High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

Based on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem. The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices. By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite element formulations of strain gradient theory for microstructures and the C0–1 patch test

Based on finite element formulations for the strain gradient theory of microstructures, a convergence criterion for the C^0–1 patch test is introduced, and a new approach to devise strain gradient finite elements that can pass the C^0–1 patch test is proposed. The displacement functions of several plane triangular elements, which satisfy the C⁰ continuity and weak C¹ continuity conditions are evaluated by the C^0–1 patch test. The difference between the proposed C^0–1 patch test and the C⁰ constant stress and C¹ constant curvature patch tests is elucidated. An 18-DOF plane strain gradient triangular element (RCT9+RT9), which passes the C^0–1 patch test and has no spurious zero energy modes, is proposed. Numerical examples are employed to examine the performance of the proposed element by carrying out the C^0–1 patch test and eigenvalue test. The proposed element is found to be without spurious zero energy modes, and it possesses higher accuracy compared with other strain gradient elements. Copyright © 2004 John Wiley & Sons, Ltd.     

A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity

C^0?1 patch test (Int. J. Numer. Meth. Engng 2004; 61 :433–454) proposed by Soh and Chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress/strain gradient elasticity. The C^0?1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations. To pass the C^0?1 patch test, the element displacement functions used to calculate strains must satisfy C⁰ continuity (or weak C⁰ continuity) and quadratic completeness. In this paper, a 24‐DOF (degrees of freedom) quadrilateral element (CQ12+RDKQ) for the couple stress/strain gradient elasticity is developed by combining the refined thin plate element RDKQ and the nonconforming element CQ12. The element RDKQ, which satisfies weak C¹ continuity, is used to calculate strain gradients, whereas strains are computed by the element CQ12, which is established based on an extended variational functional and satisfies weak C⁰ continuity and quadratic completeness. Numerical examples show that the element (CQ12+RDKQ) passes the C^0?1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects. Copyright © 2010 John Wiley & Sons, Ltd.     

A three‐dimensional C1 finite element for gradient elasticity

In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C¹ continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C¹ finite elements and their perceived computational cost. In this context, the lack of three‐dimensional C¹ elements is of particular concern. In this paper we present a new C¹ hexahedral element which, to the best of our knowledge, is the first three‐dimensional C¹ element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C¹ elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good‐quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite elements for materials with strain gradient effects

A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first‐order strain gradients and the associated work‐conjugate higher‐order stresses. In conventional displacement‐based approaches, the interpolation of displacement requires C¹‐continuity in order to ensure convergence of the finite element procedure for higher‐order theories. Mixed‐type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C⁰‐continuous shape functions and can achieve the same convergence as C¹ elements. These C⁰ elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

ASSUMED STRESS C0 QUADRILATERAL/TRIANGULAR PLATE ELEMENTS BY INTERRELATED EDGE DISPLACEMENTS

This paper presents two simple quadrilateral C⁰ plate bending elements with explicit element stiffness matrix. The element formulation is based on assumed element stress fields and the interrelated transverse displacement and rotation along element boundaries. The interrelated edge displacements not only can result in higher-order displacements interpolations for higher accuracy element and overcome the shear locking in thin plate analysis encountered by C⁰ plate elements, but can also unify the four-noded quadrilateral element and its corresponding three-noded triangular element. The latter cannot be achieved by the assumed displacement formulation. The numerical examples demonstrate the accuracy and robustness of the present assumed stress C⁰ plate elements.     

An n-sided polygonal finite element for nonlocal nonlinear analysis of plates and laminates

In this study, a locking-free n-sided C¹ polygonal finite element is presented for nonlinear analysis of laminated plates. The plate kinematics is based on Reddy's third-order shear deformation theory (TSDT). The in-plane displacements are approximated using barycentric form of Lagrange shape functions. The weak-form Galerkin formulation based on the kinematics of TSDT requires the C¹ approximation of the transverse displacement over the polygonal element. This is achieved by embedding the C⁰ Lagrange interpolants over a cubic Bernstein-Bezier patch defined over the n-sided polygonal element. Such an approach ensures the continuity of the derivative field at the inter-element edges. In addition, Eringen's stress-gradient nonlocal constitutive equations are used in the present formulation to account for nonlocality. The effect of geometric nonlinearity is taken by considering the von Kármán geometric nonlinearity. Examples are presented to show the effect of nonlocality, geometric nonlinearity, and the lamination scheme on the bending behavior of laminated composite plates. The results are compared with analytical solutions, conventional FEM results, and with those available in the literature. Shear locking is addressed considering reduced integration and consistent interpolation techniques. The patch test is used to check the convergence of the element developed.     

Static Response of Geometrically Nonlinear Laminated Composite Plates Having Uncertain Material Properties

The effect of uncertainty in material properties on the transverse bending of laminated composite plate is investigated. The transverse shear and large rotations have been included in the system equation in the framework of higher order shear deformation theory. The analysis uses Green–Lagrange nonlinear strain displacement equations to model geometric nonlinearity. The stochastic finite element analysis is performed using a direct iteration approach to handle deterministic geometric nonlinearity and perturbation approach to handle the randomness in the material properties. Mean and variance of the transverse deflection have been obtained by employing a C⁰ isoparametric nonlinear finite element model.     

