基于中心差分方案的显式三维非连续变形分析法
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摘要
为提高传统三维非连续变形分析法(DDA)求解大规模非连续问题时的效率,提出一种基于中心差分方案的显式三维DDA,求解过程简单省时且更便于实现高效的并行计算。基于最先侵入和最短退出路径原则,推导时间步内2种基本接触侵入形式的侵入点对位置的估算公式,明确侵入信息的参数种类,从而保证时域离散下块体间接触约束的时空连贯性、降低接触问题的非线性、提高接触力学计算的精度并减少重复计算量。数值算例表明,显式三维DDA具有足够的计算精度且计算效率更高,实现并行计算后有望应用于大型岩体工程非连续数值分析。
An explicit three-dimensional discontinuous deformation analysis(3 D DDA) method based on the central difference scheme is proposed to improve the computational efficiency of the traditional 3 D DDA method in solving large-scale discontinuous problems. With the explicit 3 D DDA method,it is not only easier and faster to obtain the solutions,but also more convenient to achieve the efficient parallel computing. Based on the first entrance and shortest exit principle,formulae are derived to estimate the positions of entrance points during the entrance process regarding two basic entrance mode and the entrance information is determined. These contact treatment measures ensure the computational accuracy and the tempo-spatial coherence of the contact constraints. Numerical examples demonstrate that the explicit 3 D DDA has sufficient computational accuracy and higher computational efficiency. The explicit 3 D DDA is promising for the discontinuous numerical analysis in large scale rock engineering after achieving parallel computing.
An explicit three-dimensional discontinuous deformation analysis(3 D DDA) method based on the central difference scheme is proposed to improve the computational efficiency of the traditional 3 D DDA method in solving large-scale discontinuous problems. With the explicit 3 D DDA method,it is not only easier and faster to obtain the solutions,but also more convenient to achieve the efficient parallel computing. Based on the first entrance and shortest exit principle,formulae are derived to estimate the positions of entrance points during the entrance process regarding two basic entrance mode and the entrance information is determined. These contact treatment measures ensure the computational accuracy and the tempo-spatial coherence of the contact constraints. Numerical examples demonstrate that the explicit 3 D DDA has sufficient computational accuracy and higher computational efficiency. The explicit 3 D DDA is promising for the discontinuous numerical analysis in large scale rock engineering after achieving parallel computing.
引文
[1]SHI G H.Discontinuous deformation analysis:a new numerical model for the statics and dynamics of block systems[Ph.D.Thesis][D].Berkeley:University of California,1988.
[2]栾茂田,黎勇,杨庆.非连续变形计算力学模型在岩体边坡稳定性分析中的应用[J].岩石力学与工程学报,2000,19(3):289–294.(LUAN Maotian,LI Yong,YANG Qing.Discontinuous deformation computational mechanics model and its application in stability analysis of rock slope[J].Chinese Journal of Rock Mechanics and Engineering,2000,19(3):289–294.(in Chinese))
[3]殷坤龙,姜清辉,汪洋.新滩滑坡运动全过程的非连续变形分析与仿真模拟[J].岩石力学与工程学报,2002,21(7):959–962.(YIN Kunlong,JIANG Qinghui,WANG Yang.Numerical simulation on the movement process of Xintan landslide by DDA method[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(7):959–962.(in Chinese))
[4]付晓东,盛谦,张勇慧.水电站地下洞室群分步开挖的非连续变形分析[J].岩土力学,2013,34(2):568–574.(FU Xiaodong,SHENG Qian,ZHANG Yonghui.Stepwise excavation process of underground caverns of hydropower station using DDA[J].Rock and Soil Mechanics,2013,34(2):568–574.(in Chinese))
[5]方杰,秦小军,蔡永建,等.基于DDA和FEM的砌体结构震动分析[J].大地测量与地球动力学,2016,36(6):520–524.(FANG Jie,QIN Xiaojun,CAI Yongjian,et al.Earthquake response analysis of masonry structural by discontinuous deformation analysis[J].Journal of Geodesy and Geodynamics,2016,36(6):520–524.(in Chinese))
[6]SHI G H.Three dimensional discontinuous deformation analysis[C]//Proceedings of the Fourth International Conference on Discontinuous Deformation Analysis(ICADD–4).Scotland:[s.n.],2001:1–21.
