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 mixed solid‐shell element for the analysis of laminated composites

This paper presents a versatile low order locking‐free mixed solid‐shell element that can be readily employed for a wide range of linear elastic structural analyses, that is, from thick isotropic structures to multilayer anisotropic composites. This solid‐shell element has eight nodes with only displacement degrees of freedom and few assumed stress parameters that provide very accurate interlaminar stress calculations through the element thickness. These elements can be stacked on top of each other to model multilayer structures, fulfilling the interlaminar stress continuity at the interlayer surfaces and zero traction conditions on the top and bottom surfaces of the laminate. The element formulation is based on the well‐known Fraeijs de Veubeke–Hu–Washizu mixed variational principle with enhanced assumed strains formulation and assumed natural strains formulation to alleviate the different types of locking phenomena in solid‐shell elements. The distinct feature of the present formulation is its ability to accurately calculate the interlaminar stress field in multilayer structures, which is achieved by the introduction of a constraint equation on the interlaminar stresses in the Fraeijs de Veubeke–Hu–Washizu principle‐based enhanced assumed strains formulation. The intelligent computer coding of the present formulation makes the present element appropriate for a wide range of structural analyses. To assess the present formulation's accuracy, a variety of popular numerical benchmark examples related to element convergence, mesh distortion, and shell and laminated composite analyses are investigated and the results are compared with those available in the literature. These benchmark examples reveal that the proposed formulation provides very good results for the structural analysis of shells and multilayer composites. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite volume analysis of laminated composite plates

A numerical procedure for analysis of general laminated plates under transverse load is developed utilizing the Mindlin plate theory, the finite volume discretization, and a segregated solution algorithm. The force and moment balance equations with the laminate constitutive relations are written in the form of a generic transport equation. In order to obtain discrete counterparts of the governing equations, the plate is subdivided into N control volumes by a Cartesian numerical mesh. As a result, five sets of N linear equations with N unknowns are obtained and solved using the conjugate gradient method with preconditioning. For the method validation, a number of test cases are designed to cover thick and thin laminated plates with aspect ratio (width to thickness) from 4 to 100. Simply supported orthotropic, symmetric cross‐ply, and angle‐ply laminated plates under uniform and sinusoidal pressure loads are solved, and results are compared with available analytical solutions. The shear correction factor of 5/6 is utilized throughout the procedure, which is consistent with test cases used in the reviewed literature. Comparisons of the finite volume method results for maximum deflections at the center of the plate and the Navier solutions obtained for aspect ratios 10, 20, and 100 shows a very good agreement. Copyright © 2016 John Wiley & Sons, Ltd.     

Transverse stress profiles reconstruction for finite element analysis of laminated plates

The present work focuses on a posteriori, equilibrium based, reconstruction of transverse stress profiles in the finite element analysis of FSDT laminated plates. The accuracy of this reconstruction depends on accuracy of the first and second-order derivatives of the plate stress resultants, which is not guaranteed by most available low-order plate finite elements. To cure this trouble, two different strategies, based on the Recovery by Compatibility in Patches procedure, are here proposed and compared. Numerical results of typical reconstructed transverse stress profiles are presented showing the effectiveness of the proposed approach.     

A novel 3D mixed finite‐element model for statics of angle‐ply laminates

Analysis of angle‐ply laminates becomes critical and computationally involved because of the presence of extension–shear coupling. A refined three‐dimensional, mixed, 18‐node finite element (FE) model has been developed to analyse angle‐ply laminates under static loading. The minimum potential energy principle has been used for the development of the mixed FE model, where the transverse stress components (τ_xz, τ_yz and σ_z, where z is the thickness direction) have been incorporated as the nodal degrees of freedom, in addition to the three displacement fields. Further, continuity of transverse stress fields through the thickness of the plate and layerwise continuity of displacement fields have been enforced in the formulation. Because all the constitutive and the compatibility conditions have been ensured within the continuum, the present formulation is unique amongst the family of mixed FE models. Results have been obtained for various angle‐ply laminates and compared with analytical and finite‐element solutions, which have been found to be in good agreement with them. Some new results on angle‐ply with clamped–clamped support condition have also been presented to serve as benchmark results. Copyright © 2003 John Wiley & Sons, Ltd.     

