# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 2 Preliminaries
### Model Problems  
Let us consider a wide range of problems of linear elasticity covered by the standard local (strong) formulation  
\[\left. \begin{array}{c}\sigma_{,j}^{ij}+f^{i}=0\\ \sigma^{ij}=D^{ijkl}\varepsilon_{kl}\\ \varepsilon_{kl}=\frac{1}{2}(u_{k,l}+u_{l,k})\end{array} \right\},\quad\mathbf{x}\in V \tag{1}\]  
composed of the equilibrium, constitutive, and geometrical equations, respectively. Above, \(i,j,k,l=1,2,3\), while the smooth components \(f^{i}\in L^{2}(V)\) stand for the known body loading vector, with \(V\) representing volume of the body. The tensor components \(\sigma^{ij}\) and \(\varepsilon_{kl}\) stand for stresses and strains, while the unknown vector components \(u_{k}\) denote displacements. The terms \(D^{ijkl}\) are components of the fourth-order tensor of elasticities. The elasticities are symmetric \(D^{ijkl}=D^{ijkl}=D^{klij}\) and satisfy the strong and uniform ellipticity condition: \(D^{ijkl}\xi_{ij}\xi_{kl}\geq\alpha\xi_{ij}\xi_{kl}\), \(\xi_{ij}=\xi_{ji}\), where \(\xi_{ij}\) is any real-valued tensor of the second rank and \(\alpha\) is a positive constant. Note that such elasticities may correspond to three-dimensional stress and strain states or to the constrained, plane stress or plane strain, states. Note also that, for the specific case of isotropy, the elasticities can be expressed by the Young's modulus \(E\) and [PERSON]'s ratio \(\
u\). The set (1) has to be completed with the displacement and stress boundary conditions of the form  
\[ \begin{array}{rcl}\sigma^{ij}n_{j}&=&p^{i},\quad\mathbf{x}\in S_{P}\\ u_{i}&=&w_{i},\quad\mathbf{x}\in S_{W}\end{array}  \tag{2}\]  
where \(i,j=1,2,3\), while the smooth components \(p^{i}\in L^{2}(S)\) and the terms \(w_{i}\) stand for the known values of stress vector components and displacement vector components on the parts \(S_{P}\) and \(S_{W}\) of the surface \(S\equiv\partial V\) of the body \(V\), with \(S=S_{P}\cup S_{W}\) and \(S_{P}\cap S_{W}=\emptyset\). The vector components \(n_{j}\) represent unit outward normal to the surface part \(S_{P}=S_{T}\cup S_{B}\) composed of the top \(S_{T}\) and bottom \(S_{B}\) surfaces of the body. The way they are determined is explained below. Equation (1) is valid within the body volume \(V\). Such a volume may represent thick- or thin-walled structures or complex structures containing such parts. Due to the model character of the considerations of this section, we limit ourselves to the first case. The corresponding thin-walled domain of the volume \(V\) is sufficiently smooth (Lipschitzian or smoother), open, and bounded region. As we apply the 3D Cartesian description of the problem, i.e., we employ Cartesian coordinates \(\mathbf{x}\), the following explicit (curvilinear) and implicit (Cartesian) definitions of the volume \(V\) are appropriate:  
\[ \begin{array}{l}V=\left\{\mathbf{\eta}\!\in\!R^{3}\!\colon\,(\eta^{1},\eta^{2}) \in S_{M},\,\eta^{3}\!\in\!(-t/2,t/2)\right\}\\ \equiv\left\{\mathbf{x}(\mathbf{\eta})\!\in\!R^{3}\!\colon\,\mathbf{x}=F(\mathbf{\eta}),\,( \eta^{1},\eta^{2})\in S_{M},\,\eta^{3}\!\in\!(-t/2,t/2)\right\}\end{array}  \tag{3}\]  
where \(F\) is the reversible map converting the curvilinear coordinates \(\mathbf{\eta}\) into the Cartesian ones \(\mathbf{x}\). The function \(t=t(\eta^{1},\eta^{2})\) measures the symmetric thickness in the direction \(\eta^{3}\) normal to the mid-surface \(S_{M}\). The mid-surface \(S_{M}\), where \(\eta^{3}=0\), of the thin-walled body can be any sufficiently smooth (Lipschitzian or smoother), two-dimensional, open and bounded region. Formal definition of the thin-walled body geometry also needs introduction of the lateral part \(S_{L}\) of the body boundary \(S\) aswell as the upper (top) \(S_{T}\) and lower (bottom) \(S_{B}\) parts of this boundary. More details on the applied definition of the thin-walled geometry can be found in [48].