Quadrilateral Finite Element Research Articles

A nonconforming quadrilateral finite element approximation to nonlinear schrödinger equation

In this article, a nonconforming quadrilateral element (named modified quasi-Wilson element) is applied to solve the nonlinear schrödinger equation (NLSE). On the basis of a special character of this element, that is, its consistency error is of order O(h3) for broken H1-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover, the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.

A New Rotated Nonconforming Quadrilateral Element

In this paper, a new nonparametric nonconforming quadrilateral finite element is introduced. This element takes the four edge mean values as the degrees of the freedom and the finite element space is a subspace of $$P_{2}$$ . Different from the other nonparametric elements, the basis functions of this new element can be expressed explicitly without solving linear systems locally, which can be achieved by introducing a new reference quadrilateral. To evaluate the integration, a class of new quadrature formulae with only three equally weighted points on quadrilateral are constructed. Hence the stiffness matrix can be calculated by the same way with the parametric elements. Numerical results are shown to confirm the optimality of the convergence order for the second order elliptic problems and the Stokes problem.

Nonlinear bending and vibration analyses of quadrilateral composite plates

Linear and nonlinear analyses of shear deformable thin and thick arbitrary straight-sided quadrilateral plates are reported here using smoothed finite element technique. The Reissner-Mindlin plates are discretized with quadrilateral background cells. Then membrane and bending stiffness matrices of background quadrilateral cells are evaluated using edge-based smoothed finite element method(ES-FEM). The shear stiffness matrix is calculated based on "smoothed shear strain approach" and the performance is compared with "edge-consistent four-node quadrilateral finite element". The convergence, accuracy and sensitivity to mesh distortion of the present quadrilateral element is examined. Thereafter, the nonlinear bending and vibration analyses of trapezoidal and arbitrary straight-sided quadrilateral composite plates are presented for which only limited analytical results are available in the literature.

Robust Semi-Discrete and Fully Discrete Hybrid Stress Finite Element Methods for Elastodynamic Problems

AbstractThis paper analyzes semi-discrete and fully discrete hybrid stress quadrilateral finite element methods for 2-dimensional linear elastodynamic problems. The methods use a 4 node hybrid stress quadrilateral element in the space discretization. In the fully discrete scheme, an implicit second-order scheme is adopted in the time discretization. We derive optimal a priori error estimates for the two schemes and an unconditional stability result for the fully discrete scheme. Numerical experiments confirm the theoretical results.

A novel high-performance mixed membrane finite element for the analysis of inelastic structures

A novel high-performance eight-node quadrilateral mixed finite element is presented. Its formulation is based on a Hu-Washizu type functional, suitable to the treatment of material nonlinearities. In order to capture, even for coarse meshes, highly nonlinear spatial distribution of strain field, strain interpolation is assumed to be discontinuous, piecewise-constant over suitable element subdomains. A robust element state determination procedure is proposed to solve compatibility and constitutive equations at element level. The mixed element stability is numerically assessed. Supported by the comparison with compatible quadrilaterals, numerical simulations concerning elastoplastic and shape-memory alloy structures prove accuracy and effectiveness of the proposed formulation.

New convergence analysis for assumed stress hybrid quadrilateral finite element method

New error estimates are established for Pian and Sumihara's (PS) 4-node assumed stress hybrid quadrilateral element [T.H.H. Pian, K. Sumihara, Rational approach for assumed stress finite elements, Int. J. Numer. Methods Engrg., 20 (1984), 1685-1695], which is widely used in engineering computation. Based on an equivalent displacement-based formulation to the PS element, we show that the numerical strain and a postprocessed numerical stress are uniformly convergent with respect to the Lame constant \begin{document} $λ$ \end{document} on the meshes produced through the uniform bisection procedure. Within this analysis framework, we also show that both the numerical strain and stress are uniformly convergent on meshes which are stable for the \begin{document} $Q_1-P_0$ \end{document} Stokes element.

The minimal angle condition for quadrilateral finite elements of arbitrary degree

We study W1,p Lagrange interpolation error estimates for general quadrilateralQk finite elements with k≥2. For the most standard case of p=2 it turns out that the constant C involved in the error estimate can be bounded in terms of the minimal interior angle of the quadrilateral. Moreover, the same holds for any p in the range 1≤p<3. On the other hand, for 3≤p we show that C also depends on the maximal interior angle. We provide some counterexamples showing that our results are sharp.

