General Quadrilateral Element Research Articles

Convergence in the incompressible limit of new discontinuous Galerkin methods with general quadrilateral and hexahedral elements

Standard low-order finite elements, which perform well for problems involving compressible elastic materials, are known to under-perform when nearly incompressible materials are involved, commonly exhibiting the locking phenomenon. Interior penalty (IP) discontinuous Galerkin methods have been shown to circumvent locking when simplicial elements are used. The same IP methods, however, result in locking on meshes of quadrilaterals. The authors have shown in earlier work that under-integration of specified terms in the IP formulation eliminates the locking problem for rectangular elements. Here it is demonstrated through an extensive numerical investigation that the effect of using under-integration carries over successfully to meshes of more general quadrilateral elements, as would likely be used in practical applications, and results in accurate displacement approximations. Uniform convergence with respect to the compressibility parameter is shown numerically. Additionally, a stress approximation obtained here by postprocessing shows good convergence in the incompressible limit.

Anisotropic interpolation error estimate for arbitrary quadrilateral isoparametric elements

The aim of this paper is to show that, for any $$p \in [1,\infty )$$ , the $$W^{1,p}$$ -anisotropic interpolation error estimate holds on quadrilateral isoparametric elements verifying the maximum angle condition (MAC) and the property of comparable lengths for opposite sides (clos), i.e., on all those quadrilaterals with interior angles uniformly bounded away from $$\pi $$ and with both pairs of opposite sides having comparable lengths. For rectangular elements our interpolation error estimate agrees with the usual one whereas for perturbations of rectangles (the most general quadrilateral elements previously considered as far as we know) our result has some advantages with respect to the pre-existing ones: the interpolation error estimate that we proved is written by using two neighboring sides of the element instead of the sides of the unknown perturbed rectangle and, on the other hand, conditions MAC and clos are much simpler requirements to verify and with a clearer geometrical sense than those involved in the definition of perturbations of a rectangle.

Erratum to: On the Stability of the Flux Reconstruction Schemes on Quadrilateral Elements for the Linear Advection Equation

The flux reconstruction (FR) approach to high-order methods has proved to be a promising alternative to traditional discontinuous Galerkin (DG) schemes since they facilitate the adoption of explicit time-stepping methods suitable for parallel architectures like GPUs. The FR approach provides a parameterized family of schemes through which various classical schemes like nodal-DG and spectral difference methods can be recovered. Further, the parameters can be varied to obtain schemes with a maximum stable time-step, or minimum dispersion or dissipation errors etc., providing us a single powerful framework unifying high-order discontinuous Finite Element Methods. There have been various studies on the accuracy and the stability of these schemes and in particular, a subset of the FR schemes known as ESFR or VCJH schemes have been shown to be stable in 1D and on simplex elements in 2D and 3D for the linear advection as well as the advection---diffusion equations. However, the stability of the FR schemes on tensor product quadrilateral elements has remained an open question. Although it is the most natural extension of the 1D FR approach, it has posed a significant challenge, especially for general quadrilateral elements. In this paper, we investigate the stability of the VCJH-type FR schemes for linear advection on Cartesian quadrilateral meshes and show that the schemes could become unstable under certain conditions. However, we find that the VCJH scheme recovering the DG method is stable on all Cartesian meshes. Although we restrict ourselves to Cartesian meshes in order to circumvent the algebraic complexity posed by the variation of the Jacobian matrix inside general tensor-product quadrilateral elements, our analysis offers significant insight into the possible origins of instability in the FR approach on general quadrilaterals.

