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
𝜌𝜕𝑡 𝒗 =
∑
𝜕𝜃 (𝑴𝜃 𝝈) + 𝒇
𝜃∈{𝑥,𝑦,𝑧}
6
116
𝜦𝜕𝑡 𝝈 =
∑
𝜕𝜃 (𝑵𝜃 𝒗) + 𝜦𝜕𝑡 𝝈0
𝜃∈{𝑥,𝑦,𝑧}
115
(1)
117
𝑡
118
where 𝒗 = (𝑣𝑥 , 𝑣𝑦 , 𝑣𝑧 ) is the velocity vector and 𝝈 = (𝜔, 𝜔′ , 𝜔′′ , 𝜎𝑥𝑦 , 𝜎𝑥𝑧 , 𝜎𝑦𝑧 )𝑡 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
𝜔 = (𝜎𝑥𝑥 + 𝜎𝑦𝑦 + 𝜎𝑧𝑧 )
3
123
1
𝜔′ = (2𝜎𝑥𝑥 − 𝜎𝑦𝑦 − 𝜎𝑧𝑧 )
3
124
𝜔′′ = 3 (−𝜎𝑥𝑥 + 2𝜎𝑦𝑦 − 𝜎𝑧𝑧 ),
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, 𝜌, and the medium matrix
129
𝜦 = diag (3𝜆+2𝜇 , 2𝜇 , 2𝜇 , 𝜇 , 𝜇 , 𝜇), which is composed by the Lamé parameters 𝜆 and 𝜇. The
130
external forces and the initial stresses are defined as 𝒇 and 𝝈0 , respectively, while the terms 𝑴𝜃
131
and 𝑵𝜃 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 𝐾 elements, so that
137
𝐾
139
Ω ≃ Ωℎ = ∑ 𝐷𝑖
𝑖=1
138
(2)
140
141
where each 𝐷𝑖 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 𝑖-tetrahedron,
145
146
1
𝑛+
2
𝜌𝑖 (𝑰3 ⊗ 𝑲𝑖 )
147
(3)
148
(𝚲𝑖 ⊗ 𝑲𝑖 )
149
𝒗𝑖
𝑛−
− 𝒗𝑖
𝛥𝑡
1
2
=−
𝝈𝑛+1
− 𝝈𝑛𝑖
𝑖
=−
𝛥𝑡
∑ (𝑴𝜃 ⊗ 𝑬𝑖𝜃 )𝝈𝑛𝑖 +
𝜃∈{𝑥,𝑦,𝑧}
𝑘∈𝑁𝑖
1
𝑛+
2
∑ (𝑵𝜃 ⊗ 𝑬𝑖𝜃 )𝒗𝑖
𝜃∈{𝑥,𝑦,𝑧}
1
∑ [(𝑷𝑖𝑘 ⊗ 𝑭𝑖𝑘 )𝝈𝑛𝑖 + (𝑷𝑖𝑘 ⊗ 𝑮𝑖𝑘 )𝝈𝑛𝑘 ]
2
+
1
1
1
𝑛+
𝑛+
∑ [(𝑸𝑖𝑘 ⊗ 𝑭𝑖𝑘 )𝒗𝑖 2 + (𝑸𝑖𝑘 ⊗ 𝑮𝑖𝑘 )𝒗𝑘 2 ]
2
𝑘∈𝑁𝑖
(4)
150
151
where 𝑁𝑖 is the group of adjacent elements to the 𝑖-tetrahedron and ⊗ represents the tensor
152
product. The mass matrix, 𝑲𝑖 , the stiffness matrices, 𝑬𝑖𝜃 , for all 𝜃 ∈ {𝑥, 𝑦, 𝑧}, the flux matrices,
153
𝑭𝑖𝑘 and 𝑮𝑖𝑘 , and the auxiliary flux matrices, 𝑸𝑖𝑘 and 𝑷𝑖𝑘 , 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, 𝒏, 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, 𝒗, over Γ as
202
203
𝒗± (𝑡, 𝒙) = lim 𝒗(𝑡, 𝒙 ± 𝜖𝒏(𝒙)).
𝜖→0
204
205
We define the vector 𝑽 as the velocity discontinuity across Γ, where its fault tangential
206
component, namely the fault slip rate, is
207
208
𝑽 𝑇 ≔ ⟦𝒗𝑇 ⟧ = 𝒗+𝑇 − 𝒗−𝑇 ,
(5)
209
210
and its fault normal component is
211
212
−
𝑽𝑁 ≔ ⟦𝒗𝑁 ⟧ = 𝒗+
𝑁 − 𝒗𝑁 .
(6)
213
214
The slip magnitude at any time 𝑡 is thus defined as the integral of the modulus of 𝑽 𝑇 over time:
215
216
𝑡
𝑈(𝑡) = ∫0 ‖𝑽 𝑇 (𝑠, 𝒙)‖𝑑𝑠.
