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基于适合分析T样条的高阶数值流形方法

刘登学 张友良 刘高敏

刘登学, 张友良, 刘高敏. 基于适合分析T样条的高阶数值流形方法[J]. 力学学报, 2017, 49(1): 212-222. doi: 10.6052/0459-1879-16-217
引用本文: 刘登学, 张友良, 刘高敏. 基于适合分析T样条的高阶数值流形方法[J]. 力学学报, 2017, 49(1): 212-222. doi: 10.6052/0459-1879-16-217
Liu Dengxue, Zhang Youliang, Liu Gaomin. HIGHER-ORDER NUMERICAL MANIFOLD METHOD BASED ON ANALYSIS-SUITABLE T-SPLINE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 212-222. doi: 10.6052/0459-1879-16-217
Citation: Liu Dengxue, Zhang Youliang, Liu Gaomin. HIGHER-ORDER NUMERICAL MANIFOLD METHOD BASED ON ANALYSIS-SUITABLE T-SPLINE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 212-222. doi: 10.6052/0459-1879-16-217

基于适合分析T样条的高阶数值流形方法

doi: 10.6052/0459-1879-16-217
基金项目: 

国家重点基础研究发展计划(973计划) 2014CB047100

国家自然科学基金 11272330

详细信息
    通讯作者:

    刘登学, 在读博士, 主要研究方向:计算岩石力学.E-mail:liudengxue123@sina.cn

  • 中图分类号: O302

HIGHER-ORDER NUMERICAL MANIFOLD METHOD BASED ON ANALYSIS-SUITABLE T-SPLINE

  • 摘要: 数值流形方法是一种非常灵活的数值计算方法,连续体的有限单元方法和块体系统的非连续变形分析方法只是这一数值方法的特例.数值流形方法中高阶位移函数的构造可通过提高权函数的阶次来实现,这种方法往往需要沿单元边界配置适当的边内节点,这些结点的出现增加了前处理的复杂性,特别是对于大型复杂的空间问题.另一方面,在数值流形方法中可通过缩小单元尺寸(h加密)来提高求解精度.当模拟裂纹扩展时,这种细化策略可用来克服裂纹尖端的奇异性.一个传统的解决方案是细化整个网格,但这会导致计算效率的显著降低.将适合分析的T样条(analysis-suitable T-spline,AST)引入数值流形方法中来建立高阶数值流形方法的分析格式,有效的避免了该问题的出现.AST样条基函数具有线性无关,单位分解,局部加密等许多重要性质,使得其非常适合用于工程设计及分析.在引入AST样条后,可通过改变数学覆盖的构造形式建立不同阶次的数值流形方法分析格式;AST样条自身的局部加密性质也使得数值流形方法中的数学网格局部加密更容易实现.算例结果表明:随着AST样条基函数阶次的提高,数值流形方法的计算结果有了明显的改善;基于AST样条基函数的数值流形方法在保持计算精度的前提下降低了自由度的数量.

     

  • 图  1  数值流形方法中的覆盖和单元

    Figure  1.  A simple example to illustrate the basic concepts of NMM

    图  2  裂纹尖端处的奇异物理覆盖

    Figure  2.  Singular physical covers at crack tip in NMM

    图  3  一个T网格

    Figure  3.  A T-mesh

    图  4  锚点$A$对应的局部节点向量

    Figure  4.  Inferring local vectors for anchor $A$

    图  5  锚点$D$对应的局部节点向量

    Figure  5.  Inferring local vectors for anchor $D$

    图  6  扩展T网格

    Figure  6.  The extended T-mesh. Face extensions are represented by dotted arrows and edge extensions are represented by dashed arrows. The T-junctions are denoted by black dots

    图  7  (a)适合分析的T网格(b)扩展T网格

    Figure  7.  (a) An analysis-suitable T-mesh (b) the extended T-mesh

    图  8  采用AST网格的数值流形方法

    Figure  8.  Numerical manifold method with AST mesh

    图  9  不同阶次的数学覆盖

    Figure  9.  Mathematical cover with different order

    图  10  $p =1$时权值在一个数学覆盖中的分布 \$\{x_{1},x_{2},x_{3}\}=\{0,1, 2\}$,$\{ y_{1},y_{2},y_{3}\}=\{0,1,2\}$

