3D hp-Adaptative Discontinuous Galerkin Method for Modeling Earthquake Dynamics Josué Tago1, Víctor M. Cruz-Atienza1, Jean Virieux2, Vincent Etienne3 and Francisco J. Sánchez-Sesma4 1Instituto 2Institut 3UMR de Geofísica, Universidad Nacional Autónoma de México, Mexico de Sciences de la Tèrre, Université Joseph Fourier, France Géoazur, Université de Nice – Sophia Antipolis, France 4Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico Submitted to Journal of Geophysical Research March 14, 2012 1 1 Abstract 2 3 We introduce a novel scheme, the DGCrack, to simulate the dynamic rupture of earthquakes in 4 three dimensions based on an hp-adaptative discontinuous Galerkin method. We solve the 5 velocity-stress weak formulation of elastodynamic equations in an unstructured tetrahedral 6 mesh with arbitrary mesh refinements (h-adaptivity) and local approximation orders (p- 7 adaptivity). Our scheme considers second-order fault elements (P2) where dynamic-rupture 8 boundary conditions are verified through ad hoc fluxes across the fault. To model the Coulomb 9 slip-dependent friction law, we introduce a predictor-corrector scheme for estimating shear fault 10 tractions, in addition to a special treatment of the normal tractions that guarantees the 11 continuity of fault normal velocities. We verify the DGCrack by comparison with several methods 12 for two spontaneous rupture tests and find excellent results (i.e. rupture times RMS errors 13 smaller than 1.0%) provided that one or more fault elements resolve the fault cohesive zone. To 14 make this comparison clear, we introduce a methodology based on cross-correlation 15 measurements that provide a simple way to quantify the similarity between solutions. Our 16 verification tests include a 60o dipping normal fault reaching the free surface. The DGCrack 17 method reveals power convergence rates close to those of well-established methods and a 18 numerical efficiency significantly higher than that of similar discontinuous Galerkin approaches. 19 We apply the method to the 1992 Landers-earthquake fault system in a layered medium, 20 considering heterogeneous initial stress conditions and mesh adaptivites. Our results show that 21 previously proposed dynamic models for the Landers earthquake require a reevaluation in 22 terms of the initial stress conditions. 23 2 24 Key points: Dynamic rupture, discontinuous Galerkin, unstructured mesh, Landers earthquake 25 26 1. Introduction 27 28 The availability of high-quality near-field records of large subduction earthquakes in the last few 29 years makes it possible, like never before, to test and validate physics-based rupture models. The 30 development of sophisticated models to explain such aggregate of observations is now largely 31 justified. Huge efforts have been made by the seismological community in the last ten years to 32 overcome technical limitations preventing most dynamic source models, at the end of the 90s, 33 from integrating the effect of fault geometry in the spontaneous rupture of earthquakes. Because 34 of both its simplicity and efficiency, the finite difference (FD) method has been one of the first 35 and most persistent approaches to simulate rupture dynamics along planar faults (e.g. Andrews, 36 1976; Madariaga, 1976; Miyatake, 1980; Day, 1982; Virieux and Madariaga, 1982; Harris and 37 Day, 1993; Madariaga et al., 1998; Peyrat et al., 2001; Day et al., 2005; Dalguer and Day, 2007). 38 Although different strategies have been proposed in recent years to integrate complex fault 39 geometries into this kind of method (Cruz-Atienza and Virieux, 2004; Kase and Day, 2006; Cruz- 40 Atienza et al., 2007; Kozdon et al., 2011) today, most common approaches handle numerical 41 lattices (meshes) that are adaptable to the problem geometry (i.e. fault geometry). One group of 42 methods is based on boundary integral equations (BIE) and have also been used for longtime 43 (e.g. Das and Aki, 1977; Andrews, 1985; Cochard and Madariaga, 1994; Kame and Yamashita, 44 1999; Aochi et al., 2000; Lapusta et al., 2000; Hok and Fukuyama, 2011). However, since these 45 methods discretize only boundaries and require semi-analytical approximations of Green 46 functions, they have difficulties integrating heterogeneities of the bulk properties where the fault 3 47 is embedded. The other group consists of domain methods based on weak formulations of the 48 elastodynamic equations, and can be separated into two subgroups depending on the way the 49 lattice boundaries are treated. On one hand the continuous finite element methods (FEM), whose 50 formulations require continuity between the mesh elements except where special treatments of 51 boundary conditions are imposed (e.g. Oglesby and Day, 2000; Ampuero, 2002; Festa and Vilotte, 52 2006; Ma and Archuleta, 2006; Kaneko et al., 2008; Ely et al., 2009; Barall, 2009) and, on the 53 other, the discontinuous finite element methods, better known as the discontinuous Galerkin 54 (DG) methods, which only consider fluxes between elements and, therefore, do not impose any 55 field continuity across their boundaries. When studying the earthquakes source physics, the 56 discontinuity produced across the fault by the rupture process must be accurately treated, so 57 that the DG strategy is naturally suitable for tackling this problem. 58 59 The first dynamic rupture model based on a DG approach was introduced in two dimensions 60 (2D) by Benjemaa et al. (2007) for low-order (P0) interpolation functions. In this case, where the 61 basis functions are constants, the DG schemes are also known as finite volume (FV) methods 62 (LeVeque, 2002) and provide computationally efficient algorithms as fast as second order FD 63 schemes (i.e. they are equivalent in efficiency on rectangular meshes). However, the extension to 64 three dimensions (3D) of this model (Benjemaa et al., 2009) revealed to have convergence 65 problems for unstructured tetrahedral grids (e.g. non-planar faults) (Tago et al., 2010). In these 66 irregular grids, P0 elements have zero-order convergence for wave propagation modeling due to 67 the centered flux approximation (Brossier et al, 2009; Remaki et al, 2011), so increasing the 68 element interpolation order to achieve a proper numerical convergence of wave propagation 69 with a DG scheme is mandatory. Nevertheless, it is also known that the convergence rate for the 4 70 dynamic-rupture numerical problem is insensitive to high interpolation orders (i.e. 4th order or 71 higher), and that second order interpolation methods are often the most accurate and efficient 72 approximations for applying the corresponding fault boundary conditions (Cruz-Atienza et al., 73 2007; Moczo et al., 2007; Rojas et al., 2009; Kozdon et al., 2011). A notable case where the 74 convergence rate is essentially not sensitive to increments in the interpolation order is the 75 ADER-DG discontinuous Galerkin method for 2D and 3D geometries by de la Puente et al. (2009) 76 and Pelties et al. (2012), respectively, despite its spectral convergence for the wave propagation 77 problem (Dumbser and Kaser, 2006). The ADER-DG is based on a modal interpolation 78 formulation, instead of the nodal interpolation we consider here. Both formulations are 79 mathematically equivalent but computationally different (Hesthaven and Warburton, 2008). Our 80 choice of using the nodal approximation essentially relies in the fact that the evaluation of fluxes 81 requires fewer computations than in a modal scheme, as we shall explain on Section 4.1. 