Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 198390, 14 pages

http://dx.doi.org/10.1155/2015/198390

## Accurate, Efficient, and Robust Q4-Like Membrane Elements Formulated in Cartesian Coordinates Using the Quasi-Conforming Element Technique

Department of Mechanics, Tianjin University, Tianjin 300072, China

Received 19 September 2014; Revised 13 January 2015; Accepted 27 January 2015

Academic Editor: Chenfeng Li

Copyright © 2015 G. Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By using the quasi-conforming element technique, two four-node quadrilateral membrane elements with 2 degrees of freedom at each node (Q4-like membrane element) are formulated in rectangular Cartesian coordinates. One of the four-node quadrilateral membrane elements is based on the assumed strain field with only five independent strain parameters and accounting for the Poisson effect explicitly. There are no independent internal parameters and numerical integration involved in the evaluation of the strain parameters in these four-node quadrilateral membrane elements, and their element stiffness matrices are computed explicitly in Cartesian coordinates. Consequently, the formulation of these four-node quadrilateral membrane elements is extremely simple, and the resulting elements are very computationally efficient. These two quasi-conforming quadrilateral membrane elements pass the patch test and are free from shear locking and insensitive to the element distortion in the range of practical application. The numerical result comparison with other four-node quadrilateral membrane elements, including Q4-like plane elements with drilling degrees of freedom and the Q6-type isoparametric elements with very complicated nonconforming modes, shows that the present quasi-conforming quadrilateral membrane elements are not only reliable and robust, but also very accurate in both displacement and stress evaluations in the analysis of practical plane elasticity problems.

#### 1. Introduction

The Q4-type membrane elements, which are the quadrilateral membrane elements with four corner nodes and only 2 degrees of freedom at each node as well no internal parameters involved, is one of the most useful finite elements since they are used not only in the displacement and stress analysis of membranes, but also in the four-node quadrilateral flat-shell elements [1, 2]. And four-node quadrilateral flat-shell elements are regarded as the most efficient shell element in the dynamic analysis involving surface contacts of panel-like structures [3]. Because the original Q4 element in which each in-plane displacement component is approximated by a bilinear displacement interpolation in terms of four-nodal displacement variables suffers the shear locking when a membrane undergoes in-plane bending, Wilson et al. [4] pioneered the use of the higher-order displacement interpolations defined in terms of internal parameters to improve the performance of four-node quadrilateral membrane elements. The four-node quadrilateral membrane elements with two displacement-like internal parameters can be designated as Q6-type (or Q6-like) elements. The displacement interpolations in the original Q6 element proposed by Wilson et al. possesses nonconforming modes and Q6 fails to pass patch test although it can give good numerical results in many cases. Many researchers have devoted tremendous efforts to improve the properties and performance of Q6-type elements in the past five decades (vide the reference papers listed in [5]). And the employment of the nonconforming modes is still a major approach to improve the membrane elements based on assumed displacement fields up to now [5, 6].

The use of drilling degrees of freedom can efficiently remove the shear locking and improve the computational accuracy. Liu et al. [7] and Chen and Li [8] developed the quasi-conforming membrane elements with drilling degrees of freedom. Based on a functional treating drilling rotations as independent variables, Iura and Atluri [9] developed a reliable and accurate four-node membrane element with drilling degrees of freedom. Sze et al. [10] presented a four-node hybrid stress membrane element with drilling degrees of freedom. Some other papers on the membrane elements with drilling degrees of freedom using the conventional displacement approach can be found in [11]. These membrane elements with drilling degrees of freedom are more accurate indeed than the membrane elements with only translational degrees of freedom, but they are more computationally expensive since more nodal degrees of freedom are used.

Various hybrid stress methods and assumed strain methods have been used to develop improved membrane elements. Based on the earlier work, Pian and Sumihara [12] proposed a rational approach for assumed stress finite elements in 1983 and presented a Q4-like hybrid stress membrane element. In 1992, Sze et al. [10] used orthogonal stress modes in the four-node quadrilateral membrane elements with drilling degrees of freedom. The enhanced assumed strain method and the assumed natural strain methods are still been employed now by many researchers as an efficient scheme to remove shear locking in the membrane part of reliable and accurate shell elements [6, 13–15]. The numerical integrations are used in all these improved membrane elements.