Numerical modeling of toxic nonaqueous phase liquid removal from contaminated groundwater systems: mesh effect and discretization error estimation

SummaryNumerical modeling has now become an indispensable tool for investigating the fundamental mechanisms of toxic nonaqueous phase liquid (NAPL) removal from contaminated groundwater systems. Because the domain of a contaminated groundwater system may involve irregular shapes in geometry, it is necessary to use general quadrilateral elements, in which two neighbor sides are no longer perpendicular to each other. This can cause numerical errors on the computational simulation results due to mesh discretization effect. After the dimensionless governing equations of NAPL dissolution problems are briefly described, the propagation theory of the mesh discretization error associated with a NAPL dissolution system is first presented for a rectangular domain and then extended to a trapezoidal domain. This leads to the establishment of the finger‐amplitude growing theory that is associated with both the corner effect that takes place just at the entrance of the flow in a trapezoidal domain and the mesh discretization effect that occurs in the whole NAPL dissolution system of the trapezoidal domain. This theory can be used to make the approximate error estimation of the corresponding computational simulation results. The related theoretical analysis and numerical results have demonstrated the following: (1) both the corner effect and the mesh discretization effect can be quantitatively viewed as a kind of small perturbation, which can grow in unstable NAPL dissolution systems, so that they can have some considerable effects on the computational results of such systems; (2) the proposed finger‐amplitude growing theory associated with the corner effect at the entrance of a trapezoidal domain is useful for correctly explaining why the finger at either the top or bottom boundary grows much faster than that within the interior of the trapezoidal domain; (3) the proposed finger‐amplitude growing theory associated with the mesh discretization error in the NAPL dissolution system of a trapezoidal domain can be used for quantitatively assessing the correctness of computational simulations of NAPL dissolution front instability problems in trapezoidal domains, so that we can ensure that the computational simulation results are controlled by the physics of the NAPL dissolution system, rather than by the numerical artifacts. Copyright © 2014 John Wiley & Sons, Ltd.

A finite volume method on distorted quadrilateral meshes for discretization of the energy equation's conduction term

AbstractA finite volume method (FVM) on distorted meshes for discretizing the energy equation's conduction term is presented. In this method, it is possible to compose the computational mesh of general quadrilateral elements (cells), namely, the cells are not required to be rectangular. The gradient of temperature on the cell's surface is computed to be second‐order accurate. Therefore, the error of numerical results by this method is smaller than using the traditional multilateral element method (MEM). The error does not depend on the degree of mesh distortion. The formulation based only on Taylor's theorem is straightforward. These are advantageous features to revise the fluid flow computation programs (based on FVM) that neglected the heat conduction term of the energy equation. The test calculations show that the convergence tendency of the numerical error using this method with the distorted mesh is the same as using an ordinary 2‐node central difference scheme on a constant‐interval rectangular mesh. By this method a conduction term was added to the energy equation of a SALE [1] program which had neglected that term originally, and z numerical calculation of a fluid flow with a heat transfer problem was performed. The numerical result of the present method with the distorted mesh well agrees with the analytical solution and the result of REM with a rectangular mesh. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.20375

On the smoothed finite element method

AbstractRecently, Liu et al. proposed the smoothed finite element method by using the non‐mapped shape functions and then introducing the strain smoothing operator when evaluating the element stiffness in the framework of the finite element method. However, the theories and examples by Liu et al. are not sufficient for general quadrilateral elements. This paper shows that the non‐mapped shape functions used in the smoothed finite element have disadvantages in existence, linearity, non‐negativity and patch test. Copyright © 2008 John Wiley & Sons, Ltd.

Finite elements with embedded strong discontinuities for the modeling of failure in solids

AbstractThis paper presents new finite elements that incorporate strong discontinuities with linear interpolations of the displacement jumps for the modeling of failure in solids. The cases of interest are characterized by a localized cohesive law along a propagating discontinuity (e.g. a crack), with this propagation occurring in a general finite element mesh without remeshing. Plane problems are considered in the infinitesimal deformation range. The new elements are constructed by enhancing the strains of existing finite elements (including general displacement based, mixed, assumed and enhanced strain elements) with a series of strain modes that depend on the proper enhanced parameters local to the element. These strain modes are designed by identifying the strain fields to be captured exactly, including the rigid body motions of the two parts of a splitting element for a fully softened discontinuity, and the relative stretching of these parts for a linear tangential sliding of the discontinuity. This procedure accounts for the discrete kinematics of the underlying finite element and assures the lack of stress locking in general quadrilateral elements for linearly separating discontinuities, that is, spurious transfers of stresses through the discontinuity are avoided. The equations for the enhanced parameters are constructed by imposing the local equilibrium between the stresses in the bulk of the element and the tractions driving the aforementioned cohesive law, with the proper equilibrium operators to account for the linear kinematics of the discontinuity. Given the locality of all these considerations, the enhanced parameters can be eliminated by their static condensation at the element level, resulting in an efficient implementation of the resulting methods and involving minor modifications of an existing finite element code. A series of numerical tests and more general representative numerical simulations are presented to illustrate the performance of the new elements. Copyright © 2007 John Wiley & Sons, Ltd.