(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
𝜏𝑐 − ‖𝑻𝑇 ‖ ≥ 0
222
𝜏𝑐 𝑽 𝑇 − 𝑻𝑇 ‖𝑽 𝑇 ‖ = 0,
(8)
(9)
11
223
224
and a third jump condition applied to (6)
225
226
𝑽𝑁 = 0,
(10)
227
228
where 𝑻𝑇 is the fault tangential traction vector, and 𝜏𝑐 is the fault frictional strength. The fault
229
strength is determined by the Coulomb friction law, which depends on the fault normal stress,
230
𝜎𝑁 , the friction coefficient, 𝜇, and the fault cohesion, 𝐶, as
231
232
𝜏𝑐 = 𝐶 − 𝜎𝑁 𝜇.
(11)
233
234
Condition (8) upper bounds the magnitude of 𝑻𝑇 to the fault strength, 𝜏𝑐 , that always acts
235
opposite to sliding on Γ. The second condition (9) forces the slip rate to be parallel to 𝑻𝑇 , 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 𝜇
244
linearly depend on the slip as
245
12
246
𝑈
𝑈
0
0
𝜇(𝑈) = 𝜇0 + (𝜇𝑠 − 𝜇𝑑 ) (1 − 𝛿 ) 𝐻 (1 − 𝛿 ),
(12)
247
248
where 𝜇𝑠 and 𝜇𝑑 are the static and dynamic friction coefficients, respectively, and 𝛿0 is the
249
critical slip weakening distance.
250
251
A series of studies have tried to estimate 𝛿0 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 𝛿0 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 𝛿0 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 𝐷𝑖 . The physical domain, Ω, is then
269
decomposed as follow
270
271
Ω ≃ Ωℎ = ∑𝐾
𝑖=1 𝐷𝑖 such that if Γ ⊂ Ω then {∀𝐷𝑖 ∶ (𝐷𝑖 ∖ 𝑆𝑖 ) ∩ Γ = ∅}
272
273
where each 𝐷𝑖 is a straight-sided tetrahedron with surface 𝑆𝑖 and the union of all 𝐾 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 𝑖-tetrahedron within the plus-side of the domain, and the other to the
284
adjacent 𝑘-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 𝒗±
𝑇 =
288
{𝒗𝑖𝑇 , 𝒗𝑘𝑇 }. 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
𝑸𝑖𝑘 ≠ (𝑷𝑖𝑘 )𝑇 ,
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
𝑺, defined in Benjemaa et al. (2009), and then isolate the updated field values to get the
307
equivalent symmetric system
308
309
1
𝑛+
2
𝒗𝑖
1
𝑛−
2
= 𝒗𝑖
311
+
∆𝑡
(𝑰 ⊗ 𝑲𝑖 )−1 [−
𝜌𝑖 3
+
∑ (𝑴𝜃 ⊗ 𝑬𝑖𝜃 )𝝈𝑛𝑖
𝜃∈{𝑥,𝑦,𝑧}
1
∑ [(𝑷𝑖𝑘 ⊗ 𝑭𝑖𝑘 )𝝈𝑛𝑖 + (𝑷𝑖𝑘 ⊗ 𝑮𝑖𝑘 )𝝈𝑛𝑘 ]]
2
𝑘∈𝑁𝑖
310
(13)
15
312
1
𝑛+
2
𝝈𝑛+1
= 𝝈𝑛𝑖 + ∆𝑡(𝚲𝑠𝑖 ⊗ 𝑲𝑖 )−1 − [ ∑ (𝑵𝜃𝑠 ⊗ 𝑬𝑖𝜃 )𝒗𝑖
𝑖
𝜃∈{𝑥,𝑦,𝑧}
314
+
1
1
1
𝑛+
𝑛+
∑ [(𝑷𝑡𝑖𝑘 ⊗ 𝑭𝑖𝑘 )𝒗𝑖 2 + (𝑷𝑡𝑖𝑘 ⊗ 𝑮𝑖𝑘 )𝒗𝑘 2 ]
2
𝑘∈𝑁𝑖
313
(14)
315
316
where 𝚲si = 𝑺𝚲i , 𝑵sθ = 𝑺𝑵θ and 𝑷𝑡𝑖𝑘 = 𝑺𝑸𝑖𝑘 .
