# Improvement of the Zienkiewicz-Zhu Error Recovery Technique Using a Patch Configuration
## 6 Square Plate Example  
The quality of the stress (ZZ) and displacement recovery is demonstrated through the finite element analysis of a plane square plate example. The example was used by [PERSON] and [PERSON] [4] to validate the effectiveness of the ZZ error estimation. The example was a 1 \(\times\) 1 square domain subjected to the action of body forces (\(b_{x}\), \(by\)). The exact solution for displacement (\(u\), \(v\)) and body forces in the form of polynomials, are given in Equations (20) to (22):  
\[u=0;\ y=-x\ y(1-x)\ \times\ (1-y), \tag{20}\]  
\[b_{x}=(\alpha+\beta)\ \times\ (1-2x)\ \times\ (1-2y), \tag{21}\]  
\[b_{y}=-2\ \beta\ y\ \times(1-y)-(\alpha+2\ \beta)\ 2x\ \times(1-x), \tag{22}\]  
The constants \(\alpha\) and \(\beta\) are given as:  
\[\alpha=E\ \
u/[(1-2\
u)\ (1+\
u)];\ \beta=E/[2(1+\
u)], \tag{23}\]  
where \(E\) and \(\
u\) are the modulus of elasticity and [PERSON]'s ratio, respectively, with values of 1.0 N/mm\({}^{2}\) and 0.3, respectively. A two-dimensional computer software is developed, incorporating the adaptive finite element procedures explained above, to simulate the elastic and plastic plate problems. The square plate domain is discretized as regular and irregular meshes using triangular/quadrilateral elements. The triangular/quadrilateral element meshes for plate domain are shown in Figure 3. The error convergence with different orders of mesh in the finite element analysis and effectivity of error estimation using a standard/modified patch-based ZZ stress recovery and a modified patch-based displacement recovery, for triangular elements, are listed in Tables 1-4. The analysis results for error convergence and effectivity using quadrilateral elements are shown in Tables 5 and 6. The solution errors were measured in the energy norm. The convergence rate of error with regular subdivision for the linear/quadratic triangular elements and the linear quadrilateral element in the original solution, the recovered solution using a standard patch-based ZZ recovery, a modified patch-based ZZ recovery, and a modified patch-based displacement recovery are obtained, respectively, as (0.97875, 1.37313, 1.87252, and 1.88603), (1.97848, 2.50622, 3.19768, and 2.88449), and (1.00169, 1.65603, 2.35230, and 2.02723). The error convergence rates are obtained in finite element results as (0.58530, 0.848150, 0.95638, and 1.11017), (1.07415, 1.26349, 1.57042, and 1.61414), and (0.54163, 0.98011, 0.94495, and 0.98600), respectively, for linear/quadratic triangular elements and linear quadrilateral element considering irregular subdivisions. Figure 3: Regular/irregular subdivided element meshes for square plate domain. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Mesh Size (1/h)**} & \multirow{2}{*}{
\begin{tabular}{c} **FEM Error** \\ **(\(\times\)10\({}^{-3}\))** \\ \end{tabular} } & \multicolumn{2}{c}{**ZZ (Standard Patch)**} & \multicolumn{2}{c}{**ZZ (Modified Patch)**} & \multicolumn{2}{c}{**LS (Modified Patch)**} \\ \cline{3-8}  & & **Error (Stress)** & **Error (Stress)** & **Error (\(\times\)10\({}^{-3}\))** & **Effectivity** & **Error (Stress)** & **Error (Displacement)** (\(\times\)10\({}^{-3}\))** & **Effectivity** \\ \hline
1/4 & 93.7491 & 144.2486 & 1.4313 & 50.5738 & 0.97347 & 58.4784 & 0.929995 \\
1/16 & 24.4409 & 22.6413 & 1.2449 & 3.2715 & 0.98795 & 4.4621 & 0.989587 \\
1/32 & 12.2480 & 8.29942 & 1.14508 & 1.0301 & 0.99498 & 1.1581 & 0.997014 \\ \hline Rate of Conv. & 0.97875 & 1.37313 & 1.87252 & & 1.88603 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Solution error and global effectivity for the plate problem using stress/displacement recovery and patch configuration (regular linear triangular element).