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Conclusions We perform extensive numerical simulations of 3D rupture propagation governed by the linear slip-weakening in order to define the optimal parameters that ensure spontaneous rupture propagation. We consider square, elliptical and circular shapes of the initiation zone and find that the area of the initiation zone, for a given value of S, determines whether the rupture will spontaneously propagate or not regardless the particular shape of initiation zone. The transition between propagating and non-propagating ruptures is inconsistent with published theoretical estimates of the critical half-length in 2D and 3D cases. We find that results obtained by FEM and ADER-DG methods converge. This indicates that the results of this study are method independent. Additionally, we observe that adjusted over-shoot value is a good way to minimize the size of initiation zone without introducing numerical artifacts. This might be useful in simulations with heterogeneous faults. For numerical simulations we used parameters in the table (right). To obtain different values of the strength parameter, S, we used different values of initial traction and fixed values of static and dynamic coefficients of friction. Abstract Numerical simulations of earthquake ruptures require artificial procedures to initiate spontaneous dynamic propagation under linear slip-weakening friction. A frequently applied technique uses a stress asperity for which effects of the forced initiation depend on size and geometry of the asperity, the spatial distribution of stress in and around the asperity, and the maximum stress-overshoot value. To properly study the physics of earthquake rupture processes, it is mandatory to understand, and then minimize, the effects of the artificial initiation on the subsequent spontaneous rupture propagation. The model space of possible parameterizations for rupture initiation is large, but a detailed parametric study of their effects on the resulting dynamic crack propagation has not yet been conducted. We therefore extend our previous work to define the optimal initiation parameterization that ensures spontaneous rupture propagation. We consider different sizes and shapes of the initiation region as well as different stress-overshoot values and different spatial sampling to investigate the conditions for spontaneous rupture propagation. This study does not examine in detail the stress distribution within the initiation patch as the required additional parameters greatly expand the model space; this is left for future work. We perform extensive numerical simulations of 3D rupture propagation governed by linear slip-weakening friction. We find that the boundary delimiting regions of the spontaneous rupture propagation and premature rupture arrest appears to be determined by the area of the initiation patch and over-shoot value. Our results indicate that using over-shoot values appropriately scaled to the size of the initiation zone generate correct results without numerical artifacts. Additionally, we compare our numerical results with 2D and 3D analytical estimates (Andrews, 1976 a,b; Day, 1982; Campillo & Ionescu, 1997; Favreau et al., 1999; Uenishi & Rice, 2003, 2004), and find that in 3D numerical simulations the transition between propagating and non-propagating ruptures is incompatible with theoretical estimates of the critical half- lengths. We conjecture that these discrepancies are due to 3D effects and less idealized boundary/initial conditions in the numerical simulations. Our study thus guides the appropriate choice of the rupture initiation zone, to reduce unsuccessful but costly dynamic rupture simulations, and to minimize the effects of forced initiation on the subsequent spontaneous rupture propagation. Initiating spontaneous rupture propagation in dynamic models with linear slip-weakening friction – a parametric study REFERENCES Andrews 1976a JGR Andrews 1976b JGR Campillo & Ionescu 1997 JGR Day 1982 BSSA Day et al. 2005 JGR Dunham 2007 JGR Favreau et al. 1999 BSSA Galis et al. 2010 GJI Pelties et al. 2012 JGR Uenishi & Rice 2003 JGR Uenishi & Rice 2004 EOS (AGU