A consecutive-interpolation quadrilateral element (CQ4): Formulation and applications

• An efficient, smooth and accurate quadrilateral element (CQ4) is formulated.
• The consecutive-interpolation procedure is integrated into the CQ4 successfully.
• The CQ4 holds the continuous nodal gradients and continuous nodal stresses without smoothing operation.
• High accuracy and convergence rate, insensitive mesh distortions, free volume locking, etc. are found for the CQ4.
• The new CQ4 is applied to stress analysis of simple and complicated structures in 2D.

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.