82 83 In this work we introduce a novel discontinuous Galerkin approach, namely the DGCrack 84 method, to model the dynamic rupture of earthquakes in 3D geometries. The numerical platform 85 of our model is the GeoDG3D code (Etienne et al., 2009), so that it accounts for multitask 86 computing using the Message Passing Interface (MPI) with ~85 % scalability, free surface 87 boundary conditions with arbitrary geometry and Convolutional Perfectly Matching Layer 88 (CPML) absorbing boundary conditions along the external edges of the computation domain 89 (Etienne et al., 2009 and references therein). In this approach, intrinsic attenuation via the rock 90 quality factors Q has been introduced as well, but will not be discussed further in this work 91 (Tago et al., 2010). To maximize both the efficiency and the accuracy of the scheme depending on 92 the model properties and geometry, the method handles unstructured mesh refinements (i.e. h- 5 93 adaptivity) and locally adapts the order of the nodal interpolations (i.e. within every grid 94 element; p-adaptivity) (Etienne et al., 2009). We first introduce the mathematical and 95 computational concepts for the 3D dynamic rupture problem, and then assess both its accuracy 96 and convergence rate by comparing calculated solutions with those yielded by finite difference 97 (DFM), finite element (FEM), spectral boundary integral (MDSBI) and spectral element 98 (SPECFEM3D) methods for two spontaneous rupture benchmark tests (Harris et al., 2009). Since 99 one of our major goals in the near future is the investigation of dynamic rupture propagation 100 along realistic (nonplanar) fault geometries, we take special care to verify the accuracy of the 101 highly singular normal stress field across the fault during rupture propagation, as the fault 102 normal tractions strongly determine the radiated energy throughout the Coulomb failure 103 criterion. We finally illustrate the capabilities of the DGCrack method through a spontaneous 104 rupture simulation along the 1992 Landers earthquake fault system, which is a geometrically 105 intricate and physically interesting study case. 106 107 2. Elastodynamic equations 108 109 Velocities and stresses induced by the propagation of waves in a homogeneous elastic medium 110 can be modeled with a first order hyperbolic system (Madariaga, 1976). Following the 111 transformation proposed by Benjemaa et al. (2009), a pseudo-conservative form of the system is 112 given by 113 114 = ∑ ( ) + ∈{,,} 6 116 = ∑ ( ) + 0 ∈{,,} 115 (1) 117 118 where = ( , , ) is the velocity vector and = (, ′ , ′′ , , , ) is the stress 119 vector. To avoid physical properties in the right hand side of (1), a change of variables is done in 120 the stress vector leading to its first three components 121 122 1 = ( + + ) 3 123 1 ′ = (2 − − ) 3 124 ′′ = 3 (− + 2 − ), 1 125 126 which involve the mean and the deviatoric stresses. 127 128 In this model, the physical properties of the medium are the density, , and the medium matrix 129 = diag (3+2 , 2 , 2 , , , ), which is composed by the Lamé parameters and . The 130 external forces and the initial stresses are defined as and 0 , respectively, while the terms 131 and are constant real matrices whose definition can be found in Benjemaa et al. (2009). 3 3 3 1 1 1 132 133 2.1 hp-adaptive Discontinuous Galerkin scheme 134 7 135 To make the local approximation of the hyperbolic system (1), we first need to decompose the 136 domain Ω into elements, so that 137 139 Ω ≃ Ωℎ = ∑ =1 138 (2) 140 141 where each is a straight-sided tetrahedron and the mesh (i.e. decomposed domain) is 142 assumed to be geometrically conforming. By applying a Discontinuous Galerkin approach to the 143 weak formulation of (1), as proposed by Etienne et al. (2010), we obtain the following velocity- 144 stress iterative scheme in every -tetrahedron, 145 146 1 + 2 (3 ⊗ ) 147 (3) 148 ( ⊗ ) 149 − − 1 2 =− +1 − =− ∑ ( ⊗ ) + ∈{,,} ∈ 1 + 2 ∑ ( ⊗ ) ∈{,,} 1 ∑ [( ⊗ ) + ( ⊗ ) ] 2 + 1 1 1 + + ∑ [( ⊗ ) 2 + ( ⊗ ) 2 ] 2 ∈ (4) 150 151 where is the group of adjacent elements to the -tetrahedron and ⊗ represents the tensor 152 product. The mass matrix, , the stiffness matrices, , for all ∈ {, , }, the flux matrices, 153 and , and the auxiliary flux matrices, and , are defined by Etienne et al. (2010). The 154 size of these matrices depends on the order of the polynomial basis (e.g. P0, P1, P2, …, Pk) used 8 155 for the nodal interpolation. Staggered time integration is done through a second order explicit 156 leap-frog scheme, which allows the alternation of velocities and stresses computation. 157 158 One of the main features of this scheme is the h-adaptivity, which allows working with 159 unstructured tetrahedral meshes and thus to adapt the mesh geometry to both the physical 160 properties of the medium and the problem geometry (e.g. mesh refinement). Furthermore, the p- 161 adaptivity is possible thanks to the fluxes between neighboring elements, which are such that 162 two adjacent tetrahedra may have different interpolation orders. The fluxes are computed via a 163 non-dissipative centered scheme that allows choosing different degrees of freedom (DOF) in 164 every tetrahedron. As shown by Etienne et al. (2010), this property is a powerful tool to optimize 165 the domain discretization by adapting the element order to the medium waves speed. Our 166 scheme includes finite volume approximation orders, P0 (i.e. constant functions), linear 167 interpolation functions, P1, and quadratic interpolation functions, P2. Etienne et al. (2010) have 168 shown that this scheme is efficient and accurate enough for modeling wave propagation in large 169 and highly heterogeneous elastic media. The distribution of interpolation orders in the 170 computational domain is such that P0 elements describe the CPML slab while elements with 171 both P1 and P2 approximations discretize the physical domain depending on their sizes. This 172 enhances the accuracy of the scheme and minimizes the computational load. Current 173 computational developments will allow us in the near future to consider higher interpolation 174 orders away from the rupture zone for large-distance wave propagation. 175 176 3. Dynamic rupture model 177 9 178 Earthquakes are highly nonlinear phenomena produced by sliding instabilities along geological 179 faults. The stability of the rupture system depends on several physical factors, like the initial 180 state of stresses in the earth, the material properties and rheology, the sliding rate, the fault 181 geometry, and the constitutive relationship governing the mechanics of the rupture surface. 182 During rupture propagation, fault tractions evolve dynamically depending on all these factors, 183 and the accuracy of an earthquake model strongly depends on the correctness of boundary 184 conditions applied to these tractions in accordance with the fault constitutive relationship (i.e. 185 friction law), which in turn depends on the accurate energy transportation through elastic waves 186 in the medium. Insuring the accuracy of both boundary conditions (local feature) and wave 187 propagation (global feature) has longly been a difficult task for many seismologists. In the next 188 sections we will introduce both the fault boundary conditions and the friction law used in our 189 dynamic source model and then its formulation into the DG scheme. In spite of the attention they 190 deserve, we shall not discussed here the features of wave propagation since they have been 191 previously analyzed by Etienne et al (2010) and Tago et al. (2010). 192 193 3.1 Boundary conditions and friction law 194 195 The fault, Γ, is a piecewise discretized surface with a normal vector, , defined on each surface 196 element. Slip and stresses over Γ are related through a friction law so that the fault tangential 197 tractions evolve according to that law, which in turn depends, for instance, on some fault 198 kinematic parameter and wave propagation around the rupture tip. On the other hand, the strain 199 field is accommodated, in the elastic medium, through displacement (i.e. velocity) discontinuities 10 200 across Γ. It is thus convenient to split our domain into a plus- and a minus-side with respect to 201 the fault (Figure 1) and express the limiting velocity field, , over Γ as 202 203 ± (, ) = lim (, ± ()). →0 204 205 We define the vector as the velocity discontinuity across Γ, where its fault tangential 206 component, namely the fault slip rate, is 207 208 ≔ ⟦ ⟧ = + − − , (5) 209 210 and its fault normal component is 211 212 − ≔ ⟦ ⟧ = + − . (6) 213 214 The slip magnitude at any time is thus defined as the integral of the modulus of over time: 215 216 () = ∫0 ‖ (, )‖. (7) 217 218 The dynamic rupture boundary conditions on the fault are the following two jump conditions, 219 involving the tangential fields (Day et al., 2005), 220 221 − ‖ ‖ ≥ 0 222 − ‖ ‖ = 0, (8) (9) 11 223 224 and a third jump condition applied to (6) 225 226 = 0, (10) 227 228 where is the fault tangential traction vector, and is the fault frictional strength. The fault 229 strength is determined by the Coulomb friction law, which depends on the fault normal stress, 230 , the friction coefficient, , and the fault cohesion, , as 231 232 = − . (11) 233 234 Condition (8) upper bounds the magnitude of to the fault strength, , that always acts 235 opposite to sliding on Γ. The second condition (9) forces the slip rate to be parallel to , the 236 tangential traction. Another implication of these jump conditions is that, whenever the inequality 237 of (8) holds, the slip rate vector is zero, which can be easily seen through the modulus of (9). 238 Finally, condition (10) implies that there is neither fault opening nor mass interpenetration 239 across the fault during rupture propagation. 