Two refined non‐conforming quadrilateral flat shell elements

Two refined quadrilateral flat shell elements named RSQ20 and RSQ24 are constructed in this paper based on the refined non‐conforming element method, and the elements can satisfy the displacement compatibility requirement at the interelement of the non‐planar elements by introducing the common displacements suggested by Chen and Cheung. A refined quadrilateral plate element RPQ4 and a plane quadrilateral isoparametric element are combined to obtain the refined quadrilateral flat shell element RSQ20, and a refined quadrilateral flat shell element RSQ24 is constructed on the basis of a RPQ4 element and a quadrilateral isoparametric element with drilling degrees of freedom. The numerical examples show that the present method can improve the accuracy of shell analysis and that the two new refined quadrilateral flat shell elements are efficient and accurate in the linear analysis of some shell structures. Copyright © 2000 John Wiley & Sons, Ltd.     

Mixed‐enhanced formulation for geometrically linear axisymmetric problems

A four‐noded quadrilateral axisymmetric formulation in the context of a mixed‐enhanced method is presented. The strain field is represented by two sets of element parameters, which results in enhanced performance and coarse mesh accuracy in bending dominated problems and locking‐free response in the near incompressible limit. The mixed fields presented are such that variational stress recovery is permissible. In addition, the formulation is cast such that the mixed parameters are obtained explicitly yielding finite element arrays with the proper rank using standard order quadrature. In this paper our attention is restricted to the area of geometrically linear problems in solid mechanics. Representative simulations show favourable performance of the formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the Reissner–Mindlin formulation

A finite element formulation for refined linear analysis of multilayered shell structures of moderate thickness is presented. An underlying shell model is a direct extension of the first‐order shear‐deformation theory of Reissner–Mindlin type. A refined theory with seven unknown kinematic fields is developed: (i) by introducing an assumption of a zig‐zag (i.e. layer‐wise linear) variation of displacement field through the thickness, and (ii) by assuming an independent transverse shear stress fields in each layer in the framework of Reissner's mixed variational principle. The introduced transverse shear stress unknowns are eliminated on the cross‐section level. At this process, the interlaminar equilibrium conditions (i.e. the interlaminar shear stress continuity conditions) are imposed. As a result, the weak form of constitutive equations (the so‐called weak form of Hooke's law) is obtained for the transverse strains–transverse stress resultants relation. A finite element approximation is based on the four‐noded isoparametric element. To eliminate the shear locking effect, the assumed strain variational concept is used. Performance of the derived finite element is illustrated with some numerical examples. The results are compared with the exact three‐dimensional solutions, as well as with the analytical and numerical solutions obtained by the classical, the first‐order and some representative refined models. Copyright © 2000 John Wiley & Sons, Ltd.     

An improved discrete Kirchhoff quadrilateral element based on third‐order zigzag theory for static analysis of composite and sandwich plates

A new improved discrete Kirchhoff quadrilateral element based on the third‐order zigzag theory is developed for the static analysis of composite and sandwich plates. The element has seven degrees of freedom per node, namely, the three displacements, two rotations and two transverse shear strain components at the mid‐surface. The usual requirement of C¹ continuity of interpolation functions of the deflection in the third‐order zigzag theory is circumvented by employing the improved discrete Kirchhoff constraint technique. The element is free from the shear locking. The finite element formulation and the computer program are validated by comparing the results for simply supported plate with the analytical Navier solution of the zigzag theory. Comparison of the present results with those using other available elements based on zigzag theories for composite and sandwich plates establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed by comparing the finite element results of the square all‐round clamped composite plates with the converged three‐dimensional finite element solution obtained using ABAQUS. The comparisons also establish the superiority of the zigzag theory over the smeared third‐order theory having the same number of degrees of freedom. Copyright © 2006 John Wiley & Sons, Ltd.     

Behaviour of isoparametric quadrilateral family of Lagrangian fluid finite elements

This paper presents the formulation of both the consistent and inconsistent four‐, eight‐ and nine‐noded isoparametric quadrilateral fluid finite elements that are based on Lagrangian frame of reference. The mesh locking phenomenon due to simultaneous enforcement of twin constraints, namely the incompressibility and irrotationality constraints, is studied in detail. The study shows that the characteristic of the locked fluid elements is that it always generates numerous spurious acoustic (volume change) modes upon the enforcement of rotational constraints. That is, the rotational constraints change the character of certain volume change modes. The study further reinforces the necessity of rotational constraints in not only identifying the spurious pressure modes, but also in reducing the computational effort for determining the eigenvalues and eigenvectors. It is found that all fully integrated inconsistent models exhibit locking behaviour. However, the inconsistent eight‐ and nine‐noded elements, integrated with full integration of volumetric stiffness and one point integration of the rotational stiffness matrices, gives excellent performance, although they do not pass the inf–sup test. The four‐ and nine‐noded consistent models are found to give locking free performance while their eight‐noded counterpart exhibited locking behaviour. The study shows that only consistent nine‐noded element models pass the inf–sup test. The utility of these elements in the coupled fluid–structure interaction problem is also demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Smooth finite element methods: Convergence,accuracy and properties

