# Improvement of the Zienkiewicz-Zhu Error Recovery Technique Using a Patch Configuration
## 9 Discussion  
The accuracy of the recovery of stress field using the standard ZZ recovery technique has been improved through a larger patch configuration based on an element neighborhood pattern. In the so-called super-convergent patch recovery technique (ZZ technique), a nodal patch which represents the union of the element surrounding the node is defined and the recovery is made in the patch by least-squares fitting of polynomial expansion to the super-convergent stresses at some known points (usually gauss points) inside the elements. The improved recovery procedure is based on recovery of the stress field in the least-squares sense over an element patch that consists of the union of the elements surrounding the element under consideration. The proposed patch configuration provides more sampling points near the boundary and improves the performance of the ZZ recovery at boundary regions. The effectiveness and efficiency of the improved ZZ recovery approach is tested on benchmark plane elastic plate problems (given in [6]) involving four-node quadrilateral elements and three- and six-node triangular elements. The finite element analysis numerical results are also obtained using the same elemental patch for the extraction of more accurate displacement field. The computational results employing the recovery technique for plate problems are presented in Tables 1-9. The results obtained for the improved ZZ  
Figure 6: Adaptively modified meshes in sheet stretching problem using stress/displacement recovery and patch configuration (triangular elements initial mesh = 737, with 14% target error). recovery technique and displacement recovery technique are compared with the ZZ super-convergent patch recovery technique. The solution errors shown in the tables depict that the order of error obtained by employing recovery procedures is much lower than the FEM solution error and the error convergence is also higher as compared with the FEM solution error with a reduction in mesh size. The effect on error order, although with a reduced rate, is further improved taking higher order elements for domain discretization. It is clear from the tables that the ZZ recovery technique and displacement recovery technique using a modified patch configuration yield better results as compared with the standard ZZ super-convergent patch recovery technique when applied on two benchmark problems in elasticity. The rate of convergence and effectivity of the ZZ recovery technique using a modified patch configuration is higher than the displacement recovery technique using a modified patch configuration, indicating a higher efficiency and effectivity of the proposed modified ZZ recovery procedure. The optimal performance of the ZZ recovery technique using a modified patch is obtained with a four-node linear quadrilateral element. The improved ZZ recovery technique-based error estimation for guiding the error controlling strategies of adaptive analysis is also demonstrated. Adaptively improved meshes of the elastic problem analyses are obtained for the ZZ super-convergent patch recovery, ZZ recovery technique, and displacement recovery technique using a modified patch configuration at a target error of 4%. Table 10 shows the global errors, the number of elements, and the DOF in refined meshes, obtained from the triangular and quadrilateral element meshes, to bring the solution accuracy to the target error level. The number of elements required to achieve the target error depends on the global error and distribution of error. The adaptively refined meshes with target error using different recovery-based error estimations are given in Figures 7-9. The initial meshes are adaptively refined to bring the solution error within the target error limit. It is clear from the figures that  
The performance of the improved ZZ recovery technique is also assessed under large or plastic deformation using a sheet stretching process analysis.