240 241 In nature, the friction coefficient depends on the fault slip, slip rate and state variables 242 accounting for the sliding history and fault age (e.g. Dieterich, 1979; Ruina, 1983). However, we 243 shall assume a simple slip weakening law (Ida, 1972; Palmer and Rice, 1973), which makes 244 linearly depend on the slip as 245 12 246 0 0 () = 0 + ( − ) (1 − ) (1 − ), (12) 247 248 where and are the static and dynamic friction coefficients, respectively, and 0 is the 249 critical slip weakening distance. 250 251 A series of studies have tried to estimate 0 from historical earthquakes based on indirect source 252 observations through the inversion of strong motion seismograms (e.g. Ide and Takeo, 1997; 253 Mikumo and Yagi, 2003). However, due to the limited bandwidth of the seismograms, the slip 254 weakening distance was poorly resolved in these studies. Moreover, dynamic models based on 255 such indirectly inferred 0 values may be biased and not able to resolve the stress breakdown 256 process over the fault (Guatteri and Spudich, 2000; Spudich and Guatteri, 2004). Direct 257 observation of 0 from near-field data is seldom possible (Cruz-Atienza et al., 2009); except for 258 some isolated cases where rupture propagated with supershear speeds (Cruz-Atienza and Olsen, 259 2010). 260 261 The manner by which we incorporate the fault model given by both the jump conditions (8) to 262 (10), and the friction law (11) and (12), into our DG scheme is presented next. 263 264 3.2 Discrete source model 265 266 The domain decomposition introduced by (2) should account for the presence of Γ, so that the 267 fault surface is discretized with triangles lying on the faces of adjacent tetrahedra (Figure 1), i.e. 13 268 we preclude Γ to be embedded in any tetrahedron . The physical domain, Ω, is then 269 decomposed as follow 270 271 Ω ≃ Ωℎ = ∑ =1 such that if Γ ⊂ Ω then {∀ ∶ ( ∖ ) ∩ Γ = ∅} 272 273 where each is a straight-sided tetrahedron with surface and the union of all elements 274 describes a geometrically conforming mesh. The order of the polynomial basis chosen in our 275 method corresponds to P2 quadratic functions because higher approximation orders does not 276 significantly improve neither the accuracy of dynamic-rupture numerical schemes nor their 277 convergence rate (e.g. Moczo et al., 2007; Rojas et al., 2009; Pelties et al., 2012). As we shall see, 278 keeping a low approximation order (i.e. P2 interpolation functions) provides both good accuracy 279 and efficiency to our numerical scheme. 280 281 Since every tetrahedron has its own nodes in the DG method (i.e. ten independent nodes for P2 282 elements, Figure 1), a fault node is then composed of two co-located nodes (i.e. a split-node). One 283 of them belongs to the -tetrahedron within the plus-side of the domain, and the other to the 284 adjacent -tetrahedron in the minus-side (see Figure 1). This means that each split-node lies 285 between two tetrahedra sharing a fault element. Furthermore, since the DG method does not ask 286 for field continuity over the element faces, the dislocation produced by the rupture may be 287 handled naturally over the fault, Γ, through the discontinuity of the tangential velocities ± = 288 { , }. However, this should be treated carefully because most of the elastic fields must 289 remain continuous across Γ. 290 14 291 (Figure 1) 292 293 System (3) and (4) is solved everywhere inside Ω except over Γ, where jump conditions (8) to 294 (10) must be verified. However, as pointed out by Benjemaa et al. (2009), before treating the 295 fault fluxes accordingly we should notice that the system is not symmetric because of 296 297 ≠ ( ) , 298 299 which is due to the variable transformation (chg_var) required to group all the medium 300 properties on the left-hand side. Although our method does not require the system to be 301 symmetric, that condition is convenient because this way our scheme for P0 elements essentially 302 reduces to the one proposed by Benjemaa et al. (i.e. finite volume approach), which was derived 303 from energy balance consideration across the fault. 304 305 To obtain a symmetric system, we first multiply (4) by the symmetrical positive definite matrix 306 , defined in Benjemaa et al. (2009), and then isolate the updated field values to get the 307 equivalent symmetric system 308 309 1 + 2 1 − 2 = 311 + ∆ ( ⊗ )−1 [− 3 + ∑ ( ⊗ ) ∈{,,} 1 ∑ [( ⊗ ) + ( ⊗ ) ]] 2 ∈ 310 (13) 15 312 1 + 2 +1 = + ∆( ⊗ )−1 − [ ∑ ( ⊗ ) ∈{,,} 314 + 1 1 1 + + ∑ [( ⊗ ) 2 + ( ⊗ ) 2 ] 2 ∈ 313 (14) 315 316 where si = i , sθ = θ and = . 317 318 Since the fluxes across the fault elements Γ = { ⊆ Γ ∣ ik ≔ i ∩ k } must satisfy the jump 319 conditions (8) to (10), we cannot simply use the centered scheme proposed by Etienne et al. 320 (2010) for fluxes between regular elements. Introducing the fault vector fluxes i and i for the 321 velocity and stress schemes, respectively, the system (13) and (14) may be rewritten as: 322 323 1 + 2 325 1 − 2 = + 1 2 + ∆ ( ⊗ )−1 [− 3 ∑ ( ⊗ ) ∈{,,} ∑ [( ⊗ ) + ( ⊗ ) ] + Γ (3 ⊗ ) ∈ ∩ ⊈Γ ] 324 (15) 326 +1 = + ∆( ⊗ )−1 − [ ∑ ( ⊗ ) 1 + 2 ∈{,,} 328 327 + 1 2 ∑ 1 + 2 [( ⊗ ) 1 + + + ( ⊗ ) 2 ] + Γ (3 ⊗ ) 1 2 ∈ ∩ ⊈Γ (16) 329 16 330 where Γ is a Kronecker delta which is 1 if Γ ≠ ∅ and 0 otherwise. 331 332 Since the fault boundary conditions (8) and (9) must be applied to the traction vector , 333 following Benjemaa et al. (2009) we notice that the flux in the velocity scheme (13) through any 334 element surface may be expressed in terms of tractions as 335 336 1 2 [( ⊗ ) + ( ⊗ ) ] = [(3 ⊗ ) + (3 ⊗ ) ]. 1 2 337 338 By imposing continuity of over the fault element Γ (i.e. = = ) and assuming the 339 same approximation order in the two tetrahedra sharing that element (i.e. = ), we set the 340 flux vector across the fault, , to the unique traction vector , and define the flux across the 341 fault as 342 343 (3 ⊗ ) = (3 ⊗ ) . (17) 344 345 By substituting equation (17) into the velocity scheme (15) and regrouping the terms excluding 346 the flux across the fault on , we get 347 348 1 2 + 1 2 − = + + Γ ∆ (3 ⊗ )−1 (3 ⊗ ) , (18) 349 350 where is 351 17 353 = ∆ (3 ⊗ )−1 − ∑ ( ⊗ ) + [ ∈{,,} 1 2 ∑ [( ⊗ ) + ( ⊗ ) ] ∈ ∩ ⊈Γ ] 352 354 Because the fault normal stress determines the frictional strength via (11) and boundary 355 conditions are applied to the shear tractions, the fault traction vector has to be decomposed 356 into its tangential, , and normal, , components to finally rewrite the velocity scheme (18) 357 as 358 359 1 2 + 1 2 − = + + Γ ∆ (3 ⊗ )−1 (3 ⊗ )( + ). (19) 360 361 3.2.1 Fluxes across the fault for the velocity scheme (19) 362 363 To update velocities onΓ, we need to specify the flux across the fault (17) in (19). To this 364 purpose we require such that the jump conditions (8) and (9) are fulfilled, and so that 365 the continuity of the fault normal velocity is also guaranteed (condition (10)). All traction 366 conditions are verified at every fault node and for every time step. 367 368 To compute the fault tangential tractions, , we first notice that the inequality of condition (8) 369 along with the modulus of (9) implies that 370 371 = 0, (20) 372 18 373 which means that, whenever ‖ ‖ remains below the frictional strength, , the tangential 374 traction must ensure the continuity of the tangential velocities at every fault node shared by the 375 - and -tetrahedral. 376 377 Let us assume that Γ ≠ ∅ so Γ = 1 in (19), where the - and -tetrahedral lie at the plus-side 378 and the minus-side of the fault Γ, respectively. Since we only deal with fault nodes, we thus 379 construct the matrix −1 Γ , which is the inverse mass matrix whose components depend 380 exclusively on those nodes. It is simply constructed from −1 by eliminating its rows and 381 columns associated with the off-fault nodes. 382 383 To compute verifying condition (20) we require both tetrahedra sharing a fault element to 384 have the same order, so that the nodes in both sides of the fault match to each other (see Figure 385 1). Besides this, for the specific contribution of , we need to compute the volume and surface 386 integrals of −1 and , respectively, in a standard element (Zienkiewicz et al., 2005) such that Γ 387 388 1 −1 −1 Γ = Γ and = (21) 389 390 where , the i-tetrahedron volume, and , the i-tetrahedron fault surface, are the 391 corresponding Jacobians. Then by substituting (18) into definition (5) and using (21), we 392 express the slip rate vector as 393 394 + 1 + 2 ≔ 1 2 + 1 − 2 19 395 − 1 1 = 2 + − + ( + 1 −1 ) (3 ⊗ Γ ) (3 ⊗ ) . (22) 396 397 For getting the tangential traction, we finally use (20) into (22), which leaves us the following 398 expression 399 400 = ( ( )) (3 + ⊗ )−1 − 1 2 (3 ⊗ Γ ) (− − + ). (23) 401 402 This procedure ensures the continuity of the tangential velocity across the fault. However, if the 403 time-dependent frictional strength, , is overcome by the modulus of the tangential traction, 404 ‖ ‖, rupture must occur and the tangential velocity should not longer be continuous across 405 the associated fault node. In that case, the equality of condition (8) holds so that 406 407 − ‖ ‖ = 0 ⇔ ‖ ‖ = . 