Direct Approach to Treatment of Contact in Numerical Manifold Method
Publication: International Journal of Geomechanics
Volume 17, Issue 5
Abstract
The numerical manifold method (NMM) is suitable for the solution of both continuous and discontinuous problems in geotechnical engineering. In the conventional NMM, the contact between blocks is treated with the open-close iteration, which needs to fix or remove spurious springs between two blocks in contact and to assume properly the normal stiffness and the tangential stiffness (the penalty parameters). Unreasonable values of stiffness would result in numerical problems. To avoid the penalty parameters, contacts are treated in a direct way in which contact forces are primal variables. Numerical examples have confirmed the correctness and feasibility of the proposed procedure.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This study is supported by the National Basic Research Program of China (973 Program), under Grant No. 2014CB047100, and the National Natural Science Foundation of China, under Grant Nos. 11172313 and 51538001.
References
Areias, P., and Belytschko, T. (2005). “Analysis of three-dimensional crack initiation and propagation using the extended finite element method.” Int. J. Numer. Methods Eng., 63(5), 760–788.
Babuška, I., and Melenk, J. M. (1997). “The partition of unity method.” Int. J. Numer. Methods Eng., 40(4), 727–758.
Bao, H. R., Zhao, Z. Y., and Tian, Q. (2014). “On the Implementation of augmented Lagrangian method in the two-dimensional discontinuous deformation analysis.” Int. J. Numer. Anal. Methods Geomech., 38(6), 551–571.
Barla, M., Piovano, G., and Grasselli, G. (2012). “Rock slide simulation with the combined finite-discrete element method.” Int. J. Geomech., 711–721.
Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., and Krysl, P. (1996). “Meshless methods: An overview and recent developments.” Comput. Methods Appl. Mech. Eng., 139(1–4), 3–47.
Beskos, D. E. (1987). “Boundary element methods in dynamic analysis.” Appl. Mech. Rev., 40(1), 1–23.
Beskos, D. E. (1997). “Boundary element methods in dynamic analysis: Part II (1986-1996).” Appl. Mech. Rev., 50(3), 149–197.
Cai, Y. C., Zhuang, X. Y., and Zhu, H. H. (2013). “A generalized and efficient method for finite cover generation in the numerical manifold method.” Int. J. Comput. Methods, 10(05), 1350028.
Cai, Y. G., He, T., and Wang, R. (2000). “Numerical simulation of dynamic process of the Tangshan earthquake by a new method-LDDA.” Pure Appl. Geophys, 157(11), 2083–2104.
Chen, G. Q., Ohnishi, Y., and Ito, T. (1998). “Development of high-order manifold method.” Int. J. Numer. Methods Eng., 43(4), 685–712.
Cundall, P. A. (1971). “A computer model for simulating progressive, large-scale movements in blocky rock systems.” Proc., Symp. of the Int. Society of Rock Mechanics, Nancy, France, Vol. 1, International Society for Rock Mechanics, Lisbon, Portugal, Paper II?8, 129–136.
Desai, C. S., Zaman, M. M., Lightner, J. G., Siriwardane, H. J. (1984). “Thin‐layer element for interfaces and joints.” Int. J. Numer. Anal. Methods Geomech., 8(1), 19–43.
Dolbow, J., and Belytschko, T. (1999). “A finite element method for crack growth without remeshing.” Int. J. Numer. Methods Eng., 46(1), 131–150.
Doolin, D. M., and Sitar, N. (2004). “Time integration in discontinuous deformation analysis.” J. Eng. Mech., 249–258.
Duarte, C. A., Hamzeh, O. N., Liszka, T. J., and Tworzydlo, W. W. (2001). “A generalized finite element method for the simulation of three-dimensional dynamic crack propagation.” Comput. Methods Appl. Mech. Eng., 190(15), 2227–2262.