A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu–Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi‐equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one‐subcell element with a quasi‐equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one‐cell smoothed four‐noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

A refined non‐linear non‐conforming triangular plate/shell element

A refined non‐conforming triangular plate/shell element for geometric non‐linear analysis of plates/shells using the total Lagrangian/updated Lagrangian approach is constructed in this paper based on the refined non‐conforming element method for geometric non‐linear analysis. The Allman's triangular plane element with vertex degrees of freedom and the refined triangular plate‐bending element RT9 are used to construct the present element. Numerical examples demonstrate that the accuracy of the new element is quite high in the geometric non‐linear analysis of plates/shells. Copyright © 2003 John Wiley & Sons, Ltd.     

Stability and accuracy of finite element methods for flow acoustics. II: Two‐dimensional effects

This is the second of two articles that focus on the dispersion properties of finite element models for acoustic propagation on mean flows. We consider finite element methods based on linear potential theory in which the acoustic disturbance is modelled by the convected Helmholtz equation, and also those based on a mixed Galbrun formulation in which acoustic pressure and Lagrangian displacement are used as discrete variables. The current paper focuses on the effects of numerical anisotropy which are associated with the orientation of the propagating wave to the mean flow and to the grid axes. Conditions which produce aliasing error in the Helmholtz formulation are of particular interest. The 9‐noded Lagrangian element is shown to be superior to the more commonly used 8‐noded serendipity element. In the case of the Galbrun elements, the current analysis indicates that isotropic meshes generally reduce numerical error of triangular elements and that higher order mixed quadrilaterals are generally less effective than an equivalent mesh of lower order triangles. Copyright © 2005 John Wiley & Sons, Ltd.     

The effects of element shape on the critical time step in explicit time integrators for elasto‐dynamics

In this paper, the effects of element shape on the critical time step are investigated. The common rule‐of‐thumb, used in practice, is that the critical time step is set by the shortest distance within an element divided by the dilatational (compressive) wave speed, with a modest safety factor. For regularly shaped elements, many analytical solutions for the critical time step are available, but this paper focusses on distorted element shapes. The main purpose is to verify whether element distortion adversely affects the critical time step or not. Two types of element distortion will be considered, namely aspect ratio distortion and angular distortion, and two particular elements will be studied: four‐noded bilinear quadrilaterals and three‐noded linear triangles. The maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements. The critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements. Copyright © 2014 John Wiley & Sons, Ltd.     

A new triangular element to model inter‐laminar shear stress continuous plate theory

An efficient triangular element based on an inter‐laminar shear stress continuous plate theory is developed and applied to the analysis of composite and sandwich plates under different situations to study the performance of the element. The plate theory represents parabolic through thickness variation of transverse shear stresses where the continuity condition of these stresses are satisfied at the layer interfaces. It also satisfies transverse shear stress free condition at the top and bottom surfaces of the plate. The most attractive feature of the plate theory is that the basic unknowns are same as those used in first‐order shear deformation theory. The only problem lies with this elegant plate theory is found in its finite element implementation, as it requires C¹ continuity of transverse displacement at the element interfaces. This is a well‐known problem of thin plate elements, which is also found in some other refined plate theories. Although there are some elements based on these refined plate theories but the number of such elements is very few and they possess certain drawbacks in general. Keeping these aspects in view, an attempt has been made in this study to develop a six‐noded triangular element having equal degrees of freedom at each node. Copyright © 2004 John Wiley & Sons, Ltd.     

An enhanced co‐rotational approach for large displacement analysis of plates

This paper presents a new co‐rotational approach for the large displacement analysis of plates employing 4‐noded quadrilateral flat shell elements. The proposed approach benefits from (i) a simple local co‐rotational system invariant to the element nodal ordering, (ii) the choice of the two smallest components of the nodal normal vector as global rotational degrees of freedom, and (iii) the use of hierarchic freedoms, that are unaffected by the co‐rotational transformations, for higher‐order accuracy. Important additional benefits that arise from the aforementioned features include symmetry of the tangent stiffness matrix and complete insensitivity of the large displacement transformations to the size of the incremental step. The applicability of the new approach to moderately thick as well as thin plates is illustrated by considering two alternative local formulations based on the Reissner–Mindlin and discrete Kirchhoff hypotheses. Several examples are finally presented which demonstrate the accuracy, step‐insensitivity and computational benefits of the proposed co‐rotational approach for large displacement analysis of plate structures. Copyright © 2005 John Wiley & Sons, Ltd.     

