Testing the asymptotic behaviour of shell elements—Part II: New limit tests: analytical solutions and the RFNS element case

The asymptotic behaviour of classical benchmark tests was investigated in the first part of this work. In the present second part, the behaviour of some new limit problems, recently proposed as being specifically applicable to benchmark testing of the asymptotic behaviour of shell elements, is analytically and numerically investigated. Exact analytical solutions are obtained based on Flugge's theory for cylindrical shells. These analytical solutions are used along with, and in comparison to, the corresponding solutions obtained earlier by symbolic calculus using the Reissner–Mindlin shell model. The reformulated four‐node shell (RFNS) element is employed in the numerical analyses in a parallel, supportive–comparative character, next to the analytical investigation of the asymptotic behaviour of the new limit tests. As with the case of the classical benchmark tests, in the course of the numerical investigation, the reliability and efficiency of the RFNS element is re‐confirmed in all cases of the new asymptotic tests. A good agreement with the boundary layers described analytically is obtained even in very thin shell element applications. The various load‐carrying mechanisms shown numerically to be active in the cases under investigation follow closely the analytical predictions. The energy components appear to be more sensitive to the modelling of boundary layers in cases of mixed mode problems. In several cases, the solutions obtained earlier by using symbolic calculus are shown to be inadequate. Copyright © 2002 John Wiley & Sons, Ltd.     

A benchmark study of reduced‐strain shell finite elements: quadratic schemes

We study experimentally the accuracy and reliability of some low‐order shell finite element schemes based on modifying the standard displacement formulation by reduced‐strain expressions. We focus on quadrilateral elements with a quadratic displacement approximation. Three benchmark problems with different asymptotic behaviour in the limit of zero shell thickness is used in the experiments. Following the error analysis of a reduced‐strain scheme, we study two components of the total error, the approximation error and the consistency error. We demonstrate that the performance of the methods is both case and mesh dependent. When a bending dominated problem is solved, none of the methods studied can avoid the usual worst‐case locking effect of the approximation error on general meshes. For a membrane dominated problem the total error is typically dominated by the consistency error which often convergences slowly. Copyright © 2000 John Wiley & Sons, Ltd.     

On the classical shell model underlying bilinear degenerated shell finite elements

We study the shell models arising in the numerical modelling of shells by bilinear degenerated shell finite elements. The numerical model of a cylindrical shell obtained by using flat shell elements is given an equivalent formulation based on a classical two‐dimensional shell model. We use the connection between the models to explain how a parametric error amplification difficulty or locking is avoided by some elements. Copyright © 2001 John Wiley & Sons, Ltd.     

The mathematical shell model underlying general shell elements

We show that although no actual mathematical shell model is explicitly used in ‘general shell element’ formulations, we can identify an implicit shell model underlying these finite element procedures. This ‘underlying model’ compares well with classical shell models since it displays the same asymptotic behaviours—when the thickness of the shell becomes very small—as, for example, the Naghdi model. Moreover, we substantiate the connection between general shell element procedures and this underlying model by mathematically proving a convergence result from the finite element solution to the solution of the model. Copyright © 2000 John Wiley & Sons, Ltd.     

On the classical shell model underlying bilinear degenerated shell finite elements: general shell geometry

We study the shell models arising in the numerical modelling of shells by geometrically incompatible finite elements. We build a connection from the so‐called bilinear degenerated 3D FEM to the classical 2D shell theory of Reissner–Naghdi type showing how nearly equivalent finite element formulations can be constructed within the classical framework. The connection found here facilitates the mathematical error analysis of the bilinear elements based on the degenerated 3D approach. In particular, the connection reveals the ‘secrets’ that relate to the treatment of locking effects within this formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 1: The model and the approximation

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in this paper, is to provide non‐standard information on the original parts of the element algorithm. In order to do that, we present the definition of shape functions necessary for p‐adaptivity, as well as the procedure for imposing constraints corresponding to the lack of elongation of the straight lines perpendicular to the shell mid‐surface, which is the procedure necessary for q‐adaptivity. The 3D version of constrained approximation presented next is the basis for h‐adaptivity of the element. The second part of our research, devoted to methodology and results of the numerical research on application of the element to various plate and shell problems, are described in the second part of this paper. Copyright © 2006 John Wiley & Sons, Ltd.     

