Three-Dimensional Numerical Manifold Method Based on Viscoelastic Constitutive Relation
Publication: International Journal of Geomechanics
Volume 20, Issue 9
Abstract
In order to accurately and efficiently simulate the dynamic processes of coupled phenomena with viscoelastic, continuous and discontinuous deformations, a three-dimensional (3D) numerical manifold method combining Maxwell's viscoelasticity (3D-VisNMM) is proposed and implemented in this study. First, the matrix formulas of 3D-VisNMM are derived, and then its technique flowchart is presented. Second, four viscoelastic models, which represent creep characteristics, stress relaxation features, stress accumulation, and frictional deceleration, respectively, are used to verify the feasibility of 3D-VisNMM. The creep model shows that the simulated deformation at each time step is highly consistent with the analytical solution. The stress relaxation model shows that the accuracy of simulated stress mainly depends on the time step, that is, the range of the Relative Standard Deviation (RSD) is 0.3%–4.8%, which corresponds to a time length of 0.1–2.0 years. The gravity-driven stress accumulation model shows that the RSD between the simulated results and analytical solutions is less than 0.004%. The frictional deceleration simulation shows that the RSD of cumulative displacements and accelerations are less than 0.65% and 2.4%, respectively. All these numerical simulations show that 3D-VisNMM is suitable for analyzing viscoelastic deformations, stress relaxation, and frictional sliding issues in multitemporal scale (second–century) and multispatial scale (meter–hundred kilometers). Therefore, 3D-VisNMM has a good application prospect in Geoscience research.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
Special thanks to Prof. Genhua Shi and Zhengkang Shen for their enthusiastic guidance. This research was funded by the National Key R&D Program of China [2017YFC1500501], the National Science Foundation of China [41974011, 41474002] and the China Scholarship Council.
References
Chen, G. Q., O. Yuzo, and I. Takahiro. 1998. “Development of high-order manifold method.” Int. J. Numer. Methods Eng. 43 (4): 685–712. https://doi.org/10.1002/(SICI)1097-0207(19981030)43:4%3C685::AID-NME442%26gt;3.0.CO;2-7.
Demmel, J. W., S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W. Liu. 1999. “A supernodal approach to sparse partial pivoting.” SIAM J. Matrix Anal. Appl. 20 (3): 720–755. https://doi.org/10.1137/S0895479895291765.
Freed, A. M., and J. Lin. 2001. “Delayed triggering of the 1999 Hector Mine earthquake by viscoelastic stress transfer.” Nature 411 (6834): 180–183. https://doi.org/10.1038/35075548.
Ghasemzadeh, H., M. A. Ramezanpour, and S. Bodaghpour. 2014. “Dynamic high order numerical manifold method based on weighted residual method.” Int. J. Numer. Methods Eng. 100 (8): 596–619. https://doi.org/10.1002/nme.4752.
He, L., X. M. An, and Z. Y. Zhao. 2014. “Development of contact algorithm for three- dimensional numerical manifold method.” Int. J. Numer. Methods Eng. 97 (6): 423–453. https://doi.org/10.1002/nme.4591.
Lambeck, K., C. Smither, and P. Johnston. 1998. “Sea-level change, glacial rebound and mantle viscosity fornorthern Europe.” Geophys. J. Int. 134 (1): 102–144. https://doi.org/10.1046/j.1365-246x.1998.00541.x.
Li, X. S. 2005. “An overview of SuperLU: Algorithms, implementation, and user interface.” ACM Trans. Math. Software 31 (3): 302–325. https://doi.org/10.1145/1089014.1089017.
Li, X. S., J. W. Demmel, J. R. Gilbert, L. Grigori, M. Shao, and I. Yamazaki. 1999. SuperLU users’ guide. Berkeley, CA: Lawrence Berkeley National Laboratory.
Li, Y., M. Liu, Y. Li, and L. Chen. 2019. “Active crustal deformation in southeastern Tibetan plateau: The kinematics and dynamics.” Earth Planet. Sci. Lett. 523: 115708. https://doi.org/10.1016/j.epsl.2019.07.010.
Liu, M., and Y. Yang. 2003. “Extensional collapse of the Tibetan plateau: Results of three-dimensional finite element modeling.” J. Geophys. Res. 108 (B8): 2361. https://doi.org/10.1029/2002JB002248.
Luo, G., and M. Liu. 2010. “Stress evolution and fault interactions before and after the 2008 Great Wenchuan earthquake.” Tectonophysics 491 (1–4): 127–140. https://doi.org/10.1016/j.tecto.2009.12.019.
Ma, G. W., H. D. Wang, L. F. Fan, and B. Wang. 2017. “Simulation of two-phase flow in horizontal fracture networks with numerical manifold method.” Adv. Water Resour. 108: 293–309. https://doi.org/10.1016/j.advwatres.2017.08.013.
