# Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations
## 5 Illustrative Applications
### Infinite Incompressible Elastic Plate with Rigid Inclusion  
The characteristics of the mesh-free displacement/stress-recovery schemes were also investigated with the help of another example of an infinite incompressible elastic plate with rigid inclusion. An adaptive analysis and automatic mesh-update strategies were used in the example. The exact solution to the incompressible elasticity example is known [38]. The solution derivatives are continuous on the boundary and in the inside domain but with a single point or points lying outside of the domain. The exact displacements and stresses are given by the following equations.  
\[u_{r}=\bigg{(}\frac{T_{x}}{8 Gr}\bigg{)}\{(k-1)r^{2}]+2\gamma R^{2}+[\beta(k-1)R^ {2}+2r^{2}-2\delta\bigg{(}\frac{R^{4}}{r^{2}}\bigg{)}]\}cos2\theta, \tag{26}\]  
\[u_{\theta}=-\bigg{(}\frac{T_{x}}{8 Gr}\bigg{)}[\beta(k-1)R^{2}+2r^{2}-2\delta \bigg{(}\frac{R^{4}}{r^{2}}\bigg{)}]sin2\theta, \tag{27}\]  
\[\sigma_{rr}=(T_{x}/2)[1-\gamma(R^{2}/r^{2})]+(T_{x}/2)[1-2\beta(R^{2}/r^{2})+3 \delta(R^{4}/r^{4})]cos2\theta, \tag{28}\]  
\[\sigma_{\theta\theta}=\bigg{(}\frac{T_{x}}{2}\bigg{)}\bigg{[}1+\gamma\bigg{(} \frac{R^{2}}{r^{2}}\bigg{)}\bigg{]}-(T_{x}/2)[1-3\delta(R^{4}/r^{4})]cos2\theta, \tag{29}\]  
\[\tau_{r\theta}=-(T_{x}/2)[1+\beta\bigg{(}\frac{R^{2}}{r^{2}}\bigg{)}+3\delta(R ^{4}/r^{4})]sin2\theta, \tag{30}\]  
where \(r=y^{2}+x^{2}\), \(T_{x}\) is the uniaxial traction applied at infinity, and, for the plain strain case, the constants \(k\), \(\beta\), \(\gamma\), and \(\delta\) can be written in terms of the Poisson ratio \(v\) as \(k=3-4v\), \(\beta=-2/(3-4v)\), \(\gamma=-(2-4v)/2\), \(\delta=1/(3-4v)\).  
The problem domain is defined by radius (R) = 1 unit, sided (w,b) = 4 unit, as shown in Figure 6. The rigid inclusion center is a singular point. The arc AE boundary has both x- and y-displacement as zero. The lines of the CB and DE boundary have zero shear stress  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multicolumn{2}{c}{**Mesh-Dependent Patch-MLS (4-Node)**} & \multicolumn{4}{c}{**Mesh-Dependent Patch-MLS (6-Node)**} \\ \hline \multirow{2}{*}{**Mesh Size (I/h)**} & **FEM Error** & \multicolumn{2}{c}{**Displacement Recovery**} & \multicolumn{2}{c}{**Mesh Size (1/h)**} & **FEM Error** & \multicolumn{2}{c}{**Displacement Recovery**} \\ \cline{3-8}  & (\(\times\)10\({}^{-3}\)) & **Error (\(\times\)10\({}^{-3}\))** & **\(\Theta\)** & **Mesh Size (1/h)** & **FEM Error** (\(\times\)10\({}^{-3}\)) & **Error (\(\times\)10\({}^{-3}\))** & **\(\Theta\)** \\ \hline
1/4 & 29.44 & 16.54 & 0.7234 & \(\frac{1}{3}\) & 40.20 & 18.71 & 0.9409 \\
1/16 & 7.69 & 0.99 & 0.9677 & 1/12 & 14.11 & 2.94 & 0.9811 \\
1/32 & 3.85 & 0.24 & 0.9909 & 1/24 & 7.02 & 0.89 & 0.9859 \\ \hline Convergence Rate & 0.9777 & 2.0253 & & & 0.9736 & 1.7001 & \\ \hline \end{tabular}
\end{table}
Table 6: Displacement errors; convergence and effectivity (\(\theta\)) in incompressible plate with displacement recovery considering MLS-interpolation recovery with mesh-dependent patches (four-node quadrilateral and six-node triangular mesh).  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Meshing Scheme**} & \multirow{2}{*}{**Initial Mesh**} & \multicolumn{4}{c}{**Mesh-Free Patch**} & \multicolumn{4}{c}{**Mesh-Dependent Patch**} \\ \cline{3-10}  & & \multicolumn{2}{c}{**Displac. Recovery**} & \multicolumn{2}{c}{**Stress Recovery**} & \multicolumn{2}{c}{**Displac. Recovery**} & \multicolumn{2}{c}{**Stress Recovery**} \\ \cline{2-10}  & **N** & **DOF** & **N** & **DOF** & **N** & **DOF** & **N** & **DOF** & **N** & **DOF** \\ \hline Quadrilateral & 64 & 162 & 9209 & 19.766 & 2222 & 4584 & 11.283 & 22.926 & 14,658 & 29,730 \\ Triangular & 28 & 146 & 1439 & 5936 & 791 & 3280 & 2577 & 10,540 & 1767 & 7264 \\ \hline \end{tabular}
\end{table}
Table 5: Updated mesh DOF and element numbers (N) in incompressible elastic plate adaptive analysis with displacement/stress-recovery techniques considering quadrilateral/triangular elements and mesh-free/mesh-dependent patches (2% desirable error rate).