线性无关高阶数值流形法在断裂力学中的应用

doi:10.13722/j.cnki.jrme.2014.1702

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摘要数值流形方法(NMM)实现了对连续和非连续问题的统一求解，非常适合应用于求解断裂力学问题。但传统高阶NMM，例如采用一次多项式作为其局部位移函数时，存在线性相关问题，这又在一定程度上限制了NMM的进一步发展和应用。因此，通过在物理片上引入一种新的局部位移函数以及用于模拟裂纹尖端应力场奇异性的位移函数，建立新的NMM求解体系，尝试消除线性相关问题和求解线弹性断裂力学问题。数值算例结果表明：(1) 该方法有效地解决了线性相关问题。(2) 对于典型的线弹性断裂力学问题，即便在网格密度较小时也能够精确地计算出裂纹尖端的应力强度因子。(3) 研究区域内插值点处的应力是连续的。(4) 定义在非奇异物理片上自由度具有明确的物理含义，且第3到5个自由度恰好是所对应插值点处的应变分量，可直接获得此处的应力，减少了计算量。最后，所建议方法可以很容易地推广到其他基于单位分解理论的方法中。

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关键词 ：数值模拟, 数值流形法, 数学片, 数学覆盖, 物理片, 物理覆盖, 线性相关, 节点力连续

Abstract：The numerical manifold method(NMM) has succeeded in providing a unified solution to continuum and discontinuum problems and therefore it is highly suitable for solving fracture mechanics problems. However，the conventional high-order NMM using the first order polynomial as the local displacement function has the problem of linear dependence，which restricts to a certain degree its further development and application. A new NMM framework was established in this research by introducing a new localized displacement function，as well as a special displacement function for modeling the stress singularity around crack tips. A new paradigm that eliminates the problem of linear dependence is then derived to solve linear elastic fracture mechanics problems. The numerical examples show that：(1) The proposed method successfully eliminates the problem of linear dependence；(2) For classic linear elastic fracture problems，the stress intensity factors at the crack tip can be calculated accurately even if the mesh is relatively sparse；(3) The stress function at interpolation points inside the physical domain is continuous；(4) All the degrees of freedom defined on non-singular physical patches are physically meaningful，with the third to the fifth being the strain components at the interpolation point of the patch. As a result，the stress components at the interpolation point can be directly obtained. Finally，the proposed approach can be easily extended to other methods based on the theory of the partition of unity.

Key words： numerical simulation numerical manifold method mathematical patch mathematical cover physical patch physical cover linear dependence continuous nodal stress

引用本文:

徐栋栋1，杨永涛2，郑宏2，邬爱清1. 线性无关高阶数值流形法在断裂力学中的应用[J]. 岩石力学与工程学报, 2015, 34(12): 2463-2473.
XU Dongdong1，YANG Yongtao2，ZHENG Hong2，WU Aiqing1. APPLICATION OF THE LINEARLY INDEPENDENT HIGH-ORDER NUMERICAL MANIFOLD METHOD IN FRACTURE MECHANICS. , 2015, 34(12): 2463-2473.

链接本文:

http://www.rockmech.org/CN/10.13722/j.cnki.jrme.2014.1702 或 http://www.rockmech.org/CN/Y2015/V34/I12/2463

