Explicit Discontinuous Deformation Analysis Method with Lumped Mass Matrix for Highly Discrete Block System
Publication: International Journal of Geomechanics
Volume 18, Issue 9
Abstract
In the traditional discontinuous deformation analysis (DDA) method, the implicit time integration scheme is used to integrate equations of motion for modeling the mechanical behavior of a highly discrete rock block system. This requires that global equations be constantly solved. Hence, the computational efficiency of the traditional DDA method will decrease, especially when large-scale discontinuous problems are involved. Based on the explicit time integration scheme, an explicit version of the DDA (EDDA) method is proposed to improve computational efficiency of the traditional DDA method. Since a lumped mass matrix is used, there is no need to assemble global mass and stiffness matrices. More importantly, solving large-scale simultaneous algebraic equations can be avoided. The open–close iteration, which can assure the correct arrangement of constraints, is kept in the EDDA method. In addition, the simplex integration method, which is capable of conducting exact integration over an arbitrarily shaped block, is employed. Two numerical examples, including a sliding problem with an analytical solution and an underground cavern, are solved. The numerical results indicate the accuracy and robustness of the proposed EDDA method.
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Acknowledgments
This study was supported by the National Natural Science Foundation of China, under Grant 51609240, 11172313, and 51538001, and by the National Basic Research Program of China (973 Program), under Grant 2014CB047100.
References
Antolini, F., M. Barla, G. Gigli, and A. Giorgetti. 2016. “Combined finite–discrete numerical modeling of runout of the Torgiovannetto di Assisi rockslide in central Italy.” Int. J. Geomech. 16 (6): 04016019. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000646.
Bao, H. R., G. Yagoda-Biran, and Y. H. Hatzor. 2013. “Site response analysis with two-dimensional numerical discontinuous deformation analysis method.” Earthquake Eng. Struct. Dyn. 43 (2): 225–246. https://doi.org/10.1002/eqe.2340.
Bao, H. R., Z. Y. Zhao, and Q. Tian. 2014. “On the implementation of augmented Lagrangian method in the two-dimensional discontinuous deformation analysis.” Int. J. Numer. Anal. Meth. Geomech. 38 (6): 551–571. https://doi.org/10.1002/nag.2217.
Barla, M., G. Piovano, and G. Grasselli. 2012. “Rock slide simulation with the combined finite-discrete element method.” Int. J. Geomech. 12 (6): 711–721. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000204.
Beskos, D. E. 1997. “Boundary element methods in dynamic analysis: Part II (1986–1996).” Appl. Mech. Rev. 50 (3): 149–197. https://doi.org/10.1115/1.3101695.
Beyabanaki, S., A. Jafari, and M. R. Yeung. 2010. “High-order three-dimensional discontinuous deformation analysis (3-D DDA).” Int. J. Numer. Meth. Bio. 26 (12): 1522–1547.
Cai, Y. G., T. He, and R. Wang. 2000. “Numerical simulation of dynamic process of the Tangshan Earthquake by a new method—LDDA.” Pure. Appl. Geophys. 157 (11–12): 2083–2104. https://doi.org/10.1007/PL00001076.
Chen, G., M. He, and F. Fan. 2018. “Rock burst analysis using DDA numerical simulation.” Int. J. Geomech. 18 (3): 04018001. https://doi.org/10.1061/(ASCE)GM.1943-5622.0001055.
Chen, G. Q., K. Zen, Y. Ohnishi, and K. Kasama. 2001. “Extension of manifold method and its application.” In Proc., 4th International Conf. on Analysis of Discontinuous Deformation (ICADD-4), edited by N. Bićanić, 439–450. Glasgow, UK: International Conference on Analysis of Discontinuous Deformation.
Chen, N., J. Kemeny, Q. H. Jiang, and Z. W. Pan. 2017. “Automatic extraction of blocks from 3D point clouds of fractured rock.” Comput. Geosci. 109: 149–161. https://doi.org/10.1016/j.cageo.2017.08.013.
Cheng, Y. M., and Y. H. Zhang. 2000. “Rigid body rotation and block internal discretization in DDA analysis.” Int. J. Numer. Anal. Meth. Geomech. 24 (6): 567–578. https://doi.org/10.1002/(SICI)1096-9853(200005)24:6%3C567::AID-NAG83%3E3.0.CO;2-N.
