## 2 Enrichment Scheme for the Edge-Based Smoothed 2D Solid Finite Elements
### Edge-Based Smoothed 2D Finite Elements  
The geometry of the ES-FE is given by  
\[\mathbf{x}=\sum_{i=1}^{n}h_{i}^{n}(r,s)\mathbf{x}_{i}\text{ with }\mathbf{x}_{i}=[ \begin{array}{cc}x_{i}&y_{i}\end{array} ]^{T}\ \, \tag{1}\]  
where \(\mathbf{x}_{i}\) is the position vector corresponding to node \(i\) on the global Cartesian coordinate system \((x,\,y)\) shown in Figure 1a, \(n\) is the number of nodes of element, and \(h_{i}^{n}(r,s)\) are the 2D linear interpolation functions corresponding to node \(i\) defined in the natural coordinate system \((r,\,s)\) in Figure 1b. The four-node quadrilateral and three-node triangular elements have center points (\(r=s=0\) for the four-node element and \(r=s=1/3\) for the three-node element) and are divided into four or three sub-triangles (cells), respectively, using its nodes and center points. For the four-node quadrilateral element (\(n=4\)), the piecewise linear shape functions defined in each triangular cell shown in Figure 1b are [32]  
\[ \begin{array}{l}\mathrm{h}_{1}^{4}=\left(1+2\mathrm{r}+\mathrm{s}\right)/4,\, \mathrm{h}_{2}^{4}=\left(1-2\mathrm{r}+\mathrm{s}\right)/4,\mathrm{h}_{3}^{4}= \left(1-\mathrm{s}\right)/4,\,\mathrm{h}_{4}^{4}=\left(1-\mathrm{s}\right)/4 \text{ in C1},\\ \mathrm{h}_{1}^{4}=\left(1+\mathrm{r}\right)/4,\,\mathrm{h}_{2}^{4}=\left(1- \mathrm{r}+2\mathrm{s}\right)/4,\,\mathrm{h}_{3}^{4}=\left(1-\mathrm{r}-2 \mathrm{s}\right)/4,\,\mathrm{h}_{4}^{4}=\left(1+\mathrm{r}\right)/4\text{ in C2},\\ \mathrm{h}_{1}^{4}=\left(1+\mathrm{s}\right)/4,\,\mathrm{h}_{2}^{4}=\left(1+ \mathrm{s}\right)/4,\,\mathrm{h}_{3}^{4}=\left(1-2\mathrm{r}-\mathrm{s}\right)/4,\,\mathrm{h}_{4}^{4}=\left(1+2\mathrm{r}-\mathrm{s}\right)/4\text{ in C3},\\ \mathrm{h}_{1}^{4}=\left(1+\mathrm{r}+2\mathrm{s}\right)/4,\,\mathrm{h}_{2}^{4} =\left(1-\mathrm{r}\right)/4,\,\mathrm{h}_{3}^{4}=\left(1-\mathrm{r}\right)/4,\,\mathrm{h}_{4}^{4}=\left(1+\mathrm{r}-2\mathrm{s}\right)/4\text{ in C4}.\end{array}  \tag{2}\]  
The linear shape functions for the three-node triangular element (\(n=3\)) are  
\[h_{1}^{3}(r,s)=1+r+s,\,\,h_{2}^{3}(r,s)=r,\,\,h_{3}^{3}(r,s)=s, \tag{3}\]  
where the linear shape functions are the same for all cells shown in Figure 1b. The displacement of the ES-FE is interpolated by  
\[\overline{\upomega}=\sum_{i=1}^{n}h_{i}^{n}(r,s)\overline{\upomega}_{i}\text{ with }\overline{\upomega}_{i}=\left[ \begin{array}{cc}u_{i}&v_{i}\end{array} \right]^{T}\, \tag{4}\]  
in which \(\overline{\upomega}_{i}\) is the displacement vector of node \(i\) in the global Cartesian coordinate system. Employing the standard isoparametric finite element procedure with Equations (1) to (4), the strain field within a cell \(c\) of an element \(e\) is obtained using  
\[\varepsilon^{(e,c)}=\mathbb{B}^{(e,c)}\overline{\upomega}^{(e)}\text{ with }\mathbb{B}^{(e,c)}=\left[ \begin{array}{cccc}\mathbb{B}_{1}^{c}&\mathbb{B}_{2 }^{c}&\cdots&\mathbb{B}_{n}^{c}\end{array} \right]\,\,\overline{\upomega}^{(e)}=\left[ \begin{array}{cc} \overline{\upomega}_{1}&\overline{\upomega}_{2}&\cdots&\overline{\upomega}_{n }\end{array} \right]\ ^{T}, \tag{5}\]  
where \(\varepsilon^{(e,c)}=\ \left[\varepsilon_{xx}\quad\varepsilon_{yy}\quad 2 \varepsilon_{xy}\right]^{T}\,\,\mathbb{B}^{(e,c)}\) is the displacement-strain relation matrix on the cell \(c\) in the element \(e\), \(\overline{\upomega}^{(e)}\) is the nodal displacement vector of the element, and \(\mathbb{B}_{i}^{c}\) is the strain-displacement relation matrix corresponding to node \(i\) in cell \(c\), as seen in Figure 2a. Since the geometry and displacement interpolations are based on the linear shape functions in Equations (2) and (3), the strain components in Equation (5) are constants in each cell. For the ES-FEM, the smoothing domain is defined at each edge with its neighboring cells. Let us consider three-node triangular and four-node quadrilateral elements attached to the edge, colored in red (see, Figure 2).