408 409 Therefore, to compute the slip rate (22) at every fault node and for every time step, the 410 tangential traction is adjusted according to the following criterion, which depends on whether 411 the fault point has broken or not: 412 413 = { ‖ ‖ if ‖ ‖ < if ‖ ‖ ≥ (24) 414 20 415 Since the nodes within a fault element are coupled through the flux matrix, , and the mass 416 matrix, Γ , when rupture happens in a given node and the condition (24) is imposed, tractions 417 in the remaining nodes change for the same time step. Thus, to accurately and simultaneously 418 satisfy (24) in every fault node, i.e. to allow rupture propagation inside the fault elements, we 419 use an iterative predictor-corrector (PC) scheme. If is the number of nodes in a given fault 420 element (i.e. six for P2 elements) and is the number of nodes that have broken, then the 421 predictor-corrector scheme operates only if 0 < < . The PC scheme will basically 422 find the tangential tractions, , in the unbroken nodes, given the boundary condition applied 423 in the broken nodes for the same time step. Thus, the procedure will only influence the two 424 interacting elements sharing the same breakable portion of the fault surface. 425 426 Our predictor-corrector scheme is simple and converges fast: when a fault node breaks in a given 427 element, the modulus of its tangential traction is set to (condition (24)). Once applied this 428 condition, the predicted tangential tractions in the unbroken nodes of the same element 429 must be recomputed accordingly via (23). To this purpose, a new mass matrix, , must Γ 430 be constructed considering only the unbroken fault nodes while for the broken nodes the 431 tangential traction condition is set. If the magnitude of the new-predicted tractions overcomes 432 the fault strength, then it is corrected by setting it to . This updating cycle continues iteratively 433 through new predictions and corrections until no other node breaks inside the element after the 434 last correction. The maximum number of possible iterations is given by the order of the 435 approximation, i.e. DOF, used in the fault elements, and will always be smaller than (i.e. 436 five or less iterations for P2 elements). The PC procedure has to be performed locally in each 437 piece of fault surface. It is an efficient iterative verification of boundary conditions and 21 438 represents an additional reason to preserve low interpolation orders (i.e. P2 specifically) in the 439 tetrahedra sharing a fault segment. 440 441 Since the DG schemes do not enforce continuity of the fields between two adjacent tetrahedra, 442 and the accuracy of our method depends on the special treatment of velocities over the fault, we 443 must take care of the fault normal tractions as well in the same manner we do for the tangential 444 components. We now deduce a formula to compute the normal traction, , which ensures 445 continuity of the fault normal velocity field. This model constrain is given by the jump condition 446 (10). 447 448 As done for the tangential slip rate (equation (22)), definition (6) may be developed as 449 450 + 1 + 1 + 1 2 ≔ 2 − 2 − 1 1 = 2 + − + ( + 451 1 −1 ) (3 ⊗ Γ ) (3 ⊗ ) . 452 453 Using condition (10) to force continuity of the normal velocity, we then define the fault normal 454 traction 455 456 = (( + ) ) (3 ⊗ )−1 (3 ⊗ Γ ) (− 1 2 − − + ), (25) 457 458 from which the fault normal stress is given by 22 459 460 = ∙ . 461 462 The frictional strength, , can now be computed in every fault node using (11) as a function of 463 both and the friction coefficient, , which depends on the fault slip, U n (equation (7)), 464 through the slip weakening law (12). 465 466 Definitions given for the normal (equation (25)) and tangential (equation (23)) traction 467 components finally allow us to update the velocity field in every fault node via equation (19). 468 469 3.2.2 Fluxes across the fault for the stress scheme (16) 470 + 1 471 For the stress scheme and within the i-tetrahedron, the flux across the fault, 2 , is computed 472 using only the velocity field in that tetrahedron, so that the flux is given by 473 474 + 1 2 + = 1 2 . (27) 475 476 The simplicity of this flux comes from the computation of a unique traction vector on the fault 477 that guarantees either the continuity or discontinuity of the velocity field depending on whether 478 the fault has broken or not. This fault-flux approximation is equivalent to the one proposed by 479 Benjemaa et al. (2009), where the flux estimation is based on an energy balance consideration 480 across the fault. The simplicity of definition (27) is due to the continuity of the fault normal 481 velocity implicit in the computation of through (25). 23 482 483 4. Rupture model verification, convergence and efficiency 484 485 Verification of dynamic rupture models is a particularly difficult task. Since no analytical solution 486 exists for the spontaneous rupture problem (i.e. closed form equations for the resulting 487 motions), the only possible way to be confident of a given approach is the comparison of results 488 for a well-posed rupture problem between various numerical techniques based on different 489 approximations. This kind of exercise has been systematically performed in the last years by an 490 international group of modelers (Harris et al., 2009). In this section we present results for two 491 benchmark 492 http://scecdata.usc.edu/cvws/index.html), and compare them with those obtained with finite 493 difference, finite element, spectral element, discontinuous Galerkin and spectral boundary 494 integral methods. Based on these comparisons, we assess the numerical convergence rate of our 495 method, its efficiency and determine numerical criteria to guarantee its accuracy. tests proposed by this group, i.e. TPV3 and TPV10 (see: 496 497 4.1 The Problem Version 3 (TPV3) 498 499 Consider the spontaneous rupture of a vertical right-lateral strike-slip fault embedded in a 500 homogeneous fullspace with P- and S-waves speeds of 6000 m/s and 3434 km/s, respectively, 501 and density of 2670 kg/m3. The fault is rectangular and measures 30 km long by 15 km wide 502 (Figure 2). Rupture nucleation happens in a 3 km by 3 km square region, centered both along- 503 strike and along-dip, because the initial shear stress there is higher than the fault strength. The 504 friction law is linear slip-weakening with zero cohesion (Equations (11) and (12)), and both 24 505 static and dynamic friction coefficients are constant over the fault. The initial fault normal 506 traction is also constant, as are the static and dynamic fault strengths. Values for all source 507 parameters, i.e. initial stress conditions and friction parameters, are shown in Table 1. Results 508 for this problem are compared with those yielded by the DFM finite difference scheme (Day et 509 al., 2005), the ADER-DG discontinuous Galerkin scheme (Pelties et al., 2012) and the spectral 510 boundary integral equation method by Geubelle and Rice (1995) with the implementation of 511 E.M. Dunham (MDSBI: Multidimensional Spectral Boundary Integral, version 3.9.10, 2008); all of 512 them for an equivalent grid size of 50 m. 513 514 All DGCrack solutions presented in this section were calculated in the same 100 x 110 x 95 km3 515 physical domain discretized with unstructured h-adaptive meshes so that the element 516 characteristic lengths go from 1 km in the CPML slab to the desired length over the fault plane 517 (i.e. 1.0, 0.8, 0.5, 0.4, 0.3, 0.2 and 0.1 km). 518 519 (Figure 2) 520 521 Our first comparison corresponds to the rupture times on the fault plane with the DFM method 522 (Figure 3). We have used a characteristic elements size of 100 m over the fault (i.e. an effective 523 grid size (internode distance) of about 50 m in our P2 elements approximation). The fit between 524 both solutions is almost perfect. No significant difference may be seen in this comparison. Figure 525 4 compare DGCrack seismograms at fault points PI (pure in-plane deformation) and PA (pure 526 anti-plane deformation) (see Figure 2) for the slip (4a), shear traction (4b) and slip rate (4c and 527 4d) fields with those yielded by the DFM, ADER-DG and MDSBI methods. Except for weak 25 528 oscillations, the comparison is also excellent. Besides the stress build-ups, which are nicely 529 resolved at both observational points before failure, let us notice how the friction law is well 530 resolved as compared to the other solutions, with stress overshoots around 7 s and 8 s at PI, and 531 8.5 s and 10.5 s at PA. The associated slip reactivations are also well modeled and can be seen in 532 the slip rate functions at both points. Stopping phases from the fault edges strongly determine 533 the slip rate and are sharply resolved at 6.5 s and 7 s at PI, and at 4 s and 7.5 s at PA. A closer 534 comparison of slip rates at both observational points (Figure 4d) suggests that the closest two 535 solutions to each other correspond to ours and the one generated by the spectral boundary 536 integral method (MDSBI). Despite weak oscillations present in the DFM, MDSBI and DGCrack 537 waveforms, both amplitude and phase of the DGCrack and MDSBI solutions are remarkably 538 similar. 