[30]	张慧华，祝晶晶. 复杂裂纹问题的多边形数值流形方法求解[J]. 固体力学学报，2013，34(1)：38-46.(ZHANG Huihua，ZHU Jingjing. Numerical manifold analysis of complex crack problems on polygonal elements[J]. Chinese Journal of Solid Mechanics，2013，34(1)：38-46.(in Chinese)) " target="_blank">
[32]	LI S C，CHENG Y M. Enriched meshless manifold method for two-dimensional crack modeling[J]. Theoretical and Applied Fracture Mechanics，2005，44(3)：234-248. " target="_blank">
[35]	ZHANG G X，ZHU B F，LU Z C. Cracking simulation of the Wuqiangxi ship lock by manifold method[C]// Proceedings of the Sixth Internatonsl Conference on Analysis of Discontinuous Deformation (ICADD-6). [S. l.]：[s. n.]，2003：133-140. " target="_blank">
[40]	杨永涛，徐栋栋，郑宏. 动载下裂纹应力强度因子计算的数值流形元法[J]. 力学学报，2014，46(5)：730-738.(YANG Yongtao，XU Dongdong，ZHENG Hong. Evaluation on stress intensity factor of crack under dynamic load using numerical manifold method[J]. Chinese Journal of Theoretical and Applied Mechanics，2014，46(5)：730-738.(in Chinese))
[1]	张楚汉. 论岩石、混凝土离散-接触-断裂分析[J]. 岩石力学与工程学报，2008，27(2)：217-235.(ZHANG Chuhan. Discrete-contact- fracture analysis of rock and concrete[J]. Chinese Journal of Rock Mechanics and Engineering，2008，27(2)：217-235.(in Chinese))
[7]	SUKUMAR N，MOËS N，MORAN B，et al. Extended finite element method for three-dimensional crack modelling[J]. International Journal for Numerical Methods in Engineering，2000，48(11)：1 549-1 570. " target="_blank">
[10]	STROUBOULIS T，BABUŠKA I，COPPS K. The design and analysis of the generalized finite element method[J]. Computer Methods in Applied Mechanics and Engineering，2000，181(1)：43-69. " target="_blank">
[11]	STROUBOULIS T，COPPS K，BABUŠKA 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(8)：1 401-1 417. " target="_blank">
[12]	DUARTE C A，HAMZEH O N，LISZKA T J，et al. A generalized finite element method for the simulation of three-dimensional dynamic crack propagation[J]. Computer Methods in Applied Mechanics and Engineering，2001，190(15)：2 227-2 262. " target="_blank">
[13]	DUARTE C A，RENO L G，SIMONE A. A high-order generalized FEM for through-the-thickness branched cracks[J]. International Journal for Numerical Methods in Engineering，2007，72(3)：325-351. " target="_blank">
[14]	BELYTSCHKO T，KRONGAUZ Y，ORGAN D，et al. Meshless methods：an overview and recent developments[J]. Computer Methods in Applied Mechanics and Engineering，1996，139(1)：3-47. " target="_blank">
[15]	BELYTSCHKO T，GU L，LU Y Y. Fracture and crack growth by element free Galerkin methods[J]. Modelling and Simulation in Materials Science and Engineering，1994，2(3A)：519-534. " target="_blank">
[16]	BELYTSCHKO T，LU Y Y，GU L. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering，1994，37(2)：229-256. " target="_blank">
[18]	CUNDALL P A. The measurement and analysis of acceleration on rock slopes[Ph. D. Thesis][D]. London：University of London，Imperial College of Science and Technology，1971. " target="_blank">
[22]	焦玉勇，张秀丽，刘泉声，等. 用非连续变形分析方法模拟岩石裂纹扩展[J]. 岩石力学与工程学报，2007，26(4)：682-691.(JIAO Yuyong，ZHANG Xiuli，LIU Quansheng，et al. Simulation of rock crack propagation using discontinuous deformation analysis method[J]. Chinese Journal of Rock Mechanics and Engineering，2007，26(4)：682-691.(in Chinese))
[25]	姜清辉，周创兵. 四面体有限单元覆盖的三维数值流形方法[J]. 岩石力学与工程学报，2005，24(24)：4 455-4 460.(JIANG Qinghui，ZHOU Chuangbing. Three-dimensional numerical manifold method with tetrahedron finite element covers[J]. Chinese Journal of Rock Mechanics and Engineering，2005，24(24)：4 455-4 460.(in Chinese)) " target="_blank">
