# Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations
## 7 Conclusions  
The present study explored the effectiveness and applicability of the mesh-free MLS interpolation-approach-based error recovery of displacement/stress in incompressible elastic finite element analysis. The moving least-squares (MLS) interpolation technique considering radial weights over circular support domains was employed for the recovery of solution errors and the errors were quantified in energy norms. The incompressible elastic plate problems, with six-node triangular and four-node quadrilateral discretization schemes, were solved to investigate the order of the errors, the rate of the error convergence, the effectiveness (\(\theta\)), and the updated meshes for the desired error limit. It was found from the computational results that the recovered field-variable type, the types of elements, and the patch formation had strong effects on the quality of the solution-error recovery. The error convergence obtained with the help of the mesh-free recovery was found to be better than that obtained with the mesh-dependent error-recovery technique. It was also found that the error-recovery performance was better for the quadrilateral elements than for the triangular elements selected for the meshing of the problems. The error-convergence rates were computed in the original FEM solution, in the post-processed solution using mesh-free MLS based displacement, and in the mesh-dependent patch-based least-squares-based displacement, and the stress-recovery values were (0.9777, 2.2501, 2.0012, 1.6710, and 1.5436) and (0.9736, 2.0869, 1.6931, 1.8806, and 1.4973) for the four-node quadrilateral and six-node triangular meshes, respectively. These illustrative examples demonstrate the mesh-free MLS based error recovery technique's ability to extract errors effectively and efficiently in the finite element analysis of incompressible elastic problems.  
Conceptualization, M.A.; methodology, N.B.K.; validation, N.B.K. and S.A.; formal analysis, M.A. and S.A.; investigation, N.B.K.; resources, N.B.K.; writing--original draft preparation, M.A.; writing--review and editing, N.B.K.; visualization, S.A.; supervision, M.A.; project administration, S.A.; funding acquisition, M.A. All authors have read and agreed to the published version of the manuscript.  
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the Large Group Research Project under grant number (R.G.P2/424/44).  
Not applicable.  
The authors also acknowledge to the Dean of the Faculty of Engineering for his valuable support and help.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.  
\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{2-10}  & & & **Displac. Recovery** & & **Stress Recovery** & & **Displac. Recovery** & & **Stress Recovery** \\ \cline{2-10}  & **N** & **DOF** & **N** & **DOF** & **N** & **DOF** & **N** & **DOF** \\ \hline Quadrilateral & 266 & 594 & 694 & 1490 & 419 & 926 & 760 & 1622 & 609 & 1314 \\ Triangular & 124 & 558 & 151 & 676 & 121 & 548 & 218 & 962 & 188 & 842 \\ \hline \end{tabular}
\end{table}
Table 9: Updated mesh-element numbers (N) and degrees of freedom (DOF) in incompressible plate with rigid circular inclusion adaptive analysis with displacement/stress-recovery techniques considering quadrilateral/triangular elements and mesh-free/mesh-dependent patches (1% desirable error).