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We derive a non-hypersingular boundary integral equation, in a fully explicit form, for the time-domain analysis of the dynamics
of a 3-D non-planar crack, located in an infinite homogeneous isotropic medium. The hypersingularities, existent in the more
straightforward expression, are removed by way of a technique of regularization based on integra...
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Context 1
... Du i nY s the slip across the crack in the i-th direction at location n and time s as de®ned by the relative displacement of one (positive) side of C with reference to the other (negative) side, c ijpq the elastic constants, and n n the unit vector normal to the crack surface at location n pointing from the nega- tive to the positive side of C (Fig. 1). G kp xY t À sY nY 0 is the displacement Green's function, representing the displacement in the k-th direction at receiver point x and time t À s due to a unit force in the p-th direction applied at source point n and time 0. Summation over repeated indices is implied and d ab denotes the Kro- necker's ...
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Citations
... Taking the spatial deri v ati ve of eq. ( 33 ) and placing the receiver point on fault surface, the BIEs on fault surface are obtained: t − τ resulting from a slip at a unit speed of source point ξ at time 0 at direction k . Detailed deri v ation steps from eq. ( 33 ) to eq. ( 34 ), as well as the explicit expression for ˆ K i jk , can be found in Tada et al. ( 2000 ) and Tada ( 2006 ). The form of eq. ( 34 ) is the same as that in elastic mediums, only the integral kernels need to be replaced by the integral kernels in poroelastic mediums. ...
... With the modified governing equations, the conventional BIEs for the elastic medium are extended to the poroelastic case by expressing the pore pressure as a function of total stress, resulting in the BIEs of ef fecti ve stress tensors. We use triangular discretization and previous solutions from Tada ( 2000Tada ( , 2006 and Feng & Zhang ( 2017 ) to obtain the solutions of the integral kernels. Consequently, we establish a BIE method for simulating spontaneous rupture in fluid-saturated media. ...
Plenty of studies have suggested that pore fluid may play an important role in earthquake rupture processes. Establishing numerical models can provide great insight into how pore fluid may affect earthquake rupture processes. However, numerical simulation of 3D spontaneous ruptures in poroelastic mediums is still a challenging task. In this article, it is found that a closed-form time-domain Green’s function of Biot’s poroelastodynamic model can be constructed when the source frequency and source-field distance are within a certain range. The time-domain Green’s function is validated by being transformed into the frequency domain and comparing with the frequency-domain Green’s functions obtained by former papers. Poroelastic wave propagation phase diagrams for various two-phase poroelastic mediums are then plotted to show the applicable range of frequency and source-field distance for the new time-domain Green’s function. It is shown that the applicable range not only include the frequency and spatial range of concern in seismology but also overlap that in acoustics. Based on the time-domain Green’s function, the boundary integral equations for modeling dynamic ruptures in elastic mediums are extended to fluid-saturated mediums. In the meantime, a functional relationship between the effective stress tensor and the total stress tensor in fluid-saturated mediums is also obtained, which allows us to directly obtain the effective stress by boundary integral equations. The spontaneous rupture processes controlled by the slip-weakening friction law on faults in elastic mediums and in fluid-saturated mediums are compared. It is found that under the same conditions, fluid-saturated rocks are more prone to supershear rupture than dry rocks. This result suggests that pore fluid may promote the excitation of supershear rupture. The poroelastic wave propagation phase diagrams also suggest that simulating a coseismic phase in the real scale requires a certain sample length in laboratories. They also suggest that an undrained governing equation is suitable for seismic wave propagation simulation in poroelastic media.
... Due to the wide variability in time scales, the acceleration term in the elastodynamic relations is often neglected, hence seismic waves are not modelled and transient wave-mediated stress transfer is ignored. The radiation damping part of the inertia term represents the outflow of energy due to the instantaneous fault response to slip (Cochard & Madariaga 1994;Tada et al. 2000). ...
... We note that the radiation damping term appears naturally when deriving non hyper-singular boundary element methods (Cochard & Madariaga 1994;Perrin et al. 1995;Geubelle & Rice 1995;Tada et al. 2000) and when considering the exact Riemann problem at the fault, which can be used to implement provable stable fault boundary conditions for the fully dynamic equations (Duru & Dunham 2016; Uphoff 2020). ...