Although the development of four-node quadrilateral membrane elements based on various strategies has a long history and considerable achievements are accumulated, there is a renewed interest in recent years in the four-node quadrilateral membrane elements with the higher accuracy and computationally efficiency. Zhong and Ji [16] proposed a rational element approach to formulate displacement-based membrane elements. The so-called rational quadrilateral membrane element RQ4 is much better than Q4, but its computational accuracy is still not desirable enough. Cen and his coworkers [5, 17, 18] presented a number of four-node quadrilateral membrane elements by using the quadrilateral area coordinates. Element AGQ6-I [17] which is based on the quadrilateral area coordinates exhibits excellent performance and it is quite insensitive to mesh distortions. However, AGQ6-I failed in the strict patch test and can only pass the weak patch test. Therefore, its convergence raised some discussions and several techniques were adopted to make it pass the patch test [5]. Unfortunately, the accuracy of the modified versions of AGQ6-I by all three different remedies deteriorates even though they can pass the strict patch test. The quadrilateral area coordinates are also employed to couple with the enhanced assumed strain method to derive reliable and accurate Q6-type membrane elements used in the flat-shell element [6].

The quasi-conforming element technique proposed by Tang and his coworkers [19–22] is a general assumed strain method to formulate reliable and accurate elements. Although it has been primarily used for the developments of the quasi-conforming plate and shell elements that involve with the -continuity problems, a number of quasi-conforming membrane elements have been presented [2, 21–25]. Chen and Tang [21] presented a quasi-conforming quadrilateral isoparametric membrane element QC6 where two displacement-like internal parameters are used. Liu et al. [7] developed a quasi-conforming membrane element with drilling degrees of freedom QR4 in which the isoparametric element technique was also used. Based on the quasi-conforming element technique, Chen and Li [8] presented some improved quadrilateral membrane elements with drilling degrees of freedom also in terms of the natural coordinates. Numerical integration is used in the elements given by Liu et al. [7] as well as by Chen and Li [8]. Based on the assumed element strains given by the rational displacement field, Liu et al. [26] presented a quasi-conforming quadrilateral membrane element AQCE4 in the Cartesian coordinates. AQCE4 yields very good results when the coarse meshes are used, but it seems not to converge to the exact solution as the mesh is getting finer.

As a matter of fact, by using the assumed strain method in the quasi-conforming element technique, Shi and Voyiadjis derived a four-node quadrilateral cylindrical membrane element for the four-node cylindrical shell elements in 1991 [23, 27] and a four-node quadrilateral plane element for the membrane part of a four-node nonlinear quadrilateral flat-shell element in Cartesian coordinates in 1991 [2]. The element stiffness matrices of both the quadrilateral cylindrical shell element and the flat shell are evaluated explicitly. However, the performance of this four-node quadrilateral membrane element when it is used alone was not reported since the focus of these shell elements is the flexural analysis of shells.

The computational efficiency is a very critical issue in nonlinear and dynamic analysis of structures. The reliable one point quadrature is believed to be efficient for both membrane and shell elements (please refer to the related references given in [6]). Then, the quadrilateral membrane elements with explicit element stiffness, that is, there are no any numerical integrations, could be more computationally efficient and desirable. Based on the quasi-conforming element technique and the strain field derived from the displacement interpolation, Xia et al. [28] presented two quasi-conforming quadrilateral membrane elements WDC4 and WDC6 where two displacement-like internal parameters are utilized in WDC6. The element stiffness matrices of both WDC4 and WDC6 are evaluated explicitly in Cartesian coordinates. WDC4 and WDC6 pass the patch test and are very efficient. Thus it was claimed that WDC4 and WDC6 are the first quadrilateral membrane elements directly formulated in Cartesian coordinates that are able to pass patch test. Unfortunately, the aforementioned claim about the quadrilateral membrane elements formulated in Cartesian coordinates is incorrect. Furthermore, the accuracy of WDC4 and WDC6 are not competitive to other quasi-conforming quadrilateral membrane elements reported in the literature such as AQCE4 [26]. The details about the accuracy comparison will be shown later.

The objective of this paper is to develop a more accurate four-node quadrilateral quasi-conforming membrane element with two degrees of freedom per node by explicitly taking account of the Poisson effect in the assumed element strain field. This new four-node quadrilateral quasi-conforming membrane element should be not only reliable, but also more computationally efficient than the improved four-node quadrilateral membrane elements based on other methods. The performance of the four-node quadrilateral assumed strain membrane element developed in 1990 for the membrane part of a four-node quadrilateral flat-shell element [2] is also presented here for the first time. Both of these four-node quadrilateral quasi-conforming membrane elements pass the patch test, exhibit no shear locking, and are insensitive to the element distortion in the range of practical application. The numerical result comparison with other four-node quadrilateral membrane elements, including both Q4-like elements with drilling degrees of freedoms and Q6-type elements, indicates that the four-node quadrilateral quasi-conforming membrane elements presented in this paper are not only reliable and robust but also very accurate indeed.

#### 2. General Formulation of Quasi-Conforming Elements

A typical continuum considered in the boundary value problem is depicted in Figure 1, where denote the rectangular Cartesian coordinates, and represent, respectively, the force boundary and the displacement boundary on , respectively, are the given tractions on , are used to represent the direction cosines of a point on , and and signify, respectively, a subdomain and its surface in the continuum.