317
318
Since the fluxes across the fault elements 𝑆𝑖Γ = {𝑆𝑖𝑘 ⊆ Γ ∣ 𝑆ik ≔ 𝑆i ∩ 𝑆k } 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 𝓕i and 𝓖i for the
321
velocity and stress schemes, respectively, the system (13) and (14) may be rewritten as:
322
323
1
𝑛+
2
𝒗𝑖
325
1
𝑛−
2
= 𝒗𝑖
+
1
2
+
∆𝑡
(𝑰 ⊗ 𝑲𝑖 )−1 [−
𝜌𝑖 3
∑ (𝑴𝜃 ⊗ 𝑬𝑖𝜃 )𝝈𝑛𝑖
𝜃∈{𝑥,𝑦,𝑧}
∑ [(𝑷𝑖𝑘 ⊗ 𝑭𝑖𝑘 )𝝈𝑛𝑖 + (𝑷𝑖𝑘 ⊗ 𝑮𝑖𝑘 )𝝈𝑛𝑘 ] + 𝛿Γ (𝑰3 ⊗ 𝑮𝑖𝑘 )𝓕𝑛𝑖
𝑘∈𝑁𝑖
𝐷𝑖 ∩𝐷𝑘 ⊈Γ
]
324
(15)
326
𝝈𝑛+1
= 𝝈𝑛𝑖 + ∆𝑡(𝚲𝑠𝑖 ⊗ 𝑲𝑖 )−1 − [ ∑ (𝑵𝜃𝑠 ⊗ 𝑬𝑖𝜃 )𝒗𝑖
𝑖
1
𝑛+
2
𝜃∈{𝑥,𝑦,𝑧}
328
327
+
1
2
∑
1
𝑛+
2
[(𝑷𝑡𝑖𝑘 ⊗ 𝑭𝑖𝑘 )𝒗𝑖
1
𝑛+
𝑛+
+ (𝑷𝑡𝑖𝑘 ⊗ 𝑮𝑖𝑘 )𝒗𝑘 2 ] + 𝛿Γ (𝑰3 ⊗ 𝑮𝑖𝑘 )𝓖𝑖
1
2
𝑘∈𝑁𝑖
𝐷𝑖 ∩𝐷𝑘 ⊈Γ
(16)
329
16
330
where 𝛿Γ is a Kronecker delta which is 1 if 𝑆𝑖Γ ≠ ∅ and 0 otherwise.
331
332
Since the fault boundary conditions (8) and (9) must be applied to the traction vector 𝑻,
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
[(𝑷𝑖𝑘 ⊗ 𝑭𝑖𝑘 )𝝈𝑛𝑖 + (𝑷𝑖𝑘 ⊗ 𝑮𝑖𝑘 )𝝈𝑛𝑘 ] = [(𝑰3 ⊗ 𝑭𝑖𝑘 )𝑻𝑛𝑖 + (𝑰3 ⊗ 𝑮𝑖𝑘 )𝑻𝑛𝑘 ].
1
2
337
338
By imposing continuity of 𝑻𝑖 over the fault element 𝑆𝑖Γ (i.e. 𝑻𝑛𝑖𝑘 = 𝑻𝑛𝑖 = 𝑻𝑛𝑘 ) and assuming the
339
same approximation order in the two tetrahedra sharing that element (i.e. 𝑭𝑖𝑘 = 𝑮𝑖𝑘 ), we set the
340
flux vector across the fault, 𝓕𝑛𝑖 , to the unique traction vector 𝑻𝑛𝑖𝑘 , and define the flux across the
341
fault as
342
343
(𝑰3 ⊗ 𝑭𝑖𝑘 )𝓕𝑛𝑖 = (𝑰3 ⊗ 𝑭𝑖𝑘 )𝑻𝑛𝑖𝑘 .
(17)
344
345
By substituting equation (17) into the velocity scheme (15) and regrouping the terms excluding
346
the flux across the fault on 𝑹𝑛𝑖 , we get
347
348
1
2
𝑛+
𝒗𝑖
1
2
𝑛−
= 𝒗𝑖
+ 𝑹𝑛𝑖 + 𝛿Γ
∆𝑡
𝜌𝑖
(𝑰3 ⊗ 𝑲𝑖 )−1 (𝑰3 ⊗ 𝑭𝑖𝑘 )𝑻𝑛𝑖𝑘 ,
(18)
349
350
where 𝑹𝑛𝑖 is
351
17
353
𝑹𝑛𝑖 =
∆𝑡
𝜌𝑖
(𝑰3 ⊗ 𝑲𝑖 )−1 − ∑ (𝑴𝜃 ⊗ 𝑬𝑖𝜃 )𝝈𝑛𝑖 +
[
𝜃∈{𝑥,𝑦,𝑧}
1
2
∑ [(𝑷𝑖𝑘 ⊗ 𝑭𝑖𝑘 )𝝈𝑛𝑖 + (𝑷𝑖𝑘 ⊗ 𝑮𝑖𝑘 )𝝈𝑛𝑘 ]
𝑘∈𝑁𝑖
𝐷𝑖 ∩𝐷𝑘 ⊈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 𝑻𝒏𝑖𝑘 has to be decomposed
356
into its tangential, 𝑻𝑛𝑖𝑘𝑇 , and normal, 𝑻𝑛𝑖𝑘𝑁 , components to finally rewrite the velocity scheme (18)
357
as
358
359
1
2
𝑛+
𝒗𝑖
1
2
𝑛−
= 𝒗𝑖
+ 𝑹𝑛𝑖 + 𝛿Γ
∆𝑡
𝜌𝑖
(𝑰3 ⊗ 𝑲𝑖 )−1 (𝑰3 ⊗ 𝑭𝑖𝑘 )(𝑻𝑛𝑖𝑘𝑇 + 𝑻𝑛𝑖𝑘𝑁 ).