FM) ACKONWLEDGEMENTS We gratefully acknowledge the funding by the European Union through the Initial Training Network QUEST (grant agreement 238007), a Marie Curie Action within the ‘People’ Programme. C. Pelties is funded by VolkswagenStiftung (ASCETE project). ADER-DG results have been computed on the BlueGene/P Shaheen of the King Abdullah University of Science and Technology, Saudi Arabia. Comparison of theoretical estimates with numerical results Theoretical estimates of the half-length size Theoretical estimates of the area calculated from the half-length size line colorline style square circle ellipse Numerical results ellipse circle square Martin GALIS 1, Christian PELTIES 2, Jozef KRISTEK 3,4, Peter MOCZO 3,4 and P. Martin MAI 1 1) KAUST – King Abdullah University of Science and Technology, Thuwal, Saudi Arabia 2) Ludwig-Maximilians-University Munich, Munich, Germany 3) Comenius University Bratislava, Slovakia 4) Slovak Academy of Sciences, Bratislava, Slovakia ECGS Workshop 2012 - Earthquake source physics on various scales Luxembourg, October 3-5, 2012 1. SHAPE of the initiation zone Material parameters VPVP VSVS densityQ P, Q S 6 000 m/s3 464 m/s2 670 kg/m 3 Infinity Friction law parameters ( fault dimensions 30km x 15km ) static coeff. of friction dynamic coeff. of friction normal component of initial traction characteristic distance 0.677777780.525120 MPa0.4 m We present results using these non-dimensional quantities: strength parameter non-dimensional length/area of initiation zone (as used by Dunham, 2007) Model parameters 2. SIZE of the initiation zone 3. OVERSHOOT in the initiation zone Here we search for the minimal size of the initiation zone that generates spontaneous rupture propagation. We consider different shapes of initialization zone - square, circle and ellipse. For simulations we use finite-element method (FEM) with two element sizes – h =100m and h = 50m. Each gray point represents one simulation. The black point represents simulation with the largest considered initiation zone for which rupture did not propagated spontaneously over the whole fault plane, the green point represents simulation with the smallest considered initiation zone for which we observed spontaneous rupture propagation. The red lines mark the transition between propagating and non-propagating ruptures. To parameterize this boundary we use power functions and SquareCircleEllipse h = 100 m h = 50 m h = 100 m h = 50 m 3.0 2.5 2.0 1.5 1.0 0.5 0.0 00.511.520.250.751.251.75 3.0 2.5 2.0 1.5 1.0 0.5 0.0 00.511.520.250.751.251.75 3.0 2.5 2.0 1.5 1.0 0.5 0.0 00.511.520.250.751.251.75 3.0 2.5 2.0 1.5 1.0 0.5 0.0 00.511.520.250.751.251.75 3.0 2.5 2.0 1.5 1.0 0.5 0.0 00.511.520.250.751.251.75 3.0 2.5 2.0 1.5 1.0 0.5 0.0 00.511.520.250.751.251.75 3.0 2.5 2.0 1.5 1.0 0.5 0.0 00.511.520.250.751.251.75 3.0 2.5 2.0 1.5 1.0 0.5 0.0 00.511.520.250.751.251.75 non-dimensional half-length of initiation zone non-dimensional half-length of initiation zone strength parameter ellipse circle square ellipse circle square ellipse circle square ellipse circle square 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 non-dimensional area of initiation zone non-dimensional area of initiation zone 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 00.511.520.250.751.251.75 20 18 16 14 12 10 2 8 4 6 strength parameter FEM h = 100m FEM h = 50m ADER-DG h=200m, O4 To check if the results are method dependent we consider only square shaped initiation zone and we use two different numerical methods: FEM (a finite element method, Galis et al., 2010) and ADER-DG (an arbitrarily high order – discontinuous Galerkin method, Pelties et al., 2012). For each method we consider also different spatial discretizations. FEMADER-DG meshuniform hexahedral unstructured tetrahedral smallest change of initiation zone size h/2 (h = element size) 50m (fixed for all discretizations) discreti- zations h=200, 150, 100, 75, 50, 37.5, 25m for S=2.0 also 18.75m h=200m: O: 2, 3, 4, 5, 6 h=300m: O: 3, 4, 5 degrees of freedom (dof) 8 (N+1)(N+2)(N+3)/6 N is polynomial degree (N = O + 1) rupture dynamics TSN (traction-at-split- node method) inverse Riemann problem strength parameter 10100 19 18 17 16 15 