3D dynamics of discrete element systems comprising irregular discrete elements—integration solution for finite rotations in 3D

An algorithm for transient dynamics of discrete element systems comprising a large number of irregular discrete elements in 3D is presented. The algorithm is a natural extension of contact detection, contact interaction and transient dynamics algorithms developed in recent years in the context of discrete element methods and also the combined finite‐discrete element method. It complements the existing algorithmic procedures enabling transient motion including finite rotations of irregular discrete elements in 3D space to be accurately integrated. Copyright © 2002 John Wiley & Sons, Ltd.     

A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part I—solid‐shell element formulation

In the recent years, solid‐shell finite element models which possess no rotational degrees of freedom and applicable to thin plate/shell analyses have attracted considerable attention. Development of these elements are not straightforward. Shear, membrane, trapezoidal, thickness and dilatational lockings must been visioned. In this part of this paper, a novel eight‐node solid‐shell element is proposed. To resolve the shear and trapezoidal lockings, the assumed natural strain (ANS) method is resorted to. The hybrid‐stress formulation is employed to rectify the thickness and dilatational locking. The element is computationally more efficient than the conventional hybrid elements by adopting orthogonal‐assumed stress modes and enforcing admissible sparsity in the flexibility matrix. Popular benchmark tests are exercised to illustrate the efficacy of the elements. In Part II of the paper, the element will be generalized for smart structure modelling by including the piezoelectric effect. Copyright © 2000 John Wiley & Sons, Ltd.     

Stress resultant geometrically exact form of classical shell model and vector‐like parameterization of constrained finite rotations

In this work we consider the geometrically exact shell model subjected to finite rotations, making use of rotation vector parameters for handling the corresponding constrained rotation for smooth shells. A modification of such a parameterization which is based on the incremental rotation vector and thus capable of avoiding the singularity problem is also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Non‐planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model

A methodology for solving three‐dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near‐tip fields that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three‐dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems. Copyright © 2002 John Wiley & Sons, Ltd.     

Direct determination of asymptotic structural postbuckling behaviour by the finite element method

Application of the finite element method to Koiter's asymptotic postbuckling theory often leads to numerical problems. Generally it is believed that these problems are due to locking of non-linear terms of different orders. A general method is given here that explains the reason for the numerical problems and eliminates these problems. The reason for the numerical problems is that the postbuckling stresses are inaccurately determined. By including a local stress contribution, the postbuckling stresses are calculated correctly. The present method gives smooth postbuckling stresses and shows a quick convergence of the postbuckling coefficients. © 1998 John Wiley & Sons, Ltd.     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.     

Adaptive modelling of delamination initiation and propagation using an equivalent single‐layer shell approach

A major challenge for crash failure analysis of laminated composites is to find a modelling approach, which is both sufficiently accurate, for example, able to capture delaminations, and computationally efficient to allow full‐scale vehicle crash simulations. Addressing this challenge, we propose a methodology based on an equivalent single‐layer shell formulation which is adaptively through‐the‐thickness refined to capture initiating and propagating delaminations. To be specific, single shell elements through the laminate thickness are locally and adaptively enriched using the extended finite element method such that delaminations can be explicitly modelled without having to be represented by separate elements. Furthermore, the shell formulation is combined with a stress recovery technique which increases the accuracy of predicting delamination initiations. The paper focuses on the parameters associated with identifying, introducing and extending the enrichment areas; especially on the impact of these parameters on the resulting structural deformation behaviour. We conclude that the delamination enrichment must be large enough to allow the fracture process to be accurately resolved, and we propose a suitable approach to achieve this. The proposed methodology for adaptive delamination modelling shows potential for being computationally efficient, and thereby, it has the potential to enable efficient and accurate full vehicle crash simulations of laminated composites. Copyright © 2017 John Wiley & Sons, Ltd.     

