Assessment of the finite element solutions for 3D spontaneous rupture using GeoFEM
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2011
Received: 15 October 2010
Accepted: 22 June 2011
Published: 2 December 2014
Numerical simulations of spontaneous shear rupture on a planar fault using a slip-weakening model in a three-dimensional uniform elastic medium were conducted using a parallel FE code, GeoFEM, which was originally developed for the solid-Earth simulations on the Earth Simulator. The aim was to evaluate the accuracy and applicability of GeoFEM to earthquake rupture. The present numerical results are compared with published results obtained using a finite-difference (FD) method and a boundary integral (BI) method for rupture times and time functions of slip, slip rate, and shear stress at two particular points on the fault plane. The effects of mesh size and damping on the attenuation of spurious oscillations were also examined in a range of simulations. The appropriate degree of damping depends on mesh size and must be introduced in order to obtain reliable numerical solutions; weak damping leads to significant oscillation and strong damping to artificially low rupture speeds and low slip rates. Our results indicate that mesh size should be sufficiently small to allow the inclusion of a few grids in the cohesive zone, as shown in other numerical methods. The solutions by GeoFEM that have the appropriate mesh size and damping are quite similar to those obtained using the FD and BI methods, the difference between them being generally less than 2% in terms of the rupture time and 5.2% in terms of the peak slip rate.
Understanding spontaneous rupture is of fundamental importance for understanding the mechanisms involved in earthquakes, and must be investigated numerically because it cannot be solved analytically. FD and BI methods have often been used for the numerical simulation of spontaneous rupture, and the validity and accuracy of these methods as a function of grid size in numerical computations has been extensively studied, for example, by Day et al. (2005) and Dalguer and Day (2006, 2007). At the same time, an alternative and sophisticated simulation method has been developed using a finite element (FE) method, and this has also been used for earthquake rupture problems, as referred to by, for example, Oglesby and Day (2001), Oglesby and Archuleta (2003), Ma et al. (2008), and Barall (2009). The FE method, established using the variational principle, assembles the whole system from individual elements in which the basic shape functions are used to integrate the field variables. Integration in space over the nodes that belong to the different elements is not therefore necessary, and the restriction of using structured grids in FD no longer applies. The FE method is more flexible when dealing with complicated geometries by using an unstructured mesh. Such a feature is attractive in real simulations of rupture that involve complicated fault geometries and heterogeneous materials. Although the FE method has been developed to a considerable degree, and significant achievements have been seen in the fields of analysis of solids and structures, it is important to compare the accuracy of simulations by the FE method with those by the FD and BI methods for simulations of spontaneous rupture (Moczo et al., 2007; Galis et al., 2009, 2010).
We herein describe our numerical simulation of spontaneous shear rupture on a planar fault in a three-dimensional (3D) uniform elastic medium using the FE method. Using the same problem as that of Day et al. (2005), we attempt to evaluate the accuracy of the FE method and quantitatively compare this with that obtained by the FD and BI methods. This problem was originally defined for the SCEC (Southern California Earthquake Center/U.S. Geological Survey) Dynamic Earthquake Rupture Code Verification Exercise (Harris et al., 2009).
We used a parallel FE code, GeoFEM, which was originally developed for the solid-Earth simulations on the Earth Simulator (Iizuka et al., 2000), and has been applied to quasi-static viscoelastic deformation (e.g. Hyodo and Hirahara, 2004). The original GeoFEM was capable of large-scale static structural simulations with a constant frictional contact using a master-slave method. In its use of a master-slave method, our approach is different from the split-node technique (Andrews, 1999) that has been used in fault modeling. In our study, we decided to add some new features to the existing GeoFEM scheme in order to solve dynamic rupture problems. Firstly, we introduced a slip-weakening friction law in order to model the variation in friction that occurs during fault rupture, by which we could vary the friction as a linear function of slip, rather than using a constant value. Secondly, because fault rupture is a nonlinear dynamic problem, a modified version of the conventional Newmark time integration scheme was adopted to ensure the proper conversion of energy and momentum in time for nonlinear dynamic analysis. After giving details of the FE formulation, we present an extensive comparison between the accuracy obtained in our FE simulations and that obtained in the benchmark results of the FD and BI models as reported by Day et al. (2005). Finally, we also describe the results of our investigations into the dependency of the FE results on grid size and damping parameters.
2. Finite Element Formulations
In the following section, we describe the details of our FE formulations. We address the problem of contact using a slip-weakening friction law, we use an energy-momentum-conserving time integration method for dynamic nonlinear analysis, and we select damping parameters that reduce the oscillation of high-frequency modes.
