断裂问题的插值型无单元伽辽金比例边界法与有限元法的耦合研究
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摘要
插值型无单元伽辽金比例边界法是一种只需在计算域的边界上采用插值型无单元伽辽金法离散且无需基本解的半解析数值方法,特别适用于求解包含无限域和奇异物理场的问题.本文提出了一种用于断裂分析的插值型无单元伽辽金比例边界法与有限元法的耦合分析方法,更好地发挥插值型无单元伽辽金比例边界法和有限元法各自的优势.裂尖周边一定范围的计算域采用插值型无单元伽辽金比例边界法模拟,而其余区域则采用有限元法模拟.在这两个区域内,分别采用各自相应的位移模式,两者相互独立.利用交界面两侧位移的连续条件,可以方便地建立耦合求解方程,简明有效,易于编程计算.最后给出了两个数值算例验证本文方法的有效性.
The interpolating element-free Galerkin scaled boundary method(IEFG-SBM) is a semi-analytical method which only requires to discretize the boundary by the interpolating element-free Galerkin(EFG) method without fundamental solution.This method is ideally suited to solve problems containing infinite domain and singular physical fields. This study develops a novel method that couples the IEFG-SBM and the finite element method(FEM) for crack analysis in order to take the full advantages of both IEFG-SBM and FEM. The IEFG-SBM is adopted to model the domain close to the crack tip and the FEM is adopted in the rest of the domain. The corresponding displacement interpolation models are employed for each sub-domain respectively. Through continuity conditions on the interface between IEFG-SBM sub-domain and FEM sub-domain, the coupled formula of the proposed method can be easily derived. The proposed method is simple, effective,and easy to be programmed. Finally, two numerical examples are presented to demonstrate the validity of the proposed method.
The interpolating element-free Galerkin scaled boundary method(IEFG-SBM) is a semi-analytical method which only requires to discretize the boundary by the interpolating element-free Galerkin(EFG) method without fundamental solution.This method is ideally suited to solve problems containing infinite domain and singular physical fields. This study develops a novel method that couples the IEFG-SBM and the finite element method(FEM) for crack analysis in order to take the full advantages of both IEFG-SBM and FEM. The IEFG-SBM is adopted to model the domain close to the crack tip and the FEM is adopted in the rest of the domain. The corresponding displacement interpolation models are employed for each sub-domain respectively. Through continuity conditions on the interface between IEFG-SBM sub-domain and FEM sub-domain, the coupled formula of the proposed method can be easily derived. The proposed method is simple, effective,and easy to be programmed. Finally, two numerical examples are presented to demonstrate the validity of the proposed method.
引文
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2 Yu T,Shi L.Determination of sharp V-notch stress intensity factors using the extended finite element method.J Strain Anal Eng Des,2012,47:95-103
3 Fan J L,Guo X L.Numerical simulation on elastic-plastic fatigue crack growth behavior(in Chinese).Chin J Mech,2015,51:33-40[樊俊铃,郭杏林.弹塑性疲劳裂纹扩展行为的数值模拟.机械工程学报,2015,51:33-40]
4 Xu D D,Zheng H,Yang Y T,et al.Multiple crack propagation based on the numerical manifold method(in Chinese).Acta Mech Sin,2015,47:471-481[徐栋栋,郑宏,杨永涛,等.多裂纹扩展的数值流形法.力学学报,2015,47:471-481]
5 Tian R,Wen L F.Recent progresses on improved XFEM(in Chinese).Chin J Comput Mech,2016,33:469-477[田荣,文龙飞.改进型XFEM进展.计算力学学报,2016,33:469-477]
6 Li L X,Wang T J.The extended finite element method and its applications-a review(in Chinese).Adv Mech,2005,35:5-20[李录贤,王铁军.扩展有限元法(XFEM)及其应用.力学进展,2005,35:5-20]
7 Li Q,Chen S,Luo X.Steady heat conduction analyses using an interpolating element-free Galerkin scaled boundary method.Appl Math Comput,2017,300:103-115