A new one‐point quadrature enhanced assumed strain (EAS) solid‐shell element with multiple integration points along thickness: Part I—geometrically linear applications

Accuracy and efficiency are the main features expected in finite element method. In the field of low‐order formulations, the treatment of locking phenomena is crucial to prevent poor results. For three‐dimensional analysis, the development of efficient and accurate eight‐node solid‐shell finite elements has been the principal goal of a number of recent published works. When modelling thin‐ and thick‐walled applications, the well‐known transverse shear and volumetric locking phenomena should be conveniently circumvented. In this work, the enhanced assumed strain method and a reduced in‐plane integration scheme are combined to produce a new eight‐node solid‐shell element, accommodating the use of any number of integration points along thickness direction. Furthermore, a physical stabilization procedure is employed in order to correct the element's rank deficiency. Several factors contribute to the high computational efficiency of the formulation, namely: (i) the use of only one internal variable per element for the enhanced part of the strain field; (ii) the reduced integration scheme; (iii) the prevention of using multiple elements' layers along thickness, which can be simply replaced by any number of integration points within a single element layer. Implementation guidelines and numerical results confirm the robustness and efficiency of the proposed approach when compared to conventional elements well‐established in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Adding transverse normal stresses to layered shell finite elements for the analysis of composite structures

This work presents a formulation developed to add capabilities for representing the through thickness distribution of the transverse normal stresses, σ_z, in first and higher order shear deformable shell elements within a finite element (FE) scheme. The formulation is developed within a displacement based shear deformation shell theory. Using the differential equilibrium equations for two-dimensional elasticity and the interlayer stress and strain continuity requirements, special treatment is developed for the transverse normal stresses, which are thus represented by a continuous piecewise cubic function. The implementation of this formulation requires only C⁰ continuity of the displacement functions regardless of whether it is added to a first or a higher order shell element. This makes the transverse normal stress treatment applicable to the most popular bilinear isoparametric 4-noded quadrilateral shell elements.

To assess the performance of the present approach it is included in the formulation of a recently developed third order shear deformable shell finite element. The element is added to the element library of the general nonlinear explicit dynamic FE code DYNA3D. Some illustrative problems are solved and results are presented and compared to other theoretical and numerical results.     

A hybrid stress ANS solid‐shell element and its generalization for smart structure modelling. Part II—smart structure modelling

In Part I of the paper, a hybrid‐stress‐assumed natural strain eight‐node solid‐shell element immune to shear, membrane, trapezoidal, thickness and dilatational lockings has been developed. Moreover, the element computational cost is reduced by enforcing admissible sparsity in the flexibility matrix. In this part of the paper, the solid‐shell element is generalized to a piezoelectric solid‐shell element. Using the two solid‐shell elements, smart structures with segmented piezoelectric sensors and actuators can be conveniently modelled. A number of problems are studied and comparisons with other ad hoc element models for smart structure modelling are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

A set of pathological tests to validate new finite elements

The finite element method entails several approximations. Hence it is essential to subject all new finite elements to an adequate set of pathological tests in order to assess their performance. Many such tests have been proposed by researchers from time to time. We present an adequate set of tests, which every new finite element should pass. A thorough account of the patch test is also included in view of its significance in the validation of new elements.     

Bifurcation points and bifurcated branches by an asymptotic numerical method and Padé approximants

The aim of this work is to develop a reliable and fast algorithm to compute bifurcation points and bifurcated branches. It is based upon the asymptotic numerical method (ANM) and Padé approximants. The bifurcation point is detected by analysing the poles of Padé approximants or by evaluating, along the computed solution branch, a bifurcation indicator well adapted to ANM. Several examples are presented to assess the effectiveness of the proposed method, that emanate from buckling problems of thin elastic shells. Especially problems involving large rotations are discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

On hybrid stress,hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

The paper is concerned with the finite element formulation of a recently proposed geometrically exact shell theory with natural inclusion of drilling degrees of freedom. Stress hybrid finite elements are contrasted by strain hybrid elements as well as enhanced strain elements. Numerical investigations and comparison is carried out for a four-node element as well as a nine-node one. As far as the four-node element is concerned it is shown that the stress hybrid element and the enhanced strain one are equivalent. The hybrid strain formulation corresponds to the hybrid stress formulation only in shear dominated problems, that is the case of the plate. © 1998 John Wiley & Sons, Ltd.