# Improvement of the Zienkiewicz-Zhu Error Recovery Technique Using a Patch Configuration
## 10 Conclusions  
In this study, the classical ZZ recovery technique based on the standard node neighborhood patch configuration was improved using a larger element neighborhood patch configuration. The derivatives at any point in the element domain or node domain are calculated through better approximated stress/displacement fields through the least-squares fit of computed nodal stress/displacement using a higher order polynomial over an element patch that consists of the union of the elements surrounding the element under consideration. It is demonstrated through elastic and plastic plate examples involving four-node quadrilateral elements and three- and six-node triangular elements that the performance of the ZZ recovery technique is greatly improved using a modified patch configuration. It is observed that the relative error decreases and the convergence rate increases with mesh refinement. The quality of error estimation using the ZZ recovery of the derivative field, considering the standard and a modified patch configuration, was also compared with that of error estimation using extracted stress from recovery of a more accurate displacement field using the element neighborhood patch configuration in terms of effectivity, rate of convergence, and adaptively refined meshes. The quality of error  
Figure 8: Adaptively modified mesh in benchmark plate problem using stress/displacement recovery and patch configuration (triangular elements initial mesh = 395.4% target error).  
Figure 9: Adaptively modified mesh in benchmark plate problem using stress/displacement recovery and patch configuration (quadrilateral elements initial mesh = 99.4% target error).  
estimation using the ZZ stress recovery technique employing an element neighborhood patch configuration is better as compared with the quality of error estimation using the displacement recovery technique. It is concluded that the improved ZZ recovery technique-based finite element analysis is very effective for converging a predefined accuracy in a solution with a significantly smaller number of degrees of freedom. The numerical results also show that the improved ZZ recovery technique using an element neighborhood patch configuration can effectively be incorporated into the adaptive computations, as performance is good with both coarser and finer element meshes. It is concluded that the improved ZZ recovery technique captures the large deformation errors more reliably than the standard ZZ recovery technique.  
Conceptualization, M.A. and D.S.; methodology, M.A.; software, D.S.; validation, S.A. and M.A.A.; formal analysis, M.A. and S.A.; investigation, M.A.; resources, S.A.; data curation, M.A.A.; writing--original draft preparation, M.A.; writing--review and editing, S.A. and M.A.A.; visualization, S.A.; supervision, D.S.; project administration, M.A.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 General Research Project under grant number (R.G.P2 /73/41). The authors acknowledge to the Dean of the Faculty of Engineering for his valuable support and help.  
Not applicable.  
Exclude.  
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.