539 540 (Figure 3) 541 542 (Figure 4) 543 544 Since no analytic solution exists for this problem, quantitative comparisons between all the 545 approximations may give insights about their correctness. Figure 5b presents a quantitative 546 comparison of all numerical solutions based on cross-correlation (cc) measurements of the slip 547 rate time series on both PI and PA observational points. Each colored square of the Cross- 548 Correlation Matrix (CCM) corresponds to a cc-based metric between two solutions: for a given 549 method, its metrics with respect to the other approaches are those corresponding to its 550 associated raw and colon of the matrix. Values in the upper triangular part of CCM are given by 26 551 552 = − 1− , 553 554 where is the maximum cross-correlation coefficient between the i and j solutions, and 555 subscript min reads for the smallest coefficient of all possible combinations. Values in the lower 556 triangular part of CCM are given by 557 558 = − , 559 560 where is the delay in seconds between the i and j solutions for the maximum correlation 561 coefficient, and the max subscript means the maximum delay of all possible combinations. Both 562 measures provide a quantitative mean to assess the similarity between solutions relative to the 563 worst comparison found between all combinations. So they do not provide absolute cross- 564 correlation information except for the auto-correlations along the CCM diagonal. While the 565 ADER-DG solution is the closest to both the DGCrack and MDSBI solutions in terms of correlation 566 coefficients (see in Figure 5b), the smallest phase error is found between the DGCrack 567 and MDSBI solutions (see ). As it may also be seen on Figure 4d, the solution with the 568 lowest correlation metrics with respect to the other ones is that from DFM (i.e. first column and 569 first row). By averaging both metrics per solutions couple, we could better see which time series 570 are the closest to each other. Figure 6 presents the results of this exercise, where the two 571 discontinuous Galerkin solutions reveal to be the more similar, although very close to the one 572 yielded by the Boundary Integral method. Measures provided by this method should be 27 573 interpreted carefully since they do not account for the computational cost required by each 574 method to achieve its solution. 575 576 (Figure 5) 577 578 One important issue in dynamic rupture modeling is the control of spurious oscillations 579 produced by the advance of the crack tip throughout the discrete lattice. Using either artificial 580 viscosity or intrinsic dissipation procedures in rupture models is a delicate matter because the 581 associated damping does not distinguish between numerical and physical contributions. In other 582 words, if badly handled, dissipation may absorb frequencies belonging to the physical-problem 583 bandwidth affecting, for instance, peak slip rates and rupture speeds, as shown by Knopoff and 584 Ni (2001). This is probably why the ADER-DG scheme (de la Puente, 2009), which uses 585 intrinsically dissipative Godunov fluxes (González-Casanova, 2006; LeVeque, 2002), requires 586 high interpolation orders to achieve good accuracy (e.g. compare O2 with O3 or higher order 587 solutions in Pelties et al., 2012). In the DGCrack scheme, in contrast, the centered flux scheme we 588 use is conservative so that the energy is not intrinsically dissipated (Hesthaven and Warburton, 589 2008) but, because it is dispersive, our method presents spurious oscillations. However, since 590 these oscillations remain reasonable small and we expect them to be even smaller for physically 591 attenuating media or different friction laws (e.g. rate- and state-dependent), we have decided not 592 to integrate an artificial viscosity. 593 594 (Figure 6) 595 28 596 To assess the convergence rate of the DGCrack scheme and to determine a quantitative criterion 597 that guaranties its accuracy, Figures 5a and 5c present two different error metrics, defined by 598 Day et al. (2005) as the relative root mean square difference between a given solution and the 599 reference one (i.e. the DFM solution for h = 50 m), as a function of the characteristic elements 600 size on the fault. These metrics correspond to the rupture times over the fault and the peak slip 601 rates at fault points PA and PI (Figure 2). Both figures reveal a power convergence rate of the 602 DGCrack method with regression exponents reported in Table 2 and compared to those for other 603 methods. We also report in that table the exponent for the final slip on both observational points 604 (not shown in the figure). It is important to notice that all DGCrack solutions presented in the 605 manuscript correspond to unstructured meshes with refinements around the rupture surfaces. 606 As mentioned by de la Puente et al. (2009), this is a critical issue since the accuracy of solutions 607 significantly depends on the quality of the tetrahedral lattice built independently using standard 608 tools (in this work we have used the Gmsh software developed by Geuzaine and Remacle, 2009), 609 as can been seen in the error dispersion on both Figures 5a and 5c with respect to the regression 610 lines. 611 612 The accuracy of dynamic rupture models depends on the resolution of the cohesive zone, which 613 may vary during rupture evolution. This zone is defined as the fault region where the stress 614 breakdown process takes place. The amount of fault elements inside this region behind the 615 rupture front, Nc, is thus the numerical criterion that guaranties a given accuracy level. Figures 616 5a and 5c also reveal that one or more fault elements inside the cohesive zone (i.e. ≥ 1) is 617 enough to achieve errors smaller than 1 % and 10 % for rupture times and peak slip rates, 29 618 respectively. In the TPV3 test case, this condition implies fault element sizes smaller or equal to 619 ~450 m. 620 621 We finally address a fundamental question in computational sciences: the numerical efficiency. 622 Pelties et al. (2012) have recently introduced a method to solve the dynamic rupture problem 623 based on a discontinuous Galerkin scheme that incorporates a sophisticated strategy allowing 624 arbitrarily high order approximations in space and time, i.e. the ADER-DG method. Since both the 625 DGCrack and ADER-DG methods share many different capabilities linked to the DG 626 approximation, it is worthy to see differences and to estimate the computational cost of each 627 method to achieve the same accuracy level. Since the ADER-DG method is based on a modal 628 approximation, the fluxes across the element faces are sensitive to all modes across the element; 629 while in a nodal approximation, like in the DGCrack method, they only depend on the nodes lying 630 on the element face where the flux is computed. This difference translates into fewer 631 computations in the nodal approximations (Hesthaven and Warburton, 2008). Figure 5d 632 presents a quantitative comparison of rupture times errors as a function of total computing 633 times (i.e. the CPU time, which is given by the duration of each simulation multiplied by the 634 number of cores) for both methods. The DGCrack simulations were run in Pohualli, the parallel 635 computer with 172 cores (i.e. 2.33 GHz quad-core Xeon processors) of the Department of 636 Seismology at UNAM. Since the computational cost is not reduced by the ADER-DG method 637 when using high approximation orders (Pelties et al., 2012), values reported in Figure 5d 638 correspond to the ADER-DG O3 solution. Both regression lines have about the same slope; so 639 similar CPU time differences between the methods are expected for any grid size. If we take the 640 same accuracy level we used to establish the Nc condition in the last paragraph (i.e. 1 % error for 30 641 rupture times) then the CPU time of the ADER-DG O3 method is about 30 times larger than the 642 one required by the DGCrack method in our computing platform. Since the simulation times 643 reported by Pelties et al. were obtained using a 0.85 GHz BlueGene parallel computer, the actual 644 CPU time difference would be reduced to a factor of ~10 if both methods were run in the same 645 platform, which still is a significant factor, especially if multiple large scale simulations are 646 required for an inversion modeling purpose, for example. This comparison was made with the 647 available data reported by Pelties et al. (2012), however it would be interesting to design a 648 dynamic-rupture speed test that could better reflect the CPU time required by both schemes. 