta_{i}\ \ \ \eta_{i}^{2}\ \ \big{]}, \tag{15}\] \[\widetilde{\mathbf{u}}_{i}^{u}=\Big{[}\ \ \widetilde{u}_{i}^{\zeta}\ \ \ \widetilde{u}_{i}^{\eta}\ \ \ \widetilde{u}_{i}^{\zeta^{2}}\ \ \ \widetilde{u}_{i}^{\zeta\eta}\ \ \widetilde{u}_{i}^{\eta^{2}}\ \ \Big{]}^{T},\ \widetilde{\mathbf{u}}_{i}^{v}=\Big{[}\ \ \widetilde{v}_{i}^{\zeta}\ \ \ \widetilde{v}_{i}^{\eta}\ \ \ \widetilde{v}_{i}^{\zeta^{2}}\ \ \ \widetilde{v}_{i}^{\zeta\eta}\ \ \ \widetilde{v}_{i}^{\eta^{2}}\ \ \Big{]}^{T}.\]  
For the cubic cover function used (\(d=3\)), the components of the interpolation matrix and the enriched DOFs vector become  
\[\widetilde{\mathbf{h}}_{i}^{n}(r,s)=\mathbf{h}_{i}^{n}(r,s)\big{[}\ \xi_{i}\ \ \ \eta_{i}\ \ \ \xi_{i}^{2}\ \ \ \cdots\ \ \ \eta_{i}^{3}\ \big{]}, \tag{16}\] \[\widetilde{\mathbf{u}}_{i}^{u}=\Big{[}\ \ \widetilde{u}_{i}^{ \zeta}\ \ \ \widetilde{u}_{i}^{\eta}\ \ \ \widetilde{u}_{i}^{\zeta^{2}}\ \ \ \cdots\ \ \widetilde{u}_{i}^{\eta^{3}}\ \Big{]}^{T},\ \widetilde{\mathbf{u}}_{i}^{\zeta^{2}}\ \ \ \cdots\ \ \ \widetilde{v}_{i}^{\eta^{3}}\ \Big{]}^{T}.\]  
Using the displacement-strain relation, the strain vector within cell \(c\) of the target element \(e\) is given by  
\[\bar{\mathbf{g}}^{(e,\varepsilon)} =\bar{\mathbf{B}}^{(e,\varepsilon)}\bar{\mathbf{o}}^{(e)} \tag{17}\] \[=\mathbf{B}^{(e,\varepsilon)}\overline{\mathbf{u}}^{(e)}+\bar{ \mathbf{B}}^{(e,\varepsilon)}\widetilde{\mathbf{u}}^{(e)}\] \[=\varepsilon^{(e,\varepsilon)}+\bar{\varepsilon}^{(e,\varepsilon)}\]  
with  
\[\bar{\  
error indicator, appropriate cover functions are selected for each node, and re-analysis is performed to obtain an improved analysis result.  
The error indicator that estimates errors at each node is defined by  
\[M_{i}=\frac{1}{2}\bigg{\{}\frac{I_{i}}{\overline{J}}+\bigg{(}\frac{\overline{I}} {\overline{\tau}}\bigg{)}\frac{\tau_{i}}{\overline{\tau}}\bigg{\}}\bigg{(}\frac{ \chi_{i}}{L_{c}}\bigg{)} \tag{19}\]  
with  
\[\overline{J}=\frac{1}{N}\sum_{i=1}^{N}J_{i},\ \overline{\tau}=\frac{1}{N}\sum_{i=1 }^{N}\overline{\tau}_{i},\ J_{i}=  
The presented procedure improves the solution accuracy of edge-based smoothed finite elements, which are known to have better performance than standard finite elements, while the automatic procedure proposed by [PERSON] and [PERSON] [35] improves the solution for standard finite elements. Iterative analysis is also possible by adopting a higher degree cover function or by applying a cover function to a node to which the cover function is not applied. Repeating this re-analysis requires parameter adjustment in Equation (24) at each iteration. The most accurate solution can be obtained when applying the highest order cover function to all nodes in a given finite element mesh. In this paper, we present a procedure that can effectively obtain an accurate solution through single re-analysis automatically without parameter adjustment.