Physics-based simulations provide a path to overcome the lack of observational data hampering a holistic understanding of earthquake faulting and crustal deformation across the vastly varying space-time scales governing the seismic cycle. However, simulations of sequences of earthquakes and aseismic slip (SEAS) including the complex geometries and heterogeneities of the subsurface are challenging. We present a symmetric interior penalty discontinuous Galerkin (SIPG) method to perform SEAS simulations accounting for the aforementioned challenges. Due to the discontinuous nature of the approximation, the spatial discretisation natively provides a means to impose boundary and interface conditions. The method accommodates two- and three-dimensional domains, is of arbitrary order, handles sub-element variations in material properties and supports isoparametric elements, i.e., high-order representations of the exterior boundaries, interior material interfaces and embedded faults. We provide an open-source reference implementation, Tandem, that utilises highly efficient kernels for evaluating the SIPG linear and bilinear forms, is inherently parallel and well suited to perform high resolution simulations on large scale distributed memory architectures. Additional flexibility and efficiency is provided by optionally defining the displacement evaluation via a discrete Green’s function approach, exploiting advantages of both the boundary integral and volumetric methods. The optional discrete Green’s functions are evaluated once in a pre-computation stage using algorithmically optimal and scalable sparse parallel solvers and preconditioners. We illustrate the characteristics of the SIPG formulation via an extensive suite of verification problems (analytic, manufactured, and code comparison) for elastostatic and quasi-dynamic problems. Our verification suite demonstrates that high-order convergence of the discrete solution can be achieved in space and time and highlights the benefits of using a high-order representation of the displacement, material properties, and geometries. We apply Tandem to realistic demonstration models consisting of a 2D SEAS multi-fault scenario on a shallowly dipping normal fault with four curved splay faults, and a 3D intersecting multi-fault scenario of elastostatic instantaneous displacement of the 2019 Ridgecrest, CA, earthquake sequence. We exploit the curvilinear geometry representation in both application examples and elucidate the importance of accurate stress (or displacement gradient) representation on-fault. This study entails several methodological novelties. We derive a sharp bound on the smallest value of the SIPG penalty ensuring stability for isotropic, elastic materials; define a new flux to incorporate embedded faults in a standard SIPG scheme; employ a hybrid multi-level preconditioner for the discrete elasticity problem; and demonstrate that curvilinear elements are specifically beneficial for volumetric SEAS simulations. We show that our method can be applied for solving interesting geophysical problems using massively parallel computing. Finally, this is the first time a discontinuous Galerkin method is published for the numerical simulations of sequences of earthquakes and aseismic slip, opening new avenues to pursue extreme scale 3D SEAS simulations in the future.
... They further pointed out that this inconsistency extends to the previous analytical formulations, such as that of Jeyakumaran & Keer (1994), as well as to their own results. Although Tada & Yamashita (1996) first recognized this problem in two-dimensional cases, Aochi et al. (2000) and Tada et al. (2000) reported that this inconsistency persists in three-dimensional modeling as well. This inconsistency has been considered to be a paradox of elastic fault-modeling (Tada & Yamashita 1996;Aochi et al. 2000), called the "paradox of smooth and abrupt bends" (Tada & Yamashita 1996). ...
... In this paper, we study the paradox of smooth and abrupt bends from both analytical and numerical viewpoints in order to investigate the adequacy of the choice between smooth and discretized faults. In previous studies, the paradox of smooth and abrupt bends was considered by using only the analytically obtained non-hypersingular stress Green's functions, as in Jeyakumaran & Keer (1994); Tada (1996); Tada & Yamashita (1997); Tada et al. (2000). Because of the length of the expressions obtained in those studies, the cause of the paradox of smooth and abrupt bends does not appear clearly or become disentangled in a unified manner. ...
... We assume small displacements, and express the fault as the buried boundary that constitutes the interface between two sufficiently adjacent faces (Aki & Richards 2002, p38). The scope of our derivation includes those of the previous studies that have obtained non-hypersingular stress Green's functions for isotropic elasticity (e.g., Tada & Yamashita 1997;Tada et al. 2000). We consider the three-dimensional dynamic case, as it reduces to the other cases (static or two-dimensional) in certain limits. ...
In a dislocation problem, a paradoxical discordance is known to occur between an original smooth curve and an infinitesimally discretized curve. To solve this paradox, we have investigated a non-hypersingular expression for the integral kernel (called the stress Green's function) which describes the stress field caused by the displacement discontinuity. We first develop a compact alternative expression of the non-hypersingular stress Green's function for general two- and three-dimensional infinite homogeneous elastic media. We next compute the stress Green's functions on a curved fault and revisit the paradox. We find that previously obtained non-hypersingular stress Green's functions are incorrect for curved faults although they give correct answers when the fault geometries are discretized with flat elements.
