摘要：為克服等參坐標(biāo)的缺點，采用四邊形面積法構(gòu)造一系列4節(jié)點12自由度的廣義協(xié)調(diào)任意面積坐標(biāo)四邊形薄板(Area-coordinate Quadrilateral Plate，AQP)單元.對廣義協(xié)調(diào)條件、邊界位移場和試函數(shù)的選取規(guī)律進行深入探討.該系列單元克服等參單元易網(wǎng)格畸變的缺點，在推導(dǎo)過程中引入廣義位移的特征矩陣、節(jié)點位移的特征矩陣以及轉(zhuǎn)換矩陣的概念，以使推導(dǎo)過程簡單化、通用化.數(shù)值算例表明，此方法通用性強、易于程序化，所構(gòu)造的單元自由度少、效率和精度較高.

關(guān)鍵詞：等參坐標(biāo)；四邊形面積坐標(biāo)；薄板單元；廣義協(xié)調(diào)條件；網(wǎng)格畸變

中圖分類號：TU312；O242.21文獻(xiàn)標(biāo)志碼：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

采用等參坐標(biāo)(ξ，η)構(gòu)造四邊形薄板單元已經(jīng)取得很大進展，以DKQ[1](Discrete Kirchoff Quadrilateral)為代表的等參四邊形薄板單元已具有許多優(yōu)異性能.在DKQ基礎(chǔ)上對其進行改進，構(gòu)造出一批改良的等參四邊形薄板單元，如IDKQ[2](Improved Discrete Kirchoff Quadrilateral).這些單元的優(yōu)點是形式簡單、性能穩(wěn)定、精度較高；缺點是在計算固支邊界問題時精度不盡如人意，且網(wǎng)格的抗畸變能力不夠高.

面積坐標(biāo)最早應(yīng)用于三角形單元，DKT[3](Discrete Kirchoff Triangllar)是三角形面積坐標(biāo)薄板單元的典型代表.文獻(xiàn)[4,5]建立四邊形面積坐標(biāo)理論，文獻(xiàn)[6-8]采用四邊形面積坐標(biāo)構(gòu)造性能優(yōu)異的薄板彎曲元，然而這些單元構(gòu)造方法不具有通用性，且試函數(shù)復(fù)雜.

本文結(jié)合廣義協(xié)調(diào)理論和四邊形面積坐標(biāo)方法，構(gòu)造出一系列新型4節(jié)點12自由度的面積坐標(biāo)四邊形薄板單元(Area-Coordinate Quadrilateral Plate, AQP)，并總結(jié)出一套構(gòu)造薄板單元的通用方法.數(shù)值算例表明，所構(gòu)造的單元自由度少、列式簡單、精度高；該構(gòu)造方法通用性、程序化強，在此基礎(chǔ)上可以發(fā)展出更多優(yōu)秀單元.

1AQP系列單元的推導(dǎo)

3結(jié)論

采用面積坐標(biāo)法與廣義協(xié)調(diào)元法，構(gòu)造出四邊形薄板單元系列AQP，分別對AQP系列單元做純彎和純扭檢驗[11]，數(shù)值結(jié)果表明這2個單元均可以給出精確解，說明這3個單元嚴(yán)格通過分片檢驗，保證單元收斂的可靠性.

(1)這些單元解決了網(wǎng)格畸變問題：無論網(wǎng)格劃分程度如何嚴(yán)重，這些單元大部分都對網(wǎng)格畸變不敏感，體現(xiàn)出面積坐標(biāo)的優(yōu)越性能.

(2)對廣義位移協(xié)調(diào)單元的協(xié)調(diào)條件進行深入探討，認(rèn)為單元的協(xié)調(diào)條件都由節(jié)點協(xié)調(diào)、非節(jié)點協(xié)調(diào)、邊點協(xié)調(diào)方案組成，并對這些協(xié)調(diào)方案得到的單元計算精度進行比較.

(3)選擇試函數(shù)對單元剛度的影響是本質(zhì)性的，通過對薄板單元性能的研究，得到如下選擇試函數(shù)的規(guī)律：在單元自由度范圍內(nèi)所假設(shè)的任意1個試函數(shù)不能使選擇的廣義位移均為0，即廣義位移的特征矩陣不能出現(xiàn)1行或者1列均為0，也即所有協(xié)調(diào)條件要相互獨立.

(4)得到1套運用面積坐標(biāo)和廣義位移構(gòu)造板單元的通用方法，該方法包含所有位移協(xié)調(diào)條件，其優(yōu)點在于可以方便計算機編程，并可以推廣運用到膜單元的構(gòu)造和開發(fā)中去.在此過程中，提出節(jié)點位移的特征矩陣，廣義位移的特征矩陣以及轉(zhuǎn)換矩陣.

參考文獻(xiàn)：

[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]龍馭球, 李聚軒, 龍志飛, 等. 四邊形單元面積坐標(biāo)理論[J]. 工程力學(xué), 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]陳曉明, 岑松, 龍馭球. 采用面積坐標(biāo)和基于假設(shè)轉(zhuǎn)角的薄板元[J]. 工程力學(xué), 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, 吳新躍. 有限元網(wǎng)格劃分相關(guān)問題分析研究[J]. 計算機輔助工程, 2005, 14(2): 75-78.

[11]田中旭. 分片試驗與有限元法[J]. 應(yīng)用力學(xué)學(xué)報, 2000, 17(2): 25-31.

(編輯廖粵新)

注：“本文中所涉及到的圖表、注解、公式等內(nèi)容請以PDF格式閱讀原文。”

推薦訪問:薄板 廣義 坐標(biāo) 單元 協(xié)調(diào)