线性无关高阶数值流形法

徐栋栋; 郑宏; 杨永涛

doi:10.11779/CJGE201403011

线性无关高阶数值流形法 English Version

徐栋栋^{1, 2},
郑宏^{1, 2},
杨永涛^{1, 2}

1. 岩土力学与工程国家重点实验室,湖北武汉 430071; 2. 中国科学院武汉岩土力学研究所,湖北武汉 430071

基金项目: 国家自然科学基金项目（11172313）; 国家973项目资助课题（2011CB013505）

详细信息

作者简介:
徐栋栋(1986- )，男，山东聊城人，博士研究生，主要从事计算岩土力学等方面的研究。E-mail: xdhappy717@163.com。

中图分类号: O3；TU43
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出版历程
- 收稿日期: 2013-08-05
- 发布日期: 2014-03-19

Linearly independent higher-order numerical manifold method

XU Dong-dong^{1, 2},
ZHENG Hong^{1, 2},
YANG Yong-tao^{1, 2}

1. State Key Laboratory of Geomechanics and Geotechnical Engineering, Wuhan 430071, China; 2. Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China

摘要

摘要: 对高阶数值流形方法来说,若采用1阶局部位移函数显然提高了计算精度,但又不可避免地使总体刚度矩阵亏秩,出现线性相关问题。针对这种情况,提出局部位移函数采用1阶泰勒展开形式,使得定义在物理覆盖上的自由度具有明确的物理意义。当基函数所对应自由度取为应变分量时,定义物理覆盖为PC-u-ε型；使用局部坐标系下的应力分量来代替应变分量,进而发展了PC-u-σ型。这样方便了位移和应力边界条件的施加。数值算例表明,PC-u-ε型显著地减少了亏秩数；PC-u-σ型的施加完全地消除了亏秩数,同样保持了很高的计算精度。
- 数值流形法 /
- 泰勒展开 /
- 线性相关 /
- 亏秩 /
- Cook斜梁 /
- 1/4圆孔 /
- 边坡
Abstract: The adoption of the first-order local approximation functions has improved the accuracy, but it has also led to rank deficiency of the global stiffness matrix for the higher-order numerical manifold method, meaning the existence of the linear dependence. The first-order Taylor’s expansions with regard to the interpolation point are adopted as the local displacement functions, which makes the degrees of freedom defined on the physical cover have definite physical meanings. Then the first-order partial differential derivatives are expressed by the strain components, leading to the PC-u-ε. The strain components are further replaced by the stress components in the local framework, creating the PC-u-σ. In this way, both the displacement and the stress boundary conditions are easily applied. Numerical examples show that the PC-u-ε alone significantly causes a drastic decrease in rank deficiency, while deploying the PC-u-σ along the stress boundary completely eliminates the rank deficiency and retains higher accuracy.
- numerical manifold method /
- Taylor's expansion /
- linear dependence /
- rank deficiency /
- Cook skew beam /
- 1/4 circular hole /
- slope

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参考文献(25)