8 Chen S S,Tong G S,Wan Y.An interpolating element-free Galerkin scaled boundary method for the elasticity problem(in Chinese).Sci Sin-Phys Mech Astron,2017,47:034601[陈莘莘,童谷生,万云.弹性力学问题的插值型无单元伽辽金比例边界法.中国科学:物理学力学天文学,2017,47:034601]
9 Chen S S,Wang J.Analysis of interpolating element-free galerkin scaled boundary method for piezoelectric cracks(in Chinese).Chin J Mech,2017,53:53-59,67[陈莘莘,王娟.压电裂纹的插值型无单元伽辽金比例边界法分析.机械工程学报,2017,53:53-59,67]
10 Song C.A matrix function solution for the scaled boundary finite-element equation in statics.Comp Methods Appl Mech Eng,2004,193:2325-2356
11 Hu Z Q,Lin B,Wang Y,et al.A Hamiltonian-based derivation of scaled boundary finite element method for elasticity(in Chinese).Chin JComput Mech,2011,28:510-516[胡志强,林皋,王毅,等.基于Hamilton体系的弹性力学问题的比例边界有限元法.计算力学学报,2011,28:510-516]
12 Long X Y,Jiang C,Han X,et al.The scaled boundary finite element second order sensitivity design and fracture mechanics analysis(in Chinese).Chin J Solid Mech,2015,36:42-54[龙湘云,姜潮,韩旭,等.比例边界有限元二阶灵敏度设计及其断裂力学分析.固体力学学报,2015,36:42-54]
13 Zhong H,Mou H,Zhang W X.Modelling of crack propagation by finite fracture mechanics and scaled boundary finite element method(in Chinese).Chin J Comput Mech,2017,34:168-174[钟红,牟昊,张文宣.基于有限断裂法和比例边界有限元法的裂纹扩展模拟.计算力学学报,2017,34:168-174]
14 Belytschko T,Lu Y Y,Gu L.Element-free Galerkin methods.Int J Numer Meth Engng,1994,37:229-256
15 Qin Y X,Cheng Y M.Reproducing kernel particle boundary element-free method for temperature field problems(in Chinese).Chin J Mech,2008,44:95-100[秦义校,程玉民.温度场分析的重构核粒子边界无单元法.机械工程学报,2008,44:95-100]
16 Zhang Q D,Zhang B Y,Li B,et al.Element free Galerkin method coupled flow function for plane strain strip rolling process(in Chinese).Chin J Mech,2015,51:48-54[张清东,张勃洋,李博,等.板带轧制过程二维流函数-无网格伽辽金求解方法及其应用.机械工程学报,2015,51:48-54]
17 Wang J F,Sun F X,Cheng Y M.An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems.Chin Phys B,2012,21:090204
18 Wang J,Wang J,Sun F,et al.An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems.Int J Comput Methods,2013,10:1350043
19 Lancaster P,Salkauskas K.Surfaces generated by moving least squares methods.Math Comp,1981,37:141
20 Yin S D,Yin S,Zhou Y H.Coupled SBFEM and FEM for crack analysis based on virtual discontinuous surface method(in Chinese).Chin JComput Mech,2014,31:735-741,748[殷德胜,尹栓,周宜红.裂缝分析的比例边界有限元与有限元耦合的虚拟结构面模型.计算力学学报,2014,31:735-741,748]
21 Yang Z J,Wang X F,Yin D S,et al.A non-matching finite element-scaled boundary finite element coupled method for linear elastic crack propagation modelling.Comp Struct,2015,153:126-136
22 Tada H,Paris P C,Irwin R.The Stress Analysis of Cracks(Handbook).Hellertown:Del Research Corporation,1973
23 Kitagawa H,Yuuki R.Analysis of arbitrarily shaped crack in a finite plate using conformal mapping.First report:Construction of analysis procedure and its applicability.Trans Jpn Soc Mech Eng,1977,43:4354-4362
24 Tsang D K L,Oyadiji S O.Super singular element method for two-dimensional crack analysis.Proc R Soc A-Math Phys Eng Sci,2008,464:2629-2648
2 Yu T,Shi L.Determination of sharp V-notch stress intensity factors using the extended finite element method.J Strain Anal Eng Des,2012,47:95-103
3 Fan J L,Guo X L.Numerical simulation on elastic-plastic fatigue crack growth behavior(in Chinese).Chin J Mech,2015,51:33-40[樊俊铃,郭杏林.弹塑性疲劳裂纹扩展行为的数值模拟.机械工程学报,2015,51:33-40]
4 Xu D D,Zheng H,Yang Y T,et al.Multiple crack propagation based on the numerical manifold method(in Chinese).Acta Mech Sin,2015,47:471-481[徐栋栋,郑宏,杨永涛,等.多裂纹扩展的数值流形法.力学学报,2015,47:471-481]