2.1 Contact element with friction
2.2 Dynamic time integration scheme
A particular case of the Newmark method is the trapezoidal rule scheme, which may be used as one of the implicit time integration methods, and is also popular and unconditionally stable for linear dynamic analysis (Zienkiewicz and Taylor, 2002). However, when being applied to nonlinear dynamic problems, such as the dynamic fault rupture discussed herein, the trapezoidal rule scheme becomes rather unstable. Such instability comes from the uncontrolled oscillation of the sum of the potential energy and the momentum. The trapezoidal rule scheme does not guarantee the conservation of energy-momentum throughout the duration of the model run. In order to overcome these adverse characteristics, a generalized energy method, originally proposed by Chung and Hulbert (1993), and systematically reviewed by Kuhl and Crisfield (1999), was adopted and implemented in our own code. In simple terms, the generalized energy method modifies the Newmark method by applying the equation of motion to a general mid-point instead of to the end point. The internal force is then calculated using the displacements at the mid-point. Such a modification leads to a time integration that is based on a low numerical dissipation at lower frequencies and a high numerical dissipation at higher ones, so that the oscillation of the total energy is suppressed. Further details of this method can be found in Appendix B.
2.3 Use of damping
In all our simulations of rupture, we aimed to achieve optimal damping such that the high-frequency modes that generate spurious oscillations were reduced, while, at the same time, minimizing any unfavorable damping effects that might reduce any physically-meaningful high-frequency motion. To this end, we assigned different values of stiffness-proportional damping coefficients to different positions. Further details of this technique are given in the following section.
3. Fault Model Description
3.1 Geometry of fault
Material properties of elastic isotropic space.
Initial values of stresses and fault constitutive parameters.
Within fault 30-by-15 km
Normal stress σ n (MPa)
Initial friction coefficient μ0
Initial shear stress τ0 = μ0σ n (MPa)
Fault constitutive parameters
Static shear friction coefficient μ s
Dynamic friction coefficient μ d
Static shear yielding stress τ d = μ s σ n (MPa)
Dynamic shear stress τ d = μ d σ n (MPa)
Slip distance d0 (m)
3.2 Quasi-static loading procedure for initial conditions
Boundary conditions and static loadings for initial stress states.
Type of boundary conditions
Quasi-static loading step for initial normal/shear stresses
z = +20 km
z = −20 km
y = ±20 km
x = ±30 km, z > 0
u x 0
x = ±30 km, z > 0
−u x 0
3.3 Parameters for dynamic analysis
The dynamic rupture was triggered by a sudden drop in frictional coefficient from μ0 to μ s within the 3 km × 3 km nucleation area. The dynamic rupture then started to propagate in both the x- and y-directions. The time increment for dynamic analysis was chosen as 0.005 s per step. The duration of the whole simulation was set to 7.5 s because by this time the prescribed 30 km × 15 km rupture area was completely ruptured. The stiffness-proportional damping described in the previous section was assigned to the following three parts of the mesh: (1) a layer of elements on either side of the fault; (2) the outermost layer of elements on the constraint boundaries; (3) the rest of the homogeneous medium. We applied damping to Part (1) in an attempt to reduce the spurious oscillations in slip and slip rate, using three different cases of stiffness-proportional damping coefficient α = 5α0, α0 and 0.625α0, where α0 = 1/(100π) = 3.183 × 10−3 (yielding α = 1.592 × 10−2, 3.183 × 10−3 and 1.989 × 10−3), corresponding to damping ratios ξ = 5%, 1% and 0.625%, respectively, at a frequency of 1 Hz from Eq. (14). For Part (2), a relatively large stiffness-proportional damping coefficient α = 100 α0 = 1/π = 0.3183 was used to reduce the amplitudes of the waves reflected from the model boundaries. A very small stiffness-proportional damping coefficient α = 0.005 α0(1.592 × 10−5) was used for Part (3), in order to minimize the effect of damping throughout the whole system given that Part (3) occupies most of the model space. Although the sudden change of damping coefficients between Parts (2) and (3) may result in some degree of non-physical reflection, such a reflection will not affect the results within the simulation duration because the outermost boundaries are far away from the rupture area.
4. Numerical Results
We performed numerical simulations for several cases using different mesh sizes (100 m, 300 m, and 500 m) and different stiffness-proportional damping coefficients (α = 0.625α0, α0 and 5α0 at the fault-layer elements), in order to examine the effects of these conditions on the numerical results. Among these, the case with the mesh size of 100 m and α = 0.625α0 gives the best result, and accordingly, we now discuss the result of this case (the reference case) in some detail, in order to compare it with the benchmark results of Day et al. (2005). We then compare the simulation results obtained for different mesh sizes and different α-values of the damping coefficient. The simulations were all performed on cluster machines in an MPI-based parallel style. For the finest mesh size (100 m) with 2,450,000 elements, the simulations were run in parallel on 32 processors with a total required memory of 14.6 GB. It took about 120 hours per CPU to complete the 7.5 s simulation.