14 9 13 11 12 10 100 10 9 8 7 6 5 0 4 2 3 1 non-dimensional area of initiation zone S = 0.1S = 2.0 FEM ADER-DG Inside the stress asperity the stress is chosen larger than the yield stress. The stress overshoot is the difference between the prescribed stress and the yield stress. In this part of the study we search for the minimal overshoot that ensures spontaneous rupture propagation for prescribed initiation zone. We also compare the effects of larger overshoot on the rupture propagation. We express overshoot as percentage of dynamic stress drop, because if overshoot is expressed as a percentage of yield stress, proper scaling of over-shoot with stress-drop and strength excess is not guaranteed. For example, configurations with equal stress drops and strength excesses should provide the same results. But if overshoot depends on yield stress that will not always be true. 25 overshoot [% of dynamic stress drop] 20 15 10 5 0 00.20.40.60.81 S = 0.1 250 200 150 100 50 overshoot [% of dynamic stress drop] 00.20.40.60.81 0 S = 2.0 different over-shoot values with fixed initiation zone size adjusted over-shoot values to the size of initiation zone (green points from the left-most figure) S = 0.1 receiver in the middle of initiation zone in-plane receiver ( 15L fric from hypocenter) receiver in the middle of initiation zone in-plane receiver ( 15L fric from hypocenter) non-dimensional half-length of initiation zone strength parameter non-dimensional area of initiation zone strength parameter email: martin.galis@kaust.edu.sa www: ces.kaust.edu.sa www.nuquake.eu Note: time 0 s means slip-rate reached value 1 m/s.Note: time 0 s means slip-rate reached value 0.5 m/s. 0.0027 0.0045 0.4545 4.5455 11.3636 27.7273 45.4545 over-shoot 0.0027 0.0045 0.4545 4.5455 11.3636 27.7273 45.4545 over-shoot 0.0027 0.0333 0.3333 3.3333 13.333 3 20.000 0 33.333 3 over-shoot 0.0027 0.0333 0.3333 3.3333 13.333 3 20.000 0 33.333 3 over-shoot 12.95 0.19 8.86 0.25 6.36 0.32 4.09 0.39 2.55 0.47 1.56 0.56 0.91 0.66 0.36 0.77 0.09 0.88 0.05 1.00 over-shoot 12.95 0.19 8.86 0.25 6.36 0.32 4.09 0.39 2.55 0.47 1.56 0.56 0.91 0.66 0.36 0.77 0.09 0.88 0.05 1.00 over-shoot 250.00 0.22 196.67 0.25 163.33 0.29 136.67 0.32 116.67 0.36 81.00 0.46 63.33 0.51 50.00 0.56 43.33 0.62 33.33 0.67 23.33 0.73 16.67 0.80 13.33 0.86 3.33 0.93 0.33 1.00 over-shoot 250.00 0.22 196.67 0.25 163.33 0.29 136.67 0.32 116.67 0.36 81.00 0.46 63.33 0.51 50.00 0.56 43.33 0.62 33.33 0.67 23.33 0.73 16.67 0.80 13.33 0.86 3.33 0.93 0.33 1.00 different over-shoot values with fixed initiation zone size adjusted over-shoot values to the size of initiation zone (green points from the left-most figure) S = 2.0 non-dimensional area of initiation zone 20 18 16 14 12 10 2 8 4 6 20 18 16 14 12 10 2 8 4 6 non-dimensional area of initiation zone non-dimensional area of initiation zone FEM + ADER-DG non-dimensional area of initiation zone To explain observed differences between FEM and ADER-DG we perform a detailed study of the dependency of the minimal initiation zone size on the spatial discretization. We focus only on two limit cases – S = 0.1 (where differences between methods are smaller) and S = 2.0 (where differences between methods are larger). To compare FEM and ADER-DG results we used quantity h/ ∛ dof. This quantity was then scaled using the non-dimensional estimate of breakdown zone width at the time of rupture initiation (Day et al., 2005). The different behavior of FEM and ADER-DG with respect to the size of initiation zone and resolution is consequence of the differences between the methods. Probably the most important is the implementation of dynamic rupture - TSN method is used in FEM while inverse Riemann problem is used in ADER-DG. Some other differences between methods are summarized in the table below. non-dimensional area of initiation zone 10100 10 9 8 7 6 5 0 4 2 3 1 100 non-dimensional area of initiation zone 10100 19 18 17 16 15 14 9 13 11 12 10 100 Note: Gray squares indicate possible initiation zone sizes for each resolution. Note: Gray squares indicate possible initiation zone sizes for each resolution. FEMADER-DGFEMADER-DG FEM + ADER-DG

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