(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 𝑻𝑛𝑖𝑘𝑇 such that the jump conditions (8) and (9) are fulfilled, and 𝑻𝑛𝑖𝑘𝑁 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, 𝑻𝑛𝑖𝑘𝑇 , we first notice that the inequality of condition (8)
369
along with the modulus of (9) implies that
370
371
𝑽𝑖𝑘𝑇 = 0,
(20)
372
18
373
which means that, whenever ‖𝑻𝑛𝑖𝑘𝑇 ‖ remains below the frictional strength, 𝜏𝑐 , the tangential
374
traction must ensure the continuity of the tangential velocities at every fault node shared by the
375
𝑖- and 𝑘-tetrahedral.
376
377
Let us assume that 𝑆𝑖Γ ≠ ∅ so 𝛿Γ = 1 in (19), where the 𝑖- and 𝑘-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 𝑲−1
𝑖Γ , which is the inverse mass matrix whose components depend
380
exclusively on those nodes. It is simply constructed from 𝑲−1
by eliminating its rows and
𝑖
381
columns associated with the off-fault nodes.
382
383
To compute 𝑻𝑛𝑖𝑘𝑇 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 𝑻𝑛𝑖𝑘𝑇 , we need to compute the volume and surface
386
integrals of 𝑲𝑖 −1
and 𝑭𝑖𝑘 , respectively, in a standard element (Zienkiewicz et al., 2005) such that
Γ
387
388
1
−1
𝑲−1
𝑖Γ = 𝑉 𝑲𝑒Γ and 𝑭𝑖𝑘 = 𝑆𝑖𝑘 𝑭𝑒
𝑖
(21)
389
390
where 𝑉𝑖 , the i-tetrahedron volume, and 𝑆𝑖𝑘 , 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
𝑛+
1
𝑛+
𝑽𝑖𝑘𝑇2 ≔ 𝒗𝑖𝑇
1
2
𝑛+
1
− 𝒗𝑘𝑇 2
19
395
𝑛−
1
1
= 𝑽𝑖𝑘𝑇2 + 𝑹𝑛𝑖𝑇 − 𝑹𝑛𝑘𝑇 + 𝛥𝑡𝑆𝑖𝑘 (𝜌 𝑉 + 𝜌
𝑖 𝑖
1
𝑘 𝑉𝑘
−1
) (𝑰3 ⊗ 𝑲𝑒Γ ) (𝑰3 ⊗ 𝑭𝑒 )𝑻𝑛𝑖𝑘𝑇 .
(22)
396
397
For getting the tangential traction, we finally use (20) into (22), which leaves us the following
398
expression
399
400
𝑻𝑛𝑖𝑘𝑇
=
𝜌𝜌 𝑉𝑉
( ( 𝑖 𝑘 𝑖 𝑘 )) (𝑰3
𝛥𝑡 𝜌𝑖 𝑉𝑖 +𝜌𝑘 𝑉𝑘
⊗ 𝑭𝑒
)−1
𝑛−
1
2
(𝑰3 ⊗ 𝑲𝑒Γ ) (−𝑽𝑖𝑘 − 𝑹𝑛𝑖𝑇 + 𝑹𝑛𝑘𝑇 ).
𝑇
(23)
401
402
This procedure ensures the continuity of the tangential velocity across the fault. However, if the
403
time-dependent frictional strength, 𝜏𝑐𝑛 , is overcome by the modulus of the tangential traction,
404
‖𝑻𝑛𝑖𝑘𝑇 ‖, 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
𝜏𝑐𝑛 − ‖𝑻𝑛𝑖𝑘𝑇 ‖ = 0 ⇔ ‖𝑻𝑛𝑖𝑘𝑇 ‖ = 𝜏𝑐𝑛 .