649 Besides, it is important to notice that ADER-DG shows smooth time series on the fault and 650 accurate wave propagation away from the fault (Dumbser and Kaser, 2006). Theses 651 characteristics are not quantified here and may be essential for long-range wave propagation 652 problems, for instance. 653 654 4.2 The Problem Version 10 (TPV10) 655 656 Our last verification test consists of a 60o dipping normal fault reaching the free surface of a 657 homogeneous halfspace (Figure 7) with P- and S-waves speeds of 5716 m/s and 3300 km/s, 658 respectively, and density of 2700 kg/m3. The fault has the same dimensions as in TPV3 (i.e. 30 659 km long by 15 km width) but the center of its 3 x 3 km2 nucleation patch is located deeper, at 12 660 km along dip. Frictional and initial stress conditions on the fault plane are reported in Table 3, 661 where the cohesion term of equation (11) is not zero, and both pre-stress conditions are 662 dependent on the along-dip distance, hd. The unstructured h-adaptive tetrahedral mesh used to 31 663 obtain the DGCrack solution in shown in Figure 7, which has a characteristic element size of 100 664 m over the fault plane. 665 666 (Figure 7) 667 668 This test case has interesting features that are essential to verify a dynamic rupture model for 669 non-planar faults. Since we deal with a dipping normal fault reaching the Earth’s surface, 670 reflected waves are bounced back to the source, inducing transient variations of the fault 671 traction vector that significantly affect rupture propagation via the Coulomb failure criterion 672 (equation (11)) (Oglesby et al., 1998). Figure 8 presents a comparison of rupture times over the 673 fault obtained with three different methods (Harris et al., 2009): a finite element (FaulMod; 674 Barall, 2010), a spectral element (SPECFEM3D; Kaneko et al., 2008) and the DGCrack methods. 675 The three solutions were computed with a fault elements size of 100 m. Despite the complexity 676 of the rupture model, the match between all solutions is remarkably good. Dynamic effects on 677 rupture propagation due to the presence of the free surface are clearly seen in the upper most 2 678 km, where a secondary rupture front propagating down-dip is initiated about 2.5 s after 679 nucleation. 680 681 (Figure 8) 682 683 Figure 9 shows on- and off-fault waveforms yielded by the same three methods with (lower 684 traces) and without (upper traces) low-pass filtering at 3 Hz. From top to bottom, the left column 685 presents the time evolution of the slip rate, shear stress and normal stress at fault point FP 32 686 (Figure 7), which is aligned along strike with the center of the nucleation patch and located 1.5 687 km from the free surface along dip. On-fault solutions reveal that the best fits along the entire 688 waveforms correspond to the DGCrack and SEM signals during the first 10 s, and to the DGCrack 689 and FEM signals during the remaining 5 s. This suggests that, despite the spurious oscillations 690 present in the latter part of the DGCrack waveforms, this method provides the most accurate 691 solution for this problem compared to the SEM and FEM methods. The right column of Figure 9 692 shows, from top to bottom, the two horizontal components (i.e. fault parallel, FP, and fault 693 normal, FN, components) of the ground velocity and vertical displacements at the ground point 694 GP (Figure 7), which is located 3 km in the along strike direction from the fault extremity and 3 695 km away from the fault trace on the hanging wall. In the ground motion synthetics the situation 696 is slightly different. The closest two solutions along the entire records are those produced by the 697 DGCrack and FEM methods (see filtered velocities and displacements up to 10 s). Although both 698 the DGCrack and SEM solutions present spurious oscillations, the 3 Hz low pass filter did not 699 eliminate longer-period oscillations in the DGCrack seismograms, particularly present after 10 s. 700 Since similar noise is found in the on-fault seismograms, this inaccuracy is probably due to long- 701 range numerical dispersion associated with the centered fluxes across the fault. 702 703 (Figure 9) 704 705 5. Rupture along the 1992 Landers-Earthquake Fault System 706 707 The 28 June 1992 Landers earthquake (Mw7.3) in southern California produced one of the most 708 valuable data sets ever recorded. The amount and diversity of geophysical observations allowed 33 709 constraining the earthquake rupture history, revealing a large complexity of the slip pattern in a 710 wide frequency range (< 0.5 Hz) (e.g. Campillo and Archuleta, 1993; Wald and Heaton, 1994; 711 Olsen et al., 1997; Hernandez et al., 1999). The large rupture size (70 km long by 15 km wide), 712 the long rupture duration (> 20 s) and the intricate fault-system geometry bring an exceptional 713 opportunity to test our discontinuous Galerkin rupture model. However, the rupture process of 714 the Landers earthquake deserves an extensive analysis that goes beyond the purpose of this 715 work. For this reason we shall mainly use this study case for illustrating the capabilities of the 716 DGCrack approach in realistic conditions, and additionally to elucidate some physical aspects of 717 such an event. 718 719 (Figure 10) 720 721 The Landers earthquake ran over three main right-lateral faults, namely the Johnson Valley, 722 Homestead Valley and Emerson/Camp Rock faults, which are connected through jogs and step- 723 overs forming a complex nonplanar fault system (Figure 10). The system geometry we adopted 724 was taken from the Community Fault Model for Southern California (Plesch et al., 2007), which 725 essentially consists of several strike-slip vertical segments, as shown in Figure 10. To discretize 726 the model we considered a 1D layered medium (Wald and Heaton, 1994) (see L1, L2 and L3 727 layer boundaries on Figure 10) and took advantage of the hp-adaptivity. This means that we 728 refined the mesh around the rupture surface and simultaneously adapted the bulk elements 729 sizes to resolve 0.5 Hz waves according to each layer properties, so that the wave-propagation 730 accuracy criterion determined by Etienne et al. (2009) is satisfied (i.e. five elements per 731 minimum wavelength). The gradual mesh coarsening from 300 m over the fault to 1100 m 34 732 throughout the CPML region produced a discrete lattice with 4.48 millions tetrahedra, from 733 which 48.6 % are P1 elements and belong to the CPML slab (Figure 10), and the rest (51.4 %) are 734 P2 elements and discretize the physical domain. Dimensions of the computational volume are 735 67.0 x 101.6 x 29.0 [km] in the x, y and z directions, respectively, so that to complete a 20 s 736 simulation, the DGCrack model spent a total CPU time of 1.73 x 106 s, which correspond to 9.6 737 hours in 50 cores of our Pohualli parallel platform. 738 739 The initial stress conditions in the fault (i.e. pre-stress conditions) correspond to the initial shear 740 tractions, 0, determined by Peyrat et al. (2001) on a planar fault, which are an improved version 741 of those used by Olsen et al. (1997) (right column, Figure 11). To estimate these initial 742 conditions, they computed the static stress change in a planar fault associated with the total slip 743 found by Wald and Heaton (1994) (Figure 12a), reversed its sign and added a homogeneous 744 tectonic field of 5 MPa. Peyrat et al. (2001) has also considered a constant static fault strength s 745 = 0*s = 12.5 MPa, d = 0.0, and upper bounded 0 so that it is lower than 0.95*s everywhere on 746 the fault. We proceed the same way except in the uppermost 2.5 km, where we obtained 747 unreasonable large slips in our first simulations. For keeping realistic the simulation results, we 748 thus lowered the upper bound for 0 to 0.9*s in the whole fault and multiplied 0 by a linear 749 taper going from 1 at 2.5 km depth to 0 at the free surface. Instead of searching suitable friction 750 coefficients to explain the observed ground motions as done in previous studies (Aochi and 751 Fukuyama, 2002; Aochi et al., 2003) we simply set them homogeneous over the fault so that both 752 the strength excess and the dynamic stress drop are exactly those considered by Peyrat et al. 753 (2001), except in the shallow part where the taper was applied and within those small regions 754 where the initial shear stress was upper bounded to 0.9*s. As expected when including the real 35 755 fault geometry, to allow spontaneous rupture propagation we had to reduce by a factor of two 756 the fracture energy by setting 0, the stress breakdown slip (i.e. the slip-weakening distance, 757 equation (12)), equal to 40 cm instead of 80 cm as they assumed. Rupture was nucleated in a 7 758 km deep circular patch with 1 km of radius (Olsen et al., 1997) located 67 km from the northern 759 fault edge in the along-strike direction (Wald and Heaton, 1994). To initiate a sustained rupture, 760 we raised the initial shear stress 5% above the fault strength in that patch. As a consequence, an 761 initial stress-drop-kick of about 0.6 MPa initiated the earthquake. Figure 11 shows a series of 762 snapshots of the slip rate (left column) and shear stress (right column) on the fault during 763 rupture propagation. Different and interesting rupture patterns may be seen in that figure, as 764 reflected waves in the layers interfaces (2.9 s snapshot), rupture front jumps (5.1 s snapshot) 765 and bifurcations (7.9 s snapshot) and supershear fault segments (11.3 s snapshot), among 766 others. 