Cheng, Y. M., and Y. H. Zhang. 2002. “Coupling of FEM and DDA Methods.” Int. J. Geomech. 2 (4): 503–517. https://doi.org/10.1061/(ASCE)1532-3641(2002)2:4(503).
Choo, L. Q., Z. Zhao, H. Chen, and Q. Tian. 2016. “Hydraulic fracturing modeling using the discontinuous deformation analysis (DDA) method.” Comput. Geotech. 76: 12–22. https://doi.org/10.1016/j.compgeo.2016.02.011.
Cundall, P. A. 1971. “A computer model for simulating progressive, large-scale movements in blocky rock systems.” In Vol. 1 of Proc., Symp. of the Int. Society of Rock Mechanics, 1–8. Nancy, France: International Society of Rock Mechanics.
Desai, C. S., M. M. Zaman, J. G. Lightner, and H. J. Siriwardane. 1984. “Thin-layer element for interfaces and joints.” Int. J. Numer. Anal. Meth. Geomech. 8 (1): 19–43. https://doi.org/10.1002/nag.1610080103.
Dong, X., A. Wu, and F. Ren. 1996. “A preliminary application of discontinuous deformation analysis (DDA) to the Three Gorges Dam project.” In Proc., First Int. Forum on Discontinuous Deformation Analysis (DDA) and Simulations of Discontinuous Media, edited by M. R. Salami, and D. Banks, 310–317. Albuquerque, NM: TSI Press.
Doolin, D. M., and N. Sitar. 2004. “Time integration in discontinuous deformation analysis.” J. Eng. Mech. 130 (3): 249–258. https://doi.org/10.1061/(ASCE)0733-9399(2004)130:3(249).
Elmo, D., D. Stead, E. Eberhardt, and A. Vyazmensky. 2013. “Applications of finite/discrete element modeling to rock engineering problems.” Int. J. Geomech. 13 (5): 565–580. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000238.
Fan, L. F., X. W. Yi, and G. W. Ma. 2013. “Numerical manifold method (NMM) simulation of stress wave propagation through fractured rock mass.” Int. J. Appl. Mech. 5 (2): 1350022. https://doi.org/10.1142/S1758825113500221.
Fries, T. P., and T. Belytschko. 2010. “The extended/generalized finite element method: An overview of the method and its applications.” Int. J. Numer. Methods. Eng. 84 (3): 253–304.
Goodman, R. E., and St. John. 1977. “Chapter 4: Finite element analysis for discontinuous rocks.” In Numerical methods in geotechnical engineering, edited by C. S. Desai and J. T. Christian, 148–175. New York: McGraw-Hill Book Company.
Grayeli, R., and K. Hatami. 2008. “Implementation of the finite element method in the three-dimensional discontinuous deformation analysis (3D-DDA).” Int. J. Numer. Anal. Meth. Geomech. 32 (15): 1883–1902. https://doi.org/10.1002/nag.704.
Hsiung, S. M. 2001. “Discontinuous deformation analysis (DDA) with nth order polynomial displacement functions.” In Proc., 38th US Rock Mechanics Symp.: Rock Mechanics in the National Interest, edited by D. Elsworth, J. P. Tinucci, and K. A. Heasley, 1413–1420. Washington, DC: American Rock Mechanics Association.
Jiang, Q. H., Y. F. Chen, C. B. Zhou, and M. R. Yeung. 2013. “Kinetic energy dissipation and convergence criterion of discontinuous deformations analysis (DDA) for geotechnical engineering.” Rock Mech. Rock Eng. 46 (6): 1443–1460. https://doi.org/10.1007/s00603-012-0356-5.
Jiang, Q. H., and M. R. Yeung. 2004. “A model of point-to-face contact for three-dimensional discontinuous deformation analysis.” Rock Mech. Rock Eng. 37 (2): 95–116. https://doi.org/10.1007/s00603-003-0008-x.
Jiang, Q. H., and C. B. Zhou. 2017. “A rigorous solution for the stability of polyhedral rock blocks.” Comput. Geotech. 90: 190–201. https://doi.org/10.1016/j.compgeo.2017.06.012.
Jiang, W., and H. Zheng. 2015. “An efficient remedy for the false volume expansion of DDA when simulating large rotation.” Comput. Geotech. 70: 18–23. https://doi.org/10.1016/j.compgeo.2015.07.008.