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
𝑻𝑛𝑖𝑘𝑇 = {
𝑻𝑛𝑖𝑘𝑇
𝑻𝑛
𝑖𝑘
𝑇
‖𝑻𝑛
𝑖𝑘 ‖
if ‖𝑻𝑛𝑖𝑘𝑇 ‖ < 𝜏𝑐𝑛
𝜏𝑐𝑛
if ‖𝑻𝑛𝑖𝑘𝑇 ‖ ≥ 𝜏𝑐𝑛
(24)
𝑇
414
20
415
Since the nodes within a fault element are coupled through the flux matrix, 𝑭𝑖𝑘 , and the mass
416
matrix, 𝑲𝑖Γ , 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 𝑛𝑓𝑎𝑢𝑙𝑡 is the number of nodes in a given fault
420
element (i.e. six for P2 elements) and 𝑛𝑏𝑟𝑜𝑘𝑒𝑛 is the number of nodes that have broken, then the
421
predictor-corrector scheme operates only if 0 < 𝑛𝑏𝑟𝑜𝑘𝑒𝑛 < 𝑛𝑓𝑎𝑢𝑙𝑡 . The PC scheme will basically
422
find the tangential tractions, 𝑻𝑛𝑖𝑘𝑇 , 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 𝜏𝑐𝑛 (condition (24)). Once applied this
428
condition, the predicted tangential tractions 𝑻𝑛𝑖𝑘𝑇 in the unbroken nodes of the same element
429
must be recomputed accordingly via (23). To this purpose, a new mass matrix, 𝑲𝑢𝑛𝑏𝑟𝑜𝑘𝑒𝑛
, must
𝑒Γ
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 𝜏𝑐𝑛 . 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 𝑛𝑓𝑎𝑢𝑙𝑡 (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, 𝑻𝑛𝑖𝑘𝑁 , 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
𝑛+
1
𝑛+
1
𝑛+
1
𝑽𝑖𝑘𝑁2 ≔ 𝒗𝑖𝑁 2 − 𝒗𝑘𝑁 2
𝑛−
1
1
= 𝑽𝑖𝑘𝑁2 + 𝑹𝑛𝑖𝑁 − 𝑹𝑛𝑘𝑁 + 𝛥𝑡𝑆𝑖𝑘 (𝜌 𝑉 + 𝜌
451
𝑖 𝑖
1
𝑘 𝑉𝑘
−1
) (𝑰3 ⊗ 𝑲𝑒Γ ) (𝑰3 ⊗ 𝑭𝑒 )𝑻𝑛𝑖𝑘𝑁 .
452
453
Using condition (10) to force continuity of the normal velocity, we then define the fault normal
454
traction
455
456
𝑻𝑛𝑖𝑘𝑁
𝜌𝑖 𝜌𝑘 𝑉𝑖 𝑉𝑘
= (𝛥𝑡(𝜌 𝑉 +𝜌
𝑖 𝑖
𝑘 𝑉𝑘 )
) (𝑰3 ⊗ 𝑭𝑒
)−1
(𝑰3 ⊗ 𝑲𝑒Γ ) (−𝑽
1
2
𝑖𝑘𝑁
𝑛−
− 𝑹𝑛𝑖𝑁 + 𝑹𝑛𝑘𝑁 ),
(25)
457
458
from which the fault normal stress is given by
22
459
460
𝜎𝑁𝑛 = 𝑻𝑛𝑖𝑘𝑁 ∙ 𝒏.
461
462
The frictional strength, 𝜏𝑐𝑛 , can now be computed in every fault node using (11) as a function of
463
both 𝜎𝑁𝑛 and the friction coefficient, 𝜇, 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
𝑛+
1
471
For the stress scheme and within the i-tetrahedron, the flux across the fault, 𝓖𝑖 2 , is computed
472
using only the velocity field in that tetrahedron, so that the flux is given by
473
474
𝑛+
𝓖𝑖
1
2
𝑛+
= 𝑷𝑡𝑖𝑘 𝒗𝑖
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 𝑻𝑛𝑖𝑘𝑁 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
𝑢𝑝𝑝𝑒𝑟
𝐶𝐶𝑀𝑖𝑗
=
𝑐𝑐𝑖𝑗 −𝑐𝑐𝑚𝑖𝑛
1−𝑐𝑐𝑚𝑖𝑛
,
553
554
where 𝑐𝑐𝑖𝑗 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
𝑙𝑜𝑤𝑒𝑟
𝐶𝐶𝑀𝑖𝑗
=
𝑑𝑡𝑖𝑗 −𝑑𝑡𝑚𝑎𝑥
𝑑𝑡𝑚𝑎𝑥
,
559
560
where 𝑑𝑡𝑖𝑗 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 𝐶𝐶𝑀𝑢𝑝𝑝𝑒𝑟 in Figure 5b), the smallest phase error is found between the DGCrack
567
and MDSBI solutions (see 𝐶𝐶𝑀𝑙𝑜𝑤𝑒𝑟 ). 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. 𝑁𝑐 ≥ 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. (2002) Etude physique et numérique de la nucléation des séismes. Ph.D. thesis,
900
Université Paris VII.
901
902
Ampuero, J.P. (2012) SCEC-USGS Dynamic Rupture Code Validation Workshop. University of
903
Southern California, February 6.
904
905
Andrews, D. J. (1976). Rupture velocity of plane-strain shear cracks, J. Geophys. Res. 81, 5679–
906
5687.
907
908
Andrews, D.J., 1985. Dynamic plane-strain shear rupture with a slipweakening friction law
909
calculated by a boundary integral method, Bull. seism. Soc. Am., 75, 1–21.
910
911
Aochi, H., Fukuyama, E. and Matsu’ura, M. Spontaneous rupture propagation on a non-planar
912
fault
913
doi:10.1007/PL00001072, 2000.
in
3-D
elastic
medium,
Pure
appl.
