弹性力学有限元分析中的平衡与协调理论

傅向荣; 陈璞; 孙树立; 袁明武

doi:10.6052/j.issn.1000-4750.2021.08.0647

弹性力学有限元分析中的平衡与协调理论

doi: 10.6052/j.issn.1000-4750.2021.08.0647

傅向荣^1, ,,
陈璞^2,,
孙树立^2,,
袁明武^2,

1.
中国农业大学土木工程系，北京 100083
2.
北京大学力学与工程科学系，北京 100871

基金项目: 国家自然科学基金项目(11672362，11472014)

详细信息

作者简介:
陈　璞(1962−)，男，重庆人，教授，博士，主要从事计算力学、结构动力学方面的研究(E-mail: chenpu@pku.edu.cn)

孙树立(1968−)，男，河北人，教授，博士，主要从事计算力学方面的研究(E-mail: SUNSL@mech.pku.edu.cn)

袁明武(1939−)，男，江苏人，教授，博士，主要从事计算力学方面的研究(E-mail: yuanmw@pku.edu.cn)

通讯作者:
傅向荣(1972−)，男，湖南人，教授，博士，主要从事计算力学研究(E-mail: fuxr@cau.edu.cn)

中图分类号: O343
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- 文章访问数: 165
- HTML全文浏览量: 40
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- 被引次数: 0
出版历程
- 收稿日期: 2021-08-20
- 修回日期: 2022-03-24
- 网络出版日期: 2022-04-01
- 刊出日期: 2023-02-01

RESEARCH ON EQUILIBRIUM AND CONFORMING THEORY OF THE FINITE ELEMENT METHOD IN ELASTICITY

FU Xiang-rong^{1
, ,},
CHEN Pu^2
,,
SUN Shu-li^2
,,
YUAN Ming-wu^2
,

1.
Department of Civil Engineering, China Agricultural University, Beijing 100083, China
2.
Department of Mechanics and Engineering Science, Peking University, Beijing 100871, China

摘要

摘要: 有限元分析中的单元可以遵循不同的方法构造，该文提出以单元模型的平衡性与协调性进行分类，并对弹性力学平面问题中的几种经典单元进行了分析比较，总结了协调元、非协调元和超协调元的协调性方法，以及基于解析试函数法的平衡型方法。单元的协调性理论思路包含单纯形格式协调元和非单纯形格式协调元，以及相应的非协调和超协调元格式，关注的重点是单元边界的协调。单元的平衡性理论思路包含解析试函数和权函数的高阶完备性，关注的重点研究是单元内部及边界平衡性。研究表明：针对弹性力学中平衡性和协调性要求，两类理论给出的不同单元格式各具特点，而既能保证单元内部平衡性，又能考虑单元界面协调性的单元类型给出了更精确、合理的计算结果。
- 特征微分方程解法 /
- 平衡型单元 /
- 协调型单元 /
- 超协调元 /
- 解析试函数法
Abstract: Element formulations in finite element analysis may follow different approaches. A classification of elements in equilibrium and compatibility is presented. Analysis and comparison of some classic plane elements show that, non-conforming, conforming and super conforming elements can be constructed based on the compatibility theory, and analytical trial functions can be employed to help the construction of equilibrium elements. The researches of compatibility focus on the boundary conforming between elements, which conform bases of simplex conforming, non-simplex conforming, non-conforming and super conforming elements. The researches of equilibrium focus on the equilibrium principle in element or between elements, which emphasize high-precise completeness of analytical trial functions and weight functions. The benchmark tests show that although all kinds of elements have their own advantages and shortcomings, elements that satisfy both the equilibrium inside elements and the compatibility between elements deliver generally better performance.
- solution of characteristic differential equation /
- equilibrium theory /
- compatibility theory /
- super conforming element /
- analytical trial function method

HTML全文

图 1 纯弯梁问题

Figure 1. Pure bending beam

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图 2 三角形元网格

Figure 2. Triangular Meshes

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图 3 网格畸变敏感性测试

Figure 3. Distorted mesh test

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图 4 网格畸变敏感性测试

Figure 4. Distorted mesh test

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表 1 三角形网格下各类单元的挠度精度分析(精确解为1.0000)

