Abstract
The Zienkiewicz–Zhu (ZZ) super-convergent patch recovery technique based on a node neighborhood patch configuration is used most widely for recovery of the stress field of a finite element analysis. In this study, an improved ZZ recovery technique using element neighborhood patch configuration is proposed. 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 and improves the performance of the standard ZZ recovery technique. The effectiveness and reliability of the improved ZZ recovery approach is demonstrated through plane elastic and plastic plate problems. The problem domain is discretized with triangular and quadrilateral elements of different sizes. A comparison of the quality of error estimation using the ZZ recovery of derivative field and recovery of the displacement field using similar element neighborhood patch configurations is also presented. The numerical results show that the ZZ recovery technique and the displacement recovery technique, using a modified patch configuration, yield better results, convergence rate, and effectivity as compared with the standard ZZ super-convergent patch recovery technique. It is concluded that the improved ZZ recovery technique-based adaptive finite element analysis is very effective for converging a predefined accuracy with a significantly smaller number of degrees of freedom, especially in an elastic problem. It is also concluded that the improved ZZ recovery technique captures the plastic deformation problem solution errors more reliably than the standard ZZ recovery technique.
Highlights
The finite element method is the most used numerical tool for solving industrial problems
The quality of recovery procedure, i.e., error convergence, effectivity, and adaptively improved meshes are obtained by adaptive finite element analysis of two plane elastic plate problems, for which an exact solution is available in [4], employing the ZZ stress recovery method with the standard and a modified patch configuration, and a modified patch-based displacement recovery method
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
Summary
The finite element method is the most used numerical tool for solving industrial problems. The classical super-convergent patch recovery (SPR) technique or ZZ recovery technique [4] has been proposed to recover the lost accuracy and continuity of the stress field by interpolating from a stress surface fitted to the super-convergent stress points in a node neighborhood patch They emphasized that the error estimation would be asymptotically exact if the recovery of the finite element solution was super convergent. It is clear that the quality of recovery-based error estimation depends on the approach for recovering the stress or displacement field that satisfies the equilibrium and boundary conditions. The ZZ recovery technique uses the least-squares fitting of stress field by the same order polynomial as that present in the basis function over a patch of neighborhood nodes.
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