    Figure  10.  Distribution of weights in a mathematical cover when $p =1$ with $\{x_{1},x_{2},x_{3}\}=\{0,1, 2\}$,$\{ y_{1},y_{2},y_{3}\}=\{0,1,2\}$

    图  11  $p =2$时权值在一个数学覆盖中的分布$ \{x_{1},x_{2},x_{3}, x_{4}\}=\{0,1,2,3\}$,$\{y_{1},y_{2},y_{3},y_{4}\}=\{0,1,2,3\}$

    Figure  11.  Distribution of weights in a mathematical cover when $p =2$ with $\{x_{1},x_{2},x_{3}, x_{4}\}=\{0,1,2,3\}$,$\{y_{1},y_{2},y_{3}, y_{4}\}=\{0,1,2,3\}$

    图  12  $p =3$时权值在一个数学覆盖中的分布 $\{x_{1},x_{2},x_{3},x_{4},x_{5}\}=\{0, 1,2,3,4\}$,$\{y_{1},y_{2},y_{3},y_{4}, y_{5}\}=\{0,1,2,3,4\}$

    Figure  12.  Distribution of weights in a mathematical cover when $p=3$ with $\{x_{1},x_{2},x_{3},x_{4},x_{5}\}=\{0,1,2,3,4\}$,$\{y_{1},y_{2},y_{3},y_{4},y_{5}\}=\{0,1,2,3,4\}$

    图  13  Timoshenko 悬臂梁

    Figure  13.  Timoshenko cantilever beam

    图  14  数学覆盖网格图

    Figure  14.  Mathematical covers

    图  15  $y=0$时$y$方向位移变化图

    Figure  15.  $y$ displacement versus position when $y=0$

    图  16  $ y=0$时$\sigma _{x}$变化图

    Figure  16.  $\sigma _{x }$ versus position when $y=0$

    图  17  悬臂梁应变能误差

    Figure  17.  Error in strain energy of the cantilever beam

    图  18  含单边裂纹的有限板

    Figure  18.  Finite plate with crack

    图  19  数学覆盖网格加密

    Figure  19.  Mathematical mesh refinement

    图  20  全局数学网格加密时裂纹强度因子的计算误差

    Figure  20.  Error of SIFs using global refinement mesh

    图  21  局部数学网格加密时裂纹强度因子的计算误差

    Figure  21.  Error of SIFs using local refinement mesh

    表  1  采用一次AST样条数值流形方法计算结果

    Table  1.   SIFs results for NMM_T1

    表  2  采用二次AST样条数值流形方法计算结果

    Table  2.   SIFs results for NMM_T2

    表  3  采用三次AST样条数值流形方法计算结果

    Table  3.   SIFs results for NMM_T3

  • [1] Shi GH. Manifold method//The 1st Int. Forum on DDA Simulation of Discontinuous Media. Bekerley, California, USA. 1996:52-204 http://www.oalib.com/references/16761556
    [2] 石根华, 裴觉民. 数值流形方法与非连续变形分析. 北京:清华大学出版社, 1997

    Shi Genhua, Pei Juemin. Numerical Manifold Method and Discontinue Deformation Analysis. Beijing:Tsinghua University Press, 1997(in Chinese)
    [3] Tal Y, Hatzor YH, Feng XT. An improved numerical manifold method for simulation of sequential excavation in fractured rocks. International Journal of Rock Mechanics and Mining Sciences, 2014, 65:116-128 doi: 10.1016/j.ijrmms.2013.10.005
    [4] 焦健, 乔春生, 徐干成. 开挖模拟在数值流形方法中的实现. 岩土力学, 2010, 31(9):2951-2957 http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201009052.htm

    Jiao Jian, Qiao Chunsheng, Xu Gancheng. Simulation of excavation in numerical manifold method. Rock and Soil Mechanics, 2010, 31(9):2951-2957(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX201009052.htm
    [5] 朱爱军, 邓安富, 曾祥勇. 数值流形方法对岩土工程开挖卸荷问题的模拟. 岩土力学, 2006, 27(2):179-183 http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200602001.htm

