A consecutive-interpolation quadrilateral element (CQ4): Formulation and applications Tinh Quoc Bui a,n , Dam Quang Vo b , Chuanzeng Zhang a , Du Dinh Nguyen c a Department of Civil Engineering, University of Siegen, Paul-Bonatz-Straße 9-11, 57076 Siegen, Germany b Piping Department, Petrovietnam Engineering Company, Ho Chi Minh, Vietnam c Department of Civil Engineering, Lac Hong University, Dong Nai Province, Vietnam article info Article history: Received 3 April 2013 Received in revised form 13 February 2014 Accepted 19 February 2014 Available online 12 March 2014 Keywords: FEM Consecutive-interpolation finite element Stress analysis Numerical methods Quadrilateral element abstract An efficient, smooth and accurate quadrilateral element with four-node based on the consecutive- interpolation procedure (CIP) is formulated. The CIP is developed recently by Zheng et al. (Acta Mech Sin 26 (2010) 265–278) for triangular element with three-node. In this setting the approximation functions handle both nodal values and averaged nodal gradients as interpolation conditions. Two stages of the interpolation are required; the primary stage is carried out using the same procedure of the standard finite element method (FEM), and the interpolation is further reproduced in the secondary step according to both nodal values and average nodal gradients derived from the previous interpolation. The new consecutive-interpolation quadrilateral element with four-node (CQ4) deserves many desirable characteristics of an efficient numerical method, which involves continuous nodal gradients, continuous nodal stresses without smoothing operation, higher-order polynomial basis, without increasing the degree of freedom of the system, straightforward to implement in an existing FEM computer code, etc. Four benchmark and two practical examples are considered for the stress analysis of elastic structures in two-dimension to show the accuracy and the efficiency of the new element. Detailed comparison and some other aspects including the convergence rate, volumetric locking, computational efficiency, insensitivity to the mesh, etc. are investigated. Numerical results substantially indicate that the consecutive-interpolation finite element method (CFEM) with notable features pertains to high accuracy, convergence rate, and efficiency as compared with the standard FEM. & 2014 Elsevier B.V. All rights reserved. 1. Introduction Design procedures of improving and enhancing the perfor- mance of engineering structures through stress analysis are often time-consuming and expensive. Nowadays, simulation technolo- gies using advanced numerical methods in engineering and science are popular and have been emerged rapidly. The motiva- tions are to accurately model practical problems as exact as the techniques can. The finite element method (FEM) [1–3] and the boundary element method (BEM) [4] have become very powerful and versatile numerical methods, which are the most common and extensively used methods in a broad range of engineering applications. Owing to the simplicity, the three-node triangular and four-node quadrilateral finite elements are often introduced and applied to solve engineering problems in two-dimensions (2D). Because of the linear approximations, the spatial derivatives of the field variables are constant within each element [5]. Such constant-strain finite elements are easily formulated and imple- mented but their performance in practical applications is often unsatisfactory and, frequently low accuracy is obtained due to their low-order trial functions [5,6]. Moreover, the gradients on element-edges in both constant elements and mapped elements are discontinuous, and demanding smoothing operation in post- processing step is rigorous [7]. Other relevant issues involving volumetric locking and sensitivity to mesh, etc. for such elements can be found in Refs. [1–3,5–8] for instance. A number of advanced numerical methods have been devel- oped in order for improving the accuracy and efficiency of the conventional FEM methods. For instance, Hansbo proposed a non- conforming rotated Q1 tetrahedral element for linear elastic [9] and elastodynamic problems [10]. By containing the bilinear terms, the Q1 element performs substantially better than the standard constant-strain one in bending and allows for under- integration in nearly incompressible situations. Papanicolo- pulos and Zervos [11] presented a means for creating a class of triangular C 1 finite element particularly suitable for model- ing problems where the underlying partial differential equation is of fourth-order (e.g., beam and plate bending, deformation of Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/finel Finite Elements in Analysis and Design http://dx.doi.org/10.1016/j.finel.2014.02.004 0168-874X & 2014 Elsevier B.V. All rights reserved. n Corresponding author. Tel.: þ49 2717402836; fax: þ49 2717404074. E-mail address: tinh.buiquoc@gmail.com (T.Q. Bui). Finite Elements in Analysis and Design 84 (2014) 14–31