[27]	ZHENG H，LIU Z J，GE X R. Numerical manifold space of Hermitian form and application to Kirchhoff?s thin plate problems[J]. International Journal for Numerical Methods in Engineering，2013，95(9)：721-739.
[28]	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)：7-12.(WANG Shuilin，GE Xiurun. Application of manifold method in simulating crack propagation[J]. Chinese Journal of Rock Mechanics and Engineering，1997，16(5)：7-12.(in Chinese)) " target="_blank">
[29]	AN X M. Extended numerical manifold method for engineering failure analysis[Ph. D. Thesis][D]. Singapore：Nanyang Technology University Singapore，2010. " target="_blank">
[31]	苏海东，祁勇峰，龚亚琦. 裂纹尖端解析解与周边数值解联合求解应力强度因子[J]. 长江科学院院报，2013，30(6)：83-89.(SU Haidong，QI Yongfeng，GONG Yaqi. Compute stress intensity factors via combining analytical solutions around crack tips with surrounding numerical solutions[J]. Journal of Yangtze River Scientific Research Institute，2013，30(6)：83-89.(in Chinese))
[34]	ZHU H，ZHUANG X，CAI Y，et al. High rock slope stability analysis using the enriched meshless Shepard and least squares method[J]. International Journal of Computational Methods，2011，8(2)：209-228.
[5]	BELYTSCHKO T，BLACK T. Elastic crack growth in finite elements with minimal remeshing[J]. International Journal for Numerical Methods in Engineering，1999，45(5)：601-620. " target="_blank">
[6]	DOLBOW J，BELYTSCHKO T. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering，1999，46(1)：131-150. " target="_blank">
[9]	GRAVOUIL A，MOËS N，BELYTSCHKO T. Non-planar 3D crack growth by the extended finite element and level sets-Part II： Level set update[J]. International Journal for Numerical Methods in Engineering，2002，53(11)：2 569-2 586. " target="_blank">
[2]	GERRARD C M. Joint compliances as a basis for rock mass properties and the design of supports[J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts，1982，19(6)：285-305. " target="_blank">
[3]	GOODMAN R E，JOHN C ST. Finite element analysis for discontinuous rocks[J]. Numerical Methods in Geotechnical Engineering，1977：148-175. " target="_blank">
[4]	BABUŠKA I，MELENK J M. The partition of unity method[J]. International Journal for Numerical Methods in Engineering，1997，40(4)：727-758. " target="_blank">
[8]	MOËS N，GRAVOUIL A，BELYTSCHKO T. Non-planar 3D crack growth by the extended finite element and level sets-part I：Mechanical model[J]. International Journal for Numerical Methods in Engineering，2002，53(11)：2 549-2 568. " target="_blank">
[19]	SHI G H，GOODMAN R E. Two-dimensional discontinuous deformation analysis[J]. International Journal for Numerical and Analytical Methods in Geomechanics，1985，9(6)：541-556. " target="_blank">
[20]	SHI G H，GOODMAN R E. Generalization of two-dimensional discontinuous deformation analysis for forward modelling[J]. International Journal for Numerical and Analytical Methods in Geomechanics，1989，13(4)：359-380. " target="_blank">
[23]	NING Y，YANG J，MA G，et al. Modelling rock blasting considering explosion gas penetration using discontinuous deformation analysis[J]. Rock Mechanics and Rock Engineering，2011，44(4)：483-490.
[17]	CUNDALL P A. A computer model for simulating progressive large-scale movements in blocky rock systems[C]// Proceedings of the International Symposium Rock Fracture. Nancy，France：[s.n.]，1971：No. II-8. " target="_blank">
[21]	CAMONES L A M，VARGAS JR E A，DE FIGUEIREDO R P，et al. Application of the discrete element method for modeling of rock crack propagation and coalescence in the step-path failure mechanism[J]. Engineering Geology，2013，153：80-94. " target="_blank">
