# Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations
## 5 Illustrative Applications  
### Incompressible Infinite Square Elastic Plate  
The effectiveness of the proposed mesh-free displacement or stress-recovery scheme is examined through the error-convergence rate of the error recovery of the displacement or stress in case of an incompressible elastic body under self-weights. The problem is used by [PERSON] et al. (2016). The example does not present singularity, so the theoretical convergence rate of the recovery scheme can be compared with other recovery schemes. The exact solutions for the example are included in Equations (21)-(25).  
Domain and boundary conditions: \(\Omega\left[0\times 0\right]\times\left[1\times 1\right]\), \(u=v=0\) on \(\Gamma\)  
Exact solutions:  
\[u=2x^{2}y(1-x)^{2}(1-y)(1-2y) \tag{21}\]  
\[v=-2 xy^{2}(1-x)(1-2x)(1-y)^{2}, \tag{22}\]  
\[p=x^{2}-y^{2}, \tag{23}\]  
Figure 1: Radial support domains for MLS (mesh-free) recovery approach.  
Figure 2: Flow chart for MLS-based recovery technique coupled with adaptive analysis of incompressible elastic problems.  
Body forces:  
\[b_{x}=4y\Big{(}1-6x+6x^{2}\Big{)}\Big{(}1-3y+2y^{2}\Big{)}+12x^{2} \Big{(}1-2x+x^{2}\Big{)}\left(-1+2y\right)-2x \tag{24}\] \[b_{y}=-4x\Big{(}1-6y+6y^{2}\Big{)}\Big{(}1-3x+2x^{2}\Big{)}+12y^{2 }\Big{(}1-2y+y^{2}\Big{)}\left(-1+2x\right)+2y, \tag{25}\]  
The meshes generated using triangular and quadrilateral elements are shown in Figure 3. The plate problem was analyzed considering one-point reduced integration with quadrilateral elements (four-node) and two-point reduced integration with triangular elements (six-node) for volumetric strain term. The desired error limit for four-node quadrilateral and six-node triangular elements was kept as 2%. The finite-element-analysis results for convergence rate and effectivity of solution with different error-recovery approaches in energy norms considering meshfree patches and mesh-dependent patches are tabulated in Tables 1-4. The updated mesh plots for desired accuracy of 2% with various recovery procedures and meshing schemes is given in Figures 4 and 5. The element numbers (N) and degrees of freedom (DOF) after mesh update for desired accuracy of 2% in adaptive analysis with displacement/stress-error-recovery techniques for quadrilateral/triangular discretization are presented in Table 5.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Mesh-Free Patch**} & \multicolumn{4}{c}{**Mesh-Dependent Patch**} \\ \cline{3-10} \multirow{-2}{*}{**Elements**} & \multirow{2}{*}{**DOF**} & \multirow{2}{*}{\begin{tabular}{c} **FEM Error** \\ (\(\times\)**10\({}^{-3}\))** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Displacement Recovery** \\ **Recovery** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Stress Recovery \\ **(\(\times\)10\({}^{-3}\))** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Displacement Recovery** \\ **Recovery** \\ \end{tabular} } & \multicolumn{2}{c}{\begin{tabular}{c} **Stress Recovery (ZZ)** \\ **(\(\times\)10\({}^{-3}\))** \\ \end{tabular} } \\ \cline{2-10} \multirow{-2}{*}{**Elements**} & \multirow{2}{*}{**DOF**} & \multirow{2}{*}{\
\begin{tabular}{c} **FEM Error** \\ (\(\times\)**10\({}^{-3}\))** \\ \end{tabular} } & \multicolumn{2}{c}{
\begin{tabular}{c} **Error** \\ (\(\times\)**10\({}^{-3}\))** \\ \end{tabular}