... we can derive the boundary-integral equations after some algebra (Tada and Yamashita (1997); Tada et al. (2000)): ...
... (3) by using integration by parts (Sladek and Sladek (1984); Bonnet (1999) p 175). More details are given in appendix A. For the connection with the local coordinate system that has frequently been adopted in the rock mechanics literature (e.g., Tada and Yamashita (1997); Tada et al. (2000)) please refer to Sato et al. 2019. ...
... The work of Tada and Yamashita (1997); Tada et al. (2000) was done independently of the regularization process using the tangential differential operator. They regularized the boundary-integral equations by projecting the derivative into the local coordinate system and then performing an integration by parts. ...
Many recent studies have tried to determine the influence of geometry of faults in earthquake mechanics. In this paper, we suggest a new interpretation of the effect of geometry on the stress on a fault. Starting from the representation theorem, which links the displacement in a medium to the slip distribution on its boundary, and assuming homogeneous infinite medium, a regularized boundary-element equation can be obtained. Using this equation, it is possible to separate the influence of geometry, as expressed by the curvatures and torsions of the field line of a dislocation on the fault surface, which multiply the slip, from the effect of the gradient of slip. This allows us to shed new light on the mechanical effects of geometrical complexities on the fault surface, with the key parameters being the curvatures and torsions of the slip field on the fault surface. We have used this new approach to explain further the false paradox between smooth-and-abrupt-bends (see Sato et al. (2019)) as well as to re-interpret the effect of roughness on a fault.
... The kernel K has preferable natures ascribable to those of the Green's function of the medium, later mentioned in §2.2. The explicit forms of K [29] are not relevant for the algorithms proposed in this paper. We use the discretized expression of Eq. (1) for numerical analyses, as in previous studies for both two-dimensional (2D) problems [28] and three-dimensional (3D) problems [23]. ...
We present a fast algorithm with small memory storage to compute the spatiotemporal boundary integral equation method (ST-BIEM) particularly for the elastodynamic problem. The time complexity of the spatiotemporal convolution and memory consumption to store the integral kernel and convolved variables are originally of $\mathcal O(N^2M)$ in ST-BIEM for a given number of discretized fault elements $N$ and time steps $M$. Such huge costs of ST-BIEM are reduced to be of $\mathcal O(N \log N)$ by our methods, called the fast domain partitioning hierarchical matrices (FDP=H-matrices). FDP=H-matrices are natural extensions of the fast domain partitioning method (FDPM) and the hierarchical matrices (H-matrices), combined with newly developed two algorithms. After developing new methods, we test the cost and accuracy of FDP=H-matrices both analytically and numerically.
... In the medium of interest, analytical solutions exist for Equation (2) and therefore one can rewrite it in the form of integral eqation. The stress field at any position (x) in the medium (V ) including the interface (Σ) is [18] ...
A problem of quasi-static growth of an arbitrary shaped-crack along an interface requires many times of iterations not only for finding a spatial distribution of discontinuity but also for determining the crack tip. This is crucial when refining model resolution and also when the phenomena progresses quickly from one step to another. We propose a mathematical reformulation of the problem as a nonlinear equation and adopt different numerical methods to solve it efficiently. Compared to a previous work of the authors, the resulting code shows a great improvement of performance. This gain is important for further application of aseismic slip process along the fault interface, in the context of plate convergence as well as the reactivation of fault systems in reservoirs.
... where G is a response function of the medium (Green's function). In this study, we assume the fault plane is embedded in a homogeneous, infinite elastic medium, such that the discrete forms of G are analytically defined (Tada et al. 2000). The contribution of w pl over the whole infinite fault plane is zero, indicating a rigid body movement. ...