Jiao, Y. Y., H. Q. Zhang, X. L. Zhang, H. B. Li, and Q. H. Jiang. 2015. “A two-dimensional coupled hydromechanical discontinuum model for simulating rock hydraulic fracturing.” Int. J. Numer. Anal. Meth. Geomech. 39 (5): 457–481. https://doi.org/10.1002/nag.2314.
Katona, M. G. 1983. “A simple contact–friction interface element with applications to buried culverts.” Int. J. Numer. Anal. Meth. Geomech. 7 (3): 371–384. https://doi.org/10.1002/nag.1610070308.
Ke, T. C. 1995. “Modification of DDA with respect to rigid body rotation.” In Proc., 1st Int. Conf. on Analysis of Discontinuous Deformation, edited by J. C. Li, C. Y. Wang, and J. Sheng, 260–273. Chungli, Taiwan, China: National Central Univ.
Khan, M. S. 2010. “Investigation of discontinuous deformation analysis for application in jointed rock masses.” Ph.D. thesis, Department of Civil Engineering, Univ. of Toronto.
Kottenstette, J. T. 1999. “DDA analysis of the RCC modification for Pueblo Dam.” In Proc., ICADD-3: Third Int. Conf. on Analysis of Discontinuous Deformation-From Theory to Practice, edited by B. Amadei, 127–132. Washington DC: American Rock Mechanics Association, Balkema.
Langston, P. A., U. Tüzün, and D. M. Heyes. 1995. “Discrete element simulation of granular flow in 2D and 3D hoppers: Dependence of discharge rate and wall stress on particle interactions.” Chem. Eng. Sci. 50 (6): 967–987. https://doi.org/10.1016/0009-2509(94)00467-6.
Lin, C. T., B. Amadei, J. Jung, and J. Dwyer. 1996. “Extensions of discontinuous deformation analysis for jointed rock masses.” Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 33 (7): 671–694. https://doi.org/10.1016/0148-9062(96)00016-2.
Ma, G., W. Zhou, X. L. Chang, and W. Yuan. 2014. “Combined FEM/DEM modeling of triaxial compression tests for rockfills with polyhedral particles.” Int. J. Geomech. 14 (4): 04014014. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000372.
MacLaughlin, M. M., and N. Sitar. 1996. “Rigid body rotations in DDA.” In Proc., 1st Int. Forum on DDA and Simulations of Discontinuous Media, edited by M. R. Salami, and D. Banks, 620–636. Berkeley, CA: TSI Press.
Mahabadi, O. K., A. Lisjak, A. Munjiza, and G. Grasselli. 2012. “Y-Geo: New combined finite-discrete element numerical code for geomechanical applications.” Int. J. Geomech. 12 (6): 676–688. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000216.
Mikola, R. G., and N. Sitar. 2013. “Explicit three dimensional discontinuous deformation analysis for blocky system.” In Proc., 47th US Rock Mechanics/Geomechanics Symp. 2013, edited by L. J. Pyrak-Nolte, A. Chan, W. Dershowitz J. Morris, and J. Rostami, 1320–1327. San Francisco: American Rock Mechanics Association.
Munjiza, A. 2004. The combined finite-discrete element method. Hoboken, NJ: John Wiley & Sons.
Ning, Y. J., X. M. An, and G. W. Ma. 2011. “Footwall slope stability analysis with the numerical manifold method.” Int. J. Rock Mech. Min. Sci. 48 (6): 964–975. https://doi.org/10.1016/j.ijrmms.2011.06.011.
Qu, X. L., G. Y. Fu, and G. W. Ma. 2014. “An explicit time integration scheme of numerical manifold method.” Eng. Anal. Bound. Elem. 48: 53–62. https://doi.org/10.1016/j.enganabound.2014.06.005.
Sasaki, T., I. Hagiwara, K. Sasaki, R. Yoshinaka, Y. Ohnishi, S. Nishiyama, and T. Koyama. 2011. “Stability analyses for ancient masonry structures using discontinuous deformation analysis and numerical manifold method.” Int. J. Comput. Meth. 8 (2): 247–275. https://doi.org/10.1142/S0219876211002575.
Shi, G. H. 1988. “Discontinuous deformation analysis: A new numerical model for the statics and dynamics of block systems.” Ph.D. thesis. Dept. of Civil Engineering, Univ. of California, Berkeley. https://doi.org/10.1108/eb023855.
