# Mesh-Free MLS-Based Error-Recovery Technique for Finite Element Incompressible Elastic Computations
## 6 Discussion
In the mesh-dependent stress-recovery technique, there is a chance of a reduction in the precision of the recovery for the boundary nodes, since fewer points are available. The mesh-free recovery technique eliminates such difficulties, since more points are available in the support domain (mesh-free patch). It is evident from the tables that the error-recovery performance of the four-node-element discretization of the incompressible elastic problem was better than that of the performance of the six-node-element discretization. The tables also show that the performance of the displacement-based recovery was better than that of the stress-based recovery for both mesh-free and mesh-dependent patch formation. The error-convergence rates were computed in the original FEM solution, the post-processed solution using the mesh-free MLS-based displacement/stress-recovery scheme, and the mesh-dependent patch LS-based displacement/stress-recovery scheme, yielding values of (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. The mesh-dependent patch (mesh-dependent-element patch) for the MLS-based displacement-recovery scheme was also considered with four-node and six-node elements in order the study the effect of patch configuration on MLS-based recovery. The analysis results for the standard patch for the MLS interpolation are given in Table 6. It is evident from the tables that the recovery-based error estimations were affected by the patch formation and that the mesh-dependent MLS interpolation had a lower convergence rate than and was effective compared to the mesh-free MLS interpolation for solution-error recovery. The convergence rates in the mesh-less and mesh-dependent MLS interpolation for field-variable-error recovery were 2.2501 and 2.0253 and 2.0869 and 1.7001, respectively, for the four-node quadrilateral and six-node triangular meshes However, the MLS interpolation-based error recovery using mesh-dependent patches provided better error-recovery quality than the least-squares interpolation-based error recovery using mesh-dependent patches. Tables 7 and 8 show the error convergence and effectiveness of the displacement/stress-recovery procedures considering mesh-free and mesh-dependent patches in the incompressible elastic plate problem with rigid circular inclusion. The computational results showed similar trends for the incompressible elastic plate problem, i.e., the performance of the displacement-based recovery was better than that of the stress-based recovery for both mesh-free and mesh-dependent patch formation. The error-recovery performance was better for the quadrilateral elements than for the triangular elements selected for the meshing of the problems. In all the adaptive finite element analyses, irregular meshes were considered. The robustness of the mesh-free error-estimation technique was also tested using irregular meshes, so initial irregular meshes were considered in the adaptive finite element analysis. The updated mesh plots for the desired accuracy with various recovery procedures and meshing schemes are shown in Figures 4, 5, 8 and 9. The DOF and element numbers after the mesh improvement for the desired accuracy with the displacement and stress-recovery techniques and quadrilateral/triangular meshes are presented in Tables 5 and 9. It was observed from the mesh plots that the mesh-free error-estimation scheme was more efficient than the mesh-dependent patch-based error estimation. It was inferred that in general, the number of elements (N) necessary to attain the desired accuracy with the displacement-error-recovery scheme was lower than with the stress-error-recovery scheme with both mesh-free and standard patches and with all the types of element mesh. However, the lowest number of elements was required for the desired accuracy in the mesh-free stress-recovery technique, which suggests that this technique is highly effective in the recovery of stress.