[7]SHI G H.Producing joint polygons,cutting rock blocks and finding removable blocks for general free surfaces[J].Chinese Journal of Rock Mechanics and Engineering,2006;25(11):2 161–2 170.
[8]SHI G H.Contact theory[J].Science China-Technological Sciences,2015;58(5):1 450–1 498.
[9]姜清辉,周创兵,罗先启,等.三维DDA与有限元的耦合分析方法及其应用[J].岩土工程学报,2006,28(8):998–1 001.(JIANG Qinghui,ZHOU Chuangbin,LUO Xianqi,et al.Coupling method of three-dimensional DDA-FEM and its application[J].Chinese Journal of Geotechnical Engineering,2006,28(8):998–1 001.(in Chinese))
[10]ZHU H H,WU W,CHEN J Q,et al.Integration of three dimensional discontinuous deformation analysis(DDA)with binocular photogrammetry for stability analysis of tunnels in blocky rock mass[J].Tunnelling and Underground Space Technology,2016,51(1):30–40.
[11]ZHAO G F,LIAN J J,RUSSELL A R,et al.Three-dimensional DDA and DLSM coupled approach for rock cutting and rock penetration[J].International Journal of Geomechanics,2016,17(5):E4016015.
[12]WANG W,CHEN G Q,ZHANG H,et al.Analysis of landslide-generated impulsive waves using a coupled DDA-SPH method[J].Engineering Analysis with Boundary Elements,2016,64:267–277.
[13]ZHANG H,LIU S G,WANG W,et al.A new DDA model for kinematic analyses of rockslides on complex 3D terrain[J].Bulletin of Engineering Geology and the Environment,2018,77(2):555–571.
[14]WU W,ZHU H H,LIN J S,et al.Tunnel stability assessment by 3D DDA-key block analysis[J].Tunnelling and Underground Space Technology,2018,71:210–214.
[15]LIU S G,LI Z J,ZHANG H,et al.A 3D DDA damage analysis of brick masonry buildings under the impact of boulders in mountainous areas[J].Journal of Mountain Science,2018,15(3):657–671.
[16]姜清辉,周创兵,漆祖芳.基于Newmark积分方案的DDA方法[J].岩石力学与工程学报,2009,28(增1):2 778–2 783.(JIANG Qinghui,ZHOU Chuangbing,QI Zufang.Discontinuous deformation analysis method based on Newmark integration algorithm[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(Supp.1):2 778–2 783.(in Chinese))
[17]付晓东,盛谦,张勇慧.非连续变形分析方法中的阻尼问题研究[J].岩土力学,2015,36(7):2 057–2 062.(FU Xiaodong,SHENG Qian,ZHANG Yonghui.Damping problem in discontinuous deformation analysis method[J].Rock and Soil Mechanics,2015,36(7):2 057–2 062.(in Chinese))
[18]DOOLIN D M,SITAR N.Time integration in discontinuous deformation analysis[J].Journal of Engineering Mechanics,2004,130(3):249–258.
[19]江巍,郑宏.非连续变形分析方法中人为参数的影响[J].岩土力学,2007,28(12):2 603–2 606.(JIANG Wei,ZHENG Hong.Influences of artificial parameters in DDA method[J].Rock and Soil Mechanics,2007,28(12):2 603–2 606.(in Chinese))
[20]邬爱清,冯细霞,卢波.非连续变形分析中时间步及弹簧刚度取值研究[J].岩土力学,2015,36(3):891–897.(WU Aiqing,FENG Xixia,LU Bo.Parametric research on time step and spring stiffness in DDA[J].Rock and Soil Mechanics,2015,36(3):891–897.(in Chinese))
[21]YAGODA-BIRAN G,HATZOR Y H.Benchmarking the numerical discontinuous deformation analysis method[J].Computers and Geotechnics,2016,71(1):30–46.
