Some Recent Advances on the Quadrilateral Area Coordinate Method

Since the Quadrilateral Area Coordinate Method (QACM) was systematically established at the end of last century [1], some successful applications of this new tool have been achieved by various researchers [2, 3]. Compared with the usual isoparametric coordinate method, the QACM can make a quadrilateral finite element model less sensitive to mesh distortion, and simplifies the copmputational procedures (such as no Jacobi inverse is needed). Recently, a family of quadrilateral membrane elements with 4 to 8 nodes are developed by employing the QACM and the generalized conforming conditions. 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 and can pass the strict form patch test for straight-side cases. Numerical results show that these elements exhibit excellent performance in all benchmark problems, especially for MacNeal’s beam, thin curved beam problems, etc., which the traditional isoparametric elements can not easily achieve. The QACM coordinate system contains four components (L 1, L 2, L 3, L 4), which may bring complexities to element formulae and their construction procedure. Recently, a new category of quadrilateral area coordinate method QACMII, containing only two components Z 1 and Z 2, is systematically established. This new coordinate system (QACMII) not only has simpler form, but also keeps the most important advantages of the previous one (QACM). Numerical examples demonstrate that the new quadrilateral area coordinate methods, QACM and QACMII, are both powerful tools for constructing high-performance quadrilateral finite element models.