Maxwell, J. C. 1867. “II. On the dynamical theory of gases.” Proc. R. Soc. London 15: 167–171. https://doi.org/10.1098/rspl.1866.0039.
Shi, G. H. 1991. “Manifold method of material analysis.” In Proc., Transactions of the 9th Army Conf. on Applied Mathematics and Computing, 57–76. Minneapolis: US Army Research Office.
Shi, G. H. 1996. “Manifold method.” In Proc. of the 1st Int. Forum on DDA Simulation of Discontinuous Media, 52–204. Albuquerque, NM: TSI Press.
Shi, G. H. 1997. Numerical manifold method and discontinuous deformation analysis. Beijing: Tsinghua University Press.
Shi, G. H. 2013. “Basic equations of two dimensional and three dimensional contacts.” In Proc., 47th US Rock Mechanics Geomechanics Symp., 1–8. Alexandria, VA: American Rock Mechanics Association.
Turcotte, D., and G. Schubert. 2014. Geodynamics. Cambridge, UK: Cambridge University Press.
Wang, K. 2007. “Elastic and viscoelastic models of crustal deformation in subduction earthquake cycles.” In The seismogenic zone of subduction thrust faults, edited by T. Dixon, and J. C. Moore, 540–575. New York: Columbia University Press.
Wang, R., F. Lorenzo-Martín, and F. Roth. 2006. “PSGRN/PSCMP—a new code for calculating co-and post-seismic deformation, geoid and gravity changes based on the viscoelastic-gravitational dislocation theory.” Comput. Geosci. 32 (4): 527–541. https://doi.org/10.1016/j.cageo.2005.08.006.
Wang, Y., M. S. Hu, Q. L. Zhou, and J. Rutqvist. 2016. “A new second-order numerical manifold method model with an efficient scheme for analyzing free surface flow with inner drains.” Appl. Math. Modell. 40 (2): 1427–1445. https://doi.org/10.1016/j.apm.2015.08.002.
Wei, W., Q. H. Jiang, and C. B. Zhou. 2014. “Study on numerical manifold method based on finite defeormation theory.” [In Chinese.] Chin. J. Theor. Appl. Mech. 46 (1): 78–86.
Wu, Y. 2012. “Research on the three-dimensional manifold method and its preliminary application to geosciences.” [In Chinese.] Ph.D. thesis, Institute of Geology, China Seismological Bureau.
Wu, Y., G. Chen, Z. Jiang, L. Zhang, X. Liu, and J. Zhao. 2012. “The algorithm of simplex integration in three-dimension and its characteristic analysis.” Int. J. Adv. Comput. Technol. 4 (10): 246–256. https://doi.org/10.4156/ijact.vol4.issue10.29.
Wu, Y., G. Chen, Z. Jiang, L. Zhang, H. Zhang, F. Fan, and L. Li. 2017. “Research on fault cutting algorithm of the three-dimensional numerical manifold method.” Int. J. Geomech. 17 (5): E4016003.
Yang, S., G. Ma, X. Ren, and F. Ren. 2014. “Cover refinement of numerical manifold method for crack propagation simulation.” Eng. Anal. Boundary Elem. 43: 37–49. https://doi.org/10.1016/j.enganabound.2014.03.005.
Yang, Y., X. Tang, H. Zheng, Q. Liu, and L. He. 2016. “Three-dimensional fracture propagation with numerical manifold method.” Eng. Anal. Boundary Elem. 72: 65–77. https://doi.org/10.1016/j.enganabound.2016.08.008.
Yin, X. C. 2011. Solid mechanics. Beijing: Earthquake Press.
Yin, Y. Q. 1987. Introduction to nonlinear finite element method in solid mechanics. Beijing: Peking University Press.
Zhang, G., X. Li, and H. Li. 2015. “Simulation of hydraulic fracture utilizing numerical manifold method.” Sci. China Technol. Sci. 58 (9): 1542–1557. https://doi.org/10.1007/s11431-015-5901-5.
Zhang, H. H., and G. W. Ma. 2014. “Fracture modeling of isotropic functionally graded materials by the numerical manifold method.” Eng. Anal. Boundary Elem. 38: 61–71. https://doi.org/10.1016/j.enganabound.2013.10.006.
Zheng, H., F. Liu, and C. Li. 2015. “Primal mixed solution to unconfined seepage flow in porous media with numerical manifold method.” Appl. Math. Modell. 39 (2): 794–808. https://doi.org/10.1016/j.apm.2014.07.007.
Zheng, H., and 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.
Information & Authors
Information
Published In
Copyright
© 2020 American Society of Civil Engineers.
History
Received: Aug 23, 2019
Accepted: Apr 28, 2020
Published online: Jul 9, 2020
Published in print: Sep 1, 2020
Discussion open until: Dec 9, 2020
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.