Quadrilateral Area Coordinate Research Articles

A Universal Quadrilateral Shell Element for the Absolute Nodal Coordinate Formulation

Abstract A simple quadrilateral shell element is proposed in this work to study large deformations and large rotations of membrane/plate/shell structures. There are three merit characters in this element: locking-free; immune to mesh distortions; and robust to surface tessellations. Numerical issues in plates/shell elements such as shear-locking and thickness-locking problems are resolved, and quadrilateral area coordinates are adopted to solve the mesh distortion issues. This element can be adopted to curved shell structures, and warped deformations can be well described. Moreover, even if a shell structure cannot be easily tessellated by high quality quadrilateral polygons, it can still be discretized by a mesh consisting of high-quality triangular and quadrilateral elements, then this element can work together with a corresponding triangular element to provide accurate results on this combined mesh, and the degree-of-freedom for the discretized system is no more than several times of the number of nodes. Numerical tests validate the effectiveness, efficiency, and universality of this element in engineering scenarios.

An unsymmetric 4-node, 8-DOF plane membrane element perfectly breaking through MacNeal's theorem

Summary Among numerous finite element techniques, few models can perfectly (without any numerical problems) break through MacNeal's theorem: any 4-node, 8-DOF membrane element will either lock in in-plane bending or fail to pass a C0 patch test when the element's shape is an isosceles trapezoid. In this paper, a 4-node plane quadrilateral membrane element is developed following the unsymmetric formulation concept, which means two different sets of interpolation functions for displacement fields are simultaneously used. The first set employs the shape functions of the traditional 4-node bilinear isoparametric element, while the second set adopts a novel composite coordinate interpolation scheme with analytical trail function method, in which the Cartesian coordinates (x,y) and the second form of quadrilateral area coordinates (QACM-II) (S,T) are applied together. The resulting element US-ATFQ4 exhibits amazing performance in rigorous numerical tests. It is insensitive to various serious mesh distortions, free of trapezoidal locking, and can satisfy both the classical first-order patch test and the second-order patch test for pure bending. Furthermore, because of usage of the second form of quadrilateral area coordinates (QACM-II), the new element provides the invariance for the coordinate rotation. It seems that the behaviors of the present model are beyond the well-known contradiction defined by MacNeal's theorem. Copyright © 2015 John Wiley & Sons, Ltd.

Four-Node Generalized Conforming Membrane Elements with Drilling DOFs Using Quadrilateral Area Coordinate Methods

Two 4-node generalized conforming quadrilateral membrane elements with drilling DOF, named QAC4θand QAC4θM, were successfully developed. Two kinds of quadrilateral area coordinates are used together in the assumed displacement fields of the new elements, so that the related formulations are quite straightforward and will keep the order of the Cartesian coordinates unchangeable while the mesh is distorted. The drilling DOF is defined as the additional rigid rotation at the element nodes to avoid improper constraint. Both elements can pass the strict patch test and exhibit better performance than other similar models. In particular, they are both free of trapezoidal locking in MacNeal’s beam test and insensitive to various mesh distortions.

A four‐node plane parametric element based on quadrilateral area coordinate and its application to coupled solid‐deformation/fluid‐flow simulation for porous geomaterials

SummaryA four‐node plane parametric element AQGβ6‐I is constructed on the basis of the quadrilateral area coordinate, the generalized conforming principle and the projection technique with a penalty factor β within an interval of 0–1. When β = 0, the element has excellent bending performance. When β = 1, the element can pass patch test strictly; its performance is as good as many famous elements. When β value is between 0 and 1, such as β = 0.5, the element can arrive at a compromise between (relatively) low sensitivity to mesh distortion and perfect convergence. The work provides an illuminating method to alleviate a difficult problem in finite element modelling using the four‐node quadrilateral element, which can pass the strict patch test, but has poor performance in bending dominated problem; on the contrary, it has excellent performance in bending dominated problem but cannot pass the strong patch test.The AQGβ6‐I with the convergence formulation (β = 1) is then applied to coupled solid‐deformation/fluid‐flow simulation for porous geomaterials. The computational examples are carried out to demonstrate that the AQGβ6‐I (β = 1) element is not only stable, reliable and efficient but also of high accuracy. The present study provides a good applicable element for finite element simulations of solid‐deformation/fluid‐flow for porous geomaterials. Copyright © 2014 John Wiley & Sons, Ltd.

Making sense of the quadrilateral area coordinate membrane elements

The introduction of the quadrilateral area coordinate (QAC) membrane elements has generated more heat than light on the subject of why distorted membrane elements perform very poorly. The canonical guidelines for construction of robust elements are continuity and completeness. It is known that isoparametric (or parametric) quadrilateral membrane elements satisfy the continuity requirements always. In rectangular form, they perform extremely well, except in cases where parasitic shear is involved (plane stress modeling of bending of thin beams). So their poor performance when the elements are used in general (i.e. distorted) quadrilateral form is now attributed to the failure to accommodate the completeness requirements. Completeness is understood to be maintained in physical (i.e. Cartesian, or metric) space and not in the natural (i.e. parametric) space. The QAC approach is a compromise that tries to use shape functions which are in physical space, but as these cannot ensure exact continuity, require a relaxed generalized continuity to be imposed. However, here, the patch test cannot be ensured in the strictest sense as a careful examination of the results from various QAC elements in this paper will reveal. Even this is not enough to remove locking in the simpler 4-node elements and additional displacements fields have to be introduced through internal parameters to ensure what Prathap calls the consistency requirement. These issues are carefully studied in this investigation.

