**1. Introduction**

This paper concerns application of the algorithms for detection, assessment and resolution of numerical locking in the *hpq*-adaptive finite element elastic analysis of thin-walled structures or complex structures which include thin-walled, solid, and transition parts. We consider all cases when the influence of the locking phenomenon on the problem solution is significant. We focus on the theoretical and methodological aspects such as the idea and justification of the elaborated algorithms for a posteriori detection and assessment of the phenomenon. In addition, the necessary modification of the applied *hpq*-adaptive algorithms is of our interest. The employed model- and *hpq*-adaptive method allows for different element size *h*, different element longitudinal and transverse orders of approximation, *p* and *q*, and different element model *M* in each finite element. The paper also presents application of the introduced detection, assessment, and resolution algorithms in the *hpq*-adaptive analysis of structural elements. These algorithms are investigated in the contexts of their generality, reliability, and effectiveness.

## *1.1. State-of-the-Art Issues*

We address two specific issues dealt with the locking phenomena. The first one concerns some basic research on the nature of this phenomenon, while the second issue is the existing methods of removal of the numerical consequences of the phenomenon.

#### 1.1.1. Theoretical and Numerical Research of Locking Phenomena

The numerical locking phenomena concern thin-walled structures in which the true solution is characterized by bending strains dominance over shear and/or membrane strains. The phenomenon does not appear in the case of the membrane strains dominance. If, due to poor discretization of the problem, the shear and/or membrane strains are not equal to zero, as it results from the thin-walled theories for the thickness *t* tending to 0, then the shear, membrane, or shear–membrane strain energy numerically dominates over the bending energy, leading to numerical over-stiffening of the structure, which in turn results in too low (zero or almost zero) values of displacements (compare the work [1]). This phenomenon is called the shear, membrane, or shear–membrane locking and is typical for the displacement finite element method.

Theoretical and numerical studies of the locking phenomena concern one- and two-dimensional problems, including beams, arcs, plates, and shells. Different kinds of locking are investigated: volume (Poisson's) locking present in nearly incompressible materials (*ν* → 0.5), deformational locking present in bending-dominated thin-walled structures within the displacement formulation of the finite element method, and finally the trapezoidal locking present in hybrid-stress finite elements. The deformational locking, which is the subject of this work, may be shear (plates), membrane, or shear–membrane (shells). The significant exemplary theoretical and numerical research results concerning deformational locking are presented in [1–6], respectively. These works refer to the first-order, higher-order, and hierarchical models of plates and shells.

#### 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*<sup>1</sup> 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 [7–9]. 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 [12–14]. 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*<sup>0</sup> element. The prominent examples 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 [23–25]. 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]. The hexahedral elements corresponding to three-dimensional elasticity, equipped with independent transverse and longitudinal approximations of the higher order, and assigned for plate and shell analysis, are proposed in [31] and applied in *hp*-adaptive version in [32], for example. The recent applications of the higher-order models and approximations to locking removal are presented in the works [33,34]. These proposals are not consistent with the hierarchical approach.

Let us conclude the above survey of the methods of overcoming the locking phenomena in the context of needs of the *hpq*-adaptive method for complex structures analysis. Firstly, it should be noticed that only low-order longitudinal approximations and first-order plate and shell models are possible in the cases of the mixed and hybrid methods, the uniform or selective reduced integration, and the consistent field method. In the case of the discrete Kirchhoff constraints, only shear locking can be removed effectively. Additionally, this method of removal leads to large variability of elements. In the case of the assumed strains approach, the claimed generalization of the method for high-order in-plane approximations has not been proved in practice. For the reduced integration and assumed stress method, there are problems with their application to triangular or prismatic elements. In the case of the discrete Kirchhoff constraint and consistent field methods of removal, one deals with different number and location of translational and rotational degrees of freedom. Note that the higher-order elements are free from all these defects. Due to our earlier choices concerning the applied hierarchical models and *hpq*-approximations, the displacement formulation of such elements becomes our obvious choice.

#### 1.1.3. Detection and Assessment of the Locking

The basic method of detection and assessment of the locking phenomena is the a priori theoretical analysis of the numerical solutions of the model problems potentially suspected to be a subject of locking (see Section 1.1.1 of this literature survey). Another interesting method of detection and assessment of the locking in the one- and two-dimensional elements is proposed in [35,36]. It is based on application of the finite difference operators corresponding to the problem local formulation. In [37–39], the numerical methods of detection and assessment of the phenomenon are proposed. These methods are based on the sensitivity analysis, i.e., two or a sequence of local problems are solved for each element of the potentially affected domain.

### *1.2. The Applied Methodology*

Two issues are addressed here. The first one deals with the best choice of the method of effective detection and assessment of the locking phenomena. The second issue is related to the numerical methods of removing the phenomena.

It results from the above literature survey that the available knowledge on the locking phenomena allows understanding of the nature and sources of appearance of the phenomena. The accumulated knowledge on the phenomenon allows also for a priori determination of the solution convergence of the problems where the phenomena appear. The main difficulty in the direct application of these results in the numerical analysis of any arbitrary thin-walled structure is that the available results concern the specific model problems which may differ to the arbitrary problem under consideration.

The second conclusion from the literature survey is that the most effective way of removing the locking phenomena lies in application of the higher-order longitudinal *p*-approximation of the displacement field in the analyzed thin-walled structure or a thin-walled part of the complex structure.

It also results from the literature that some detection methods of the phenomena exist. Among these methods, the approach proposed in [37–39] seems to be best suited for adaptive analysis. This approach is based on the same numerical techniques that are applied in the error-controlled adaptivity.

The main feature of the proposed a posteriori detection, assessment and removal of the locking phenomena is that the adaptation process requires four steps, instead of the standard three steps of the error-controlled *hp*-adaptivity proposed in [40]. The additional step of adaptation, which lies in initial mesh modification, incorporates not only the automatic removal of the locking phenomena but also the automatic resolution of the boundary layers [38]. The main idea standing behind the additional adaptation step is to move the numerical solution, obtained with use of *hp*-approximations [41,42], to the asymptotic convergence range. Within this range, the standard *h*- and *p*-adaptation steps can be made based on the *hp*-convergence theorem and upper-bounding values of the global error estimates from the equilibrated residual method [43,44].

Finally, it should be noted that our detection and assessment tools are based on sensitivity analysis, not on the estimated error values themselves. Thanks to this, requirements concerning the error estimation can be relaxed.

#### *1.3. Novelty of the Paper*

The novelty of this work consists in the new algorithms for a posteriori detection and assessment of the numerical locking phenomena. This refers to shear, membrane, or shear–membrane locking. With these new algorithms, one is able to detect the presence of the phenomenon and assess its strength so that the adequate numerical means can be used to remove the phenomenon. The proposed numerical means of the removal consist in introduction of the new adaptation step, called the modification one, into the existing three-step, model- and *hpq*-adaptive finite element procedure for analysis of complex structures. The new step employs the mentioned detection and assessment algorithms and performs modification of the initial mesh through *p*-enrichment. The numerical cost of this new step is low as the detection and assessment is performed on the initial, usually coarse mesh.

The applied adaptive method [38,45,46], the new algorithms are incorporated in, takes advantage of the hierarchical models proposed in [31,47], hierarchical *hp*-approximations elaborated in [41,42], a posteriori error estimation from [30,43,44], and error-controlled adaptive procedure given in [40].