The maximum principle violations of the mixed‐hybrid finite‐element method applied to diffusion equations

The abundant literature of finite‐element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial–boundary value problems may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behaviour of the time‐dependent solutions for such problems during small time duration obtained by using a non‐conforming mixed‐hybrid finite‐element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin finite element (FE) as well as the finite‐difference (FD) methods are checked. Owing to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non‐physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial–boundary value problem will be presented. One of the propositions given to avoid any non‐physical oscillations is to use the mass‐lumping techniques. Copyright © 2002 John Wiley & Sons, Ltd.     

Corotational non‐linear analysis of thin plates and shells using a new shell element

A new three‐node triangular shell element is developed by combining the optimal membrane element and discrete Kirchhoff triangle (DKT) plate bending element, and is modified for laminated composite plates and shells so as to include the membrane‐bending coupling effect. Using appropriate shape functions for the bending and membrane modes of the element, the ‘inconsistent’ stress stiffness matrix is formulated and the tangent stiffness matrix is determined. Non‐linear analysis of thin‐walled structures with geometric non‐linearity is conducted using the corotational method. The new element is thoroughly tested by solving few popular benchmark problems. The results of the analysis are compared with those obtained using existing membrane elements. Copyright © 2006 John Wiley & Sons, Ltd.     

A finite element formulation for the simulation of propagating delaminations in layered composite structures

This paper presents a finite element method to simulate growing delaminations in composite structures. The delamination process, using an inelastic material law with softening, takes place within an interface layer having a small, but non‐vanishing thickness. A stress criterion is used to detect the critical points. To prevent mesh‐dependent solutions a regularization technique is applied. The artificial viscosity leads to corresponding stiffness matrices which guarantee stable equilibrium iterations. The essential material parameter which describes the delamination process is the critical energy release rate. The finite element calculations document the robustness and effectivity of the developed model. Extensive parameter studies are performed to show the influence of the introduced geometrical and material quantities. Copyright © 2001 John Wiley & Sons, Ltd.     

A systematic construction of B‐bar functions for linear and non‐linear mixed‐enhanced finite elements for plane elasticity problems

In a previous paper a modified Hu–Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2. For the mixed element QE2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case. In this paper an alternative approach is discussed. The new formulation leads to the same accuracy for linear elastic problems as the QE2 element; however it turns out to be more efficient in numerical simulations, especially for large deformation problems. Using orthogonal stress and strain functions we derive B̄ functions which avoid numerical inversion of matrices. The B̄ ‐strain matrix is sparse and has the same structure as the strain matrix B obtained from a compatible displacement field. The implementation of the derived mixed element is basically the same as the one for a compatible displacement element. The only difference is that we have to compute a B̄ ‐strain matrix instead of the standard B ‐matrix. Accordingly, existing subroutines for a compatible displacement element can be easily changed to obtain the mixed‐enhanced finite element which yields a higher accuracy than the Q4 and QM6 elements. Copyright © 1999 John Wiley & Sons, Ltd.     

An adaptive non‐conforming finite‐element method for Reissner–Mindlin plates

Adaptive algorithms are important tools for efficient finite‐element mesh design. In this paper, an error controlled adaptive mesh‐refining algorithm is proposed for a non‐conforming low‐order finite‐element method for the Reissner–Mindlin plate model. The algorithm is controlled by a reliable and efficient residual‐based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non‐optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficient 3D‐finite element formulation for thin mechanical and piezoelectric structures

It is the aim to compute efficiently the deformations of the mechanical structures that are excited or damped by piezoelectric actuators. They are treated as 3D structures to have much flexibility. Special design of the finite element concept is required since the structures are thin walled and locking effects have to be avoided. Although the computations are performed in the framework of 3D elasticity, we use ideas from modern plate elements and mixed methods. Copyright © 2007 John Wiley & Sons, Ltd.     

Stress recovery and error estimation for the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. This paper develops a stress recovery procedure based on a modal interpretation of the scaled boundary finite‐element method solution process, using the superconvergent patch recovery technique. The recovered stresses are superconvergent, and are used to calculate a recovery‐type error estimator. A key feature of the procedure is the compatibility of the error estimator with the standard recovery‐type finite element estimator, allowing the scaled boundary finite‐element method to be compared directly with the finite element method for the first time. A plane strain problem for which an exact solution is available is presented, both to establish the accuracy of the proposed procedures, and to demonstrate the effectiveness of the scaled boundary finite‐element method. The scaled boundary finite‐element estimator is shown to predict the true error more closely than the equivalent finite element error estimator. Unlike their finite element counterparts, the stress recovery and error estimation techniques work well with unbounded domains and stress singularities. Copyright © 2002 John Wiley & Sons, Ltd.     