Table 1. Precision of triangular elements

网格数	2×2	2×4	2×8	2×16
CST	0.0128	0.0489	0.1654	0.4058
GT9	0.1045	0.3156	0.6371	0.8510
GT9M	0.1049	0.3191	0.6504	0.8692
TR3	0.9906	0.9903	0.9911	0.9961

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表 2 网格畸变敏感性测试精度分析

Table 2. Precision in distorted mesh

d	5β	7β	P-S	ATF-GCQ4	ATF-Q4
0.0	1.0000	0.3151	1.0000	1.0000	1.0000
1.0	0.2670	0.1900	0.6290	0.5512	1.0000
2.0	0.1876	0.1574	0.5500	0.4386	1.0000
3.0	0.1911	0.1681	0.5470	0.4431	1.0000
4.0	0.2118	0.1701	0.5310	0.4297	1.0000
4.9	0.3496	0.1413	0.4980	0.4267	1.0000

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表 3 网格畸变敏感性测试计算结果

Table 3. Precision in distorted mesh

网格	结果	Q8	HSF-Q8
1	$ {\sigma _x} $(A)	56.447	120.000
	$ {\sigma _x} $(B)	−74.863	−120.000
	$w ({\rm C})$	−2.328	−12.000
2	$ {\sigma _x} $(A_上)	13.665	120.000
	$ {\sigma _x} $(A_下)	5.262	120.000
	$ {\sigma _x} $(B_上)	−5.665	−120.000
	$ {\sigma _x} $(B_下)	−14.299	−120.000
	$w ({\rm C})$	−0.477	−12.000

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参考文献(56)