    Zhu Aijun, Deng Anfu, Zeng Xiangyong. Numerical manifold method for simulation of excavation unloading in geotechnical engineering. Rock and Soil Mechanics, 2006, 27(2):179-183(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200602001.htm
    [6] 姜清辉,王书法. 锚固岩体的三维数值流形方法. 岩石力学与工程学报,2006,25(3):528-532 http://www.cnki.com.cn/Article/CJFDTOTAL-YSLX200603017.htm

    Jiang Qinghui,Wang Shufa. Three dimensional numerical manifold method simulation of anchor bolt supported rock mass. Chinese Journal of Rock Mechanics and Engineering,2006, 25(3):528-532(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-YSLX200603017.htm
    [7] Wei W, Jiang QH, Peng J. New rock bolt model and numerical implementation in numerical manifold method. International Journal of Geomechanics 2016, E4016004 http://cn.bing.com/academic/profile?id=bf8bc2a56a8e61488636abe226108026&encoded=0&v=paper_preview&mkt=zh-cn
    [8] 王水林, 葛修润. 流形方法在模拟裂纹扩展中的应用. 岩石力学与工程学报, 1997, 16(5):405-410 http://www.cnki.com.cn/Article/CJFDTOTAL-YSLX705.001.htm

    Wang Shuilin, Ge Xiuyun. Application of manifold method in simulating crack propagation. Chinese Journal of Rock Mechanics and Engineering, 1997, 16(5):405-410(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-YSLX705.001.htm
    [9] Chiou YJ, Lee YM, Tesay RJ. Mix mode fracture propagation by manifold method. International Journal of Fracture, 2002, 114:327-347 doi: 10.1023/A:1015713428989
    [10] Wu ZJ, Wong LNY. Friction crack initiation and propagation analysis using the numerical manifold method. Computers & Geotechnics, 2012, 39:38-53 http://cn.bing.com/academic/profile?id=388d550fc312b545c2546e455c3aa557&encoded=0&v=paper_preview&mkt=zh-cn
    [11] Yang SK, Ma GW, Ren XH, et al. Cover refinement of numerical manifold for crack propagation simulation. Engineering Analysis with Boundary Elements, 2014, 43:37-49 doi: 10.1016/j.enganabound.2014.03.005
    [12] Yang YT, Tang XH, Zheng H. Three-dimensional fracture propagation with numerical manifold method. Engineering Analysis with Boundary Elements, 2016, 72:65-77 doi: 10.1016/j.enganabound.2016.08.008
    [13] Jiang QH,Deng SS,Zhou CB. Modeling unconfined seepage flow using three-dimensional numerical manifold method. Journal of Hydrodynamics,2010,22(4):554-561 doi: 10.1016/S1001-6058(09)60088-3
    [14] 姜清辉, 邓书申, 周创兵. 三维高阶数值流形方法研究. 岩土力学, 2006, 27(9):1471-1474 http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200609006.htm

    Jiang Qinghui, Deng Shushen, Zhou Chuangbing. Study of three-dimensional high-order numerical manifold method. Rock and Soil Mechanics, 2006, 27(9):1471-1474(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-YTLX200609006.htm
    [15] 苏海东, 崔建华, 谢小玲. 高阶数值流形方法的初应力公式. 计算力学学报, 2010, 27(2):270-274 http://www.cnki.com.cn/Article/CJFDTOTAL-JSJG201002017.htm

    Su Haidong, Cui Jianhua, Xie Xiaoling. Initial stress equation for high-order manifold method. Chinese Journal of Computational Mechanics, 2010, 27(2):270-274(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-JSJG201002017.htm
    [16] 苏海东. 固定网格的数值流形方法研究. 力学学报, 2011, 43(1):169-178 http://lxxb.cstam.org.cn/CN/abstract/abstract142011.shtml

