采用面积坐标的四边形二次膜元

文献［１］［２］建立了四边形单元的面积坐标体系，本文在此基础上，利用优选的广义协调条件，构造了两个广义协调四边形单元，算例表明这两个单元是收敛的，可靠的。     

四边形单元面积坐标理论

本文建立了四边形单元面积坐标的系统理论，包括：（１）给出四边形单元两个特征参数ｇ１，ｇ２的定义以及四边形退化为平行四边形（含矩形），梯形，三角形时相应的特征条件；（２）给出四边形单元面积坐标的定义及其与直角坐标和四边形等参坐标之间的变换关系；（３）给出四边形单元四个面积坐标分量之间应满足的两个恒等式并予以证明；（４）给出相关的一些重要公式。可以看出，四边形面积坐标是构造四边形单元的有效工具。它既是自然坐标，具有不变性；同时它与直角坐标之间为线性关系，易于得出单元刚度矩阵的积分显式，无需依赖于数值积分。     

含两个分量的四边形单元面积坐标理论

为了便于构造抗畸变的四边形单元,建立了一套新的四边形单元面积坐标理论(QAC-2),并给出了相关的积分和微分公式。该坐标系作为自然坐标,具有明确的物理意义,且只含有两个相互独立的坐标分量,因此易于实现与直角坐标和等参坐标的沟通,便于理解和应用;两个坐标分量与直角坐标之间满足线性变换,在构造单元时易于选择完备的多项式序列,且多项式的完备次数不会随着网格的畸变而下降,因此可以保证单元的精度和抗畸变性能。     

两个采用面积坐标的四边形八结点膜元

本文采用文献（１）和（２）提出的四边形面积坐标法，并应用广义协调的概念，构造了两个新型四边形八结点膜元，数值算例表明：本文所提出的单元具有良好的性态，尤其当网络畸变时，单元依然保持良好的精度，其性能优于通常的八结点等参单元。     

混合应用三类四边形面积坐标构造八结点四边形膜元

三类四边形面积坐标已先后提出。如果对三类面积坐标加以混合应用,这将使构造四边形单元的工作更加灵活多样,具有更加广阔的优选空间。该文混合应用三类四边形面积坐标构造一个8结点四边形膜元。新单元具有如下优点:1)新单元具有优异性能,特别是对网格畸变不敏感,优于8结点等参元,显示出三类四边形面积坐标的共同优点;2)新单元的推导过程和主要列式都非常简洁。这是由于巧妙地混合应用三类面积坐标并进行优选而取得的结果。     

六面体单元体积坐标方法

基于二维问题四边形单元面积坐标法的成功思路,建立了三维六面体单元体积坐标的系统方法,包括:1)六面体单元特征参数的定义及单元退化模式研究;2)六面体单元体积坐标定义;3)六面体单元的体积坐标与直角坐标、等参坐标之间的关系;4)六面体体积坐标的微分公式。可以看到,六面体体积坐标保持了局部自然坐标的优点,并且与直角坐标始终保持线性关系。它为构造对网格畸变不敏感的新型六面体有限元模型提供了新工具。     

采用面积坐标的四边形板弯曲单元

本文采用四边形面积坐标，并应用广义协调法构造出一个具有１２个自由度的四边形板弯曲单元。单元的挠度场以面积坐标多项式表示，对应于直角坐标ｘ，ｙ的完全三次式和部分四次式，因而单元是完备的广义协调的板单元。应用的１２个协调条件为挠度的四个点协调条件和四个边协调条件，以及法向转角的四个边协调条件。由于面积坐标和直角坐标之间为线性变换关系，因此单元刚度矩阵的推导相当简单。数值算例表明：本文单元具有高精度、收敛性、可靠性和对网格畸变不敏感的优点     

三维内参型附加非协调位移基本项

文献［１２］在平面问题基础上推导出附加非协调位移基本项的通用公式。依据这一公式,在构造新型非协调位移元时,可以主动地来选择基本项,减少了过去盲目试凑的现象。本文在文献［１２］的基础上,推广至三维情况,分别推导出以等参坐标和直角坐标表述的附加非协调位移基本项通用公式。依据这些公式,本文分别以Ｈ８和Ｈ２０单元为例,发展了两个新的非协调元,数值试验表明它们能够保证收敛,有较高精度,对畸变不敏感,从而证明了本文方法的可行性。     

有限元新型自然坐标方法研究进展

网格畸变敏感问题一直是当前有限元法难以解决的问题,而新型自然坐标方法的诞生可以在一定程度上对解决这个难题有所帮助。该文介绍了有限元新型自然坐标方法研究的新近进展。包括第一类四边形面积坐标及其应用(单元构造,解析刚度矩阵的建立,以及在几何非线性问题中的应用等);第二类四边形面积坐标及其应用;六面体体积坐标及其应用。数值算例表明:无论网格如何扭曲畸变,这些基于新型自然坐标方法的有限元模型仍然保持高精度,对网格畸变不敏感。这显示了新型自然坐标方法是构造高性能单元模型的有效工具。     