[1]	BOUSSINESQ J. Application des potentiels a l'equilibre et des mouvements des solides elastiques [M]. Paris: Gauthier-Villars, 1883.
[2]	王敏中. 高等弹性力学[M]. 北京: 北京大学出版社, 2002. WANG Minzhong. Advanced elasticity [M]. Beijing: Peking University Press, 2002. ( in Chinese)
[3]	NAGHDI P M , HSU C S. On a representation of displacements in linear elasticity in terms of three stress functions [J]. J Math Mech, 1961, 10(2): 233 − 245.
[4]	LUR'E A I . Three-dimensional problem of the theory of elasticity [M]. New York: Interscience, 1964.
[5]	WANG M Z , XU B X , ZHAO Y T. General representations of polynomial elastic fields [J]. Journal of Applied Mechanics, 2012, 79(2): 021017. doi: 10.1115/1.4005556
[6]	FU X R, YUAN M W, CEN S, et al. Characteristic equation solution strategy for deriving the fundamental analytical solutions of 3D isotropic elasticity [J]. Applied Mathematics and Mechanics(English Edition), 2012, 33(10): 1253 − 1264. doi: 10.1007/s10483-012-1619-7
[7]	KELVIN L. Note on the integration of the equations of equilibrium of an elastic solid [J]. Mathematical and Physical Papers in Cambridge University Press, 1882: 97 − 99.
[8]	WILLIAMS M L. On the stress distribution at the base of a stationary crack [J]. Journal of Applied Mechanics, 1957, 24(1): 109 − 114. doi: 10.1115/1.4011454
[9]	陆明万, 罗学富. 弹性理论基础[M]. 北京: 清华大学出版社, 2000. LU Mingwan, LUO Xuefu. Basic theory of elasticity [M]. Beijing: Tsinghua University Press, 2000. ( in Chinese)
[10]	钟万勰. 弹性力学求解新体系[M]. 大连: 大连理工大学出版社, 1995. ZHONG Wanxie. A new systematic methodology for theory of elasticity [M]. Dalian: Press of Dalian University of technology, 1995. ( in Chinese)
[11]	AIRY G B. On the strains in the interior of beams [J]. London: Philosophical Transactions of the Royal Society, 1863, 153(1): 49 − 80.
[12]	傅向荣, 田歌. 基于解析试函数有限单元法的研究进展[J]. 工程力学, 2012, 29(suppl 2): 78 − 84. doi: 10.6052/j.issn.1000-4750.2012.05.S039 FU Xiangrong, TIAN Ge. Advances in the finite element method based on the analytical trial functions [J]. Engineering Mechanics, 2012, 29(Suppl 2): 78 − 84. (in Chinese) doi: 10.6052/j.issn.1000-4750.2012.05.S039
[13]	CLOUGH R W. The finite element method in plane stress analysis [C]. Pittsburgh, Pa, USA: Proceedings of the 2nd ASCE Conference on Electronic Computation, 1960, 9(1): 8 − 9.
[14]	COURANT R. Variational methods for the solution of problems of equilibrium and vibrations [J]. Bull Am Math Soc, 1943, 49(1): 1 − 23. doi: 10.1090/S0002-9904-1943-07818-4
[15]	ARGYRIS J. Energy theorems and structural analysis [M]. London: Butterworth, 1954.
[16]	ZIENKIEVICZ O C, TAYLOR R L. The finite element method [M]. London: McGraw-Hill Book Company, 1991.
[17]	龙驭球, 包世华. 结构力学[M]. 北京: 高等教育出版社, 2000. LONG Yuqiu, BAO Shihua. Structural mechanics [M]. Beijing: Higher Education Press, 2000. (in Chinese)
[18]	CLOUGH R W. Original formulation of the finite element method [J]. ECCM99. Finite Elem Anal Design, 1990, 7: 89 − 101.
[19]	CHARLTON T M. Energy principles in theory of structures [M]. Oxford: Oxford University Press, 1973.
[20]	钱令希. 余能原理 [J]. 中国科学, 1950, 1(1): 1 − 449. QIAN Lingxi. The complementary energy principle [J]. Science China, 1950. (in Chinese)
[21]	PIAN T H H. Finite element formulation by variational principles with relaxed continuity requirements [C]// Edited by AZIZ A K. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, 1972, 1(1): 671 − 687.
[22]	胡海昌. 弹性力学的变分原理及其应用[M]. 北京: 科学出版社, 1981: 465 − 476. HU Haichang. Variational principle of the elastics and its applications [M]. Beijing: Science Press, 1981: 465 − 476. (in Chinese)
[23]	FU X R, CEN S, LI C F, et al. Analytical trial function method for development of new 8-node plane element based on the variational principle containing airy stress function [J]. Engineering Computations, 2010, 27(1): 442 − 463.