5 Tian R,Wen L F.Recent progresses on improved XFEM(in Chinese).Chin J Comput Mech,2016,33:469-477[田荣,文龙飞.改进型XFEM进展.计算力学学报,2016,33:469-477]
6 Li L X,Wang T J.The extended finite element method and its applications-a review(in Chinese).Adv Mech,2005,35:5-20[李录贤,王铁军.扩展有限元法(XFEM)及其应用.力学进展,2005,35:5-20]
7 Li Q,Chen S,Luo X.Steady heat conduction analyses using an interpolating element-free Galerkin scaled boundary method.Appl Math Comput,2017,300:103-115
8 Chen S S,Tong G S,Wan Y.An interpolating element-free Galerkin scaled boundary method for the elasticity problem(in Chinese).Sci Sin-Phys Mech Astron,2017,47:034601[陈莘莘,童谷生,万云.弹性力学问题的插值型无单元伽辽金比例边界法.中国科学:物理学力学天文学,2017,47:034601]
9 Chen S S,Wang J.Analysis of interpolating element-free galerkin scaled boundary method for piezoelectric cracks(in Chinese).Chin J Mech,2017,53:53-59,67[陈莘莘,王娟.压电裂纹的插值型无单元伽辽金比例边界法分析.机械工程学报,2017,53:53-59,67]
10 Song C.A matrix function solution for the scaled boundary finite-element equation in statics.Comp Methods Appl Mech Eng,2004,193:2325-2356
11 Hu Z Q,Lin B,Wang Y,et al.A Hamiltonian-based derivation of scaled boundary finite element method for elasticity(in Chinese).Chin JComput Mech,2011,28:510-516[胡志强,林皋,王毅,等.基于Hamilton体系的弹性力学问题的比例边界有限元法.计算力学学报,2011,28:510-516]
12 Long X Y,Jiang C,Han X,et al.The scaled boundary finite element second order sensitivity design and fracture mechanics analysis(in Chinese).Chin J Solid Mech,2015,36:42-54[龙湘云,姜潮,韩旭,等.比例边界有限元二阶灵敏度设计及其断裂力学分析.固体力学学报,2015,36:42-54]
13 Zhong H,Mou H,Zhang W X.Modelling of crack propagation by finite fracture mechanics and scaled boundary finite element method(in Chinese).Chin J Comput Mech,2017,34:168-174[钟红,牟昊,张文宣.基于有限断裂法和比例边界有限元法的裂纹扩展模拟.计算力学学报,2017,34:168-174]
14 Belytschko T,Lu Y Y,Gu L.Element-free Galerkin methods.Int J Numer Meth Engng,1994,37:229-256
15 Qin Y X,Cheng Y M.Reproducing kernel particle boundary element-free method for temperature field problems(in Chinese).Chin J Mech,2008,44:95-100[秦义校,程玉民.温度场分析的重构核粒子边界无单元法.机械工程学报,2008,44:95-100]
16 Zhang Q D,Zhang B Y,Li B,et al.Element free Galerkin method coupled flow function for plane strain strip rolling process(in Chinese).Chin J Mech,2015,51:48-54[张清东,张勃洋,李博,等.板带轧制过程二维流函数-无网格伽辽金求解方法及其应用.机械工程学报,2015,51:48-54]
17 Wang J F,Sun F X,Cheng Y M.An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems.Chin Phys B,2012,21:090204
18 Wang J,Wang J,Sun F,et al.An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems.Int J Comput Methods,2013,10:1350043
19 Lancaster P,Salkauskas K.Surfaces generated by moving least squares methods.Math Comp,1981,37:141
20 Yin S D,Yin S,Zhou Y H.Coupled SBFEM and FEM for crack analysis based on virtual discontinuous surface method(in Chinese).Chin JComput Mech,2014,31:735-741,748[殷德胜,尹栓,周宜红.裂缝分析的比例边界有限元与有限元耦合的虚拟结构面模型.计算力学学报,2014,31:735-741,748]
21 Yang Z J,Wang X F,Yin D S,et al.A non-matching finite element-scaled boundary finite element coupled method for linear elastic crack propagation modelling.Comp Struct,2015,153:126-136
22 Tada H,Paris P C,Irwin R.The Stress Analysis of Cracks(Handbook).Hellertown:Del Research Corporation,1973
23 Kitagawa H,Yuuki R.Analysis of arbitrarily shaped crack in a finite plate using conformal mapping.First report:Construction of analysis procedure and its applicability.Trans Jpn Soc Mech Eng,1977,43:4354-4362
24 Tsang D K L,Oyadiji S O.Super singular element method for two-dimensional crack analysis.Proc R Soc A-Math Phys Eng Sci,2008,464:2629-2648