# A Posteriori Detection of Numerical Locking in \(hpq\)-Adaptive Finite Element Analysis
## 1 Introduction
### State-of-the-Art Issues
#### 1.1.2 Overcoming the Locking Phenomena  
We limit this survey to the methods related to the shear and/or membrane locking. This type of locking results from the thick- or thin-walled character of the plate and shell structures. The first method of overcoming the locking is based on application of the mixed or hybrid formulations of the finite element method instead of the displacement formulation. This leads to elements of the class \(C^{1}\) instead of the class \(C^{0}\). The elements of this group are usually of low-order and may need stabilization. The examples of the elements of this group are presented in [7; 8; 9]. More recent examples of the mixed and hybrid finite elements resistant to locking are published in [10; 11]. The second approach takes advantage of the so-called reduced or reduced selective numerical integration, sometimes enriched with the stabilization matrix which removes deformation modes of zero energy. The reduced integration consists in integration of the stiffness matrix with the numerical integration parameters as for the elements described with the polynomial interpolation of one order lower. In the selective version of the reduced integration the lower order is applied to a part of the stiffness matrix, responsible for the locking, i.e., the part corresponding to shear and/or membrane strain energy. The examples of application of this approach to plate and shell elements can be found in [12; 13; 14]. The recent works dealing with the reduced and reduced selective integration concern either the isogeometric analysis or the standard finite element methods, for example, in [15; 16]. The third way to overcome the phenomenon lies in introduction of the discrete Kirchhoff constraints into the elements of the class \(C^{0}\). The method is directed towards removal of the shear locking and requires that the Kirchhoff constraints are imposed on the selected points or lines within \(C^{0}\) element. The prominentexamples of plate or thick shell elements of this type can be found in [17; 18]. Recent examples of the discrete Kirchhoff and Kirchhoff-Love constraints are presented in the works [19; 20]. The fourth method consists in application of the consistent (interdependent) fields of the transverse displacement and rotations, one order higher in the case of the mentioned displacement. The method leads to different numbers of unknowns of both types within an element. The surplus displacement degrees of freedom (dofs) are removed from the model based on the condition of zero transverse shear strains and/or zero membrane strain condition. The prominent examples of this method of the locking removal can be found in [21; 22]. The fifth approach is based on the assumed shear or membrane strains consistent with the interpolated transverse displacement at some points. In this method, bending strains result from the interpolated displacement field, while the transverse shear and/or membrane (in-plane) strains possess the assumed form resulting from the interpolation based on some chosen points. The significant works, leading to the current state of this method in relation to quadrilateral plate and shell elements, are found in [23; 24; 25]. Two versions of the presented approach, based on either the enhanced assumed strains or assumed natural strains, are still being developed, for example, in [26; 27]. The sixth method lies in application of higher-order elements conforming to the displacement finite element formulation. The examples of application of such elements in the case of the classical (non-adaptive) finite element methods, can be found in [28; 29]. In the non-adaptive methods, plate or shell elements conforming to the first-order or higher-order theories are applied. The fixed longitudinal order of \(h\)-approximation up to the fifth order is usually applied within elements of this type. The adaptive quadrilateral elements, conforming to hierarchical approximations and higher-order shell models, are used in [30].