4.1 Results in comparison with benchmarks
The fault rupture that nucleated in the central square started to propagate spontaneously on the fault, following the slip-weakening frictional law. The rupture accelerated rapidly to reach subsonic velocities of propagation both in the in-plane direction along the x-axis and in the anti-plane direction along the y-axis.
For the damping coefficient α in our simulation, it is found there may exist an approximating relation to the dimensionless damping parameter , used in Day et al. (2005) and Dalguer and Day (2007), as , based on dimensional analysis, where Δt is the time step. Day et al. (2005) suggested with Δt = 0.008 s for a grid spacing of 100 m, leading to an equivalent α = 0.8 × 10−3 which was much smaller than that which we used 0.625α0 = 1.989 × 10−3. It is probably because that Day et al. (2005) applied the same level of damping globally. On the other hand, Dalguer and Day (2007) limited the damping to the fault-adjacent elements only and found the preferred higher values of the damping parameter in the range 0.2 ~ 0.4, which is close to ours.
4.2 Effects of mesh size and damping coefficient on FE results
In the following subsection, we discuss the dependence of the FE solution on mesh size and the stiffness-proportional damping coefficient used in the fault layer elements.
From the present study, we conclude that the mesh size has a significant influence on the accuracy of our FE simulations of fault rupture. For the present fault test model, the use of a mesh size of 100 m, among our other simulation cases, yields the best accuracy. The proper selection of damping coefficient for a particular frequency can reduce spurious high-frequency oscillation while having little effect on accuracy, and can further improve the performance of the model. For the case of mesh size 100 m with α = 0.625α0, it is found that although the oscillations affect less the results of rupture times (Fig. 6) and shear traction time histories as shown Figs. 7 and 8, they do affect the accuracy of the slip rate so that it would cause difficulties for further parametric investigation. It would be helpful to adopt some smoothing algorithms (e.g., Galis et al., 2010) in our FE formulation in the future.
We have obtained finite element (FE) solutions of spontaneous shear rupture in a planar fault in a 3D uniform elastic medium by using a parallel FE code, GeoFEM, with some modifications for applying a slip-weakening friction law and adopting a time integration scheme that guarantees the conservation of energy and momentum. The fault model used in our study was the same as that used by Day et al. (2005) for evaluating the accuracy of a finite difference (FD) method and a boundary integral (BI) method, and permitted us to compare our results with those obtained by Day et al. (2005). Several simulations were performed in order to examine the effects of mesh size and damping parameter necessary to avoid spurious oscillations. Comparing the time histories of the slip rate and the shear stresses, and the rupture time distribution obtained in the present FE simulations with those obtained by Day et al. (2005), we found that almost the same results were obtained by the present FE method as those by the FD and BI methods, provided that the appropriate mesh size and damping coefficients were used. In the simulation of dynamic faulting with slip-weakening friction, the size of the cohesive zone must encompass more than a few grid points to express a breakdown process properly and thereby to yield reliable numerical results. The ratio of the size of the cohesive zone to mesh size for stable numerical solutions under the present FE scheme should be greater than 5, which is almost equal to that of the FD scheme and a little larger than that of BI reported by Day et al. (2005). Because the size of the cohesive zone depends on the slip-weakening parameters, the mesh size required in the numerical computations depends on these parameters. The value of the appropriate damping coefficient is determined by the mesh size, because artificial oscillations of a longer period are generated for coarser meshes. Stronger damping artificially decreases rupture velocities, peak slip rates, and the amount of slip. Through the numerical study described herein, we confirm that our FE method provides solutions to fault rupture that are almost as accurate as those given by the FD and BI methods. Spurious oscillations observed in slip rates in the present method may be further improved by introducing sophisticated smoothing algorithms as discussed in Section 4.2. The FE methods are flexible in that they may be used with an unstructured mesh discretization, and, accordingly, they are capable of simulating complicated problems even with a non-conformed mesh on a fault plane, by using a formulation that involves master-slave contact. Our study has provided standards for the required mesh and damping conditions, and confirms the accuracy of this approach for use in possible future applications.
The authors thank two reviewers of this article, Steven Day and Peter Moczo for valuable comments that led to an improvement of the manuscript. The computations were carried out by the parallel computer of the Earthquake and Volcano Information Center in the Earthquake Research Institute, University of Tokyo. This study was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, under its Observation and Research Program for the Prediction of Earthquakes and Volcanic Eruptions.
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