# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 3 Locking Detection, Assessment, and Resolution
### The Idea and Algorithm of a Posteriori Phenomenon Detection
#### 3.1.1 Solutions from the Equilibrated Residual Method  
Let us recall now the equilibrated residual method of a posteriori error estimation, invented by [PERSON] and [PERSON] [43]. All relations of this section are taken from the cited work. Here, we apply this method to the estimation of the numerical solutions of the 3D-based hierarchical shell problems [46]. The principal relation of the method is [44]  
\[ \begin{array}{rcl}B(\mathbf{e},\mathbf{v})&=&B(\mathbf{u}^{HPQ}-\mathbf{u}^{hpq},\mathbf{v})\,=\,B( \mathbf{u}^{HPQ},\mathbf{v})-B(\mathbf{u}^{hpq},\mathbf{v})\\ &=&L(\mathbf{v})-B(\mathbf{u}^{hpq},\mathbf{v}),\,\,\,\forall\,\mathbf{v}\in\mathcal{V}^{HPQ}(V) \end{array}  \tag{14}\]  
Above, \(\mathbf{u}^{HPQ}\) stands for either the exact solution of the problem or sufficiently accurate approximation of such a solution. The quantity \(\mathbf{e}\) is the error vector corresponding to the assessed numerical solution \(\mathbf{u}^{hpq}\). The kinematically admissible displacements \(\mathbf{v}\equiv\mathbf{v}^{HPQ}\) belong to the space \(\mathcal{V}^{HPQ}\) defined in analogy to the spaces introduced for (4) and (6). The discretization parameters \(H\), \(P\), and \(Q\) are the counterparts of \(h\), \(p\), and \(q\) from the relation (6). In the case of the exact solution (\(\mathbf{u}^{HPQ}\equiv\mathbf{u}\)), one deals with \(1/H,P,Q\rightarrow\infty\), while in the case of the approximation, one deals with the finite values of the discretization parameters \(H\), \(P\) and \(Q\). After noticing that \(\mathbf{e}=\mathbf{u}^{HPQ}-\mathbf{u}^{hpq}\) and introduction of this definition into the error functional (14), the latter simplifies to:  
\[B(\mathbf{u}^{HPQ},\mathbf{v})\,\,\,=\,\,\,L(\mathbf{v}),\,\,\,\forall\,\mathbf{v}\in\mathcal{ V}^{HPQ}(V) \tag{15}\]  
This way one searches for the approximation of the exact solution (displacements) of the problem instead of the problem error itself. However, once the displacements are determined, the error can be calculated from the above error definition. The presented approach is applied in the works [30,46], for example. Let us now follow the work [43] and divide the domain \(V\) into \(E\) subdomains \(V_{e}\), corresponding to finite elements \(e=1,2,\ldots,E\). The global functional (15) and its left- and right-hand side components can now be presented as a sum of the element contributions:  
\[ \begin{array}{rcl}B(\mathbf{u}^{HPQ},\mathbf{v})&=&\sum_{e=1}^{E}\overset{e}{B}(\bm {u}^{HPQ},\overset{e}{\
u})\\ \\ L(\mathbf{v})&=&\sum_{e=1}^{E}\overset{e}{L}(\overset{e}{\
u})=\sum_{e=1}^{E} \overset{e}{L}(\overset{e}{\
u})-\sum_{e=1}^{E}\overset{e}{\beta}(\overset{e }{\
u})\\ &=&\sum_{e=1}^{E}\left[\overset{e}{L}(\overset{e}{\
u})+\int_{S_{e}\backslash (S_{P}\cup S_{W})}\overset{e}{\
u}^{T}\left\langle\overset{e}{\mathbf{r}}(\mathbf{u} ^{hpq})\right\rangle\,dS_{e}\right]\end{array}  \tag{16}\]  
where the element admissible displacements are defined as the global admissible displacements projected onto elements \(V_{e}\): \(\overset{e}{\
u}\equiv\mathbf{v}|_{V_{e}}\). As demonstrated by [PERSON] and [PERSON] [43], the sum of the auxiliary element functionals \(\overset{e}{\beta}(\overset{e}{\
u})\) is equal to 0, as the internal load consistency condition must hold. Above, the vectors \(\left\langle\overset{e}{\mathbf{r}}(\mathbf{u}^{hpq})\right\rangle\) represent the interelement loading due to equilibrated stresses. These stresses are defined in [43,44], for example. Their definition is as follows  
\[\left\langle\overset{e}{\mathbf{r}}(\mathbf{u}^{hpq})\right\rangle\,\,\,=\,\,\,\, \overset{fe\,e}{\mathbf{r}}(\mathbf{u}^{hpq})+\overset{ef\,f}{\mathbf{r}}(\mathbf{u}^{hpq}) \tag{17}\]  
where  
\[ \begin{array}{rcl}\overset{e}{\mathbf{r}}(\mathbf{u}^{hpq})&=&\mathbf{H}(\overset{e}{ \
u})\overset{e}{\sigma}(\mathbf{u}^{hpq}),\\ \overset{f}{\mathbf{r}}(\mathbf{u}^{hpq})&=&\mathbf{H}(\overset{e}{\
u})\overset{f}{\sigma }(\mathbf{u}^{hpq}),\end{array}  \tag{18}\]and  
\[H(\overset{e}{\boldsymbol{v}})=\left[