[22]BAKUN-MAZOR D,HATZOR Y H,GLASER S D.Dynamic sliding of tetrahedral wedge:The role of interface friction[J].International Journal for Numerical and Analytical Methods in Geomechanics,2012,36(2):327–343.
[23]CUNDALL P A.A computer model for simulating progressive large scale movements in blocky rock systems[C]//Proceeding of the Symposium of the International Society of Rock Mechanics.Rotterdam:[s.n.],1971:11–18.
[24]CUNDALL P A.UDEC:A Generalized distinct element program for modelling jointed rock[R].U.S.Army:[s.n.],1980.
[25]CUNDALL P A.Formulation of a three-dimensional distinct element model—part I:a scheme to detect and represent contacts in a system composed of many polyhedral blocks[J].International Journal of Rock Mechanics and Mining Sciences,1988,25(3):107–116.
[26]刘凯欣,高凌天.离散元法研究的评述[J].力学进展,2003,33(4):483–490.(LIU Kaixin,GAO Lingtian.A review on the discrete element method[J].Advances in Mechanics,2003,33(4):483–490.(in Chinese))
[27]QU X L,FU G Y,MA G W.An explicit time integration scheme of numerical manifold method[J].Engineering Analysis with Boundary Elements,2014,48:53–62.
[28]QU X L,WANG Y,FU G Y.Efficiency and accuracy verification of the explicit numerical manifold method for dynamic problems[J].Rock Mechanics and Rock Engineering,2015,48:1 131–1 142.
[29]QU X L,MA G W,FU G Y.Coupled time domain integration algorithm for numerical manifold method[J].International Journal of Geomechanics,2017,17(5):E4016005.
[30]QU X L,QI C,FU G Y.Effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method[J].International Journal for Numerical Methods in Engineering,2017,112:1 614–1 635.
[31]SHI G H.Simplex integration for manifold method,FEM,DDA and analytical analysis[C]//Proceedings of the First International Forum on Discontinuous Deformation Analysis(DDA)and Simulations of Discontinuous Media.California:[s.n.],1996:205–262.
[32]WU J H.Compatible algorithm for integrations on a block domain of any shape for three-dimensional discontinuous deformation analysis[J].Computers and Geotechnics,2010,37:153–163.
[33]姜清辉,丰定祥.三维非连续变形分析方法中的锚杆模拟[J].岩土力学,2001,22(2):176–178.(JIANG Qinghui,FENG Dingxiang.Modeling of rockbolts in three-dimensional discontinuous deformation analysis[J].Rock and Soil Mechanics,2001,22(2):176–178.(in Chinese))
[34]姜清辉,杨文柱,丰定祥.三维块体系统非连续变形正分析模型[J].岩石力学与工程学报,2002,21(12):1 765–1 769.(JIANG Qinghui,YOUNG Wenzhu,FENG Dingxiang.Three-dimensional discontinuous deformation analysis for forward modeling[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(12):1 765–1 769.(in Chinese))
[35]姜清辉,朱传云,丰定祥.三维非连续变形分析方法的变分原理及应用[J].岩石力学与工程学报,2002,21(增):2 090–2 093.(JIANG Qinghui,ZHU Chuanyun,FENG Dingxiang.Variational principle and application of three-dimensional discontinuous deformation analysis method[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(Supp.):2 090–2 093.(in Chinese))
[36]ZHANG H,LIU S G,HAN Z,et al.A new algorithm to identify contact types between arbitrarily shaped polyhedral blocks for three-dimensional discontinuous deformation analysis[J].Computers and Geotechnics,2016,80:1–15.