Geophys.,
157(11-12),
2003–2027,
914
915
Aochi, H., and E. Fukuyama, Three-dimensional nonplanar simulation of the 1992 Landers
42
916
earthquake, J. Geophys. Res., 107(B2), 2035, doi:10.1029/2000JB000061, 2002.
917
918
Aochi, H., R. Madariaga, and E. Fukuyama, Constraint of fault parameters inferred from
919
nonplanar fault modeling, Geochem. Geophys. Geosyst., 4(2), 1020, doi:10.1029/2001GC000207,
920
2003.
921
922
Barall, M. A grid-doubling finite-element technique for calculating dynamic three-dimensional
923
spontaneous rupture on an earthquake fault. Geophysical Journal International, 178: 845–859.
924
doi: 10.1111/j.1365-246X.2009.04190.x, 2009.
925
926
Benjemaa M., N. Glinsky-Olivier, V.M. Cruz-Atienza and J.Virieux. 3D Dynamic rupture
927
simulations by a finite volume method. Geophys. J. Int., doi: 10.1111/j.1365-246X. 2009.04088.x,
928
2009.
929
930
Benjemaa M., N. Glinsky, Cruz-Atienza V.M., J. Virieux and S. Piperno. Dynamic non-planar crack
931
rupture by a finite-volume method. Geophys. J. Int., 171, 271-285, doi:10.1111/j.1365-
932
246X.2006.03500.x, 2007.
933
934
Brossier, R., S. Operto and J. Virieux (2009) Seismic imaging of complex onshore structures by 2D
935
elastic frequency-domain full-waveform inversión. Geophysics, 74, doi:10.1190/1.3215771.
936
937
Campillo M. and R. Archuleta. A rupture model for the 28 June 1992 Landers, California,
938
earthquake. Geophys. Res. Lett., 20, 647 – 650, 1993.
43
939
940
Cianetti, S., C. Giunchi, and M. Cocco. Three-dimensional finite element modeling of stress
941
interaction: An application to Landers and Hector Mine fault systems, J. Geophys. Res., 110,
942
B05S17, doi:10.1029/2004JB003384, 2005.
943
944
Cochard, A. and Madariaga, R., 1994. Dynamic faulting under rate-dependent friction. Pure appl.
945
Geophys., 142, 419–445.
946
947
Cruz-Atienza, V. M. and J. Virieux (2004). Dynamic rupture simulation of non-planar faults with a
948
finite-difference
949
246X.2004.02291.x
approach,
Geophys.
J.
Int.
158,
939–954,
doi:
10.1111/j.1365-
950
951
Cruz-Atienza, V. M. (2006). Rupture dynamique des faille non-planaires en différences finies:
952
Ph.D. thesis, p. 382, University of Nice – Sophia Antipolis.
953
954
Cruz-Atienza, V. M., J. Virieux, and H. Aochi (2007). 3D finite-difference dynamic-rupture
955
modelling along non-planar faults, Geophysics 72, SM123, doi 10.1190/1.2766756.
956
957
Cruz-Atienza, V.M., Olsen, K.B., Dalguer, L.A., (2009). Estimation of the breakdown slip from
958
strong-motion seismograms: insights from numerical experiments. Bull. Seismol. Soc. Am. 9,
959
3454–3469. doi:10.1785/0120080330.
960
961
Cruz-Atienza V.M. and K. B. Olsen, (2010). Supershear Mach-Waves Expose the Fault Breakdown
44
962
Slip. Special Issue in 'Earthquake Supershear Rupture Speeds', ed. S. Das and M. Bouchon,
963
Elsevier, Tectonophysics, 493, 285-296, doi:10.1016/j.tecto. 2010.05.012.
964
965
Dalguer, L. A., and S. Day (2007). Staggered-grid split-node method for spontaneous rupture
966
simulation, J. Geophys. Res. 112, B02302, doi 10.1029/2006JB004467.
967
968
Das, S. and Aki, K., 1977. Fault plane with barriers: a versatile earthquake model, J. geophys. Res.,
969
82, 5658–5670.
970
971
Day, S. M. (1982). Three-dimensional simulation of spontaneous rupture: The effect of
972
nonuniform prestress, Bull. Seismol. Soc. Am. 72, 1881–1902.
973
974
Day, S. M., L. A. Dalguer, N. Lapusta, and Y. Liu (2005). Comparison of finite difference and
975
boundary integral solutions to three-dimensional spontaneous rupture, J. Geophys. Res. 110,
976
B12307, http://dx.doi.org/10.1029/2005JB003813.
977
978
de la Puente, J., J.-P. Ampuero, and M. Kaser (2009), Dynamic rupture modeling on unstructured
979
meshes using a discontinuous Galerkin method, J. Geophys. Res., 114, B10,302, doi:
980
doi:10.1029/2008JB006271.
981
982
Dieterich, J. (1979). Modeling of rock friction. 1. Experimental results and constitutive equations.
