Abstract
We consider the deformation of a geological structure with non-intersecting faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and state-depending friction conditions along the common interfaces. We derive a mathematical model that contains classical Dieterich- and Ruina-type friction as special cases and accounts for possibly large tangential displacements. Semi-discretization in time by a Newmark scheme leads to a coupled system of nonsmooth, convex minimization problems for rate and state to be solved in each time step. Additional spatial discretization by a mortar method and piecewise constant finite elements allows for the decoupling of rate and state by a fixed point iteration and efficient algebraic solution of the rate problem by truncated nonsmooth Newton methods. Numerical experiments with a spring slider and a layered multiscale system illustrate the behavior of our model as well as the efficiency and reliability of the numerical solver.
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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request
References
Ampuero, J.P., Rubin, A.: Earthquake nucleation on rate and state faults - aging and slip laws. J. Geophys. Res. 113(B1), B01302 (2008)
Bank, R.E., Sherman, A.H., Weiser, A.: Some refinement algorithms and data structures for regular local mesh refinement. Sci. Comput. Appl. Math. Comput. Phys. Sci. 1, 3–17 (1983)
Barbot, S., Lapusta, N., Avouac, J.-P.: Under the hood of the earthquake machine: Toward predictive modeling of the seismic cycle. Science 336(6082), 707–710 (2012)
Bastian, P., Blatt, M., Dedner, A., Dreier, N.-A., Engwer, C., Fritze, R., Gräser, C., Grüninger, C., Kempf, D., Klöfkorn, R., Ohlberger, M., Sander, O.: The DUNE framework: Basic concepts and recent developments. Comput. Math, Appl (2020)
Bastian, P., Buse, G., Sander, O.: Infrastructure for the coupling of Dune grids. In Numerical Mathematics and Advanced Applications 2009, pp 107–114. Springer, (2010)
Beeler, N.M., Tullis, T.E., Weeks, J.D.: The roles of time and displacement in the evolution effect in rock friction. Geophys. Res. Lett. 21(18), 1987–1990 (1994)
Ben-Zion, Y.: Collective behavior of earthquakes and faults: Continuum-discrete transitions, progressive evolutionary changes, and different dynamic regimes. Reviews Geophys. 46(4) (2008)
Cochard, A., Madariaga, R.: Dynamic faulting under rate-dependent friction. Pure Appl. Geophys. 142(3), 419–445 (1994)
Dal Zilio, L., van Dinther, Y., Gerya, T.V., Pranger, C.C.: Seismic behaviour of mountain belts controlled by plate convergence rate. Earth Planetary Sci. Lett. 482, 81–92 (2018)
de la Puente, J., Ampuero, J.-P., Käser, M.: Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method. J. Geophys. Res. Solid Earth 114(B10) (2009)
Fagereng, A., Toy, V.: Geology of the earthquake source: an introduction. Geological Soc. London Special Pub. 359(1), 1–16 (2011)
Fonseca, I., Leoni, G.: Modern Methods in the Calculus of Variations: L\(\hat{~}\)p Spaces. Springer Science & Business Media, (2007)
Glowinski, R.: Numerical methods for non-linear variational problems. Springer Verlag, (1984)
Gräser, C.: Convex Minimization and Phase Field Models. PhD thesis, Freie Universität Berlin, (2011)
Gräser, C., Sack, U., Sander, O.: Truncated nonsmooth Newton multigrid methods for convex minimization problems. In Bercovier, M., Gander, M., Kornhuber, R., Widlund, O. (Eds.) Domain Decomposition Methods in Science and Engineering XVIII, vol 70 of LNCSE, pp 129–136 Springer, (2009)
Gräser, C., Sander, O.: Truncated nonsmooth Newton multigrid methods for block-separable minimization problems. IMA J. Numer. Anal. 39, 454–481 (2019)
Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31, 2nd edn. Springer-Verlag, Berlin Heidelberg (2006)
Heinecke, A., Breuer, A., Rettenberger, S., Bader, M., Gabriel, A.-A., Pelties, C., Bode, A., Barth, W., Liao, X.-K., Vaidyanathan, K., et al.: Petascale high order dynamic rupture earthquake simulations on heterogeneous supercomputers. SC’14: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp 3–14. IEEE, (2014)
Herrendörfer, R., Van Dinther, Y., Gerya, T., Dalguer, L.A.: Earthquake supercycle in subduction zones controlled by the width of the seismogenic zone. Nature Geosci. 8(6), 471–474 (2015)
Kačur, J.: Method of Rothe in evolution equations. In Equadiff 6, pp 23–34. Springer, (1986)
Kato, N.: Interaction of slip on asperities: Numerical simulation of seismic cycles on a two-dimensional planar fault with nonuniform frictional property. J. Geophys. Res. Solid Earth 109(B12) (2004)
Krause, R.H.: A nonsmooth multiscale method for solving frictional two-body contact problems in 2d and 3d with multigrid efficiency. SIAM J. Sci. Comput. 31(2), 1399–1423 (2009)
Krause, R.H., Walloth, M.: Presentation and comparison of selected algorithms for dynamic contact based on the Newmark scheme. Appl. Num. Math. 62(10), 1393–1410 (2012)
