The third form of the quadrilateral area coordinate method (QACM-III): Theory, application, and scheme of composite coordinate interpolation

Abstract

Two quadrilateral area coordinate methods (hereinafter referred to as QACM-I and QACM-II) have already been proposed in 1999 and 2008, respectively. Many successful applications demonstrate that these new coordinate systems are efficient tools for developing robust quadrilateral finite element models. In this paper, another new quadrilateral area coordinate method (hereinafter referred to as QACM-III), containing two components T 1 and T 2, is systematically established. This new coordinate system (QACM-III) not only keeps the most important advantages of the QACM-I and QACM-II, but also possesses other distinguishing characters. Then, the QACM-III is applied to formulate two new 4-node membrane elements with internal parameters. The first new element, completely formulated by the QACM-III and denoted as QACIII6, exhibits much better performance than that of Wilson's Q6 element in benchmark problems, and has the simplest formulations when compared with the similar models formulated by the QACM-I and -II. The second element, denoted as CQAC6, is developed by a new scheme of composite coordinate interpolation, in which the QACM-I and -III are both used. It can strictly pass Iron's patch test and exhibits better accuracy than the Taylor's QM6 element. Finally, the new scheme of composite coordinate interpolation is employed again. Three different quadrilateral area coordinate methods, QACM-I, -II and -III, are used simultaneously to develop a new 8-node membrane element CQAC-Q8. On the one hand, element CQAC-Q8 is insensitive to mesh distortion, which the traditional isoparametric element Q8 cannot achieve; on the other hand, its construction procedure and formulations are quite simpler than the models formulated only by any single area coordinate method. The new coordinate method, the element models, and the scheme of composite coordinate interpolation presented in this paper provide new tools and ideas for high-performance finite element analysis.