983
J. Geophys. Res. 84, 2161-2168.
984
45
985
Duan, B. and S. Day (2008). Inelastic strain distribution and seismic radiation from ruptura of a
986
fault kink. J. Geophys. Res., VOL. 113, B12311, doi:10.1029/2008JB005847.
987
988
Dumbser, M., and M. Kaser (2006), An Arbitrary High Order Discontinuous Galerkin Method for
989
Elastic Waves on Unstructured Meshes II: The Three-Dimensional Isotropic Case, Geophysical
990
Journal International, 167(1), 319-336.
991
992
Ely, G., S.M. Day, and J-B. Minster (2009). A support-operator method for 3D rupture dynamics,
993
Geophys. J. Int. , 177, pp. 1140-1150, DOI: 10.1111/j.1365-246X.2009.04117.x
994
995
Etienne, V., E. Chaljub, J. Virieux and N. Glinsky (2010) An hp-adaptive Discontinuous Galerkin
996
finite-element method for 3-D elastic wave modelling. Geophys. J. Int. (2010) 183, 941–962. doi:
997
10.1111/j.1365-246X.2010.04764.x.
998
999
Festa, G., and J.-P. Vilotte (2006), Influence of the rupture initiation on the intersonic transition:
1000
Crack-like
versus
pulse-like
1001
doi:10.1029/2006GL026378.
modes,
Geophys.
Res.
Lett.,
33,
L15320,
1002
1003
Fialko, Y., Probing the mechanical properties of seismically active crust with space geodesy:
1004
Study of the co-seismic deformation due to the 1992 Mw7.3 Landers (southern California)
1005
earthquake, J. Geophys. Res., 109, 10.1029/2003JB002,756, 2004.
1006
1007
Geubelle, P., and J. Rice (1995), A spectral method for threedimensional elastodynamic fracture
46
1008
problems, J.Mech. Phys. Solids, 43, 1791–1824.
1009
1010
Geuzaine, C. and J.-F. Remacle (2009). Gmsh: a three-dimensional finite element mesh generator
1011
with built-in pre- and post-processing facilities. International Journal for Numerical Methods in
1012
Engineering, Volume 79, Issue 11, pages 1309-191.
1013
1014
González-Casanova Henríquez, P. (2006). Solución Numérica de Leyes de Conservación
1015
Hiperbólicas. Notas de Modelación y Métodos Numéricos I, CIMAT y CIMNE. ISBN 978-84-96736-
1016
15-3.
1017
1018
Harris, R. A., and S. M. Day (1993), Dynamics of fault interaction: parallel strike-slip faults, J.
1019
Geophys. Res., 98, 4461– 4472.
1020
1021
Harris R.A., M. Barall, R. Archuleta, E. Dunham, B. Aagaard, J.P. Ampuero, H. Bhat, V.M. Cruz-
1022
Atienza, L. Dalguer, P. Dawson, S. Day, B. Duan, G. Ely, Y. Kaneko, Y. Kase, N. Lapusta, Y. Liu, S. Ma,
1023
D. Oglesby, K. Olsen, A. Pitarka, S. Song and E. Templeton (2009). The SCEC/USGS Dynamic
1024
Earthquake-Rupture
1025
doi:10.1785/gssrl.80.1.119.
Code
Verification
Exercise.
Seismol.
Res.
Lett.,
80,
119-126,
1026
1027
Hernandez, B., F. Cotton and M. Campillo. Contribution of radar interferometry to a two-step
1028
inversion of kinematic process of the 1992 Landers earthquake. J. Geophys. Res., 104, 13083 –
1029
13099, 1999.
1030
47
1031
Hesthaven, J. S. and T. Warburton, (2008) Nodal Discontinuous Galerkin Methods: Algorithms,
1032
Analysis, and Applications. Springer Texts in Applied Mathematics 54, Springer Verlag, New
1033
York.
1034
1035
Hok S. and E. Fukuyama. A new BIEM for rupture dynamics in half-space and its application to
1036
the 2008 Iwate-Miyagi Nairiku earthquake. Geophys. J. Int., 184, 301–324 doi: 10.1111/j.1365-
1037
246X.2010.04835.x, 2011.
1038
1039
Ide, S. and Takeo, M., (1997). Determination of constitutive relations of fault slip based on
1040
seismic waves analysis. J. Geophys. Res. 102 (27), 379–27,391.
1041
1042
Kame, N. and Yamashita, T., 1999. Simulation of the spontaneous growth of a dynamic crack
1043
without constraints on the crack tip path, Geophys. J. Int., 139(2), 345–358, doi:10.1046/j.1365-
1044
246x.1999.00940.x.
1045
1046
Kase, Y., and S. M. Day (2006). Rupture Processes on a bending fault, Geophys. Res. Lett. , Vol. 33,
1047
L10601, 10.1029/2006GL025865.
1048
1049
Kaneko, Y., N. Lapusta, and J.-P. Ampuero (2008) Spectral element modeling of spontaneous
1050
earthquake rupture on rate and state faults: Effect of velocity-strengthening friction at shallow
1051
depths. Journal of Geophysical Research, v. 113, B09317, doi:10.1029/2007JB005553.