A method for creating a class of triangular C1 finite elements

Finite elements providing a C¹ continuous interpolation are useful in the numerical solution of problems where the underlying partial differential equation is of fourth order, such as beam and plate bending and deformation of strain‐gradient‐dependent materials. Although a few C¹ elements have been presented in the literature, their development has largely been heuristic, rather than the result of a rational design to a predetermined set of desirable element properties. Therefore, a general procedure for developing C¹ elements with particular desired properties is still lacking. This paper presents a methodology by which C¹ elements, such as the TUBA 3 element proposed by Argyris et al., can be constructed. In this method (which, to the best of our knowledge, is the first one of its kind), a class of finite elements is first constructed by requiring a polynomial interpolation and prescribing the geometry, the location of the nodes and the possible types of nodal DOFs. A set of necessary conditions is then imposed to obtain appropriate interpolations. Generic procedures are presented, which determine whether a given potential member of the element class meets the necessary conditions. The behaviour of the resulting elements is checked numerically using a benchmark problem in strain‐gradient elasticity. Copyright © 2011 John Wiley & Sons, Ltd.     

Twenty-four–DOF four-node quadrilateral elements for gradient elasticity

Computational analysis of gradient elasticity often requires the trial solution to be C¹, yet constructing simple C¹ finite elements is not trivial. In this paper, three four-node 24-DOF quadrilateral elements for gradient elasticity analysis are devised by generalizing some of the advanced element formulations for thin-plate analysis. These include the discrete Kirchhoff method, a relaxed hybrid-stress method, and the hybrid-stress method with equilibrating internal force modes. The first two methods start with the derivation of a C⁰ displacement, which is quadratic complete in the Cartesian coordinates. In the first method, at the midside points are derived and interpolated together with those at the nodes. Strain is derived from the displacement interpolation yet the second-order displacement derivatives are derived from the displacement-gradient interpolation. In the second method, only the assumed constant double-stress modes are employed to enforce the continuity of the normal derivative of the displacement. In the third method, the equilibrating internal force modes require the C¹ displacement to be defined only along the element boundary and the domain interpolation can be avoided. Patch test involving linear stress and constant double stress as well as other tests are presented to validate the proposed elements.     

A higher-order facet quadrilateral composite shell element

A C⁰ finite element formulation of flat faceted element based on a higher-order displacement model is presented for the analysis of general, thin-to-thick, fibre reinforced composite laminated plates and shells. This theory incorporates a realistic non-linear variation of displacements through the shell thickness, and eliminates the use of shear correction coefficients. The discrete element chosen is a nine-noded quadrilateral with five and nine degrees of freedom per node. A comparison of results is also made with the 2-D thin classical and 3-D exact analytical results, and finite element solutions with 9-noded first-order element. © 1997 John Wiley & Sons, Ltd.     

A 3‐node C0 triangular element for the modified couple stress theory based on the smoothed finite element method

In this paper, a 3‐node C⁰ triangular element for the modified couple stress theory is proposed. Unlike the classical continuum theory, the second‐order derivative of displacement is included in the weak form of the equilibrium equations. Thus, the first‐order derivative of displacement, such as the rotation, should be approximated by a continuous function. In the proposed element, the derivative of the displacement is defined at a node using the node‐based smoothed finite element method. The derivative fields, continuous between elements and linear in an element, are approximated with the shape functions in element. Both the displacement field and the derivative field of displacement are expressed in terms of the displacement degree of freedom only. The element stiffness matrix is calculated using the newly defined derivative field. The performance of the proposed element is evaluated through various numerical examples.     

Mechanically based models: Adaptive refinement for B‐spline finite element

This article presents two new methods for adaptive refinement of a B‐spline finite element solution within an integrated mechanically based computer aided engineering system. The proposed techniques for adaptively refining a B‐spline finite element solution are a local variant of np‐refinement and a local variant of h‐refinement. The key component in the np‐refinement is the linear co‐ordinate transformation introduced into the refined element. The transformation is constructed in such a way that the transformed nodal configuration of the refined element is identical to the nodal configuration of the neighbour elements. Therefore, the assembly proceeds as with classic finite elements, while the solution approximation conforms exactly along the inter‐element boundaries. For the h‐refinement, this transformation is introduced into a construction that merges the super element from the finite element world with the hierarchical B‐spline representation from the computational geometry. In the scope of developing sculptured surfaces, the proposed approach supports C⁰ as well as the Hermite B‐spline C¹ continuous shapes. For sculptured solids, C⁰ continuity only is considered in this article. The feasibility of the proposed methods in the scope of the geometric design is demonstrated by several examples of creating sculptured surfaces and volumetric solids. Numerical performance of the methods is demonstrated for a test case of the two‐dimensional Poisson equation. Copyright © 2003 John Wiley & Sons, Ltd.     