Quasi-static numerical simulations of slip along a fault interface characterized by multiscale heterogeneity (fractal patch model) are carried out under the assumption that the characteristic distance in the slip-dependent frictional law is scale-dependent. We also consider slip-dependent stress accumulation on patches prior to the weakening process. When two patches of different size are superposed, the slip rate of the smaller patch is reduced when the stress is increased on the surrounding large patch. In the case of many patches over a range of scales, the slip rate on the smaller patches becomes significant in terms of both its amplitude and frequency. Peaks in slip rate are controlled by the surrounding larger patches, which may also be responsible for the segmentation of slip sequences. The use of an explicit slip-strengthening-then-weakening frictional behavior highlights that the strengthening process behind small patches weakens their interaction and reduces the peaks in slip rate, while the slip deficit continues to accumulate in the background. Therefore, it may be possible to image the progress of slip deficit at larger scales if the changes in slip activity on small patches are detectable.
... Recently, Aochi (1999), Aochi et al. (2000a) and Tada et al. (2000) acquired explicit expressions for the stress Green's functions representing the transitory response of a 3-D infinite, homogeneous and isotropic medium on arbitrary curved fault. Tada discretized the faults into rectangular-shaped elements with the constant slip rate (Tada 2005) and triangular-shaped elements (Tada 2006) in order to simulate the complex system of the 3-D non-plane faults. ...
We present an equivalent form of the expressions first obtained by Tada (Geophys J Int 164:653–669, 2006. doi:10.1111/j.1365-246X.2006.03868.x), which represents the transient stress response of an infinite, homogeneous and isotropic medium to a constant slip rate on a triangular fault that continues perpetually after the slip onset. Our results are simpler than Tada’s, and the corresponding codes have a higher running speed.
... We compute the spontaneous rupture propagation along the faults with a dynamic approach, by using the numerical code developed by . The code is based on the boundary integral equation method (BIEM, Aochi et al. 2000;Tada et al. 2000), and is able to model 3-D non-planar and non-continuous faults, embedded in a homogeneous elastic medium. The BIEM method has been applied to study large earthquake ruptures (Aochi & Fukuyama 2002;, and dynamic propagation of rupture on several faults (Aochi et al. 2006;Aochi & Ulrich 2015;Douilly et al. 2015). ...
The Corinth rift (Greece) is made of a complex network of fault segments, typically 10–20 km
long separated by stepovers. Assessing the maximum magnitude possible in this region requires
accounting for multisegment rupture. Here we apply numerical models of dynamic rupture
to quantify the probability of a multisegment rupture in the rift, based on the knowledge of
the fault geometry and on the magnitude of the historical and palaeoearthquakes. We restrict
our application to dynamic rupture on the most recent and active fault network of the western
rift, located on the southern coast. We first define several models, varying the main physical
parameters that control the rupture propagation. We keep the regional stress field and stress
drop constant, and we test several fault geometries, several positions of the faults in their
seismic cycle, several values of the critical distance (and so several fracture energies) and two
different hypocentres (thus testing two directivity hypothesis). We obtain different scenarios
in terms of the number of ruptured segments and the final magnitude (between M = 5.8 for
a single segment rupture to M = 6.4 for a whole network rupture), and find that the main
parameter controlling the variability of the scenarios is the fracture energy. We then use a
probabilistic approach to quantify the probability of each generated scenario. To do that, we
implement a logical tree associating a weight to each model input hypothesis. Combining
these weights, we compute the probability of occurrence of each scenario, and show that
the multisegment scenarios are very likely (52 per cent), but that the whole network rupture
scenario is unlikely (14 per cent).
... Thus, it can be difficult to develop well-conditioned multi-block decompositions of geometries that arise in realistic fault systems. Boundary element methods have also been developed (e.g., [4,22,32,64,70]). Solutions given by these methods are limited to faults in a uniform medium and some can develop numerical instabilities. ...
We couple a node-centered finite volume method to a high order finite difference method to simulate dynamic earthquake ruptures along nonplanar faults in two dimensions. The finite volume method is implemented on an unstructured mesh, providing the ability to handle complex geometries. The geometric complexities are limited to a small portion of the overall domain and elsewhere the high order finite difference method is used, enhancing efficiency. Both the finite volume and finite differ- ence methods are in summation-by-parts form. Interface conditions coupling the nu- merical solution across physical interfaces like faults, and computational ones between structured and unstructured meshes, are enforced weakly using the simultaneous- approximation-term technique. The fault interface condition, or friction law, provides a nonlinear relation between fields on the two sides of the fault, and allows for the par- ticle velocity field to be discontinuous across it. Stability is proved by deriving energy estimates; stability, accuracy, and efficiency of the hybrid method are confirmed with several computational experiments. The capabilities of the method are demonstrated by simulating an earthquake rupture propagating along the margins of a volcanic plug.