Application of the quadrilateral area coordinate method: a new element for laminated composite plate bending problems

Recently, some new quadrilateral finite elements were successfully developed by the Quadrilateral Area Coordinate (QAC) method. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. In this paper, a new displacement-based, 4-node 20-DOF (5-DOF per node) quadrilateral bending element based on the first-order shear deformation theory for analysis of arbitrary laminated composite plates is presented. Its bending part is based on the element AC-MQ4, a recent-developed high-performance Mindlin-Reissner plate element formulated by QAC method and the generalized conforming condition method; and its in-plane displacement fields are interpolated by bilinear shape functions in isoparametric coordinates. Furthermore, the hybrid post-processing procedure, which was firstly proposed by the authors, is employed again to improve the stress solutions, especially for the transverse shear stresses. The resulting element, denoted as AC-MQ4-LC, exhibits excellent performance in all linear static and dynamic numerical examples. It demonstrates again that the QAC method, the generalized conforming condition method, and the hybrid post-processing procedure are efficient tools for developing simple, effective and reliable finite element models.

Quadrilateral membrane element family formulated by the quadrilateral area coordinate method

The quadrilateral area coordinate (QAC) method is a new tool for developing quadrilateral finite element models. Compared with those traditional elements using isoparametric coordinates, models formulated by QAC method are less sensitive to mesh distortion. In this paper, a family of quadrilateral membrane elements with 4–8 nodes are successfully developed by employing the QAC method, the generalized conforming conditions and internal parameters. All the displacement fields of these elements possess second-order completeness in Cartesian coordinates, so they are more accurate and robust in various distorted meshes. Numerical results show that the presented elements exhibit excellent performance in all benchmark problems, such as MacNeal’s beam, thin curved beam problems, etc., which the traditional isoparametric elements can not easily achieve. The efficiency of the QAC method for developing simple, effective and reliable serendipity plane membrane elements is again demonstrated.

The analytical element stiffness matrix of a recent 4‐node membrane element formulated by the quadrilateral area co‐ordinate method

AbstractThe quadrilateral area co‐ordinate (QAC) method is a new tool for developing quadrilateral finite element models. Compared with those traditional elements using isoparametric co‐ordinates, models formulated by QAC method are less sensitive to mesh distortion. Furthermore, another advantage for QAC system is that the analytical expressions for element stiffness matrix can be obtained theoretically by basic formulae of QAC integrals, which are beneficial to finite element computation procedure. However, because of the relative complexity of the derivation process, no such explicit form of the stiffness matrix has been presented before. In this paper, the analytical element stiffness matrix of a recent 4‐node quadrilateral membrane element, AGQ6‐I, is given out for the first time. This element AGQ6‐I, which was constructed by QAC method and generalized conforming conditions, successfully overcomes various locking problems and exhibits much better performances than many isoparametric models. And its formulations have also been introduced in shell analysis by other researchers. Numerical examples show that obvious higher computation efficiency can be achieved by presented explicit formulations. So the analytical form of the element stiffness matrix of element AGQ6‐I possesses significance for its further applications. Copyright © 2006 John Wiley & Sons, Ltd.

Membrane elements insensitive to distortion using the quadrilateral area coordinate method

Two 4-node quadrilateral membrane elements, denoted as AGQ6-I and AGQ6-II, have been developed in this paper. Instead of the traditional isoparametric coordinate, the quadrilateral area coordinates were used to establish the formulations of the new elements. And several generalized conforming conditions were then introduced to determine all unknown parameters. Numerical examples showed that the presented elements exhibit excellent performances in both regular and distorted mesh divisions. They could even yield exact solutions for pure bending problems under distorted meshes and provide lock-free solutions for the MacNeal’s test problem of trapezoidal locking. Besides, the weak patch test was conducted to guarantee the convergence of both new elements. It has also been demonstrated that the area coordinate method is an efficient tool for developing simple, effective and reliable serendipity plane membrane elements.

Development of eight-node quadrilateral membrane elements using the area coordinates method

Two eight-node quadrilateral elements, namely, AQ8-I and AQ8-II, have been developed using the quadrilateral area coordinate and generalized conforming methods. Some appropriate examples were employed to evaluate the performance of the proposed elements. The numerical results show that the proposed elements are superior to the standard eight-node isoparametric element, thereafter called Q8. This is because the former does not only possess the same accuracy as the latter when regular meshes are employed for analysis, but is also very insensitive to mesh distortion, for which the Q8 element can not handle. It has also been demonstrated that the area coordinate method is an efficient tool for developing simple, effective and reliable serendipity plane membrane elements.