## 5 Benchmark Example Applications  
### Application to Compressible Elastic Plate  
The finite element formulation for compressible elastic problem described by [PERSON] [20] is followed in this study. The performance of the stress and displacement recovery techniques is demonstrated through the convergence characteristics, and required meshes for intended precision in the case of a two-dimensional compressible elastic body subjected to body forces. The elements and node connectivity data are prepared as an input for regular triangular elements' discretization of the problem domain. An in-house preprocessor is developed to discretize the problem domain into irregular linear and quadratic triangular elements. The plate domain is discretized by triangular elements in regular and irregular mode as shown in Figure 1. The number of nodes for three fineness levels in regular discretization are (25, 289 and 1089) and (81, 625 and 2401) respectively for linear and quadratic triangles whereas for irregular discretization, the number of nodes are (23, 326 and 1329) and (74, 735 and 2804) respectively for linear and quadratic triangles. Figure 2 shows the node patches used in least square fitting for displacement and stress error recovery. The example was used by [PERSON] and [PERSON] [6]. It presents no singularity so that theoretical rate of convergence of the proposed recovery techniques can be easily verified with other recovery techniques. Domain = \(\Omega\) [0 \(\times\) 0] \(\times\) [1 \(\times\) 1], \(u\) = \(v\) = 0 on \(\Gamma\)  
Exact solution:  
\[u=0;v=-x\ y(1-x)\ (1-y) \tag{17}\]  
\[\varepsilon_{x}=0;\varepsilon_{y}=-x+x^{2}+2 xy-2x^{2}y, \tag{18}\]  
\[\varepsilon_{xy}=-y+2 xy+y^{2}-2 xy^{2}, \tag{19}\]  
Body forces:  
\[b_{x}=(\alpha+\beta)\cdot(1-2x)(1-2y) \tag{20}\]  
\[b_{y}=-2\ \beta\ y\ (1-y)-(\alpha+2\ \beta)\ \ 2x\ (1-x)\ \, \tag{21}\]  
The \(\alpha\) and \(\beta\) are constant and are given by the following relations:  
\[\alpha=E\cdot v/[(1-2\cdot v)\ (1+\
u)];\ \ \ \ \ \ \beta=E/[2\cdot(1+\
u)] \tag{22}\]  
where \(u\), \(v\) are the displacement in \(x\) and \(y\) direction, \(E\) and \(\
u\) is the Modulus of elasticity and Poisson's Ratio respectively with a value of \(1.0\) N/mm\({}^{2}\) and \(0.3\)[20]. Figure 1: Plate domain: regular/irregular discretization schemes with boundary conditions. The error estimation quality results, i.e., error convergence and global effectivity in finite element solution, and recovered solution employing displacement and stress error recovery with increasing order of triangular element (regular/irregular) fineness, are given in Tables 1-4. The finite element solution absolute errors of plate problem, i.e., the difference between the exact (or recovered) values of u (or \(\sigma\), \(\epsilon\)) and respective finite element analysis values, \(u^{h}\) (or \(\sigma^{h}\), \(\epsilon^{h}\)) are computed using the Equation (7) and the solution errors are represented in L\({}_{2}\)-norm of stresses, displacement, and strains using Equations (8)-(10). Figure 3 depicts the rate of convergence of error at increased order of fineness, i.e., at 1/16 and 1/36 mesh size under displacement and stress error recovery procedures in linear and quadratic triangular elements. For comparison of results of finite element analysis errors of stresses in L\({}_{2}\)-norm, they are also matched with errors of stresses in energy norm (Zienkiewicz-Zhu technique). The ZZ recovery technique-based error estimation quality results in energy norm, i.e., error convergence and global effectivity are presented in Table 5 [24]. Table 5 also shows the displacement recovery-based results in energy norm for comparison purposes [20].