An Edge-Based Smoothed Discrete Shear Gap Method Using the C0-Type Higher-Order Shear Deformation Theory for Analysis of Laminated Composite Plates

The edge-based smoothing discrete shear gap method (ES-DSG3) using three-node triangular elements is combined with a C⁰-type higher-order shear deformation theory (HSDT) to give a new linear triangular plate element for static, free vibration, and buckling analyses of laminated composite plates. In the ES-DSG3, only the linear approximation is necessary, and the discrete shear gap method (DSG) for triangular plate elements is used to avoid the shear locking and spurious zero energy modes. In addition, the stiffness matrices are calculated relying on smoothing domains associated with the edges of the triangular elements through an edge-based strain smoothing technique. Using the C⁰-type HSDT, the shear correction factors in the original ES-DSG3 can be removed and replaced by two additional degrees of freedom at each node. The numerical examples demonstrated that the ES-DSG3 show remarkably excellent performance compared to several other published elements in the literature.     

Statically admissible finite element solution of the thin plate bending problem with a posteriori error estimation

Two triangular elements of class C⁰ developed on the basis of the principle of complementary work are applied in the static analysis of a thin plate. Some techniques to widen the versatility of the equilibrium approach for the finite element method are presented. Plates of various shapes subjected to diverse types of loading are considered. The results are compared with outcomes obtained by use of the displacement-based finite element method. By use of these two dual types of solutions, the error of the approximate solution is calculated. The lower and upper bounds for the strain energy are found.     

On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

The static behaviour of spherical shallow shells bonded with piezoelectric actuators and subjected to electrical loading are studied in this paper by using the differential quadrature element method (DQEM). Geometrical non‐linear effects are considered. Detailed formulations for the DQ circular spherical shallow shell element and the DQ annular spherical shallow shell element are given for the first time. Numerical studies are performed to evaluate the effects of actuator size, thickness and boundary conditions. Very accurate results are obtained by the DQEM. Based on the results reported in this paper, one may conclude that the DQEM is a useful tool for obtaining solutions for smart materials and structures exhibiting geometric non‐linear behaviours. Thickness effects cannot be neglected when the actuator thickness is comparable to that of the base material. Snap‐through may occur when the applied voltage reaches a critical value even without mechanical loading for certain geometric configurations. Copyright © 2001 John Wiley & Sons, Ltd.     

Singular‐ended spline interpolation for two‐dimensional boundary element analysis

A method of interpolation of the boundary variables that uses spline functions associated with singular elements is presented. This method can be used in boundary element method analysis of 2‐D problems that have points where the boundary variables present singular behaviour. Singular‐ended splines based on cubic splines and Overhauser splines are developed. The former provides C²‐continuity and the latter C¹‐continuity across element edges. The potentialities of the methodology are demonstrated analysing the dynamic response of a 2‐D rigid footing interacting with a half‐space. It is shown that, for a given number of elements at the soil–foundation interface, the singular‐ended spline interpolation increases substantially the displacement convergence rate and delivers smoother traction distributions. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain‐softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C¹ continuity element. The required higher‐order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill‐posedness caused by strain‐softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post‐peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear‐band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values. Copyright © 2001 John Wiley & Sons, Ltd.     

On elimination of shear locking in the element‐free Galerkin method

In this study, a method for completely eliminating the presence of transverse shear locking in the application of the element‐free Galerkin method (EFGM) to shear‐deformable beams and plates is presented. The matching approximation fields concept of Donning and Liu has shown that shear locking effects may be prevented if the approximate rotation fields are constructed with the innate ability to match the approximate slope (first derivative of displacement) fields and is adopted. Implementation of the matching fields concept requires the computation of the second derivative of the shape functions. Thus, the shape functions for displacement fields, and therefore the moving least‐squares (MLS) weight function, must be at least C¹ continuous. Additionally, the MLS weight functions must be chosen such that successive derivatives of the MLS shape function have the ability to exactly reproduce the functions from which they were derived. To satisfy these requirements, the quartic spline weight function possessing C² continuity is used in this study. To our knowledge, this work is the first attempt to address the root cause of shear locking phenomenon within the framework of the element‐free Galerkin method. Several numerical examples confirm that bending analyses of thick and thin beams and plates, based on the matching approximation fields concept, do not exhibit shear locking and provide a high degree of accuracy for both displacement and stress fields. Copyright © 2001 John Wiley & Sons, Ltd.     

Triangular finite element for analysis of thick laminated shells

The development of a general curved triangular element based on an assumed displacement potential energy approach is presented for the analysis of arbitrarily laminated thick shells. The associated laminated shell theory assumes transverse inextensibility and layerwise constant shear angle. The present element is a quadratic triangle of C⁰-type in the curvilinear co-ordinate plane, which is then mapped onto a curved surface. Convergence of transverse displacement, moments, stresses and the effect of two Gauss quadrature schemes also form a part of the investigation. Examples of two laminated shell problems demonstrate the accuracy and efficiency of the present element. Comparison of the present LCST (layerwise constant shear-angle theory) based solutions, with those based on the CST (constant shear-angle theory) clearly demonstrates the superiority of the former over the latter, especially in the prediction of the distribution of the surface-parallel displacements and stresses through the laminate thickness.