An enriched macro finite element for the static analysis of thick general quadrilateral laminated composite plates

This article presents the formulation of an enriched macro finite element based on the trigonometric shear deformation theory for the static analysis of symmetrically laminated composite plates. Shear correction factor is not required because this theory accounts for tangential stress-free boundary conditions on the plate boundary surfaces. The macro element is obtained using the principle of virtual work and Gram-Schmidt orthogonal polynomials as enrichment functions. The implementation of the obtained algorithm is simple and efficient, and allows studying general quadrilateral plates with a single macro element. Several examples are presented to show the capability and applicability of the developed formulation.     

A two‐grid method for expanded mixed finite‐element solution of semilinear reaction–diffusion equations

We present a scheme for solving two‐dimensional semilinear reaction–diffusion equations using an expanded mixed finite element method. To linearize the mixed‐method equations, we use a two‐grid algorithm based on the Newton iteration method. The solution of a non‐linear system on the fine space is reduced to the solution of two small (one linear and one non‐linear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h^1/3). As a result, solving such a large class of non‐linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd.     

A semi-analytical finite element for laminated composite plates

This paper presents a semi-analytical finite element solution for laminated composite plates. The method is based on a mixed variational principle that involves both displacements and stresses. Finite element meshes are only used in the plane of plate, while the through thickness distributions of displacements and stresses are obtained using the method of state equations. Numerical results show that the rate of convergence of the new method is fast and the solutions can be very close to corresponding exact three-dimensional ones. The use of a recursive formulation of the state equations leads to an algebra equation system, from which solution are sought, whose dimension is independent of the numbers of layers of the plate considered.     

Interlaminar shear stresses in laminated composite plates under thermal and mechanical loading

The combined effects of thermal and mechanical loadings on the distribution of interlaminar shear stresses in composite laminated thin and moderately thick composite plates are investigated numerically using the commercially available software package MSC NASTRAN/PATRAN. The validity of the present finite element analysis is demonstrated by comparing the interlaminar shear stresses evaluated using the experimental measurement. Various parametric studies are also performed to investigate the effect of stacking sequences, length to thickness ratio, and boundary conditions on the interlaminar shear stresses with identical mechanical and thermal loadings. It is observed that the effect of thermal environment on the interlaminar shear stresses in carbon-epoxy fiber-reinforced composite laminated plates are much higher in asymmetric cross-ply laminate and anti-symmetric laminate compared to symmetric cross-ply laminate and unidirectional laminate under identical loadings and boundary conditions.     

A 10‐node composite tetrahedral finite element for solid mechanics

We propose a reformulation of the composite tetrahedral finite element first introduced by Thoutireddy et al. By choosing a different numerical integration scheme, we obtain an element that is more accurate than the one proposed in the original formulation. We also show that in the context of Lagrangian approaches, the gradient and projection operators derived from the element reformulation admit fully analytic expressions, which offer a significant improvement in terms of accuracy and computational expense. For plasticity applications, a mean‐dilatation approach on top of the underlying Hu–Washizu variational principle proves effective for the representation of isochoric deformations. The performance of the reformulated element is demonstrated by hyperelastic and inelastic calculations. Copyright © 2016 John Wiley & Sons, Ltd.     

An isostatic assumed stress triangular element for the Reissner–Mindlin plate‐bending problem

The paper describes a new assumed stress triangular element for Reissner–Mindlin plates, called TIP3, with three nodes and three degrees of freedom per node. The kinematics is constructed by means of the so‐called linked interpolation ruled by technically significant degrees of freedom (i.e. one transversal displacement and two rotations per node) without using additional bubble modes. The static representation starts from a moment–shear uncoupled polynomial approximation and is constrained to satisfy some equilibrium conditions in order to reduce the stress parameters to a minimum number. The resulting element is locking free, presents correct rank and passes the bending patch test even for very thin plates. The good performances of the element are demonstrated by several comparisons with other triangular plate elements available in the literature. Copyright © 2007 John Wiley & Sons, Ltd.