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SUMMARYThe unsymmetric FEM is one of the effective techniques for developing finite element models immune to various mesh distortions. However, because of the inherent limitation of the metric shape functions, the resulting element models exhibit rotational frame dependence and interpolation failure under certain conditions. In this paper, by introducing the analytical trial function method used in the hybrid stress-function element method, an effort was made to naturally eliminate these defects and improve accuracy. The key point of the new strategy is that the monomial terms (the trial functions) in the assumed metric displacement fields are replaced by the fundamental analytical solutions of plane problems. Furthermore, some rational conditions are imposed on the trial functions so that the assumed displacement fields possess fourthorder completeness in Cartesian coordinates. The resulting element model, denoted by US-ATFQ8, can still work well when interpolation failure modes for original unsymmetric element occur, and provide the invariance for the coordinate rotation. Numerical results show that the exact solutions for constant strain/stress, pure bending and linear bending problems can be obtained by the new element US-ATFQ8 using arbitrary severely distorted meshes, and produce more accurate results for other more complicated problems.

SUMMARYThe quadrilateral area coordinate method proposed in 1999 (hereinafter referred to as QACM-I) is a new and efficient tool for developing robust quadrilateral finite element models. However, such a coordinate system contains four components (L 1 , L 2 , L 3 , L 4 ), which may make the element formulae and their construction procedure relatively complicated. In this paper, a new category of the quadrilateral area coordinate method (hereinafter referred to as QACM-II), containing only two components Z 1 and Z 2 , is systematically established. This new coordinate system (QACM-II) not only has a simpler form but also retains the most important advantages of the previous system (QACM-I). Hence, as an application, QACM-II is used to formulate a new 4-node membrane element with internal parameters. The whole process is similar to that of the famous Wilson's Q6 element. Numerical results show that the present element, denoted as QACII6, exhibits much better performance than that of Q6 in benchmark problems, especially for MacNeal's thin beam problem. This demonstrates that QACM-II is a powerful tool for constructing high-performance quadrilateral finite element models.

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The sensitivity problem to mesh distortion and the low accuracy problem of the stress solutions are two inherent difficulties in the finite element method. By applying the fundamental analytical solutions (in global Cartesian coordinates) to the Airy stress function  of the anisotropic materials, 8-and 12-node plane quadrilateral hybrid stress-function (HS-F) elements are successfully developed based on the principle of the minimum complementary energy. Numerical results show that the present new elements exhibit much better and more robust performance in both displacement and stress solutions than those obtained from other models. They can still perform very well even when the element shapes degenerate into a triangle and a concave quadrangle. It is also demonstrated that the proposed construction procedure is an effective way for developing shape-free finite element models which can completely overcome the sensitivity problem to mesh distortion and can produce highly accurate stress solutions. finite element, hybrid stress-function (HS-F) element, shape-free, stress function, the principle of minimum complementary energy, fundamental analytical solutions, anisotropic materials PACS: 02.70.-c, 02.70.Dc, 46.15.-x, 46.25.-y

Purpose À The purpose of this paper is to propose a novel and simple strategy for construction of hybrid-''stress function'' plane element. Design/methodology/approach À First, a complementary energy functional, in which the Airy stress function is taken as the functional variable, is established within an element for analysis of plane problems. Second, 15 basic analytical solutions (in global Cartesian coordinates) of the stress function are taken as the trial functions for an 8-node element, and meanwhile, 15 unknown constants are then introduced. Third, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of the displacements along element edges, which are interpolated by element nodal displacements. Finally, the whole system can be rewritten in terms of element nodal displacement vector. Findings À A new hybrid element stiffness matrix is obtained. The resulting 8-node plane element, denoted as analytical trial function (ATF-Q8), possesses excellent performance in numerical examples. Furthermore, some numerical defects, such as direction dependence and interpolation failure, are not found in present model. Originality/value À This paper presents a new strategy for developing finite element models exhibits advantages of both analytical and discrete method.