767 768 (Figure 11) 769 770 Previous models of the Landers earthquake have shown that considering both the fault system 771 geometry and the heterogeneities of the surrounding medium is critical to explain different 772 geophysical observations (e.g. Fialko, 2004; Cianetti et al., 2005; Cruz-Atienza, 2006). Our results 773 (Figures 11 and 12) lead us to the same conclusion. If fault geometry did not play a major role 774 during the earthquake, the initial stress conditions we have used (by Peyrat et al.), which were 775 determined assuming a planar rupture surface (Olsen et al., 1997), would have been valid over 776 the real fault-system geometry. Instead, the fault geometrical barriers (i.e. kinks) strongly 777 affected the energy budget, making the spontaneous rupture propagation more difficult. Since 36 778 we had to reduce by a factor of two the fracture energy to allow rupture propagation along the 779 entire fault system, then half of the amount of energy that Peyrat et al. found to be dissipated 780 through cohesive forces without including dynamic effects on the fault strength (via the normal 781 tractions), rather seems to be related with other physical mechanisms promoting energy 782 leakage, such as off-fault anelastic processes (Duan and Day, 2008) or high frequency radiation 783 associated with changes of rupture speed and direction in the vicinity of the fault kinks (Adda- 784 Bedia and Madariaga, 2008). 785 786 Figure 12 finally shows a comparison between the final slip determined by Wald and Heaton 787 (1994) and the one yielded by our simulation. There is clearly an underestimation of the seismic 788 moment in our model, which is based on the static stress change produced by the final slip of 789 W&H in a planar fault (Olsen et al., 1997). Based on our simulation results we believe that, 790 because of the particularly intricate geometry of the fault system, a more realistic dynamic 791 model of the Landers earthquake requires to include the real rupture-surface geometry. 792 793 (Figure 12) 794 795 6. Conclusions 796 797 In this work we have introduced a novel discontinuous Galerkin method, the DGCrack, to 798 simulate the dynamic rupture propagation of earthquakes in 3D along faults with intricate 799 geometries (e.g. non-planar). The method is hp-adaptive, which means that the elements of the 800 unstructured tetrahedral mesh discretizing the simulation domain, may adapt both their sizes 37 801 (h-adaptivity) and approximation orders (p-adaptivity) depending on the problem geometry and 802 the medium properties (i.e. P0, P1 or P2 elements in the wave propagation domain, and P2 803 elements over the fault). To guarantee a high convergence rate, our scheme imposes the 804 dynamic-rupture boundary conditions through ad hoc fluxes across the fault that verify both the 805 jump conditions introduced by Day et al. (2005) and an additional condition forcing the 806 continuity of the fault-normal velocity field. The jump conditions imply the continuity of the 807 tangential fault velocities on the fault nodes where rupture did not happen, and the collinearity 808 of both the fault shear traction and the slip rate in those nodes where rupture has occurred. On 809 the other hand, the additional condition guarantees the numerical stability and accuracy of the 810 fault normal tractions required in the Coulomb slip-dependent friction law. For modeling 811 rupture propagation throughout the interior of every fault element, we have introduced an 812 efficient predictor-corrector scheme on these elements that accurately estimates the shear 813 tractions at every fault node for every time step. 814 815 A convergence analysis based on the SCEC-USGS TPV3 spontaneous-rupture benchmark have 816 revealed power convergence rates of the DGCrack method for three different fault-observable 817 RMS error metrics (Day et al., 2005), with exponents equal to 1.65 for rupture times, 1.52 for 818 final slip and 1.83 for peak slip rates. These estimates are similar to those reported for well- 819 established finite difference (DFM and SGSN), discontinuous Galerkin (ADER-DG) and boundary 820 integral (BI) methods. We have found excellent results (i.e. rupture times and peak slip rate RMS 821 errors smaller than 1% and 10%, respectively) provided that the cohesive zone is resolved by 822 one or more fault elements (i.e. Nc ≥ 1). In TPV3 test case, this condition translates into fault 823 element smaller or equal than ~450 m. Since both the DGCrack and ADER-DG methods share 38 824 many different capabilities linked to the DG approximation, we have assessed the difference in 825 computational cost between them to achieve the same accuracy level for rupture times, and find 826 that DGCrack is about 10 time more efficient than ADER-DG irrespectively of the mesh size if 827 they were run in the same computing platform. 828 829 Since no analytical solution exists for the spontaneous rupture problem, quantitative 830 comparisons between different approximated solutions may give insights about the correctness 831 of the numerical approaches to solve a given problem. We have thus introduced a simple way to 832 assess the similarity among solutions for TPV3 yielded by four different methods, based on the 833 phase and cross-correlation coefficients of slip rate time series. Results of this exercise show that 834 the DGCrack and ADER-DG discontinuous Galerkin solutions are the most similar, although very 835 close to the one yielded by the spectral boundary integral method (i.e. the MDSBI approach). As 836 suggested by J.P. Ampuero (2012), this procedure may be systematically used to quantitatively 837 compare approaches for different rupture problems like those undertaken by the international 838 group of modelers promoted by SCEC-USGS (Harris et al., 2009; see http://scecdata.usc. 839 edu/cvws/). 840 841 To complete the verification of the DGCrack method, we have also solved TPV10 (Harris et al., 842 2009), which consists of a 60o dipping normal fault reaching the free surface. This test case has 843 interesting features that are essential to verify a dynamic rupture model for non-planar faults, 844 because reflected waves in the Earth’s surface are bounced back to the source inducing transient 845 variations of the fault traction vector that significantly affect rupture propagation via the 846 Coulomb failure criterion. Comparison of rupture times and both on-fault and ground-motion 39 847 seismograms with those yielded by the SEM (Kaneko et al., 2008) and FEM (Barall, 2009) 848 approaches reveal a very good overall agreement among all solutions (especially between the 849 DGCrack and FEM solutions), including the strong dynamic perturbations on the fault normal 850 tractions close to the free surface. Numerical oscillations in the DGCrack solution are mainly 851 present after 10 s of the earthquake nucleation, which is probably due to long-range numerical 852 dispersion associated with the centered fluxes across the fault. However, since these oscillations 853 remain reasonable small and we expect them to be even smaller for physically attenuating media 854 (Tago et al., 2010) or different friction laws (e.g. rate- and state-dependent), we have decided not 855 to integrate an artificial viscosity into the scheme. 