# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 3 Locking Detection, Assessment, and Resolution
### Strain Energy Components
Note that the vector of displacement dofs is related to the displacement field \(\overset{e}{\boldsymbol{u}}^{hpq}\) with the standard interpolation formula of the element \(\overset{e}{\boldsymbol{u}}^{hpq}=\overset{e}{\boldsymbol{N}}\overset{e}{ \boldsymbol{q}}^{hpq}\), where \(\overset{e}{\boldsymbol{N}}\) stands for the shape function matrix of the element. Because of the 3D-based shell models applied in our research, further decomposition of the strain energy will be performed approximately. The exact decomposition is easy for the conventional shell models with the degrees of freedom defined on the mid-surface. Then, the dofs corresponding to the even and odd powers of the thickness contribute to the membrane and bending strains, respectively. In the 3D-based formulation, the dofs are defined along (or through) the thickness, and extracting their membrane and bending contributions needs much more complex procedure explained in [32]. The approximate procedure requires replacement of the top and bottom displacements of the symmetric thin-walled structure with their sums and differences which contribute to the mid-surface displacements and rotations. This allows for distinguishing the following components of the in-plane strains  
\[\varepsilon_{\alpha\beta}\ \approx\ \gamma_{\alpha\beta}+\kappa_{\alpha\beta} \tag{25}\]  
where \(\gamma_{\alpha\beta}\) stand for the mid-surface strains (membrane strains) and \(\kappa_{\alpha\beta}\) are the three-dimensional counterparts of the change of curvature tensor (bending strains). With the above division, the in-plane density \(v_{l}\) of strain energy can be decomposed into three parts (membrane, bending, and coupling ones):  
\[v_{l}\ \approx\ v_{m}+v_{b}+v_{c} \tag{26}\]  
that contain products of the membrane strains, bending strains, and their combination, respectively. Consequently, one also has  
\[U_{l}\ \approx\ U_{m}+U_{b}+U_{c} \tag{27}\]  
Calculation of the two components of the right-hand side of (25) on the element level needs adequate transformation of the numerical representation of element longitudinal strains \(\varepsilon_{l}\), defined as a matrix product of the corresponding part of the strain-displacement matrix and the vector of displacement dofs. For the 3D-based hierarchical shell formulation, presented in [38; 45], this transformation reads  
\[ \begin{array}{rcl}\varepsilon_{l}&=&\overset{\epsilon}{B}_{l}\overset{ \epsilon}{q}^{hpq}\\ &=&[\mathbf{B}_{t},\mathbf{B}_{b},\mathbf{B}_{o}]\times[\mathbf{q}_{t},\mathbf{q}_{b},\mathbf{q}_{o}] ^{T}\\ &=&[\mathbf{B}_{t}+\mathbf{B}_{b},\mathbf{B}_{t}-\mathbf{B}_{b},\mathbf{B}_{o}]\times[\frac{1}{2}( \mathbf{q}_{t}+\mathbf{q}_{b}),\frac{1}{2}(\mathbf{q}_{t}-\mathbf{q}_{b}),\mathbf{q}_{o}]^{T}\\ &=&[\mathbf{B}_{s},\mathbf{B}_{\phi},\mathbf{B}_{o}]\times[\mathbf{q}_{s},\mathbf{q}_{\phi},\mathbf{q} _{o}]^{T}=\mathbf{B}_{s}\,\mathbf{q}_{s}+\mathbf{B}_{\phi}\,\mathbf{q}_{\phi}+\mathbf{B}_{o}\,\mathbf{q }_{o}\\ &=&\mathbf{\gamma}+\kappa+\mathbf{r}\\ &\approx&\mathbf{\gamma}+\kappa\end{array}  \tag{28}\]  
where the sub-blocks \(\mathbf{B}_{t}\), \(\mathbf{B}_{b}\), and \(\mathbf{B}_{o}\) correspond to the top \(\mathbf{q}_{t}\), bottom \(\mathbf{q}_{b}\), and all other \(\mathbf{q}_{o}\) displacement dofs of the element dofs vector \(\overset{\epsilon}{q}^{hpq}\). Location of the dofs of three types is on the top and bottom surfaces of the shell element, and apart from these two surfaces, respectively. In turn, the sub-blocks \(\mathbf{B}_{s}\), \(\mathbf{B}_{\phi}\) correspond to the mid-surface displacement dofs \(\mathbf{q}_{s}=\frac{1}{2}(\mathbf{q}_{t}+\mathbf{q}_{b})\) and rotational dofs \(\mathbf{q}_{\phi}=\frac{1}{2}(\mathbf{q}_{t}-\mathbf{q}_{b})\). The approximate character of the performed transformation results from neglecting the mentioned other dofs in the resultant decomposition into the membrane and bending strain contributions. In the criterion for detection of the dominance of bending strains, the solutions of two local problems from the equilibrated residual method are applied.