## 3 Applications to Benchmark Examples
### Infinite Incompressible-Elastic Plate with Rigid Circular Opening  
The infinite incompressible-elastic plate with circular opening example is also analyzed to examine the recovery properties of the mesh-free RPI-based recovery schemes. The adaptive mesh-refining approach is implemented to study the error distribution behavior. The analytical solution of the example problem is known [31]. The gradients are continuous on both the boundary and inside domain, having one or more singular points outside of the domain. The following equations can be used to obtain true displacement and stresses in the problem.  
\[u_{r}=\left(\frac{T_{x}}{8 Gr}\right)\{(k-1)r^{2}\}+2\gamma R^{2}+[\beta(k-1)R^ {2}+2r^{2}-2\delta\left(\frac{R^{4}}{r^{2}}\right)]\}\ cos2\theta, \tag{21}\]  
\[u_{\theta}=-\left(\frac{T_{x}}{8 Gr}\right)\left[\beta(k-1)R^{2}+2r^{2}-2\delta \left(\frac{R^{4}}{r^{2}}\right)\right]sin2\theta, \tag{22}\]  
\[\sigma_{rr}=\left(T_{x}/2\right)\left[1-\gamma\left(R^{2}/r^{2}\right)\right] \ +\left(T_{x}/2\right)\left[1-2\beta\left(R^{2}/r^{2}\right)+3\delta\left(R^{4} /r^{4}\right)\ cos2\theta, \tag{23}\]  
\[\sigma_{\theta\theta}=\left(\frac{T_{x}}{2}\right)\left[1+\gamma\left(\frac{R ^{2}}{r^{2}}\right)\right]-\left(T_{x}/2\right)\left[1-3\delta\left(R^{4}/r^{4 }\right)\right]\ cos2\theta\, \tag{24}\]  
\[\tau_{r}=-\left(T_{x}/2\right)\left[1+\beta\left(\frac{R^{2}}{r^{2}}\right)+3 \delta\left(R^{4}/r^{4}\right)\right]\ sin2\theta, \tag{25}\]  
where \(r=y^{2}+\frac{x^{2}}{r^{2}}\), \(T_{x}=\) uniaxial traction applied at infinity, and constants \(k[=3-4v]\), \(\beta[=-2/(3-4v)]\), \(\gamma[=-(2-4v)/2]\), \(\delta[=1/(3-4v)]\) depend on [PERSON]'s ratio \(\upsilon\) only.  
The dimensions of the problem are taken as \(R=1\) unit, \(w=b=4\) unit, as shown in Figure 5. The rigid circular opening's center is a singular point. Both of the displacement components have a value of zero along the edge of the circular arc AE. Both the normal displacement  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Adaptive Analysis Results (Original Uniform Mesh with**} \\ \cline{2-5}
**Recovery Type** & \multicolumn{2}{c}{**615 Elements and 1218 DOF, 4\% Target Error)**} \\ \cline{2-5}  & **FEM** & **Projected** & **Adaptive Mesh Properties** \\ \cline{2-5}  & **Error** & **Error** & **N** & **DOF** \\ \hline LS (conventional patches) & 9.43 & 9.64 & 5629 & 11,520 \\ RPI (influence zones-MQ-Cir.) & 9.43 & 9.26 & 5872 & 12,026 \\ RPI (conventional patches-MQ) & 9.43 & 9.30 & 5606 & 11,490 \\ RPI (influence zones-TSP-Cir.) & 9.43 & 9.13 & 5950 & 12,086 \\ RPI (influence zones-TSP-Rect.) & 9.43 & 9.22 & 5454 & 11,186 \\ RPI (conventional patches-TSP) & 9.43 & 9.23 & 5587 & 11,454 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Adaptive finite element analysis results (overall errors, total number of elements (N) and DOF) for incompressible-elastic plate with 4% target error using quadrilateral elements.  
component and the shear stress are equal to zero along the CB and DE border symmetry lines. Along boundaries AB and CD, the tractions computed from Equations (23)-(25) are imposed. The numerical quadrature using 12- Gauss points per element side is used to calculate the corresponding load vectors. The plate problem is analyzed by RPI-based error estimator incorporated adaptive procedures and the target error in energy norm is kept as 2%. The uniform subdivision of four-node quadrilateral elements is used. The discretized domain is shown in Figure 5. The numerical results of the convergence of error and effectivity for four-node quadrilateral elements with one-point reduced integration for penalty term is given in Table 8. The adaptive mesh refinement results, i.e., degrees of freedom (DOF) and number of element (N) after mesh was updated for 2% target error in incompressible plate with rigid circular opening, considering meshless and conventional patches are given in Table 9. The updated mesh plots for target error of 2% are depicted in Figure 5.