Abstract
In the simulations of Burridge-Knopoff (BK) model of earthquakes, the friction force laws are important to produce earthquake-like stick-slip behaviors. Some friction force laws are set-valued and the BK model with them can produce consistent results with observed data of earthquakes in some aspects. However, it is cumbersome to simulate the BK model with set-valued laws by conventional explicit integration methods. In the presence of set-valued laws, the explicit integration methods can easily lead to the numerical chattering, violated constraints on the velocity of force laws, and the difficulty of identifying the states of blocks of the BK model. This paper employs an implicit Euler integration method to simulate the BK model with symmetric and asymmetric set-valued laws. This method removes the numerical chattering in the BK model, even in the cases of large time step sizes. It can easily detect the stuck or slipping state of a block element. Comparing to previous results integrated by explicit integration methods in the literature, the results integrated by this implicit method show smoother curves and lower irregularities in the magnitude distribution of events.
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References
Scholz, C.H.: The Mechanics of Earthquakes and Faulting. Cambridge University Press, New York (1990)
Scholz, C.H.: Earthquakes and friction laws. Nature 391, 37–42 (1998)
Burridge, R., Knopoff, L.: Model and theoretical seismicity. Bull. Seismol. Soc. Am. 57(3), 341–371 (1967)
Carlson, J.M., Langer, J.S.: Mechanical model of an earthquake fault. Phys. Rev. A 40(11), 6470–6484 (1989)
Erickson, B.A., Birnir, B., Lavallée, D.: Periodicity, chaos and localization in a Burridge-Knopoff model of an earthquake with rate-and-state friction. Geophys. J. Int. 187(1), 178–198 (2011)
Xia, J., Gould, H., Klein, W., Rundle, J.B.: Simulation of the Burridge-Knopoff model of earthquakes with variable range stress transfer. Phys. Rev. Lett. 95(24), 248,501.1–248,501.4 (2005)
Xia, J., Gould, H., Klein, W., Rundle, J.B.: Near-mean-field behavior in the generalized Burridge-Knopoff earthquake model with variable-range stress transfer. Phys. Rev. E 77(3), 031,132.1–031,132.11 (2008)
Helmstetter, A., Hergarten, S., Sornette, D.: Foreshocks and aftershocks in the Olami-Feder-Christensen model. Phys. Rev. Lett. 88(23), 238,501–238,504 (2002)
Helmstetter, A., Hergarten, S., Sornette, D.: Properties of foreshocks and aftershocks of the nonconservative self-organized critical Olami-Feder-Christensen model. Phys. Rev. E 70(4), 0461,201–04612,013 (2004)
Carlson, J.M., Langer, J.S., Shaw, B.E., Tang, C.: Intrinsic properties of a Burridge-Knopoff model of an earthquake fault. Phys. Rev. A 44(2), 884–897 (1991)
Carlson, J.M., Langer, J.S.: Properties of earthquakes generated by fault dynamics. Phys. Rev. Lett. 22(62), 2632–2635 (1989)
Marone, C.: Laboratory-derived friction laws and thier application to seismic faulting. Ann. Rev. Earth Planet. Sci. 26, 643–696 (1994)
Dicterich, J.: A constituive law for rate of earthquake production and its application to earthquake clustering. J. Geophys. Res. 99(B2), 2601–2618 (1994)
Mori, T., Kawamura, H.: Simulation study of spatiotemporal correlations of earthquakes as a stick-slip frictional instability. Phys. Rev. Lett. 94(5), 058,501.1–058,501.4 (2005)
Mori, T., Kawamura, H.: Simulation study of the one-dimensional Burridge-Knopoff model of earthquakes. J. Geophys. Res. Solid Earth 111(B7), B073,021–B0730,216 (2006)
Mori, T., Kawamura, H.: Simulation study of the two-dimensional Burridge-Knopoff model of earthquakes. J. Geophys. Res. Solid Earth 113(B6), B063,011–BB0630,116 (2008)
Dupont, P., Hayward, V., Armstrong, B., Altpeter, F.: Single state elastoplastic friction models. IEEE Trans. Autom. Control 47(5), 787–792 (2002)
Kikuuwe, R., Takesue, N., Sano, A., Mochiyama, H., Fujimoto, H.: Admittance and impedance representations of friction based on implicit Euler integration. IEEE Trans. Robot. 22(6), 1176–1188 (2006)
Xiong, X., Kikuuwe, R., Yamamoto, M.: A differential-algebraic method to approximate nonsmooth mechanical systems by ordinary differential equations. J. Appl. Math. 2013, 13 (2013). Article ID 320276
Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-smooth Mechanical Systems. Lecture Notes in Applied and Computational Mechanics, vol. 18. Springer-Verlag, Berlin (2004)
Brogliato, B., Daniilidis, A., Lemaréchal, C., Acary, V.: On the equivalence between complementarity systems, projected systems and differential inclusions. Syst. Control Lett. 55(1), 45–51 (2006)
Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer-Verlag, Berlin (2008)
Moreau, J.J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Eng. 177(3-4), 329–349 (1999)
Dirkse, S.P., Ferris, M.C.: The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems. Optimization Methods and Software 5(2), 123–156 (1995)
Acary, V., Bonnefon, O., Brogliato, B.: Time-stepping numerical simulation of switched circuits within the nonsmooth dynamical systems approach. IEEE Trans. Computer-Aided Design for Integrated Circuits and Systems 29(7), 1042–1055 (2010)
Acary, V., Brogliato, B.: Implicit Euler numerical scheme and chattering-free implementation sliding model systems. Syst. Control Lett. 59(5), 284–293 (2010)
Bastien, J., Schatzman, M.: Numerical precision for differential inclusions with uniqueness. ESAIM: Mathematical Modelling and Numerical Analysis 36(3), 427–460 (2002)
Acary, V., Brogliato, B.: Implicit Euler numerical scheme and chattering-free implementation sliding model systems. Tech. Rep. RR-6886, INRIA (2009)
Greenhalgh, S., Acary, V., Brogliato, B.: On preserving dissipativity properties of linear complementarity dynamical systems with the θ-method. Numer. Math. 125(4), 601–637 (2013)
Marques, M.M.: Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, vol. 9. Birkhauser, Basel (1993)
Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space. Journal of Differential Equations 26(3), 347–374 (1977)
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Communicated by: Silas Alben
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Xiong, X., Kikuuwe, R. & Yamamoto, M. Implicit Euler simulation of one-dimensional Burridge-Knopoff model of earthquakes with set-valued friction laws. Adv Comput Math 41, 1039–1057 (2015). https://doi.org/10.1007/s10444-014-9398-4
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DOI: https://doi.org/10.1007/s10444-014-9398-4