Krause, R.H., Wohlmuth, B.I.: A Dirichlet-Neumann type algorithm for contact problems with friction. Comput. Vis. Sci. 5(3), 139–148 (2002)
Lapusta, N., Rice, J.R., Ben-Zion, Y., Zheng, G.: Elastodynamic analysis for slow tectonic loading with spontaneous rupture episodes on faults with rate-and state-dependent friction. J. Geophys. Res. Solid Earth 105(B10), 23765–23789 (2000)
Madden, E.H., Bader, M., Behrens, J., van Dinther, Y., Gabriel, A.-A., Rannabauer, L., Ulrich, T., Uphoff, C., Vater, S., Wollherr, S.: Methods and test cases for linking physics-based earthquake and tsunami models. (2019)
Marone, C.: Laboratory-derived friction laws and their application to seismic faulting. Annual Rev. Earth Planetary Sci. 26(1), 643–696 (1998)
Meister, O., Rahnema, K., Bader, M.: Parallel memory-efficient adaptive mesh refinement on structured triangular meshes with billions of grid cells. ACM Trans. Math. Softw. (TOMS) 43(3), 1–27 (2016)
Pelties, C., De la Puente, J., Ampuero, J.-P., Brietzke, G. B., Käser, M.: Three-dimensional dynamic rupture simulation with a high-order discontinuous Galerkin method on unstructured tetrahedral meshes. J. Geophys. Res. Solid Earth 117(B2) (2012)
Pipping, E.: Dynamic problems of rate- and state friction in viscoelasticity. PhD thesis, Freie Universität Berlin, (2014)
Pipping, E.: Existence of long-time solutions to dynamic problems of viscoelasticity with rate-and-state friction. ZAMM-J. Appl. Math. Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 99(11), e201800263 (2019)
Pipping, E., Kornhuber, R., Rosenau, M., Oncken, O.: On the efficient and reliable numerical solution of rate-and-state friction problems. Geophys. J. Int. 204(3), 1858–1866 (2016)
Pipping, E., Sander, O., Kornhuber, R.: Variational formulation of rate- and state-dependent friction problems. ZAMM - J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik, 95(4), 377–395 (2015)
Podlesny, J.: Multiscale Modelling and Simulation of Deformation Accumulation in Fault Networks. PHD thesis, Freie Universität Berlin, (2021)
Ranjith, K., Rice, J.R.: Stability of quasi-static slip in a single degree of freedom elastic system with rate and state dependent friction. J. Mech. Phys. Solids 47(6), 1207–1218 (1999)
Rice, J.R., Ben-Zion, Y.: Slip complexity in earthquake fault models. Proceed. National Acad. Sci. 93(9), 3811–3818 (1996)
Rice, J.R., Lapusta, N., Ranjith, K.: Rate and state dependent friction and the stability of sliding between elastically deformable solids. J. Mech. Phys. Solids 49(9), 1865–1898 (2001)
Rosenau, M., Lohrmann, J., Oncken, O.: Shocks in a box: An analogue model of subduction earthquake cycles with application to seismotectonic forearc evolution. J. Geophys. Res. Solid Earth 114(B1) (2009)
Rothe, E.: Zweidimensionale parabolische Randwertaufgaben als Grenzfall eindimensionaler Randwertaufgaben. Math. Annalen 102(1), 650–670 (1930)
Ruina, A.: Slip instability and state variable friction laws. J. Geophys. Res. 88(B12), 10359–10370 (1983)
Sander, O.: Multidimensional Coupling in a Human Knee Model. PhD thesis, Freie Universität Berlin, (2008)
Sander, O.: The PSurface library. Comp. Vis. Sci. 14(8), 353–370 (2011)
Scholz, C.: Earthquakes and friction laws. Nature 391(6662), 37–42 (1998)
Sobolev, S.V., Muldashev, I.A.: Modeling seismic cycles of great megathrust earthquakes across the scales with focus at postseismic phase. Geochem. Geophys. Geosyst. 18(12), 4387–4408 (2017)
Ulrich, T., Gabriel, A.-A., Ampuero, J.-P., Xu, W.: Dynamic viability of the 2016 mw 7.8 Kaikōura earthquake cascade on weak crustal faults. Nature Commun. 10(1), 1–16 (2019)
Van Dinther, Y., Gerya, T.V., Dalguer, L.A., Mai, P.M., Morra, G., Giardini, D.: The seismic cycle at subduction thrusts: Insights from seismo-thermo-mechanical models. J. Geophys. Res. Solid Earth 118(12), 6183–6202 (2013)
Wohlmuth, B.I.: A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Num. Anal. 38(3), 989–1012 (2000)
Wohlmuth, B.I.: Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numerica 20, 569–734 (2011)
Wohlmuth, B.I., Krause, R.H.: Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems. SIAM J. Sci. Comput. 25(1), 324–347 (2003)
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Open Access funding enabled and organized by Projekt DEAL. This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”, Project Number 235221301, Project B01 “Fault networks and scaling properties of deformation accumulation”
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Gräser, C., Kornhuber, R. & Podlesny, J. Numerical simulation of multiscale fault systems with rate- and state-dependent friction. Comput Geosci 28, 1–21 (2024). https://doi.org/10.1007/s10596-023-10231-4
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DOI: https://doi.org/10.1007/s10596-023-10231-4