Free vibration analysis of composite and sandwich plates using an improved discrete Kirchhoff quadrilateral element based on third-order zigzag theory

The new improved discrete Kirchhoff quadrilateral element based on the third-order zigzag theory developed earlier by the present authors for the static analysis of composite and sandwich plates is extended for dynamics and assessed for its performance for the free vibration response. The element is free from the shear locking. The finite element formulation is validated by comparing the results for simply supported plates with the analytical Navier solution of the zigzag theory. Comparison of the present results for the natural frequencies with those of a recently developed triangular element based on the zigzag theory, for composite and sandwich plates, establishes the superiority of the present element in respect of simplicity, accuracy and computational efficiency. The accuracy of the zigzag theory is assessed for composite and sandwich plates with various boundary conditions and aspect ratio by comparing the finite element results with the 3D elasticity analytical and finite element solutions.     

Exponential finite elements for diffusion–advection problems

A new finite element method for the solution of the diffusion–advection equation is proposed. The method uses non‐isoparametric exponentially‐varying interpolation functions, based on exact, one‐ and two‐dimensional solutions of the Laplace‐transformed differential equation. Two eight‐noded elements are developed and tested for convergence, stability, Peclet number limit, anisotropy, material heterogeneity, Dirichlet and Neumann boundary conditions and tolerance for mesh distortions. Their performance is compared to that of conventional, eight‐ and 12‐noded polynomial elements. The exponential element based on two‐dimensional analytical solutions fails basic tests of convergence. The one based on one‐dimensional solutions performs particularly well. It reduces by about 75% the number of elements and degrees of freedom required for convergence, yielding an error that is one order of magnitude smaller than that of the eight‐noded polynomial element. The exponential element is stable and robust under relatively high degrees of heterogeneity, anisotropy and mesh distortions. Copyright © 2005 John Wiley & Sons, Ltd.     

Extended rotation‐free plate and beam elements with shear deformation effects

This paper describes a methodology for extending rotation‐free plate and beam elements to accounting for transverse shear deformation effects. The ingredients for the element formulation are a Hu–Washizu‐type mixed functional, a linear interpolation for the deflection and the shear angles over standard finite elements and a finite volume approach for computing the bending moments and the curvatures over a patch of elements. As a first application of the general procedure, we present an extension of the three‐noded rotation‐free basic plate triangle (BPT) originally developed for thin plate analysis to account for shear deformation effects of relevance for thick plates and composite‐laminated plates. The nodal deflection degrees of freedom (DOFs) of the original BPT element are enhanced with the two shear deformation angles. This allows to compute the bending and shear deformation energies leading to a simple triangular plate element with three DOFs per node (termed BPT+ element). For the thin plate case, the shear angles vanish and the element reproduces the good behaviour of the original thin BPT element. As a consequence the element is applicable to thick and thin plate situations without exhibiting shear locking effects. The numerical solution for the thick case can be found iteratively starting from the deflection values for the Kirchhoff theory using the original thin BPT element. A two‐noded rotation‐free beam element termed CCB+ applicable to slender and thick beams is derived as a particular case of the plate formulation. The examples presented show the robustness and accuracy of the BPT+ and the CCB+ elements for thick and thin plate and beam problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Refined global–local higher‐order theory and finite element for laminated plates

Based on completely three‐dimensional elasticity theory, a refined global–local higher‐order theory is presented as enhanced version of the classical global–local theory proposed by Li and Liu (Int. J. Numer. Meth. Engng. 1997; 40 :1197–1212), in which the effect of transverse normal deformation is enhanced. Compared with the previous higher‐order theory, the refined theory offers some valuable improvements these are able to predict accurately response of laminated plates subjected to thermal loading of uniform temperature. However, the previous higher‐order theory will encounter difficulty for this problem. A refined three‐noded triangular element satisfied the requirement of C¹ weak‐continuity conditions in the inter‐element is also presented. The results of numerical examples of moderately thick laminated plates and even thick plates with span/thickness ratios L/h = 2 are given to show that in‐plane stresses and transverse shear stresses can be reasonably predicted by the direct constitutive equation approach without smooth technique. In order to accurately obtain transverse normal stresses, the local equilibrium equation approach in one element is employed herein. Copyright © 2006 John Wiley & Sons, Ltd.     