1052
1053
Knopoff, L. and Ni, X.X., 2001. Numerical instability at the edge of a dynamic fracture, Geophys. J.
48
1054
Int., 147, F1–F6.
1055
1056
Kozdon J. E., E. M. Dunham and Jan Nordstrom. Simulation of Dynamic Earthquake Ruptures in
1057
Complex Geometries Using High-Order Finite Difference Methods, Journal of Scientific
1058
Computing. Submitted 2011.
1059
1060
Lapusta, N., Rice, J.R., Ben-Zion, Y. and Zheng, G., 2000. Elastodynamic analysis for slow tectonic
1061
loading with spontaneous rupture episodes on faults with rate- and state-dependent friction. J.
1062
geophys. Res., 105(B10), 23 765–23 790.
1063
1064
LeVeque, R. J. (2002) Finite Volume Methods for Hyperbolic Problems, Cambridge University
1065
Press, ISBN 0-521-81087-6.
1066
1067
Ma, S., and R. J. Archuleta (2006), Radiated seismic energy based on dynamic rupture models of
1068
faulting, J. Geophys. Res., 111, B05315, doi:10.1029/2005JB004055.
1069
1070
Madariaga, R., K. B. Olsen, and R. Archuleta (1998). Modeling dynamic rupture in a 3D
1071
earthquake fault model, Bull. Seismol. Soc. Am. 88, 1182–1197.
1072
1073
Mikumo, T., Yagi, Y., (2003). Slip-weakening distance in dynamic rupture of in-slab normal-
1074
faulting earthquakes. Geophys. J. Int. 155, 443–455.
1075
1076
Miyatake T. (1980). Numerical simulations of earthquake source process by a three- dimensional
49
1077
crack model. part I. rupture process. J. Phys. Earth, 28(6):565–598.
1078
1079
Moczo P., J. Kristek, M. Galis, P. Pazak and M. Balazovjech. The finite-difference and finite-element
1080
modeling of seismic wave propagation and earthquake motion. Acta Physica Slovaca, vol. 57, No.
1081
2, 177 – 406, 2007.
1082
1083
Oglesby, David D., Ralph J. Archuleta and Stefan B. Nielsen (1998) Earthquakes on Dipping
1084
Faults: The Effects of Broken Symmetry. Science, Vol. 280 no. 5366 pp. 1055-1059, DOI:
1085
10.1126/science.280.5366.1055
1086
1087
Oglesby, D. D., and S. M. Day, The effect of fault geometry on the 1999 Chi-Chi (Taiwan)
1088
earthquake, Geophys. Res. Lett., 28, 1831 – 1834, 2001.
1089
1090
Olsen, K. B., R. Madariaga, R. and R. Archuleta. Three-dimensional dynamic simulation of the
1091
1992 Landers earthquake. Science, 278, 834 – 838, 1997.
1092
1093
Pelties
C.,
J.
de
la
Puente,
J.-P.
Ampuero,
G.
Brietzke
and
M.
Kaeser
(2012)
1094
Three-dimensional dynamic rupture simulation with a high-order Discontinuous Galerkin
1095
method on unstructured tetrahedral meshes, J. Geophys. Res., doi:10.1029/ 2011JB008857. (in
1096
press).
1097
1098
Remaki, L, O. Hassan and K. Morgan (2011) Aerodynamic Computations Using a Finite Volume
1099
Method with an HLLC Numerical Flux Function. In Mathematical Modelling of Natural
50
1100
Phenomena, Cambridge University Press, 6 (03) : pp 189-212.
1101
1102
Peyrat, S., K. B. Olsen, and R. Madariaga (2001). Dynamic modeling of the 1992 Landers
1103
earthquake, J. Geophys. Res. 106, 26,467–26,482.
1104
1105
Rojas O. E. M. Dunham, S. M. Day, L. A. Dalguer and J. E. Castillo. Finite difference modelling of
1106
rupture
1107
10.1111/j.1365-246X.2009.04387.x, 2009.
propagation with strong velocity-weakening friction.
Geophys. J.
Int. doi:
1108
1109
Ruina, A., (1983). Slip instability and state variable friction laws, J. geophys. Res., 88, 10 359–10
1110
370.
1111
1112
Spudich, P., Guatteri, M., (2004). The effect of bandwidth limitations on the inference of
1113
earthquake slip-weakening distance from seismograms. Bull. Seismol. Soc. Am. 94, 2028–2036
1114
1115
Tago J., Cruz-Atienza V. M., Etienne V., Virieux J., E. Chaljub, M. Benjemaa and Sánchez-Sesma F. J.
1116
3D dynamic rupture with anelastic wave propagation using an hp-adaptive Discontinuous
1117
Galerkin method. American Geophysical Union, Eos Trans. AGU, 91, Fall Meet. 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