## 2 Enrichment Scheme for the Edge-Based Smoothed 2D Solid Finite Elements
### Entriched 2D Finite Elements  
Here we review the formulation of the EN-FEM. The enrichment scheme can expand the solution space of the FEM without mesh refinement or additional nodes, and the solution accuracy can be improved effectively by considering the adequate cover function for a particular problem. The geometry interpolation of the EN-FE is the same as the interpolation of the standard and smoothed finite elements and is given in Equation (1). The interpolation of displacement is given by multiplying the shape functions with cover functions defined over cover area as follows:  
\[\widetilde{\mathbf{u}}(r,s)=\sum_{i=1}^{n}h_{i}^{n}(r,s)\widetilde{\mathbf{u}} _{i}\text{ with }\widetilde{\mathbf{u}}_{i}=\left[ \begin{array}{cc} \widetilde{u}_{i}&\widetilde{v}_{i}\end{array} \right]^{T}, \tag{8}\]  
Figure 2: Strain smoothing domain for the edge-based smoothed element: (**a**) Finite element mesh. The red line corresponds to a target edge. (**b**) The elements are divided into cells. (**c**) Smoothing domains of the edge-based smoothed element and the smoothed strain in the smoothing domain. where \(\widetilde{u}_{i}\) and \(\widetilde{v}_{i}\) are cover functions for the displacements \(u\) and \(v\), respectively. The cover area consists of elements attached to node \(i\) as shown in Figure 3.  
The polynomial cover functions are  
\[\widetilde{u}_{i}=\mathbf{p}_{i}(\mathbf{x})\mathbf{u}_{i}^{u},\;\widetilde{v}_ {i}=\mathbf{p}_{i}(\mathbf{x})\mathbf{u}_{i}^{v}\text{ in }C_{i} \tag{9}\]  
with  
\[\mathbf{p}_{i}(\mathbf{x})=\left[ \begin{array}{cccccc}1&\xi_{i}&\eta_{i}& \xi_{i}^{2}&\cdots&\eta_{i}^{d}\end{array} \right],\;\xi_{i}=\frac{(x-x_{i})}{ \chi_{i}},\;\eta_{i}=\frac{(y-y_{i})}{\chi_{i}}, \tag{10}\]  
\[\mathbf{u}_{i}^{u}=\left[ \begin{array}{cccccc}\overline{u}_{i}&\widetilde{u }_{i}^{\xi}&\widetilde{u}_{i}^{\eta}&\widetilde{u}_{i}^{\xi^{2}}&\cdots& \widetilde{u}_{i}^{\eta^{d}}\end{array} \right]^{T},\;\mathbf{u}_{i}^{v}=\left[ \begin{array}{cccccc}\overline{v}_{i}&\widetilde{v}_{i}^{\xi}&\widetilde{v }_{i}^{\eta}&\widetilde{v}_{i}^{\xi^{2}}&\cdots&\widetilde{v}_{i}^{\eta^{d}} \end{array} \right]^{T}, \tag{11}\]  
where \(\mathbf{p}_{i}\) is the polynomial basis vector for node \(i\), \(d\) is the degree of the polynomial bases, \(\chi_{i}\) is the largest edge length of elements belonging to \(C_{i}\), and \(\mathbf{u}_{i}^{u}\) and \(\mathbf{u}_{i}^{v}\) are the degrees of freedom vectors corresponding to the bases of the polynomial cover functions for the displacements in the \(x\)- and \(y\)- directions, respectively. Substituting cover functions in Equation (9) into Equation (8), the displacement interpolation is represented by  
\[\widetilde{\mathbf{u}}=\overline{\mathbf{u}}+\widetilde{\mathbf{u}}\] \[=\sum\limits_{i=1}^{n}h_{i}^{n}(r,s)\mathbf{u}_{i}+\sum\limits_{ i=1}^{n}\widehat{\mathbf{H}}_{i}^{n}(r,s)\widetilde{\mathbf{u}}_{i} \tag{12}\]  
with  
\[\widetilde{\mathbf{u}}_{i}=\left[ \begin{array}{c}\widetilde{\mathbf{u}}_{i }^{u}\\ \widetilde{\mathbf{u}}_{i}^{v}\end{array} \right],\;\widetilde{\mathbf{H}}_{i}^ {n}(r,s)=\left[ \begin{array}{cc}\widehat{\mathbf{h}}_{i}^{n}(r,s)&0\\ 0&\widehat{\mathbf{h}}_{i}^{n}(r,s)\end{array} \right], \tag{13}\]  
Figure 3: Description of the cover region: (**a**) Finite element mesh and linear/higher order strain part \(\widehat{\epsilon}^{(\epsilon,\epsilon)}\) of element 1 and 2, (**b**) cover region corresponding to the node 1.  
where \(\widetilde{\mathbf{u}}_{i}\) and \(\widetilde{\mathbf{H}}_{i}^{n}\) are the enriched DOFs vector and the corresponding interpolation matrix, respectively.