Shi, G. H. 1991. “Manifold method of material analysis.” In Transactions of the 9th Army Conf. on Applied Mathematics and Computing. Rep. No. 92-1, 57–76. Minneapolis: US Army Research Office.
Shi, G. H. 2015. “Contact theory.” Sci. China Technol. Sci. 58 (9): 1450–1498. https://doi.org/10.1007/s11431-015-5814-3.
Tsesarsky, Y., and Y. H. Hatzor. 2006. “Tunnel roof deflection in blocky rock masses as a function of joint spacing and friction—A parametric study using discontinuous deformation analysis (DDA).” Tunnelling Underground Space Technol. 21 (1): 29–45. https://doi.org/10.1016/j.tust.2005.05.001.
Wei, W., Q. H. Jiang, and J. Peng. 2016. “New rock bolt model and numerical implementation in numerical manifold method.” Int. J. Geomech. 17 (5): E4016004. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000669.
Wong, L. N. Y., and Z. Wu. 2014. “Application of the numerical manifold method to model progressive failure in rock slopes.” Eng. Fract. Mech. 119: 1–20. https://doi.org/10.1016/j.engfracmech.2014.02.022.
Wu, J. H. 2007. “Applying discontinuous deformation analysis to assess the constrained area of the unstable Chiu-fen-erh-shan landslide slope.” Int. J. Numer. Anal. Methods Geomech. 31 (5): 649–666. https://doi.org/10.1002/nag.548.
Yan, C., and H. Zheng. 2017a. “FDEM-flow3D: A 3D hydro-mechanical coupled model considering the pore seepage of rock matrix for simulating three-dimensional hydraulic fracturing.” Comput. Geotech. 81: 212–228. https://doi.org/10.1016/j.compgeo.2016.08.014.
Yan, C., and H. Zheng. 2017b. “Three-dimensional hydromechanical model of hydraulic fracturing with arbitrarily discrete fracture networks using finite-discrete element method.” Int. J. Geomech. 17 (6): 04016133. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000819.
Yang, Y. T., G. H. Sun, H. Zheng, and X. D. Fu. 2016a. “A four-node quadrilateral element fitted to numerical manifold method with continuous nodal stress for crack analysis.” Comput. Struct. 177: 69–82. https://doi.org/10.1016/j.compstruc.2016.08.008.
Yang, Y. T., X. H. Tang, H. Zheng, Q. S. Liu, and L. He. 2016b. “Three-dimensional fracture propagation with numerical manifold method.” Eng. Anal. Bound. Elem. 72: 65–77. https://doi.org/10.1016/j.enganabound.2016.08.008.
Yang, Y. T., X. H. Tang, H. Zheng, Q. S. Liu, and Z. J. Liu. 2018. “Hydraulic fracturing modeling using the enriched numerical manifold method.” Appl. Math. Model. 53: 462–486. https://doi.org/10.1016/j.apm.2017.09.024.
Yang, Y. T., D. D. Xu, G. H. Sun, and H. Zheng. 2017a. “Modeling complex crack problems using the three-node triangular element fitted to numerical manifold method with continuous nodal stress.” Sci. China. Technol. Sci. 60 (10): 1537–1547. https://doi.org/10.1007/s11431-016-0733-4.
Yang, Y. T., and H. Zheng. 2016. “A three-node triangular element fitted to numerical manifold method with continuous nodal stress for crack analysis.” Eng. Fract. Mech. 162: 51–75. https://doi.org/10.1016/j.engfracmech.2016.05.007.
Yang, Y. T., and H. Zheng. 2017. “Direct approach to treatment of contact in numerical manifold method.” Int. J. Geomech. 17 (5): E4016012. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000714.
Yang, Y. T., H. Zheng, and M. V. Sivaselvan. 2017b. “A rigorous and unified mass lumping scheme for higher-order elements.” Comput. Methods Appl. Mech. Eng. 319: 491–514. https://doi.org/10.1016/j.cma.2017.03.011.
Yeung, M. R., Q. H. Jiang, and N. Sun. 2003. “Validation of block theory and three-dimensional discontinuous deformation analysis as wedge stability analysis methods.” Int. J. Rock. Mech. Min. Sci. 40 (2): 265–275. https://doi.org/10.1016/S1365-1609(02)00137-5.