[24]	龙驭球. 分区和分项混合能量原理[C]// 全国弹性和塑性力学学术交流会论文集. 重庆, 中国力学学会, 1980. LONG Yuqiu. Sub-region and sub-item mixed energy principle [C]// Proceeding of elastic and plastic mechanics in Chinese. ChongQing, Chinese Society of Theoretical and Applied Mechanics, 1980. (in Chinese)
[25]	冯康. 基于变分原理的差分格式[J]. 应用数学与计算数学, 1965, 2(4): 238 − 262. FENG Kang. Difference scheme based on variational principle [J]. Applied Mathematica and Computational Mathematics, 1965, 2(4): 238 − 262. (in Chinese)
[26]	余德浩, 冯康. 20世纪中国知名科学家学术成就概览·数学卷·第二分册(王元主编)[M]. 北京: 科学出版社. 2011. YU Dehao, FENG Kang. The achievement of famous Chinese scientists in the 20 Century, 2nd Volume of Mathematics [M]. Edited by Wang Yuan. Beijing: Science Press, 2011. (in Chinese)
[27]	黄鸿慈, 王荩贤, 崔俊芝, 等. 按位移解平面弹性问题的差分方法[J]. 应用数学与计算数学, 1966, 3(1): 54 − 60. HUANG Hongci, WANG Jinxian, CUI Junzhi, et al. Difference scheme based on displacement solution on the plane elasticity [J]. Applied Mathematics and Computation, 1966, 3(1): 54 − 60. (in Chinese)
[28]	冯康, 石钟慈. 弹性结构的数学理论[M]. 北京: 科学出版社, 2010. FENG Kang, SHI Zhongci. Mathematics theory of the elastic structure [M]. Beijing: Science Press, 2010. (in Chinese)
[29]	石钟慈. 有限元方法 [M]. 北京: 科学出版社, 2016. SHI Zhongci. The finite element method [M]. Beijing: Science Press, 2016. (in Chinese)
[30]	崔俊芝, 刘福森. 平面应力分析的变分法和有限元法[J]. 数学的认识与实践, 1972(1): 23 − 34. CUI Junzhi, LIU Fusen. Variational method and finite element method on the analysis of the plane stress problems [J]. Mathematics in Practice and Theory, 1972(1): 23 − 34. (in Chinese)
[31]	FINLAYSON B A, SCRIVEN L E. The method of weighted residuals - A review [J]. Applied Mechanics Reviews, 1966.
[32]	龙驭球, 龙志飞, 岑松. 新型有限元论[M]. 北京: 清华大学出版社, 2004. LONG Yuqiu, LONG Zhifei, CEN Song. Advanced theory of the finite element method [M]. Beijing: Tsinghua University Press, 2004. (in Chinese)
[33]	傅向荣. 基于解析试函数的广义协调元[D]. 北京: 清华大学, 2002. FU Xiangrong. The generalized conforming element based on the analytical trial functions [D]. Beijing: Tsinghua University, 2002. (in Chinese)
[34]	TURNER M J, CLOUGH R W, MARTIN H C, et al. Stiffness and deflection analysis of complex structures [J]. Journal of the Aeronautical Sciences, 1956, 23(9): 805 − 823. doi: 10.2514/8.3664
[35]	LEE N S, BATHE K J. Effects of element distortion on the performance of isoparametric elements [J]. International Journal for Numerical Methods in Engineering, 1993, 36(1): 3553 − 3576.
[36]	WILSON E L, TAYLOR R L, DOHERTY W P, et al. Incompatible displacement models [C]// Fenves S J, et al. Numerical and computer methods in structural mechanics. Academic Press, 1973.
[37]	唐立民, 陈万吉, 刘迎曦. 有限元分析中的拟协调元[J]. 大连工学院学报, 1980, 19(2): 19 − 35. TANG Limin, CHEN Wali, LIU Yingxi. Quasi conforming element in the finite element method [J]. Journal of Dalian Institute of Technology, 1980, 19(2): 19 − 35. (in Chinese)
[38]	龙驭球, 辛克贵. 广义协调元[J]. 土木工程学报, 1987, 20(1): 1 − 14. doi: 10.15951/j.tmgcxb.1988.04.004 LONG Yuqiu, XIN Kegui. Generalized conforming element [J]. Journal of Civil Engineering, 1987, 20(1): 1 − 14. (in Chinese) doi: 10.15951/j.tmgcxb.1988.04.004
[39]	陈万吉. 精化直接刚度法与不协调模式[J]. 计算结构力学及其应用, 1993, 10(4): 363 − 368. CHEN Wanji. Refined directly stiffness method and non-conforming mode [J]. Computational Structural Mechanics and Its Applications, 1993, 10(4): 363 − 368. (in Chinese)
[40]	FU X R, YUAN M W, CHEN P. A conforming triangular plane element with rotational degrees of freedom [J]. Mathematical Problems in Engineering, 2015, 1(1): 1 − 9.
[41]	PIAN T H H. Derivation of element stiffness matrices by assumed stress distributions [J]. AIAA J, 1964, 2(7): 1333 − 1336. doi: 10.2514/3.2546
[42]	傅向荣, 岑松, 蒋秀根, 等. 几种典型有限元单元模型的性能分析[C]. 姚振汉, 等, 卞学鐄先生纪念文集. 南京: 河海大学出版社, 2011, 1(1): 128 − 138. FU Xiangrong, CEN Song, JIANG Xiugen, et al. Study on some classic finite element models – some views in the teaching of the finite element method [C]. YAO Zhenhan, et al. Highly performance finite element methods & memories to T. H. H. Pian. Nanjing: Hehai University Press, 2011, 1(1): 128 − 138. (in Chinese)