    Su Haidong. Study on numerical manifold method with fixed meshs. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(1):169-178(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract142011.shtml
    [17] Wang Y, Hu MS, Zhou QL, et al. A nes second-order numerical manifold method model with an efficient scheme for analyzing free surface flow with inner drains. Applied Mathematical Modelling, 2016, 40:1427-1445 doi: 10.1016/j.apm.2015.08.002
    [18] Zheng H, Xu DD. New strategies for some issues of numerical manifold method in simulation of crack propagation. International Journal for Numerical Methods in Engineering, 2014, 97:986-1010 doi: 10.1002/nme.v97.13
    [19] 郭朝旭, 郑宏. 高阶数值流形方法中线性相关问题的研究. 工程力学, 2012, 29(12):228-232 http://cdmd.cnki.com.cn/Article/CDMD-11934-1012032231.htm

    Guo Chaoxu, Zheng Hong. Study of linear dependence problem in high-order numerical manifold method. Engineering Mechanics, 2012, 29(12):228-232(in Chinese) http://cdmd.cnki.com.cn/Article/CDMD-11934-1012032231.htm
    [20] 蔡永昌, 刘高扬. 基于独立覆盖的高阶数值流形方法. 同济大学学报(自然科学版), 2015, 43(12):1794-1799 http://www.cnki.com.cn/Article/CJFDTOTAL-TJDZ201512005.htm

    Cai Yongchang, Liu Gaoyang. High-order manifold method with independent covers. Journal of Tongji University:Natural Science, 2015, 43(12):1794-1799(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-TJDZ201512005.htm
    [21] Fan H, Zheng H, He SM, et al. A novel numerical manifold with derivative degrees of freedom and without linear dependence. Engineering Analysis with Boundary Elements, 2016, 64:19-37 doi: 10.1016/j.enganabound.2015.11.016
    [22] Hughes TJR, Cottrell JA, Bazilevs Y. et al. Isogeometric analysis:CAD, finite elements, NURBS, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194:4135-4195 http://cn.bing.com/academic/profile?id=30a971f78f528e680c4515ac8e364ed3&encoded=0&v=paper_preview&mkt=zh-cn
    [23] Piegl L, Tiller W. The NURBS Book. Springer-Verlag Berlin and Heidelberg GmbH, Berlin, 1997:292-295 http://cn.bing.com/academic/profile?id=109e4d7f2d860aea40cda56b1c430e3c&encoded=0&v=paper_preview&mkt=zh-cn
    [24] Sederberg TW, Zheng J, Bakenov A, et al. T-splines and T-nurccs. ACM Transactions on Graphic, 2003, 22:477-484 doi: 10.1145/882262
    [25] Bazilevs Y, Calo VM, Cottrell JA, et al. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering, 2010, 199:229-263 doi: 10.1016/j.cma.2009.02.036
    [26] Zhang YL, Liu DX, Tan F. Numerical manifold method based on isogeometric analysis. Science China Technological Sciences, 2015, 58:1520-1532 doi: 10.1007/s11431-015-5900-6
    [27] Li X, Zheng J, Sederberg TW, et al. On linear independence of T-splines blending functions. Computer Aided Geometric Design, 2012, 29:63-76 doi: 10.1016/j.cagd.2011.08.005
    [28] Ma GW, An XM, Zhang HH, et al. Modelling complex crack problems using the numerical manifold method. International Journal of Fracture, 2009, 156:21-35 doi: 10.1007/s10704-009-9342-7
    [29] Cox MG. The numerical evaluation of B-Splines. IMA Journal of Applied Mathematics, 1972, 10:134-149 doi: 10.1093/imamat/10.2.134
    [30] Carl DB. On calculating with B-Splines. Journal of Approximation Theory, 1972, 6:50-62 doi: 10.1016/0021-9045(72)90080-9
    [31] Sukumar N, Huang ZY, Prevost JH, et al. Partition of unity enrichment for bimaterial interface cracks. International Journal for Numerical Methods in Engineering, 2004, 59:1075-1102 doi: 10.1002/(ISSN)1097-0207
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出版历程
  • 收稿日期:  2016-08-01
  • 修回日期:  2016-11-17
  • 网络出版日期:  2016-11-23
  • 刊出日期:  2017-01-18

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