# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 3 Locking Detection, Assessment, and Resolution
### Strain Energy Components  
In this work, we apply the proposed approach to the local solutions from the equilibrated residual method and to their sum. To apply this approach, one has to take into consideration that the strain energy approximation from (8) corresponds to the structures of the infinitely small thickness, for which the first-order theory can approximate the three-dimensional description. As we intend to check the locking phenomena in the structures of the finite thickness, described with the hierarchical models corresponding to higher-order shell theories, the decomposition of the three-dimensional strain energy into component energies typical for the first-order models can only be done approximately. Let us start with the following decomposition of the strain vector into its longitudinal, transverse and shear parts  
\[\boldsymbol{\varepsilon}\ =\ [\boldsymbol{\varepsilon}_{l},\boldsymbol{ \varepsilon}_{n},\boldsymbol{\varepsilon}_{s}]^{T} \tag{21}\]where \(\varepsilon_{l}=[\ldots,\varepsilon_{n\beta},\ldots]^{T}\), \(\varepsilon_{n}=[\varepsilon_{33}]^{T}\), \(\varepsilon_{s}=[\ldots,\varepsilon_{3\alpha},\varepsilon_{\alpha 3},\ldots]^{T}\) and \(\alpha,\beta=1,2\) are the local directions tangent to longitudinal directions of the thin-walled structure, and the index 3 corresponds to the local transverse direction. The analogous decomposition of the stress vector \(\sigma\) is also possible. Such decompositions lead to the division of the density \(v\) of the total strain energy into the following components:  
\[ \begin{array}{rcl}v&=&\sigma^{ij}\varepsilon_{ij}\\ &=&\sigma^{\alpha\beta}\varepsilon_{\alpha\beta}+\sigma^{33}\varepsilon_{33}+ 2\sigma^{3\alpha}\varepsilon_{3\alpha}\\ &=&v_{l}+v_{n}+v_{s}\end{array}  \tag{22}\]  
where \(i,j=1,2,3\) stand for the global Cartesian directions. In the above relation, the symmetries: \(\sigma^{3\alpha}=\sigma^{\alpha 3}\) and \(\varepsilon_{3\alpha}=\varepsilon_{\alpha 3}\) were taken into account. The terms \(v_{l}\), \(v_{n}\), \(v_{s}\) represent longitudinal, transverse, and shear components, respectively, of the density function. Consequently, the strain energy can also be divided into the corresponding energy components \(U_{l}\), \(U_{n}\), \(U_{s}\):  
\[ \begin{array}{rcl}U&=&\int_{V}v\,dV\\ &=&U_{l}+U_{n}+U_{s}\end{array}  \tag{23}\]  
Our calculations of the strain components of (21) on the element level correspond to the 3D-based hierarchical shell formulation proposed and developed in [38; 45]. In this formulation local strains (defined in two longitudinal and the third transverse directions) and the global displacement dofs are applied. Because of that, the relation defining local strains \(\varepsilon\) (or their components) by the product of the element strain-displacement matrix \(\overset{e}{\boldsymbol{B}}\) (or its components) and the element displacement dofs vector \(\overset{e}{\boldsymbol{q}}^{hpq}\) can be written as follows  
\[ \begin{array}{rcl}\varepsilon&=&\overset{e}{\boldsymbol{B}}\overset{e}{ \boldsymbol{q}}^{hpq}\\ &=&[\overset{e}{\boldsymbol{B}}_{l},\overset{e}{\boldsymbol{B}}_{n}, \overset{e}{\boldsymbol{B}}_{s}]^{T}\overset{e}{\boldsymbol{q}}^{hpq}\\ &=&[\overset{e}{\boldsymbol{B}}_{l}\overset{e}{\boldsymbol{q}}^{hpq}, \overset{e}{\boldsymbol{B}}_{n}\overset{e}{\boldsymbol{q}}^{hpq}, \overset{e}{\boldsymbol{B}}_{s}\overset{e}{\boldsymbol{q}}^{hpq}]^{T}\\ &=&[\varepsilon_{l},\varepsilon_{n},\varepsilon_{s}]^{T}\end{array}  \tag{24}\]  
where the longitudinal \(\overset{e}{\boldsymbol{B}}_{l}\), transverse \(\overset{e}{\boldsymbol{B}}_{n}\), and shear \(\overset{e}{\boldsymbol{B}}_{s}\) blocks of the strain-displacement matrix include the latter matrix rows corresponding to the in-plane, normal and shear strain components: \(\varepsilon_{l}\), \(\varepsilon_{n}\), \(\varepsilon_{s}\).