[1]	王水林, 葛修润. 流形元方法在模拟裂纹扩展中的应用[J]. 岩石力学与工程学报, 1997, 16(5): 405-410. (WANG Shui-lin, GE Xiu-run. Application of manifold method in simulating crack propagation[J]. Chinese Journal of Rock Mechanics and Engineering, 1997, 16(5): 405-410. (in Chinese))
[2]	韩有民, 罗先启, 王水林, 等. 裂纹扩展时物理覆盖与流形单元的生成算法[J]. 岩土工程学报, 2005, 27(6): 662-666. (HAN You-min, LUO Xian-qi, WANG Shui-lin, et al. Formulation algorithm of covers and manifold elements in NMM during propagation of cracks[J]. Chinese Journal of Geotechnical Engineering, 2005, 27(6): 662-666. (in Chinese))
[3]	曹文贵, 速宝玉. 流形元覆盖系统自动形成方法之研究[J]. 岩土工程学报, 2001, 23(2): 187-190. (CAO Wen-gui, SU Bao-yu. A study on techniques of automatically forming of cover system of numerical manifold method[J]. Chinese Journal of Geotechnical Engineering, 2001, 23(2): 187-190. (in Chinese))
[4]	凌道盛, 何淳健, 叶茂. 数值流形单元法数学网格自适应[J]. 计算力学学报, 2008, 25(2): 201-205. (LING Dao-sheng, HE Chun-jian, YE Mao. Mathematical mesh adaptation of numerical manifold element method[J]. Chinese Journal of Computational Mechanics, 2008, 25(2): 201-205. (in Chinese))
[5]	朱爱军, 邓安福, 曾祥勇. 数值流形方法对岩土工程开挖卸荷问题的模拟[J]. 岩土力学, 2006, 27(2): 179-183. (ZHU Ai-jun, DENG An-fu, ZENG Xiang-yong. Numerical manifold method for simulation of excavation unloading in geotechnical engineering[J]. Rock and Soil Mechanics, 2006, 27(2): 179-183. (in Chinese))
[6]	BABUSKA I, MELENK J M. Partition of unity method[J]. International Journal for Numerical Methods in Engineering, 1997, 40(4): 727-758.
[7]	ODEN J T, DUARTE C A, ZIENKIEWICZ O C. A new cloud-based hp finite element method[J]. Computer Methods in Applied Mechanics and Engineering, 1998, 153(1/2): 117-126.
[8]	STROUBOULIS T, COPPS K, BABUSKA I. The generalized finite element method: an example of its implementation and illustration of its performance[J]. International Journal for Numerical Methods in Engineering, 2000, 47: 1401-1417.
[9]	STROUBOULIS T, BABUSKA I, COPPS K. The design and analysis of the generalized finite element method[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 181: 43-69.
[10]	SHI G H. Manifold method[C]// Proceedings of the First International Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, TSI Press, Albuquerque, New Mexico, USA, 1996: 52-204.
[11]	CHEN G, OHNISHI Y, ITO T. Development of higher-order manifold method[J]. International Journal for Numerical Methods in Engineering, 1998, 43: 685-712.
[12]	LIN J S.A mesh-based partition of unity method for discontinuity modeling[J]. Computer Methods in Applied Mechanics and Engineering, 2003, 192: 1515-1532.
[13]	TIAN R, YAGAWA G, TERASAKA H. Linear dependence problems of partition of unity-based generalized FEMs[J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195: 4768-4782.
[14]	蔡永昌, 张湘伟. 使用高阶覆盖位移函数的数值流形方法及其应力精度的改善[J]. 机械工程学报, 2000, 36(9): 20-24. (CAI Yong-chang, ZHANG Xiang-wei. Expansions to high-order cover function and improvement of the stress accuracy in numerical manifold method[J]. Chinese Journal of Mechanical Engineering, 2000, 36(9): 20-24. (in Chinese))
[15]	CAI Y C, ZHUANG X Y, AUGARDE C. A new partition of unity finite element free from the linear dependence problem and possessing the delta property[J]. Computer Methods in Applied Mechanics and Engineering, 2010, 199: 1036-1043.
[16]	TIAN R, YAGAWA G. Generalized nodes and high-performance elements[J]. International Journal for Numerical Methods in Engineering, 2005, 64: 2039-2071.
[17]	TIAN R, MASTUBARA H, YAGAWA G. Advanced 4-node tetrahedrons[J]. International Journal for Numerical Methods in Engineering, 2006, 68: 1209-1231.
[18]	RIKER C, HOLZER S M. The mixed-cell-complex partition-of-unity method[J]. Computer Methods in Applied Mechanics and Engineering, 2009, 198: 1235-1248.
[19]	RAJENDRAN S, ZHANG B R. A “FE-meshfree” QUAD4 element based on partition of unity[J]. Computer Methods in Applied Mechanics and Engineering, 2007, 197: 128-147.
[20]	AN X M, LI L X, MA G W, et al. Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes[J]. Computer Methods in Applied Mechanics and Engineering, 2011, 200(5/6/7/8): 665-674.
[21]	郭朝旭. 高阶数值流形方法中线性相关问题的研究[D]. 宜昌: 三峡大学, 2012. (GUO Chao-xu. Study on linear dependence problem in high-order numerical manifold method[D]. Yichang: China Three Gorges University, 2012. (in Chinese))
[22]	TIMOSHENKO S P, GOODIER J N. Theory of elasticity[M]. New York: McGraw-Hill, 1970.
[23]	COOK R D, MALKUS D S, PLESHA M E. Concepts and applications of finite element analysis[M]. 3rd ed. New York: John Wiley & Sons Inc, 1989.
[24]	ZHENG H, LIU D F, LI C G. Slope stability analysis based on elasto-plastic finite element method[J]. International Journal for Numerical Methods in Engineering, 2005, 64(14): 1871-1888.
[25]	杨永涛, 郑宏, 张建海. 基于三角形网格的虚多边形有限元法[J]. 岩石力学与工程学报, 2013, 32(6): 1214-1221. (YANG Yong-tao, ZHENG Hong, ZHANG Jian-hai. A virtual polygonal finite element method based on triangular mesh[J]. Chinese Journal of Rock Mechanics and Engineering, 2013, 32(6): 1214-1221. (in Chinese))