A new enhanced assumed strain quadrilateral membrane element with drilling degree of freedom and modified shape functions

SummaryA new four‐node quadrilateral membrane finite element with drilling rotational degree of freedom based on the enhanced assumed strain formulation is presented. A simple formulation is achieved by five incompatible modes that are added to the Allman‐type interpolation. Furthermore, modified shape functions are used to improve the behaviour of distorted elements. Numerical results show that the proposed new element exhibits good numerical accuracy and improved performance, and in many cases, superior to existing elements. In particular, Poisson's locking in nearly incompressible elasticity fades and the element performs well when it becomes considerably distorted even when it takes almost triangular shape. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Simulation of a New Technique for Vibration Tests, Based Upon Active Vibration Control

ABSTRACTIn this work, a novel technique is proposed for generating desired transient vibrations in a structure. Cantilevered plate structure is instrumented with one piezoelectric sensor patch and one piezoelectric actuator patch. Quadrilateral plate finite element having three degrees of freedom at each node is employed to divide the plate into finite elements. Thereafter, Hamilton's principle is used to derive equations of motion of the smart plate. Finite-element model is reduced to first three modes using orthonormal modal truncation and subsequently, the reduced finite-element model is converted into a state-space model. Optimal tracking control is then applied to the state-space model of the smart plate. Using this optimal controller, cantilevered plate is made to vibrate as per desired decay curves of first three modes. Simulation results show that presented optimal control strategy is very effective in simultaneously tracking first three vibration modes of the smart plate.

Problem-dependent cubic linked interpolation for Mindlin plate four-node quadrilateral finite elements

We employ the so-called problem-dependent linked interpolation concept to develop two cubic 4-node quadrilateral plate finite elements with 12 external degrees of freedom that pass the constant bending patch test for arbitrary node positions of which the second element has five additional internal degrees of freedom to get polynomial completeness of the cubic form. The new elements are compared to the existing linked-interpolation quadratic and nine-node cubic elements presented by the author earlier and to the other elements from literature that use the cubic linked interpolation by testing them on several benchmark examples.

Sensitivity Analysis of the Galerkin Finite Element Method Neutron Diffusion Solver to the Shape of the Elements

The purpose of the present study is the presentation of the appropriate element and shape function in the solution of the neutron diffusion equation in two-dimensional (2D) geometries. To this end, the multigroup neutron diffusion equation is solved using the Galerkin finite element method in both rectangular and hexagonal reactor cores. The spatial discretization of the equation is performed using unstructured triangular and quadrilateral finite elements. Calculations are performed using both linear and quadratic approximations of shape function in the Galerkin finite element method, based on which results are compared. Using the power iteration method, the neutron flux distributions with the corresponding eigenvalue are obtained. The results are then validated against the valid results for IAEA-2D and BIBLIS-2D benchmark problems. To investigate the dependency of the results to the type and number of the elements, and shape function order, a sensitivity analysis of the calculations to the mentioned parameters is performed. It is shown that the triangular elements and second order of the shape function in each element give the best results in comparison to the other states.

Hybrid stress quadrilateral finite element approximation for stochastic plane elasticity equations

SummaryThis paper considers stochastic hybrid stress quadrilateral finite element analysis of plane elasticity equations with stochastic Young's modulus and stochastic loads. Firstly, we apply Karhunen–Loève expansion to stochastic Young's modulus and stochastic loads so as to turn the original problem into a system containing a finite number of deterministic parameters. Then we deal with the stochastic field and the space field by k−version/p−version finite element methods and a hybrid stress quadrilateral finite element method, respectively. We derive a priori error estimates, which are uniform with respect to the Lamè constant λ∈(0,+∞). Finally, we provide some numerical results. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Field validation of a three-dimensional dynamic track-subgrade interaction model

An efficient three-dimensional dynamic track-subgrade interaction model has been formulated and then validated by field investigations at various field and traffic conditions including the effect of different train speeds and types of trains. The model contains a two-dimensional discrete support track model and three-dimensional computation-efficient finite element soil subgrade model. In the two-dimensional track model, the rail beam is modelled as an Euler-Bernoulli beam. The two-dimensional track model discretizes the tie and ballast as rigid bodies with designated spacing. The three-dimensional finite element subgrade model is simulated by plane-stress quadrilateral finite elements. The longitudinal direction of the subgrade model is expanded in the frequency domain and is assumed to be homogeneous. Therefore, the computing time could be largely reduced. A moving dynamic loading is applied on top of the rail. The model is capable of taking train speed variations and the profile change of the cross section into consideration. Multiple field instrumentation tests covering the two train types and different train speeds at the test site were then conducted to verify the accuracy of the dynamic track-subgrade interaction model. Testing site is located on the Amtrak's highest speed line (Northeast Corridor: 250 km/h) near Kingston, Rhode Island in the United States. A method to obtain the tie deflection from accelerometer data at Kingston was proposed and then validated at another site on the Northeast Corridor. Tie deflections measured in the field were compared with those predicted by the three-dimensional dynamic track-subgrade interaction model. It is concluded that this model can predict track performance accurately for the Kingston site.

Quadrilateral membrane finite elements with rotational DOFs for the analysis of geometrically linear and nonlinear plane problems

In this paper, we present the formulations of two four-node quadrilateral membrane finite elements with rotational degrees of freedom to analyze geometric linear and nonlinear plane problems. They are based on a plane adaptation of the space fiber rotation concept that considers virtual rotations of a nodal fiber within the element enhancing the displacement vector approximation of low-order elements. An updated Lagrangian approach is chosen to describe large displacement with small strain kinematics. Several geometric linear and nonlinear benchmarks are presented to assess the performance of the proposed membrane elements and the obtained results demonstrate their efficiency.