Yeung, M. R., Q. H. Jiang, and N. Sun. 2007. “A model of edge-to-edge contact for three-dimensional discontinuous deformation analysis.” Comput. Geotech. 34 (3): 175–186. https://doi.org/10.1016/j.compgeo.2006.11.001.
Yu, Y., and J. Yin. 2015. “Some modifications to the process of discontinuous deformation analysis.” J. Rock Mech. Geotech. Eng. 7 (1): 95–100. https://doi.org/10.1016/j.jrmge.2014.12.001.
Zhang, G. X., Y. Zhao, and X. C. Peng. 2010. “Simulation of toppling failure of rock slope by numerical manifold method.” Int. J. Comput. Meth. 7 (1): 167–189. https://doi.org/10.1142/S0219876210002118.
Zhang, Y. H., X. D. Fu, and Q. Sheng. 2014. “Modification of the discontinuous deformation analysis method and its application to seismic response analysis of large underground caverns.” Tunnelling Underground Space Technol. 40: 241–250. https://doi.org/10.1016/j.tust.2013.10.012.
Zheng, F., Y. Y. Jiao, X. L. Zhang, F. Tan, L. Wang, and Q. Zhao. 2017. “Object oriented contact detection approach for three-dimensional discontinuous deformation analysis based on entrance block theory.” Int. J. Geomech. 17 (5): E4016009. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000718.
Zheng, H., and W. Jiang. 2009. “Discontinuous deformation analysis based on complementary theory.” Sci. China Ser. E: Technol. Sci. 52 (9): 2547–2554. https://doi.org/10.1007/s11431-009-0256-4.
Zheng, H., and X. K. Li. 2015. “Mixed linear complementarity formulation of discontinuous deformation analysis.” Int. J. Rock Mech. Min. 75: 23–32. https://doi.org/10.1016/j.ijrmms.2015.01.010.
Zheng, H., F. Liu, and X. L. Du. 2015a. “Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method.” Comput. Methods Appl. Mech. Eng. 295: 150–171. https://doi.org/10.1016/j.cma.2015.07.001.
Zheng, H., Z. J. Liu, and X. R. Ge. 2013. “Numerical manifold space of Hermitian form and application to Kirchhoff’s thin plate problems.” Int. J. Numer. Methods Eng. 95 (9): 721–739. https://doi.org/10.1002/nme.4515.
Zheng, H., F. Liu, and C. Li. 2014a. “The MLS-based numerical manifold method with applications to crack analysis.” Int. J. Fracture. 190 (1–2): 147–166. https://doi.org/10.1007/s10704-014-9980-2.
Zheng, H., F. Liu, and C. Li. 2015b. “Primal mixed solution to unconfined seepage flow in porous media with numerical manifold method.” Appl. Math. Model. 39 (2): 794–808. https://doi.org/10.1016/j.apm.2014.07.007.
Zheng, H., and D. D. Xu. 2014. “New strategies for some issues of numerical manifold method in simulation of crack propagation.” Int. J. Numer. Methods Eng. 97 (13): 986–1010. https://doi.org/10.1002/nme.4620.
Zheng, H., and Y. T. Yang. 2017. “On generation of lumped mass matrices in partition of unity based methods.” Int. J. Numer. Methods Eng. 112 (8): 1040–1069. https://doi.org/10.1002/nme.5544.
Zhu, B. F. 2009. The finite element method, theory, and applications [In Chinese.] Beijing: China Water Resources and Hydropower Press.
Zhu, H., W. Wu, J. Chen, G. Ma, X. Liu, and X. Zhuang. 2016. “Integration of three dimensional discontinuous deformation analysis (DDA) with binocular photogrammetry for stability analysis of tunnels in blocky rock mass.” Tunnelling Underground Space Technol. 51: 30–40. https://doi.org/10.1016/j.tust.2015.10.012.
Zienkiewicz, O. C., and R. L. Taylor. 2000. The finite element method. 5th ed. Oxford, UK: Butterworth-Heinemann.
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International Journal of Geomechanics
Volume 18 • Issue 9 • September 2018
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© 2018 American Society of Civil Engineers.
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Received: Oct 13, 2017
Accepted: Mar 13, 2018
Published online: Jun 20, 2018
Published in print: Sep 1, 2018
Discussion open until: Nov 20, 2018
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