[43]	PIAN T H H, SUMIHARA K. Rational approach for assumed stress finite element [J]. International Journal for Numerical Methods in Engineering, 1984, 20: 1685 − 1695. doi: 10.1002/nme.1620200911
[44]	龙驭球, 支秉琛, 匡文起, 等. 分区混合有限元法计算应力强度因子[J]. 力学学报, 1982, 18(4): 341 − 353. LONG Yuqiu, ZHI Bingchen, KUANG Wenqi, et al. Sub-region mixed finite element method for the calculation of stress intensity factor [J]. Chinese Journal of Theoretical and Applied Mechanics, 1982, 18(4): 341 − 353. (in Chinese)
[45]	钟万勰, 纪峥. 理性有限元[J]. 计算结构力学及其应用, 1996, 13(1): 1 − 8. ZHONG Wanxie, JI Zheng. Rational finite element [J]. Computational Structural Mechanics and Applications, 1996, 13(1): 1 − 8. (in Chinese)
[46]	龙驭球, 傅向荣. 基于解析试函数的广义协调四边形厚板元[J]. 工程力学, 2002, 19(3): 10 − 15. doi: 10.3969/j.issn.1000-4750.2002.03.003 LONG Yuqiu, FU Xiangrong. Two generalized conforming quadrilateral thick plate elements based on analytical trial functions [J]. Engineering Mechanics, 2002, 19(3): 10 − 15. (in Chinese) doi: 10.3969/j.issn.1000-4750.2002.03.003
[47]	傅向荣, 龙驭球. 基于解析试函数的广义协调四边形膜元[J]. 工程力学, 2002, 19(4): 12 − 16. doi: 10.3969/j.issn.1000-4750.2002.04.003 FU Xiangrong, LONG Yuqiu. Generalized conforming quadrilateral plane elements based on analytical trial functions [J]. Engineering Mechanics, 2002, 19(4): 12 − 16. (in Chinese) doi: 10.3969/j.issn.1000-4750.2002.04.003
[48]	傅向荣, 龙驭球. 解析试函数法分析平面切口问题[J]. 工程力学, 2003, 20(4): 33 − 38. doi: 10.3969/j.issn.1000-4750.2003.04.006 FU Xiangrong, LONG Yuqiu. Analysis of plane notched problems with analytical trial functions’ method [J]. Engineering Mechanics, 2003, 20(4): 33 − 38. (in Chinese) doi: 10.3969/j.issn.1000-4750.2003.04.006
[49]	CEN S, FU X R, ZHOU M J. 8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes [J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(29/30/31/32): 2321 − 2336. doi: 10.1016/j.cma.2011.04.014
[50]	CEN S, ZHOU M J, FU X R. A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions [J]. Computers & Structures, 2011, 87(5/6): 517 − 528.
[51]	CEN S, FU X R, ZHOU G H, et al. Shape-free finite element method: The plane hybrid stress-function(HS-F) element method for anisotropic materials [J]. Science in China series G-Physics Mechanics & Astronomy, 2011, 54(4): 653 − 665.
[52]	CEN S, ZHOU G H, FU X R. A shape-free 8-node plane element by unsymmetric analytical trial function method [J]. International Journal for Numerical Methods in Engineering, 2012, 91(1): 158 − 185.
[53]	袁驷, 袁全. 固端法: 二维有限元先验定量误差估计与控制[J]. 工程力学, 2021, 38(1): 8 − 14. doi: 10.6052/j.issn.1000-4750.2020.07.ST05 YUAN Si, YUAN Quan. Fixed-end method: A prior quantitative error estimate and control for two-dimensional FEM [J]. Engineering Mechanics, 2021, 38(1): 8 − 14. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.07.ST05
[54]	黄泽敏, 袁驷. 二维有限元单元角结点位移精度修正之初探[J]. 工程力学, 2021, 38(增刊): 1 − 6. doi: 10.6052/j.issn.1000-4750.2020.07.S002 HUANG Zeming, YUAN Si. A preliminary study on accuracy improvement of nodal displacements in 2D finite element method [J]. Engineering Mechanics, 2021, 38(Suppl): 1 − 6. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.07.S002
[55]	陈晓明, 李云贵. 用广义协调方法推导的平面四节点等参单元[J]. 工程力学, 2018, 35(12): 1 − 6. doi: 10.6052/j.issn.1000-4750.2017.09.0706 CHEN Xiaoming, LI Yungui. A 4-Node isoparametric element formulated with generalized conforming conditions [J]. Engineering Mechanics, 2018, 35(12): 1 − 6. (in Chinese) doi: 10.6052/j.issn.1000-4750.2017.09.0706
[56]	岑松, 尚闫, 周培蕾, 等. 形状自由的高性能有限元方法研究的一些进展[J]. 工程力学, 2017, 34(3): 1 − 14. doi: 10.6052/j.issn.1000-4750.2016.10.0763 CEN Song, SHANG Yan, ZHOU Peilei, et al. Advances in shape-free finite element methods: A review [J]. Engineering Mechanics, 2017, 34(3): 1 − 14. (in Chinese) doi: 10.6052/j.issn.1000-4750.2016.10.0763