Application Of The Quadrilateral Area Coordinate Method: Geometrically Nonlinear Analysis For Plane Problem

The Quadrilateral Area Coordinate Method (QACM) is a powerful new tool for developing 2-D finite element models. Elements based on QACM are less sensitive to mesh distortion than the traditional isoparametric elements. Up to date, some element models formulated by QACM have already been successfully developed for linear analysis, such as the quadrilateral 4-node membrane elements AGQ6 and QACM4, and so on.Nonlinear Finite Element Method is very important and widely used in both scientific and engineering computations. However, large deformation often leads to mesh distortions, which is fatal to isoparametric elements: the accuracy may drop dramatically and the results become unreliable. In contrast, elements by QACM can keep their accuracy under severely distorted mesh. Therefore, QACM should have a splendid application future in nonlinear analysis.In this paper, the implicit geometrically nonlinear formulations of several QACM elements are constructed for the first time. Their advantage of being less sensitive to mesh distortion is proved by numerical examples. It can be seen that their performance is much better than the isoparametric elements with the same order. Furthermore, an analytical element stiffness matrix of element AGQ6 for linear analysis is also presented here, and the computational efficiency is obviously improved in comparison with models using numerical integration scheme. This advantage may be utilized in further nonlinear applications.The detailed contributions of this dissertation are as follows:The analytical element stiffness matrix of AGQ6 for linear analysis is given in a neat form, and the improvement in computational efficiency is demonstrated by numerical examples. On the basis of nonlinear continuum mechanics, the geometrically nonlinear formulations of two QACM elements, i.e. AGQ6 and QACM4, are established.Furthermore, the geometrically nonlinear formulations of three isoparametric elements, Q4, Q6 and QM6, are also presented.The user subroutines UEL of above element models are programmed with FORTRAN77 language. Then, all of them are introduced into finite element software ABAQUS.Performance of these QACM elements, conventional isoparametric elements and element models in ABAQUS are checked and compared by examples using various loads, distorted meshes and mesh densities. Numerical results show that the QACM elements possess much better accuracy and convergence for geometrically nonlinear problems.