856 857 We finally applied the DGCrack method to study some aspects of the 1992 Landers earthquake 858 considering a realistic fault-system geometry. Our simulation considered a 1D layered medium 859 and took advantage of the hp-adaptivity by refining the unstructured mesh around the rupture 860 surface and simultaneously adapting the elements sizes to resolve 0.5 Hz waves everywhere in 861 the simulation domain. 48.6% of the 4.48 millions tetrahedra in the mesh are P1 elements and 862 belong to the CPML slab, while the rest (51.4%) are P2 elements and discretize the entire 863 physical domain. Our source model considered a slightly modified version of the heterogeneous 864 initial shear stress determined by Peyrat et al. (2001) in a planar fault from the final slip found 865 by Wald and Heaton (1994). Since we had to reduce by a factor of two the fracture energy with 866 respect to the earthquake model proposed by Peyrat et al. to allow rupture propagation along 867 the entire non-planar fault system, and found a significant seismic moment underestimation, we 868 have concluded that a more realistic dynamic model of the Landers earthquake requires the 869 computation of the initial stress conditions considering the fault-system geometry. 40 870 871 Recent large subduction earthquakes have raised fundamental questions concerning the stability 872 and segmentation of the subduction zones. To study this seismogenic regions accounting for 873 more realistic physical behaviors, introducing into DGCrack both flexible friction laws (e.g. rate- 874 and state-dependent) and the presence of fluids in the fault zone (i.e. thermal pressurization) 875 would be essential in the future. 876 877 7. Acknowledgments 878 879 We especially thank Steven Day for his advise and clarifications regarding some physical model 880 considerations, for corrections of this manuscript, as well as for the cohesive zone resolution 881 data of TPV3. We thank Christian Pelties for fruitful discussions and for providing us the TPV3 882 solutions by the ADER-DG and MDSBI codes. We thank Ana Rocher for drawing Figure 1. We 883 thank Mondher Benjemaa for his comments and experience. Without the Gmsh software this 884 work could not be possible, so we are grateful to its developers. We thank Alfonso Trejo, from 885 HPC-Team, for his outstanding assistance in setting the parallel supercomputer Pohualli, in 886 which all simulations were performed. We also thank Carl Gable for his advise on meshing 887 strategies for the Landers earthquake fault-system. MDSBI solutions have been computed by 888 Gilbert B. Brietzke. This work has been possible thanks to the Mexican “Consejo Nacional de 889 Ciencia y Tecnología” (CONACyT) under the grant number 80205, and partially supported by 890 both the French “Agence Nationale de la Recherche” under the grant ANR-2011-BS56-017 and 891 the European “Marie Curie Actions—International Research Staff Exchange Scheme”, under the 892 grant 295217. 41 893 894 8. References 895 896 Adda-Bedia, M. and R. Madariaga (2008) Seismic Radiation from a Kink on an Antiplane Fault. 897 Bull. Seismol. Soc. Am., Vol. 98, No. 5, pp. 2291–2302, doi: 10.1785/0120080003. 898 899 Ampuero, J. P. 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Suppl., Abstract 1118 S51A-1915, 2010. 1119 1120 Virieux, J., and R. Madariaga (1982). Dynamic faulting studied by a finite difference method, Bull. 1121 Seismol. Soc. Am. 72, 345–369. 1122 51 1123 Wald, D. and Heaton, T. Spatial and temporal distribution of slip for the 1992 Landers, California, 1124 earthquake. Bull. Seismol. Soc. Am., 84, 668 – 691, 1994. 1125 1126 Zienkiewicz, O.C.; Taylor, R.L.; Zhu, J.Z. (2005). Finite Element Method – Its Basis and 1127 Fundamentals (6th Edition), Elsevier. 1128 1129 52 1130 Table 1. On-fault frictional and stress parameters for TPV3. The initial shear stress points to the 1131 along-strike direction. Medium properties outside the fault represent an infinite barrier. Fault Parameters Nucleation Outside Nucleation Static friction coefficient, s 0.677 0.677 Dynamic friction coefficient, d 0.525 0.525 Slip weakening distance, 0 (m) 0.40 0.40 Initial shear stress, 0 (MPa) 81.6 70.0 Initial normal stress, 0 (MPa) 120.0 120.0 1132 1133 1134 1135 1136 Table 2. TPV3 convergence rate exponents for different methods and error metrics. Method Rupture Times Final Slip Peak Slip Rate DGCrack 1.65 1.52 1.83 ADER-DG O31 2.84 0.99 0.80 DFM2 2.96 1.58 1.18 BI2 2.74 1.53 1.19 SGSN3 - 1.63 0.70 1137 1138 1139 1140 53 1141 1142 Table 3. On-fault frictional and stress parameters for TPV10. The initial shear stress points to the 1143 along-dip direction. hd is the along-dip distance measured in meters from the free surface. 1144 Medium properties outside the fault represent an infinite barrier. Fault Parameters Nucleation Outside Nucleation Cohesion, C (MPa) 0.20 0.20 Static friction coefficient, s 0.760 0. 760 Dynamic friction coefficient, d 0.448 0. 448 Slip weakening distance, 0 (m) 0.50 0.50 Initial normal stress, 0 (MPa) 0.007378 x hd 0.007378 x hd Initial shear stress, 0 (MPa) C + 0 (0.0057+s) 0.55 x 0 1145 54 1146 Figure Captions 1147 1148 Figure 1 Second-order (P2) interpolation-order tetrahedra illustrating two mesh elements 1149 shearing the fault surface . Red dots represent the five split nodes (i.e. two collocated nodes, 1150 one per element) lying on the fault from which the fluxes across are computed and the 1151 dynamic-rupture boundary conditions applied. 1152 1153 Figure 2 TPV3 rupture problem geometry. The gray square represents the nucleation patch, and 1154 the PA and PI dots represent the pure-antiplane and pure-inplane observational fault points, 1155 respectively. 1156 1157 Figure 3 Comparison of rupture times for TPV3 yielded by the DGCrack and DFM (Day et al., 1158 2005) methods, both with an effective mesh increment of 50 m (i.e. 100 m fault elements for 1159 DGCrack). 1160 1161 Figure 4 Comparison of on-fault time histories at PI (left column) and PA (right column) for a) 1162 the slip, b) the shear stress, and the slip rate (c and d) produced by four different numerical 1163 methods (inset legend: DFM, Day et al., 2005; ADER-DG and MDSBI, Pelties et al., 2012), all of 1164 them with an effective grid size of 50 m. 1165 1166 Figure 5 DGCrack convergence, accuracy and efficiency analysis based on TPV3 solutions. 1167 Frames a) and c) show regression lines with the power convergence rates for two error metrics 1168 with exponents reported on Table 2 for unstructured meshes, as a function of both the 55 1169 characteristic fault-element size (lower axis) and the cohesive zone resolution (upper axis). 1170 Frame b) presents a cross-correlation based comparison between four different methods (see 1171 text). Frame d) shows CPU total-time regressions yielded by the DGCrack and ADER-DG (Pelties 1172 et al., 2012) methods for the same accuracy range. Since the DGCrack simulations were run in a 1173 faster computing platform, time differences should be divided by a factor of ~3 to obtain actual 1174 values (see text). 1175 1176 Figure 6 Averages of the cross-correlation measurements between pairs of solutions of Figure 1177 5b yielded by four numerical methods. 1178 1179 Figure 7 TPV10 60o dipping normal-fault (orange rectangle) problem geometry discretized with 1180 an unstructured mesh. The CPML slab is discretized by the blue tetrahedra while the physical 1181 domain by the green tetrahedra. The blue square represents the nucleation patch, and the yellow 1182 crosses represent the fault point (FP) and hanging-wall free-surface ground point (GP) where 1183 solutions were obtained and compared (see Figure 9). 1184 1185 Figure 8 Comparison of rupture times for TPV10 yielded by the DGCrack, FEM (Barall, 2009) 1186 and SEM (Kaneko et al., 2008) methods, all of them obtained with an effective mesh increment 1187 over the fault of 100 m. 1188 1189 Figure 9 TPV10 on-fault at FP (left column, see Figure 7) and off-fault at GP (right column, see 1190 Figure 7) waveforms comparison for three methods (inset) with (lower traces) and without 1191 (upper traces) 3 Hz low-pass filtering. 56 1192 1193 Figure 10 1992 Landers earthquake fault-system geometry embedded in a 1D layered medium 1194 (L1, L2 and L3) and discretized with an hp-adaptive tetrahedral mesh (see text). 1195 1196 Figure 11 Landers earthquake simulation results. Slip rate (left column) and shear stress (right 1197 column) snapshots over the non-planar fault system (Figure 10) for different times of the 1198 rupture evolution process. 1199 1200 Figure 12 Final slip for the 1992 Landers earthquake found by Wald and Heaton (1992) 1201 compared with the final slip produced by our DGCrack simulation (see Figure 11). 57 Figures Figure 1 58 Figure 2 Figure 3 59 Figure 4 60 Figure 5 61 Figure 6 Figure 7 62 Figure 8 63 Figure 9 64 Figure 10 65 Figure 11 66 Figure 12 67

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