## 3 Least Squares Interpolation-Based Displacement Recovery Technique  
The recovery of field variable (displacement) is obtained by least squares fit of the computed nodal field variable (\(u\)) using a higher order polynomial over an element patch (Figure 1). The following functional is minimized for least square fitting:  
\[\pi_{f}(a)=\frac{1}{2}\sum_{i=1}^{np}\left[u_{i}^{h}(x_{i},\ y_{i})-u(x_{i}, \ y_{i})\right]^{2}, \tag{4}\]  
where \(u_{i}\ (x_{i},\ y_{i})=P_{i}\ (x_{i},\ y_{i})\.\)\(a\) in which \(a\) is the vector of unknown parameters \(a_{u}\) and \(a_{v}\), and \(P_{i}=\left[1,\ x_{i},\ y_{i},\ x_{i}^{2},\ x_{i}y_{i},\ y_{i}^{2},\ \dots\dots\right]\).  
Minimization condition of \(\pi_{f}(a)\) implies that \(a\) satisfies the following relation:  
\[\sum_{i=1}^{np}P_{i}^{T}(x_{i},\ y_{i})\cdot P_{i}(x_{i},\ y_{i})\cdot a=\sum_{i=1 }^{np}P_{i}^{T}(x_{i},\ y_{i})x\ u_{i}^{h}(x_{i},\ y_{i})\,, \tag{5}\]  
Equation (5), in simplified form, can be written as:  
\[a=A^{-1}\ b, \tag{6}\]  
where \(A=\sum_{i=1}^{np}P_{i}^{T}(x_{i},\ y_{i})P_{i}(x_{i},\ y_{i})\) and \(b=\sum_{i=1}^{np}P_{i}^{T}(x_{i},\ y_{i})u_{i}^{h}(x_{i},\ y_{i})\)  
**4. L\({}_{2}\)-Norm-Based Error Quantification and Mesh Improvement**  
The error in computed state variable or gradients, i.e., displacement (\(u\)), stress (\(\sigma\)) or strain (\(\epsilon\)), \(e_{u}\)* (or \(e_{\sigma}\)*, \(e_{\epsilon}\)*) is equal to the difference between the exact (or recovered) values of \(u\) (or \(\sigma\), \(\epsilon\)) and respective numerical values, \(u^{h}\) (or \(\sigma^{h}\), \(\epsilon^{h}\)), i.e.,  
\[e_{u}^{*}=u-u^{h},\ e_{\sigma}^{*}=\sigma-\sigma^{h},\ e_{\epsilon}^{*}= \epsilon-\epsilon^{h}, \tag{7}\]  
The norms are used to measure the finite element solution errors. L\({}_{2}\)-norm giving integral measure of the error directly for state variable or gradients may be defined as follows:  
\[||e||_{L_{2}}=\left[\int_{\Omega}e_{u}^{*\intercal}e_{u}^{*}d\Omega\right]^{ 1/2}, \tag{8}\]  
\[||e||_{L_{2}}=\left[\int_{\Omega}e_{\sigma}^{\intercal}e_{\sigma}^{*}d\Omega \right]^{1/2}, \tag{9}\]  
\[||e||_{L_{2}}=\left[\int_{\Omega}e_{\epsilon}^{\intercal}e_{\epsilon}^{*}d \Omega\right]^{1/2}, \tag{10}\]  
where \(\Omega\) represents problem domain.  
The effectivity (\(\theta\)) of the computed solution error is defined as the ratio of evaluated error and exact error. When the mesh size approaches to zero, the problem effectivity (\(\theta\)) converges to one for an asymptotically exact error estimator [5]. The accuracy (\(\eta\)) of a finite element solution may be defined as follows:  
\[\eta=\frac{||e^{*}||}{||\sigma^{*}||}\, \tag{11}\]  
\[||\sigma^{*}||^{2}=||\sigma^{h}||^{2}+||e||_{\epsilon}^{2}, \tag{12}\]  
The solution is satisfactory if accuracy (\(\eta\)) is within the aimed accuracy (\(\eta_{allourable}\)). If solution accuracy is more than the aimed accuracy, the mesh may be adaptively improved. The permissible global error is given as follows:  
\[||e||_{allourable}=\eta_{allourable}||^{||e||} \tag{13}\]  
The allowable error in the \(i\)th element is calculated using the following relation:  
\[||e||_{allourable(i)}=\frac{||e||_{allourable}}{\sqrt{N}}, \tag{14}\]  
The element modification parameter \(\xi_{i}\) guides the refinement:  
\[\xi_{i}=\frac{||e||_{allourable}}{||e||_{allourable}\ (i)}, \tag{15}\]If \(\xi_{i}\) is greater than 1, improvement of mesh is needed. The modified element size (\(l_{mod}\)) is found with the help of the following equation:  
\[h_{mod}=\frac{h_{old}}{\xi_{i}^{1/p}}\, \tag{16}\]