An Adaptive Hybrid Stress Transition Quadrilateral Finite Element Method for Linear Elasticity

In this paper, we discuss an adaptive hybrid stress finite element method on quadrilateral meshes for linear elasticity problems. To deal with hanging nodes arising in the adaptive mesh refinement, we propose new transition types of hybrid stress quadrilateral elements with 5 to 7 nodes. In particular, we derive a priori error estimation for the 5node transition hybrid stress element to show that it is free from Poisson-locking, in the

An Enhanced Tire Model for Dynamic Simulation based on Geometrically Exact Shells

Abstract In the present work, a tire model is derived based on geometrically exact shells. The discretization is done with the help of isoparametric quadrilateral finite elements. The interpolation is performed with bilinear Lagrangian polynomials for the mid-surface as well as for the director field. As time stepping method for the resulting differential algebraic equation a backward differentiation formula is chosen. A multilayer material model for geometrically exact shells is introduced, to describe the anisotropic behavior of the tire material. To handle the interaction with a rigid road surface, a unilateral frictional contact formulation is introduced. Therein a special surface to surface contact element is developed, which rebuilds the shape of the tire.

Mixed shell element for static and buckling analysis of variable angle tow composite plates

A mixed quadrilateral 3D finite element, obtained from the Hellinger–Reissner functional, is presented for linear static and buckling analyses of variable-angle tow (VAT) composite plates. Variable-angle tows describe curvilinear fiber paths within composite laminae and are a promising technology for tailoring the buckling and post-buckling capability of plates. Due to the variable stiffness across the planform of the VAT plates, pre-buckling stresses can be tailored and redistributed towards supported edges, thereby greatly improving the buckling load. A linear mixed element called MISS-4 is used as starting point for this work. The element presents a self-equilibrated and isostatic state of stress. The kinematics lead to element compatibility matrix calculations based solely on the interpolation along element edges. The drilling rotations do not require penalty functions or non-symmetric formulations, thus avoiding spurious energy modes. The buckling analysis is reliably performed via a co-rotational formulation. In this work VAT plates with linear fiber angle variation in one direction, and constant stiffness properties in the orthogonal direction are studied. Numerical examples of VAT plates subjected to different loads and boundary conditions are investigated herein. The convergence of displacements, stress resultants and buckling loads are presented, and comparisons with numerical results, obtained using the S4R finite element of Abaqus and the pseudo-spectral Differential Quadrature Method, are shown.

A finite element model for propagating delamination in laminated composite plates based on the Virtual Crack Closure method

In the paper, a simple and efficient algorithm to track a moving delamination front of arbitrary shape, using a laminated finite plate element model in conjunction with the Virtual Crack Closure Technique (VCCT), is proposed. The solution requires the calculation of the virtually closed area in front of the delamination, which is approximated by means of a 6-node-polygon, the delamination opening behind this front and the reaction forces in the nodes at the delamination front. These quantities are calculated in a local coordinate system (LCS) defined in the nodes along the delamination front. Using the proposed algorithm, arbitrary meshes composed of 4- and 9-node quadrilateral finite elements can be considered. The proposed model is developed in the context of a layered finite element plate model. To prevent interlaminar penetration of adjacent layers in the delaminated region, an algorithm recently proposed by Marjanović et al. (2015) is adopted. The model performance is demonstrated by re-analyses of the Double-Cantilever-Beam problem, for which analytical solutions exist, and by transient analyses of laminated composite plates with propagating delamination fronts.

Modeling of piezoelectric plates with variables separation for static analysis

In this work, the modeling of laminated composite plates with embedded piezoelectric layers is addressed through a variables separation approach. Both the displacement and electric potential fields are approximated as a sum of separated functions of the in-plane coordinates x, y and the transverse coordinate z. This choice yields to a nonlinear problem that can be solved by an iterative process. That consists of solving a 2D and 1D problem successively at each iteration. In the thickness direction, a fourth and second-order expansion in each layer is considered for the displacements and the electric potential, respectively. For the in-plane description, classical eight-node quadrilateral finite element is used. Numerical examples involving several representative laminates are addressed to show the accuracy of the present LayerWise (LW) method. It is shown that it can provide quasi-3D results less costly than classical LW computations. In particular, the estimation of the transverse stresses which is of major importance for damage analysis is very good.

Superconvergence and recovery type a posteriori error estimation for hybrid stress finite element method

Superconvergence and a posteriori error estimators of recovery type are analyzed for the 4-node hybrid stress quadrilateral finite element method proposed by Pian and Sumihara (Int. J. Numer. Meth. Engrg., 1984, 20: 1685-1695) for linear elasticity problems. Uniform superconvergence of order $O(h^{1+\min\{\alpha,1\}})$ with respect to the Lam\'{e} constant $\lambda$ is established for both the recovered gradients of the displacement vector and the stress tensor under a mesh assumption, where $\alpha>0$ is a parameter characterizing the distortion of meshes from parallelograms to quadrilaterals. A posteriori error estimators based on the recovered quantities are shown to be asymptotically exact. Numerical experiments confirm the theoretical results.