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

In simulating dynamic fault motions, a split-node technique (Andrews, 1999; Day *et al.*, 2005; Ma *et al.*, 2008) has, to date, enjoyed widespread use for modeling the boundary conditions of faults in both FD and FE calculations. However, in general, contact problems are usually modeled by means of a master-slave type of contact element, which does not require the complete overlapping of nodes at the fault, and can be applied to more general unstructured and non-conforming mesh discretization on the fault plane. A node-to-segment contact element without friction was previously used in the original version of GeoFEM. The detailed formulation of the frictionless contact and the method by which contact constraint is enforced through the use of a penalty can be found in Iizuka *et al.* (2000). We herein give the derivation of the condition of consistent stiffness of the contact element using the slip-weakening friction law proposed by Ida (1972). We assume a simple linear slip weakening and a friction coefficient *μ* expressed as

where the subscripts *s* and *d* denote static and dynamic friction, respectively, *δ* is the slip across the fault and *δ*_{0} is the critical slip distance. A frictional contact problem that followed a Coulomb plastic law was described by Peric and Owen (1992), in which the frictional coefficient was constant. In the slip-weakening friction law of Eq. (1), the frictional coefficient *μ* is no longer constant. Instead, it is a linear function of the slip *δ*, until *8* reaches a value *δ*_{0}, above which the coefficient *μ* assumes a constant value *μ*_{
d
}. Therefore, the derivation of the tangent stiffness representing the traction-displacement relation must be rewritten.

At a certain time step *T* + Δ*T*, the contact force on the surface can be written as:

where ^{T+ΔT}**f**_{
t
} and ^{T + ΔT}**f**_{
n
} are the frictional and normal force, respectively, ^{T+ΔT}**t** and ^{T+ΔT}**n** are the unit vectors in the direction of the slip and the contact surface normal, respectively. The displacement difference **u** at a contact point can be expressed explicitly in terms of the nodal displacements **u**_{
i
} on both sides of the contact surface (Appendix A). By differentiating Eq. (2) with respect to **u**, we obtain the tangential contact stiffness matrix at time *T +* Δ*T* :

where the symbol ⊗ denotes the outer product of two vectors, and we assume that the contact surfaces do not deform very much, so that the differential of the contact normal ^{T+ΔT}**n** with respect to *d***u** can be ignored. In the first term on the right-hand side of Eq. (3), the term *d*(*μ*|^{T+ΔT}**f**_{
t
}|)/*d***u** can be further separated into two terms:

As shown in Fig. 1, the slip on the contact surface *δ* can be written as

where ^{T+ΔT}**u**_{
t
} is the slip vector on the contact point between the two sides of the contact surface at time *T +* Δ*T*, after relative slip initially occurs, but not a frictional path dependent slip. Therefore, from Eq. (1), *dμ/d***u** in the second term of the right-hand side of Eq. (4) can be expressed as

where ^{T+ΔT}**t**′ denotes the unit vector in the direction of the slip increment (^{T+ΔT}**u**_{
t
} − ^{T}**u**_{
t
})/|^{T+ΔT}*u*_{
t
} − ^{T}**u**_{
t
} |, as shown in Fig. 1. The first term on the right-hand side of Eq. (4) can be written as:

where ^{T+ΔT}**D**_{
n
} = *p*_{
n
} (^{T+ΔT}**n** ⊗^{T+ΔT}**n**), and *p*_{
n
} is the penalty in the contact surface normal. A very large penalty is applied in the contact normal direction in order to limit the penetration of the slave node into the master segment. This is one of the most commonly-adopted methods for introducing constrained conditions into variational equations. Theoretically, the non-penetration condition on the contact surface requires an infinitely large penalty. However, too large a penalty would lead to an ill-conditioned matrix and cause numerical problems in the solution. Herein, the value of the penalty in the contact normal direction is chosen to be 10^{10} times the Young’s modulus of the material, a value that is sufficiently large to limit the penetration within a scale of displacement ×10^{−10} but does not lead to the occurrence of an ill-conditioned stiffness matrix. The same value of penalty is also used for the contact tangent direction *p*_{
t
}. Since the direction of slip varies continuously, it is necessary to derive the derivative of ^{T+ΔT}**t** with respect to **u**:

