# Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations
## 6 Discussion  
The use of the moving least squares (MLS) interpolation-based error-recovery approach to increase the reliability of incompressible elastic analysis is introduced in this study. The computational results were obtained for error quality, i.e., convergence rates, effectivity, and updated meshing for the intended accuracy by analyzing the incompressible plate problems. Moving least squares (MLS) interpolation considers radial weights over circular support domains and errors are estimated in energy norms. The incompressible constraints are implemented through a displacement/pressure-based mixed approach. The plate domains are discretized using triangular and quadrilateral elements. The incompressible elastic benchmark examples, including problems with singularity, were studied to demonstrate the effectiveness and efficiency of mesh-free MLS interpolation for displacement/stress-error recovery, and of mesh-dependent MLS interpolation for displacement error recovery. The MLS interpolation-based error-recovery results were also compared with the least-squares interpolation-based error-recovery results. The mesh-dependent patch formation for displacement recovery considers all the elements sur  
Figure 8: Updated meshes of incompressible plate with rigid inclusion in displacement/stress-recovery techniques considering mesh-free (MF) and mesh-dependent (MD) patches (quadrilateral with initial mesh = 266, 1% desired accuracy). (**a**) Mesh (initial); (**b**) disp. recovery (MF); (**c**) stress recovery (MF); (**d**) disp. recovery (MD); (**e**) stress recovery (MD). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Elements**} & \multirow{3}{*}{**DOF**} & \multicolumn{4}{c}{**Mesh-Free Patch**} & \multicolumn{4}{c}{**Mesh-Dependent Patch**} \\ \cline{3-10}  & & \multicolumn{2}{c}{**Displacement**} & \multicolumn{2}{c}{**Stress Recovery**} & \multicolumn{2}{c}{**Displacement Recovery**} & \multicolumn{2}{c}{**Stress Recovery**} \\ \cline{3-10}  & & \multicolumn{2}{c}{**Error (\(\times 10^{-3}\))**} & \multicolumn{2}{c}{**Recovery**} & \multicolumn{2}{c}{**Error (\(\times 10^{-3}\))**} & \multicolumn{2}{c}{**0**} & \multicolumn{2}{c}{**Error (\(\times 10^{-3}\))**} & \multicolumn{2}{c}{**0**} \\ \hline
124 & 558 & 150.98 & 104.88 & 0.7295 & 93.95 & 0.6296 & 133.92 & 0.9637 & 137.51 & 0.8937 \\
373 & 1604 & 73.53 & 40.66 & 0.7791 & 40.66 & 0.6626 & 65.97 & 1.0426 & 67.65 & 0.9273 \\
874 & 3666 & 40.52 & 36.41 & 1.0168 & 25.23 & 0.6304 & 21.14 & 0.8994 & 23.00 & 0.8181 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Displacement/stress errors’ convergence and effectivity (0) in incompressible plate with rigid inclusion in displacement/stress-rigid inclusion in displacement/stress-recovery scheme considering mesh-free and mesh-dependent patches (six-node triangular irregular mesh). Figure 9: Updated meshes of incompressible plate with rigid inclusion in displacement/stress-recovery techniques considering mesh-free (MF) and mesh-dependent (MD) patches (triangular with initial mesh = 124, 1% desired accuracy). (**a**) Mesh (initial); (**b**) disp. recovery (MF); (**c**) stress recovery (MF); (**d**) disp. recovery (MD); (**e**) stress recovery (MD). rounding the specified element, and mesh-dependent patch formation for stress recovery considers the union of the nodes surrounding the specified node. The target error limit in the energy norm is kept as 2% in the benchmark elastic plate and 1% for the benchmark plate with rigid circular inclusion. The incompressible elastic analysis results for error convergence and effectivity for various recovery procedures considering mesh-free and mesh-dependent patches are tabulated in Tables 1-4 and 6. For both the four-node and the six-node elements, the convergence obtained with the help of the mesh-free MLS-based error recovery was found to be better than that obtained with the mesh-dependent error recovery techniques. This indicates the higher efficiency of the mesh-free error-estimation scheme.