Goodman, R. E., and John, S. (1977). “Finite element analysis for discontinuous rocks.” Numerical methods in geotechnical engineering, C. S. Desai and J. T. Christian, eds., McGraw-Hill Book, New York, 148–175.
He, L., An, X. M., Ma, G. W., and Zhao, Z. Y. (2013). “Development of three-dimensional numerical manifold method for jointed rock slope stability analysis.” Int. J. Rock Mech. Min. Sci., 64, 22–35.
He, L., An, X. M., and Zhao, Z. Y. (2014). “Development of contact algorithm for three-dimensional numerical manifold method.” Int. J. Numer. Methods Eng., 97(6), 423–453.
He, L., and Ma, G. W. (2010). “Development of 3D numerical manifold method.” Int. J. Comput. Methods, 7, 107–129.
Jiang, Q. H., Chen, Y. F., Zhou, C. B., and Yeung, M. R. (2013). “Kinetic energy dissipation and convergence criterion of discontinuous deformations analysis (DDA) for geotechnical engineering.” Rock Mech. Rock Eng., 46(6), 1443–1460.
Jiang, Q. H., Deng, S. S., Zhou, C. B., and Lu, W. B. (2010). “Modeling unconfined seepage flow using three-dimensional numerical manifold method.” J. Hydrodyn., 22(4), 554–561.
Jiang, Q. H., Zhou, C. B., and Li, D. Q. (2009). “A three-dimensional numerical manifold method based on tetrahedral meshes.” Comput. Struct., 87(13–14), 880–889.
Jiang, W., and Zheng, H. (2011). “Discontinuous deformation analysis based on variational inequality theory.” Int. J. Comput. Methods, 8(2), 193–208.
Jing, L. (2003). “A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering.” Int. J. Rock Mech. Min. Sci., 40(3), 283–353.
Lin, C. T., Amadei, B., Jung, J., Dwyer, J. (1996). “Extensions of discontinuous deformation analysis for jointed rock masses.” Int. J. Rock Mech. Min. Sci., 33(7), 671–694.
Katona, M. G. (1983). “A simple contact–friction interface element with applications to buried culverts.” Int. J. Numer. Anal. Methods Geomech., 7(3), 371–384.
Ma, G., Zhou, W., Chang, X. and Yuan, W. (2014). “Combined FEM/DEM modeling of triaxial compression tests for rockfills with polyhedral particles.” Int. J. Geomech., 04014014.
Ma, G. W., An, X. M., and He, L. (2010). “The numerical manifold method: A review.” Int. J. Comput. Methods, 7(1), 1–32.
Munjiza, A. (2004). The combined finite-discrete element method, John Wiley & Sons, New York.
Ning, Y. J., An, X. M., and Ma, G. W. (2011). “Footwall slope stability analysis with the numerical manifold method.” Int. J. Rock Mech. Min. Sci., 48(6), 964–975.
Perrone, N., and Kao, R. (1975). “A general finite difference method for arbitrary meshes.” Comput. Struct., 5(1), 45–57.
Rabczuk, T., and Belytschko, T. (2004). “Cracking particles: a simplified mesh-free method for arbitrary evolving cracks.” Int. J. Numer. Methods Eng., 61(13), 2316–2343.
Reddy, J. N. (2004). An introduction to nonlinear finite element analysis, Oxford University Press, Oxford, U.K.
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, CA.
Shi, G. H. (1991). “Manifold method of material analysis.” Transactions of the 9th Army Conf. on Applied Mathematics and Computing, Rep. No. 92-1, U.S. Army Research Office, Minneapolis, MN, 57–76.
Shi, G. H. (2015). “Contact theory.” Sci. China Ser. E., 58(9), 1450–1496.
Simo, J. C., and Laursen, T. A. (1992). “An augmented Lagrangian treatment of contact problems involving friction.” Comput. Struct., 42(1), 97–116.
Spivak, M. (1965). Calculus on manifolds: A modern approach to classical theorems of advanced calculus, W. A. Benjamin, New York.