A powerful finite element for plate bending

A powerful finite element formulation for plate bending has been developed using a modified version of the variational method of Trefftz. The notion of a boundary has been generalized to include the interelement boundary. All boundary conditions and the interelement continuity requirements (displacements, slopes, internal forces) have been obtained as natural conditions on the generalized boundary. Coordinate functions have been constructed to satisfy the nonhomogeneous Lagrange equation locally within the elements. Singularities due to isolated loads have been properly taken into account. For practical use a general quadrilateral element has been developed and its accuracy illustrated on several numerical examples. Work is in progress to extend the formulation to anisotropic and moderately thick plates and to vibration analysis.

Modified Finite Strip Analysis of Stiffened Plate Bending and Buckling Problems

The formulation of a long quadrilateral finite element for plate bending is described herein. This method is a development of the finite strip method to the finite element procedure. It is well known that comparatively accurate solutions can be obtained with a few degrees of freedom by the finite strip descretization, because the displacement interpolation functions in the longitudinal direction are selected as eigenfunctions. It imposes, however, a significant geometric limitation because of its rectangular shape and the incapability to connect adjacent elements along the shorter sides.By modifying the finite strip method the authors were able to obtain the versatility contained in the finite element method.The nodal displacements are taken in this method as generalized coordinates which are amplitudes of the eigenfunctions in the finite strip method. A general quadrilateral element is mapped into the standard square element in the local coordinates for the derivation of the stiffness. The long quadrilateral element satisfies all the continuity requirements except the slope along the shorter sides.The element stiffness derivation is outlined and numerical results are presented to evaluate the accuracy and demonstrate the effectiveness of the element. As a practical example the buckling analysis is performed with respect to the corner connection of the trans. ring of an ore carrier. The results show this method to be potent especially for stiffened plates such as those of ship structures.

Two hybrid elements for analysis of thick, thin and sandwich plates

AbstractTwo general quadrilateral elements for plate bending are developed. Each has 12 degrees of freedom and accounts for transverse shear deformation effects. Each is built from four triangular elements whose properties are derived by the assumed‐stress hybrid approach. Matrices needed to generate the stiffness properties of the triangular elements are explicitly stated so as to facilitate their use. Numerical test cases, in which shear deformation effects range from negligible to very important, are used to illustrate the behaviour of the elements. It is concluded that the simpler of the two quadrilaterals is one of the best plate elements currently available.

Gauss-Legendre Numerical Integrations over a Quadrilateral Element in Closed Form

In this paper we investigate the stiffness matrix of a general quadrilateral element in closed form using n x n Gauss-Legendre quadrature rule. For this, we propose four types of nodal coordinate transformation. The terms of the matrix are divided into two groups, namely - diagonal and non-diagonal. Only one term (called leading) from each group is computed, and then the remaining fourteen terms are computed from these two leading terms exploited one of the proposed types of coordinate transformation. This leads us a great savings in computational time and memory space. In order to compute the matrix we use these transformations in two ways, and thus two algorithms are given to generate the matrix. Finally, numerical example is given to verify the effectiveness of the present formulation. Keywords: Gauss-Legendre quadrature; Numerical integration; Quadrilateral finite element; Stiffness matrix; Closed form. DOI: http://dx.doi.org/10.3329/bjsir.v46i3.9050 BJSIR 2011; 46(3): 399-405