A two-level mesh repartitioning scheme for the displacement-based lower-order finite element methods in volumetric locking-free analyses

We present an approach for repartitioning existing lower-order finite element mesh based on quadrilateral or triangular elements for the linear and nonlinear volumetric locking-free analysis. This approach contains two levels of mesh repartitioning. The first-level mesh re-partitioning is an h-adaptive mesh refinement for the generation of a refined mesh needed in the second-level mesh coarsening. The second-level mesh coarsening involves a gradient smoothing scheme performed on each pair of adjacent elements selected based on the first-level refined mesh. With the repartitioned mesh and smoothed gradient, the equivalence between the mixed finite element formulation and the displacement-based finite element formulation is established. The extension to nonlinear finite element formulation is also considered. Several linear and non-linear numerical benchmarks are solved and numerical inf-sup tests are conducted to demonstrate the accuracy and stability of the proposed formulation in the nearly incompressible applications.     

An updated Lagrangian finite element sensitivity analysis of large deformations using quadrilateral elements

A continuum parameter and shape sensitivity analysis is presented for metal forming processes using the finite element method. The sensitivity problem is posed in a novel updated Lagrangian framework as suitable for very large deformations when remeshing operations are performed during the analysis. In addition to exploring the issue of transfer of variables between meshes for finite deformation analysis, the complex problem of transfer of design sensitivities (derivatives) between meshes for large deformation inelastic analyses is also discussed. A method is proposed that is shown to give accurate estimates of design sensitivities when remeshing operations are performed during the analysis. Sensitivity analysis for the consistent finite element treatment of near incompressibility within the context of the assumed strain methods is also proposed. In particular, the performance of four‐noded quadrilateral elements for the sensitivity analysis of large deformations is studied. The results of the continuum sensitivity analysis are validated by a comparison with those obtained by a finite difference approximation (i.e. using the solution of a perturbed deformation problem). The effectiveness of the method is demonstrated by applications in the design optimization of metal forming processes. Copyright © 2001 John Wiley & Sons, Ltd.     

Two simple and efficient displacement‐based quadrilateral elements for the analysis of composite laminated plates

Two simple 4‐node 20‐DOF and 4‐node 24‐DOF displacement‐based quadrilateral elements named RDKQ‐L20 and RDKQ‐L24 are developed in this paper based on the first‐order shear deformation theory (FSDT) for linear analysis of thin to moderately thick laminates. The deflection and rotation functions of the element sides are obtained from Timoshenko's laminated composite beam functions. Linear displacement interpolation functions of the standard 4‐node quadrilateral isoparametric plane element and displacement functions of a quadrilateral plane element with drilling degrees of freedom are taken as in‐plane displacements of the proposed elements RDKQ‐L20 and RDKQ‐L24, respectively. Due to the application of Timoshenko's laminated composite beam functions, convergence can be ensured theoretically for very thin laminates. The elements are simple in formulation, and shear‐locking free for extremely thin laminates even with full integration. A hybrid‐enhanced procedure is employed to improve the accuracy of stress analysis, especially for transverse shear stresses. Numerical tests show that the new elements are convergent, not sensitive to mesh distortion, accurate and efficient for analysis of thin to moderately thick laminates. Copyright © 2004 John Wiley & Sons, Ltd.     

A study of crack-tip plastic zone by elastoplastic finite element analysis

An elastoplastic finite element analysis has been carried out on a thin centre cracked panel using plane eight noded quadratic quadrilateral isoparametric elements with nonsingular displacement formulation. The linearised total strain method has been employed for the solution and the material data including those pertaining to postyield behaviour were used for the evaluation of the crack-tip stress field. The shape and size of the plastic zones at the crack tip corresponding to four stress levels have been obtained, which compare favourably with those obtained by other researchers.     

Elasto‐plastic finite element analysis of shells with damage due to microvoids

This paper presents a non‐linear finite element analysis for the elasto‐plastic behaviour of thick/thin shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis and Woelke (Int. J. Solids Struct. 2004; 41 :3747–3769) provides a set of shell constitutive equations. Numerical implementation of the shell theory leading to the development of the C⁰ quadrilateral shell element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large rotation elasto‐plastic model for shells presented by Voyiadjis and Woelke (General non‐linear finite element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the damage effects due to microvoids, formulated within the framework of a micromechanical damage model. The evolution equation of the scalar porosity parameter as given by Duszek‐Perzyna and Perzyna (Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD‐Vol. 183/MD‐50, 9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage effects, the influence of the microcracks and elastic damage are not considered in this paper. The damage modelled through the evolution of porosity is incorporated directly into the yield function, giving a generalized and convenient loading surface expressed in terms of stress resultants and stress couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34 :1089–1104) is used to derive the large rotation, elasto‐plastic‐damage tangent stiffness matrix. Some of the important features of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells. 2006, submitted). In addition, a non‐layered model is adopted in which integration through the thickness is not necessary. Consequently, the elasto‐plastic‐damage stiffness matrix is also given explicitly and numerical integration is not performed. This makes this model consistent mathematically, accurate for a variety of applications and very inexpensive from the point of view of computer power and time. Copyright © 2006 John Wiley & Sons, Ltd.