四边形单元第三类面积坐标系统

四边形单元面积坐标系统的两种型式(QAC-Ⅰ和QAC-Ⅱ)已被建立.QAC-Ⅰ含四个坐标分量(L1,L2,L3,L4),其中只有两个是独立分量.QAC-Ⅱ只含两个独立的坐标分量(Z1,Z2).这些面积坐标系统为建立对网格畸变不敏感的新型四边形单元提供理论基础.该文系统地建立了具有两个坐标分量(T1,T2)的四边形单元第三类面积坐标系统(QAC-Ⅱ).这个新的QAC-Ⅲ系统不仅保留了QAC-Ⅰ和QAC-Ⅱ的丰要优点,而且具有其他一些优异特性:1)它是自然坐标;2)它与直角坐标系统保持线性关系;3)它只含两个坐标分量;4)由它导出的形函数具有比较简洁的形式;5)它可以直接地推广应用于曲边单元;6)采用三类系统Ⅰ、系统Ⅱ、系统Ⅲ的混合形式常可以导出优化的结果.     

采用面积坐标和基于假设转角的薄板元

采用四边形面积坐标方法,从假设转角位移场入手构造了两个广义协调四边形4结点薄板单元AΨQ-I和AΨQ-II。通过采用边界协调条件一次项与二次项分别协调使转角场实现了三次完备。与DKQ等同类单元相比,单元的精度和抗网格畸变能力都有很大提高。     

Unsymmetric extensions of Wilson's incompatible four-node quadrilateral and eight-node hexahedral elements

The unsymmetric finite element is based on the virtual work principle with different sets of test and trial functions. In this article, the incompatible four-node quadrilateral element and eight-node hexahedral element originated by Wilson et al. are extended to their unsymmetric forms. The isoparametric shape functions together with Wilson's incompatible functions are chosen as the test functions, while internal nodes at the middle of element sides/edges are added to generate the trial functions with quadratic completeness in the Cartesian coordinate system. A local area/volume coordinate frame is established so that the trial shape functions can be explicitly obtained. The key idea which avoids the matrix inversion is that the trial nodal shape functions are constructed by standard quadratic triangular/tetrahedral elements and then transformed in consistent with the quadrilateral/hexahedral elements. Numerical examples show that the present elements keep the merits of both incompatible and unsymmetric elements, that is, high numerical accuracy, insensitivity to mesh distortion, free of trapezoidal and volumetric locking, and easy implementation.     

Quadrilateral membrane elements with analytical element stiffness matrices formulated by the new quadrilateral area coordinate method (QACM‐II)

A novel strategy for developing low‐order membrane elements with analytical element stiffness matrices is proposed. First, some complete low‐order basic analytical solutions for plane stress problems are given in terms of the new quadrilateral area coordinates method (QACM‐II). Then, these solutions are taken as the trial functions for developing new membrane elements. Thus, the interpolation formulae for displacement fields naturally possess second‐order completeness in physical space (Cartesian coordinates). Finally, by introducing nodal conforming conditions, new 4‐node and 5‐node membrane elements with analytical element stiffness matrices are successfully constructed. The resulting models, denoted as QAC‐ATF4 and QAC‐ATF5, have high computational efficiency since the element stiffness matrices are formulated explicitly and no internal parameter is added. These two elements exhibit excellent performance in various bending problems with mesh distortion. It is demonstrated that the proposed strategy possesses advantages of both the analytical and the discrete method, and the QACM‐II is a powerful tool for constructing high‐performance quadrilateral finite element models. Copyright © 2008 John Wiley & Sons, Ltd.     

Development of eight-node quadrilateral membrane elements using the area coordinates method

 Two eight-node quadrilateral elements, namely, AQ8-I and AQ8-II, have been developed using the quadrilateral area coordinate and generalized conforming methods. Some appropriate examples were employed to evaluate the performance of the proposed elements. The numerical results show that the proposed elements are superior to the standard eight-node isoparametric element, thereafter called Q8. This is because the former does not only possess the same accuracy as the latter when regular meshes are employed for analysis, but is also very insensitive to mesh distortion, for which the Q8 element can not handle. It has also been demonstrated that the area coordinate method is an efficient tool for developing simple, effective and reliable serendipity plane membrane elements. Received 11 August 1999     

A new quadrilateral area coordinate method (QACM‐II) for developing quadrilateral finite element models

The 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 (L₁, L₂, L₃, L₄), 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 Z₁ and Z₂, 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. Copyright © 2007 John Wiley & Sons, Ltd.