[26]	姜清辉，邓书申，周创兵. 三维高阶数值流形方法研究[J]. 岩土力学，2006，27(9)：1 471-1 474.(JIANG Qinghui，DENG Shushen，ZHOU Chuangbing. Study of three-dimensional high-order numerical manifold method[J]. Rock and Soil Mechanics，2006，27(9)：1 471- 1 474.(in Chinese)) " target="_blank">
[33]	GAO H，CHENG Y. A complex variable meshless manifold method for fracture problems[J]. International Journal of Computational Methods，2010，7(1)：55-81. " target="_blank">
[36]	WU Z，WONG L N Y. Frictional crack initiation and propagation analysis using the numerical manifold method[J]. Computers and Geotechnics，2012，39：38-53. " target="_blank">
[37]	ZHENG H，XU D D. New strategies for some issues of numerical manifold method in simulation of crack propagation[J]. International Journal for Numerical Methods in Engineering，2014，97(13)：986-1 010. " target="_blank">
[38]	ZHENG H，LIU F，LI C G. The MLS-based numerical manifold method with applications to crack analysis[J]. International Journal of Fracture，2014，190(2)：47-166. " target="_blank">
[39]	ZHENG H，LIU F，LI C G. Primal mixed solution to unconfined seepage flow in porous media with numerical manifold method[J]. Applied Mathematical Modelling，2015，39(2)：794-808. " target="_blank">
[41]	徐栋栋，郑宏，杨永涛. 线性无关高阶数值流形法[J]. 岩土工程学报，2014，36(3)：482-488.(XU Dongdong，ZHENG Hong，YANG Yongtao. Linearly independent higher-order numerical manifold method[J]. Chinese Journal of Geotechnical Engineering，2014，36(3)：482-488.(in Chinese))
[42]	郭朝旭，郑宏. 高阶数值流形方法中的线性相关问题研究[J]. 工程力学，2012，29(12)：228-232.(GUO Chaoxu，ZHENG Hong. Study on linear dependence problem in high-order numerical manifold method[J]. Engineering Mechanics，2012，29(12)：228-232.(in Chinese))
[43]	徐栋栋，郑宏. 数值流形法在处理强奇异性问题时的网格无关性[J]. 岩土力学，2014，35(8)：2 385-2 394.(XU Dongdong，ZHENG Hong. Mesh independence of numerical manifold method in treating strong singularity[J]. Rock and Soil Mechanics，2014，35(8)：2 385-2 394.(in Chinese)) " target="_blank">
[24]	SHI G H. Manifold method of material analysis[C]// Transactions of the Ninth Army Conference on Applied Mathematics and Computing. Minneapolish，Minncsoda，USA：[s. n.]，1992：51-76. " target="_blank">
[45]	XU D D，ZHENG H，XIA K W. New Numerical Quadrature of Integrand with Singularity of 1/r and its Application[J]. Applied Mechanics and Materials，2014，444：641-649. " target="_blank">
[46]	XU D D，ZHENG H. Mesh independence test of numerical manifold method in treating strong singularity[C]// Proceedings of 11th International Conference on Analysis of Discontinuous Deformation. Japan，Fukuoka：[s. n.]，2013，91-96. " target="_blank">
[49]	SIH G C. Energy-density concept in fracture mechanics[J]. Engineering Fracture Mechanics，1973，5(4)：1 037-1 040 " target="_blank">
[44]	徐栋栋，郑宏，夏开文，等. 高阶扩展数值流形法在裂纹扩展中的应用[J]. 岩石力学与工程学报，2014，33(7)：1 375-1 387.(XU Dongdong，ZHENG Hong，XIA Kaiwen，et al. Application of higher-order enriched numerical manifold method to crack propagation[J]. Chinese Journal of Rock Mechanics and Engineering，2014，33(7)：1 375-1 387.(in Chinese)) " target="_blank">
[48]	中国航空研究院主编. 应力强度因子手册(增订版)[M]. 北京：科学出版社，1993：356-357.(Chinese Aeronautical Establishment. Handbook of the stress intensity factor(revised edition)[M]. Beijing：Science Press，1993：356-357.(in Chinese)) " target="_blank">
[47]	ZHANG H H，LI L X，AN X M，et al. Numerical analysis of 2D crack propagation problems using the numerical manifold method[J]. Engineering Analysis with Boundary Elements，2010，34(1)：41-50.