Strouboulis, T., Babuška, I., and Copps, K. (2000). “The design and analysis of the generalized finite element method.” Comput. Methods Appl. Mech. Eng., 181(1), 43–69.
Stupkiewicz, S., Lengiewicz, J., and Korelc, J. (2010). “Sensitivity analysis for frictional contact problems in the augmented Lagrangian formulation.” Comput. Methods Appl. Mech. Eng., 199(33–36), 2165–2176.
Sun, L., Zhao, G. F., and Zhao, J. (2013). “Particle manifold method (PMM): A new continuum-discontinuum numerical model for geomechanics.” Int. J. Numer. Anal. Methods Geomech., 37(12), 1711–1736.
Wei, W., Jiang, Q., and Peng, J. (2016). “New rock bolt model and numerical implementation in numerical manifold method.” Int. J. Geomech., E4016004.
Wong, L. N. Y., and Wu, Z. (2014). “Application of the numerical manifold method to model progressive failure in rock slopes.” Eng. Fract. Mech., 119, 1–20.
Yan, C. Z., Zheng, H., Sun, G. H., and Ge, X. R. (2016). “Combined finite-discrete element method for simulation of hydraulic fracturing.” Rock Mech. Rock Eng, 49(4), 1389–1410.
Yang, Y. T., Xu, D. D., and Zheng, H. (2014). “Evaluation on stress intensity factor of crack under dynamic load using numerical manifold method.” Chin. J. Theor. Appl. Mech., 46, 730–738.
Zhang, G. X., Zhao, Y., and Peng, X. C. (2010). “Simulation of toppling failure of rock slope by numerical manifold method.” Int. J. Comput. Methods, 7(1), 167–189.
Zheng, H., and Jiang, W. (2009). “Discontinuous deformation analysis based on complementary theory.” Sci. China Ser. E., 52(9), 2547–2554.
Zheng, H., and Li, J. L. (2007). “A practical solution for KKT systems.” Numer. Algorithms, 46(2), 105–119.
Zheng, H., and Li, X. K. (2015a). “Mixed linear complementarity formulation of discontinuous deformation analysis.” Int. J. Rock Mech. Min. Sci., 75, 23–32.
Zheng, H., Liu, D. F., Lee, C. F., and Yue, Z. Q. (2004). “A sophisticated node-pair model for interface problems.” Comput. Geotech., 31(2), 137–153.
Zheng, H., Liu, F., and Du, X. L. (2015b). “Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method.” Comput. Methods Appl. Mech. Eng., 295, 150–171.
Zheng, H., Liu, F., and Li, C. (2014a). “The MLS-based numerical manifold method with applications to crack analysis.” Int. J. Fract., 190(1–2), 147–166.
Zheng, H., Liu, F., and Li, C. (2015c). “Primal mixed solution to unconfined seepage flow in porous media with numerical manifold method.” Appl. Math. Model., 39(2), 794–808.
Zheng, H., Liu, Z. J., and Ge, X. R. (2013). “Numerical manifold space of Hermitian form and application to Kirchhoff’s thin plate problems.” Int. J. Numer. Methods Eng., 95(9), 721–739.
Zheng, H., and Xu, D. D. (2014b). “New strategies for some issues of numerical manifold method in simulation of crack propagation.” Int. J. Numer. Methods Eng., 97(13), 986–1010.
Zheng, H., Zhang, P., and Du, X. L. (2016). “Dual form of discontinuous deformation analysis.” Comput. Methods Appl. Mech. Eng., 305, 196–216.
Zienkiewicz, O. C., and Taylor, R. L. (2000). The finite element method, 5th Ed., Butterworth-Heinemann, Oxford, U.K.
Information & Authors
Information
Published In
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Aug 10, 2015
Accepted: Apr 7, 2016
Published online: Aug 8, 2016
Discussion open until: Jan 8, 2017
Published in print: May 1, 2017
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.