[37]ZHANG H,CHEN G Q,ZHENG L,et al.Detection of contacts between three-dimensional polyhedral blocks for discontinuous deformation analysis[J].International Journal of Rock Mechanics and Mining Sciences,2015,78(1):57–73.
[38]FAN H,HE S M.An angle-based method dealing with vertex–vertex contact in the two-dimensional discontinuous deformation analysis(DDA)[J].Rock Mechanics and Rock Engineering,2015,48(5):2 031–2 043.
[39]ZHANG H,LIU S G,ZHENG L,et al.Extensions of edge-to-edge contact model in three-dimensional discontinuous deformation analysis for friction analysis[J].Computers and Geotechnics,2016,71:261–275.
[40]SHI G H.Manifold method[C]//Proceedings of the First International Forum on Discontinuous Deformation Analysis(DDA)and Simulations of Discontinuous Media.California:[s.n.],1996:52–204.
[41]ZHU H H,WU W,ZHUANG X Y,et al.Method for estimating normal contact parameters in collision modeling using discontinuous deformation analysis[J].International Journal of Geomechanics,2017,17(5):E4016011.
[2]栾茂田,黎勇,杨庆.非连续变形计算力学模型在岩体边坡稳定性分析中的应用[J].岩石力学与工程学报,2000,19(3):289–294.(LUAN Maotian,LI Yong,YANG Qing.Discontinuous deformation computational mechanics model and its application in stability analysis of rock slope[J].Chinese Journal of Rock Mechanics and Engineering,2000,19(3):289–294.(in Chinese))
[3]殷坤龙,姜清辉,汪洋.新滩滑坡运动全过程的非连续变形分析与仿真模拟[J].岩石力学与工程学报,2002,21(7):959–962.(YIN Kunlong,JIANG Qinghui,WANG Yang.Numerical simulation on the movement process of Xintan landslide by DDA method[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(7):959–962.(in Chinese))
[4]付晓东,盛谦,张勇慧.水电站地下洞室群分步开挖的非连续变形分析[J].岩土力学,2013,34(2):568–574.(FU Xiaodong,SHENG Qian,ZHANG Yonghui.Stepwise excavation process of underground caverns of hydropower station using DDA[J].Rock and Soil Mechanics,2013,34(2):568–574.(in Chinese))
[5]方杰,秦小军,蔡永建,等.基于DDA和FEM的砌体结构震动分析[J].大地测量与地球动力学,2016,36(6):520–524.(FANG Jie,QIN Xiaojun,CAI Yongjian,et al.Earthquake response analysis of masonry structural by discontinuous deformation analysis[J].Journal of Geodesy and Geodynamics,2016,36(6):520–524.(in Chinese))
[6]SHI G H.Three dimensional discontinuous deformation analysis[C]//Proceedings of the Fourth International Conference on Discontinuous Deformation Analysis(ICADD–4).Scotland:[s.n.],2001:1–21.
[7]SHI G H.Producing joint polygons,cutting rock blocks and finding removable blocks for general free surfaces[J].Chinese Journal of Rock Mechanics and Engineering,2006;25(11):2 161–2 170.
[8]SHI G H.Contact theory[J].Science China-Technological Sciences,2015;58(5):1 450–1 498.
[9]姜清辉,周创兵,罗先启,等.三维DDA与有限元的耦合分析方法及其应用[J].岩土工程学报,2006,28(8):998–1 001.(JIANG Qinghui,ZHOU Chuangbin,LUO Xianqi,et al.Coupling method of three-dimensional DDA-FEM and its application[J].Chinese Journal of Geotechnical Engineering,2006,28(8):998–1 001.(in Chinese))
[10]ZHU H H,WU W,CHEN J Q,et al.Integration of three dimensional discontinuous deformation analysis(DDA)with binocular photogrammetry for stability analysis of tunnels in blocky rock mass[J].Tunnelling and Underground Space Technology,2016,51(1):30–40.
