摘要：为克服等参坐标的缺点，采用四边形面积法构造一系列4节点12自由度的广义协调任意面积坐标四边形薄板(Area-coordinate Quadrilateral Plate，AQP)单元.对广义协调条件、边界位移场和试函数的选取规律进行深入探讨.该系列单元克服等参单元易网格畸变的缺点，在推导过程中引入广义位移的特征矩阵、节点位移的特征矩阵以及转换矩阵的概念，以使推导过程简单化、通用化.数值算例表明，此方法通用性强、易于程序化，所构造的单元自由度少、效率和精度较高.
关键词：等参坐标；四边形面积坐标；薄板单元；广义协调条件；网格畸变
中图分类号：TU312；O242.21文献标志码：A
Generalized conforming plate elements based on quadrilateral area coordinate
KANG Lan, ZHANG Qilin, QIN Zhonghui
(College of Civil Eng., Tongji Univ., Shanghai 200092, China)
Abstract：To solve some problems occurred in the isoparametric coordinates, a series of quadrilateral generalized Area-coordinate Quadrilateral Plate (AQP) elements with four nodes and 12 degrees of freedom are successfully constructed by quadrilateral area coordinate method. Generalized conforming conditions, displacement field on boundary and selective rule of trying function are deeply discussed. The elements overcome the mesh distortion of isoparametric elements, and characteristic matrix of node displacements and generalized displacements and conversion matrix are introduced into deductive procedure to make it simple and universal. The examples show that the method is of strong generality and easy programming， and the elements have less freedom and demonstrate higher efficiency and accuracy.
Key words：isoparametric coordinate;quadrilateral area coordinate; thin plate element; generalized conforming condition; mesh distortion
0引言
采用等参坐标(ξ，η)构造四边形薄板单元已经取得很大进展，以DKQ[1](Discrete Kirchoff Quadrilateral)为代表的等参四边形薄板单元已具有许多优异性能.在DKQ基础上对其进行改进，构造出一批改良的等参四边形薄板单元，如IDKQ[2](Improved Discrete Kirchoff Quadrilateral).这些单元的优点是形式简单、性能稳定、精度较高；缺点是在计算固支边界问题时精度不尽如人意，且网格的抗畸变能力不够高.
面积坐标最早应用于三角形单元，DKT[3](Discrete Kirchoff Triangllar)是三角形面积坐标薄板单元的典型代表.文献[4,5]建立四边形面积坐标理论，文献[6-8]采用四边形面积坐标构造性能优异的薄板弯曲元，然而这些单元构造方法不具有通用性，且试函数复杂.
本文结合广义协调理论和四边形面积坐标方法，构造出一系列新型4节点12自由度的面积坐标四边形薄板单元(Area-Coordinate Quadrilateral Plate, AQP)，并总结出一套构造薄板单元的通用方法.数值算例表明，所构造的单元自由度少、列式简单、精度高；该构造方法通用性、程序化强，在此基础上可以发展出更多优秀单元.
1AQP系列单元的推导
3结论
采用面积坐标法与广义协调元法，构造出四边形薄板单元系列AQP，分别对AQP系列单元做纯弯和纯扭检验[11]，数值结果表明这2个单元均可以给出精确解，说明这3个单元严格通过分片检验，保证单元收敛的可靠性.
(1)这些单元解决了网格畸变问题：无论网格划分程度如何严重，这些单元大部分都对网格畸变不敏感，体现出面积坐标的优越性能.
(2)对广义位移协调单元的协调条件进行深入探讨，认为单元的协调条件都由节点协调、非节点协调、边点协调方案组成，并对这些协调方案得到的单元计算精度进行比较.
(3)选择试函数对单元刚度的影响是本质性的，通过对薄板单元性能的研究，得到如下选择试函数的规律：在单元自由度范围内所假设的任意1个试函数不能使选择的广义位移均为0，即广义位移的特征矩阵不能出现1行或者1列均为0，也即所有协调条件要相互独立.
(4)得到1套运用面积坐标和广义位移构造板单元的通用方法，该方法包含所有位移协调条件，其优点在于可以方便计算机编程，并可以推广运用到膜单元的构造和开发中去.在此过程中，提出节点位移的特征矩阵，广义位移的特征矩阵以及转换矩阵.
参考文献：
[1]BATOZ J L, TAHAR M B. Evaluation of a new quadrilateral thin plate bending element[J]. International J for Numerical Methods in Eng, 1982, 18 (11): 1 655-1 677.
[2]RAZAQPURA G, NOFAL M, VASILESCU A. An improved quadrilateral finite element for analysis of thin plates[J]. Finite Elements in Analysis & Design, 2003, 40 (1): 1-23.
[3]CEDERBAUM G, ABOUDI J, ELISHAKOFF I. Dynamic instability of shear-deformable viscoelastic laminated plates by Lyapunov exponents[J]. International J Solids & Structures, 1991, 28 (3): 317-327.
[4]龙驭球, 李聚轩, 龙志飞, 等. 四边形单元面积坐标理论[J]. 工程力学, 1997, 14 (3): 1-11.
[5]LONG Yuqiu, LI Juxuan, LONG Zhifei, et al. Some basic formulae for area coordinates used in quadrilateral elements[J]. Communications in Numerical Methods in Eng, 1999, 19 (12): 841-852.
[6]LONG Yuqiu, LI Juxuan, LONG Zhifei, et al. Area coordinates used in quadrilateral elements[J]. Communications in Numerical Methods in Eng, 1999, 19 (8): 533-545.
[7]SOH A K, LONG Zhifei, CEN Song. Development of a new quadrilateral thin plate element using area coordinates[J]. Computer Methods in Applied Mechanics & Eng, 2000, 190(8-10): 979-987.
[8]陈晓明, 岑松, 龙驭球. 采用面积坐标和基于假设转角的薄板元[J]. 工程力学, 2005, 22(4): 1-5.
[9]CHEN Xiaoming, CEN Song, LONG Yuqiu, et al. Membrane elements insensitive to distortion using the quadrilateral area coordinate method[J]. Computers & Structures, 2004, 82 (1): 35-54.
[10]金晶, 吴新跃. 有限元网格划分相关问题分析研究[J]. 计算机辅助工程, 2005, 14(2): 75-78.
[11]田中旭. 分片试验与有限元法[J]. 应用力学学报, 2000, 17(2): 25-31.
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