where is the trial friction force assuming that no slip occurs at time *T+*Δ*T* and **I** is a 3 × 3 unit matrix. Figure 2 gives an explanation of the return mapping procedure used to update the frictional force. When the value of exceeds that of the friction force, it is ‘pulled back’ to a frictional force that follows the slip-weakening frictional law. ^{T+ΔT}**D**_{
t
} is the derivative of with respect to **u** and can be written as

By combining Eqs. (4), (6), (7) and (8), Eq. (3) finally reduces to

The last term on the right-hand side of Eq. (10) denotes the contribution from the slip-weakening friction law, without which a simple frictional contact formulation with a constant frictional coefficient is obtained.

### 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 the dynamic analysis of a system that has many degrees of freedom, it is the high-frequency modes that are the sources of the numerical instability. In simulations of fault rupture, the high-frequency modes can be damped using numerical damping methods. The simple approach of Rayleigh damping is used in many finite-element analysis programs, such as ABAQUS (2006) and NASTRAN (2005). In these programs, reliable results for numerically sensitive structural systems may be obtained by the increased damping of the higher-frequency modes of the system, but this approach suffers from a lack of physical justification and significant errors may result. A damping term that is proportional to the velocity is introduced into the equation of motion (see Eq. (B.1) in Appendix B). In Rayleigh damping, the damping matrix *C* is expressed as a linear combination of the mass and stiffness matrices in the form

where *C*, *M* and *K* are the damping, mass and stiffness matrices, respectively, while *α*_{
M
} and *α*_{
K
} are proportional coefficients of mass and stiffness. Herein, we do not restrict our approach to the use of the global matrices of Eq. (11); instead, we assign different values of *α*_{
M
} and *α*_{
K
} to individual elements. The damping ratio, which is a dimensionless measure that describes how much the oscillations in a system decay after a disturbance (Bathe, 1996), can be expressed as a function of angular frequency *ω*(*ω = f/*2π) or in terms of a frequency *f*, using *α*_{
M
} and *α*_{
K
} in the case of Rayleigh damping:

It is well known that under mass-proportional damping, a greater degree of damping is applied to the low-frequency modes of the system, and that under stiffness-proportional damping, the damping is more effective at the higher-frequency modes. Because high-frequency artificial oscillations should be reduced in our simulations of fault rupture, only stiffness-proportional damping is applied here. The damping matrix may then be written as

where the subscript _{
K
} of *α* is omitted for simplicity. Then, Eq. (12) may be reduced to:

Because of the absence of the mass-proportional coefficient, the damping ratio *ξ* is a linear function of *ω* (or *f*). For example, if the frequency mode *f* = 100 Hz needs to be reduced by 50%, then, by using Eq. (14), the corresponding damping coefficient can be calculated as *α* = *ξ*/(*πf*) = 0.5/(100*π*) ≈ 1.592 × 10^{−3}. This value of *α* leads to a damping ratio *ξ* = 2.5% at a frequency mode of *f* = 5 Hz. It is noteworthy that Semblat (1997) investigated a rheological interpretation of Rayleigh damping and found an approximate relationship between the quality factor *Q* and the damping ratio as 1/ *Q* ≈ 2*ξ* for weak to moderate Rayleigh damping. In addition, it is also worth mentioning that a similar artificial viscosity damping was introduced in a series of FD simulations by, for instance, Day *et al.* (2005), Dalguer and Day (2007) and Rojas *et al.* (2009). Such a damping is treated as a device to suppress short-wavelength oscillations. Therefore, it is to regularize the numerical solution, rather than to represent a physical damping, just as in the case of Rayleigh 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.