[11]ZHAO G F,LIAN J J,RUSSELL A R,et al.Three-dimensional DDA and DLSM coupled approach for rock cutting and rock penetration[J].International Journal of Geomechanics,2016,17(5):E4016015.
[12]WANG W,CHEN G Q,ZHANG H,et al.Analysis of landslide-generated impulsive waves using a coupled DDA-SPH method[J].Engineering Analysis with Boundary Elements,2016,64:267–277.
[13]ZHANG H,LIU S G,WANG W,et al.A new DDA model for kinematic analyses of rockslides on complex 3D terrain[J].Bulletin of Engineering Geology and the Environment,2018,77(2):555–571.
[14]WU W,ZHU H H,LIN J S,et al.Tunnel stability assessment by 3D DDA-key block analysis[J].Tunnelling and Underground Space Technology,2018,71:210–214.
[15]LIU S G,LI Z J,ZHANG H,et al.A 3D DDA damage analysis of brick masonry buildings under the impact of boulders in mountainous areas[J].Journal of Mountain Science,2018,15(3):657–671.
[16]姜清辉,周创兵,漆祖芳.基于Newmark积分方案的DDA方法[J].岩石力学与工程学报,2009,28(增1):2 778–2 783.(JIANG Qinghui,ZHOU Chuangbing,QI Zufang.Discontinuous deformation analysis method based on Newmark integration algorithm[J].Chinese Journal of Rock Mechanics and Engineering,2009,28(Supp.1):2 778–2 783.(in Chinese))
[17]付晓东,盛谦,张勇慧.非连续变形分析方法中的阻尼问题研究[J].岩土力学,2015,36(7):2 057–2 062.(FU Xiaodong,SHENG Qian,ZHANG Yonghui.Damping problem in discontinuous deformation analysis method[J].Rock and Soil Mechanics,2015,36(7):2 057–2 062.(in Chinese))
[18]DOOLIN D M,SITAR N.Time integration in discontinuous deformation analysis[J].Journal of Engineering Mechanics,2004,130(3):249–258.
[19]江巍,郑宏.非连续变形分析方法中人为参数的影响[J].岩土力学,2007,28(12):2 603–2 606.(JIANG Wei,ZHENG Hong.Influences of artificial parameters in DDA method[J].Rock and Soil Mechanics,2007,28(12):2 603–2 606.(in Chinese))
[20]邬爱清,冯细霞,卢波.非连续变形分析中时间步及弹簧刚度取值研究[J].岩土力学,2015,36(3):891–897.(WU Aiqing,FENG Xixia,LU Bo.Parametric research on time step and spring stiffness in DDA[J].Rock and Soil Mechanics,2015,36(3):891–897.(in Chinese))
[21]YAGODA-BIRAN G,HATZOR Y H.Benchmarking the numerical discontinuous deformation analysis method[J].Computers and Geotechnics,2016,71(1):30–46.
[22]BAKUN-MAZOR D,HATZOR Y H,GLASER S D.Dynamic sliding of tetrahedral wedge:The role of interface friction[J].International Journal for Numerical and Analytical Methods in Geomechanics,2012,36(2):327–343.
[23]CUNDALL P A.A computer model for simulating progressive large scale movements in blocky rock systems[C]//Proceeding of the Symposium of the International Society of Rock Mechanics.Rotterdam:[s.n.],1971:11–18.
[24]CUNDALL P A.UDEC:A Generalized distinct element program for modelling jointed rock[R].U.S.Army:[s.n.],1980.
[25]CUNDALL P A.Formulation of a three-dimensional distinct element model—part I:a scheme to detect and represent contacts in a system composed of many polyhedral blocks[J].International Journal of Rock Mechanics and Mining Sciences,1988,25(3):107–116.
[26]刘凯欣,高凌天.离散元法研究的评述[J].力学进展,2003,33(4):483–490.(LIU Kaixin,GAO Lingtian.A review on the discrete element method[J].Advances in Mechanics,2003,33(4):483–490.(in Chinese))
[27]QU X L,FU G Y,MA G W.An explicit time integration scheme of numerical manifold method[J].Engineering Analysis with Boundary Elements,2014,48:53–62.
[28]QU X L,WANG Y,FU G Y.Efficiency and accuracy verification of the explicit numerical manifold method for dynamic problems[J].Rock Mechanics and Rock Engineering,2015,48:1 131–1 142.
[29]QU X L,MA G W,FU G Y.Coupled time domain integration algorithm for numerical manifold method[J].International Journal of Geomechanics,2017,17(5):E4016005.
[30]QU X L,QI C,FU G Y.Effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method[J].International Journal for Numerical Methods in Engineering,2017,112:1 614–1 635.
[31]SHI G H.Simplex integration for manifold method,FEM,DDA and analytical analysis[C]//Proceedings of the First International Forum on Discontinuous Deformation Analysis(DDA)and Simulations of Discontinuous Media.California:[s.n.],1996:205–262.
[32]WU J H.Compatible algorithm for integrations on a block domain of any shape for three-dimensional discontinuous deformation analysis[J].Computers and Geotechnics,2010,37:153–163.
[33]姜清辉,丰定祥.三维非连续变形分析方法中的锚杆模拟[J].岩土力学,2001,22(2):176–178.(JIANG Qinghui,FENG Dingxiang.Modeling of rockbolts in three-dimensional discontinuous deformation analysis[J].Rock and Soil Mechanics,2001,22(2):176–178.(in Chinese))
[34]姜清辉,杨文柱,丰定祥.三维块体系统非连续变形正分析模型[J].岩石力学与工程学报,2002,21(12):1 765–1 769.(JIANG Qinghui,YOUNG Wenzhu,FENG Dingxiang.Three-dimensional discontinuous deformation analysis for forward modeling[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(12):1 765–1 769.(in Chinese))
[35]姜清辉,朱传云,丰定祥.三维非连续变形分析方法的变分原理及应用[J].岩石力学与工程学报,2002,21(增):2 090–2 093.(JIANG Qinghui,ZHU Chuanyun,FENG Dingxiang.Variational principle and application of three-dimensional discontinuous deformation analysis method[J].Chinese Journal of Rock Mechanics and Engineering,2002,21(Supp.):2 090–2 093.(in Chinese))
[36]ZHANG H,LIU S G,HAN Z,et al.A new algorithm to identify contact types between arbitrarily shaped polyhedral blocks for three-dimensional discontinuous deformation analysis[J].Computers and Geotechnics,2016,80:1–15.
[37]ZHANG H,CHEN G Q,ZHENG L,et al.Detection of contacts between three-dimensional polyhedral blocks for discontinuous deformation analysis[J].International Journal of Rock Mechanics and Mining Sciences,2015,78(1):57–73.
[38]FAN H,HE S M.An angle-based method dealing with vertex–vertex contact in the two-dimensional discontinuous deformation analysis(DDA)[J].Rock Mechanics and Rock Engineering,2015,48(5):2 031–2 043.
[39]ZHANG H,LIU S G,ZHENG L,et al.Extensions of edge-to-edge contact model in three-dimensional discontinuous deformation analysis for friction analysis[J].Computers and Geotechnics,2016,71:261–275.
[40]SHI G H.Manifold method[C]//Proceedings of the First International Forum on Discontinuous Deformation Analysis(DDA)and Simulations of Discontinuous Media.California:[s.n.],1996:52–204.
[41]ZHU H H,WU W,ZHUANG X Y,et al.Method for estimating normal contact parameters in collision modeling using discontinuous deformation analysis[J].International Journal of Geomechanics,2017,17(5):E4016011.