187 10.22 Maximum displacements and velocities along an east-west line for the two fault depths.188 10.23 Time history of displacements at two locations for the two hypocenter locations. 226 10.65 Velocity-time histories at two locations for the four cases of initial heterogeneity.
Notation
Chapter 1 Introduction
- Background
- Organization
However, we need more information about the robustness of the properties of near-source ground motions. Using prescribed fractions, we assess the sensitivity of the ground motions to variations in the earthquake parameters.
Chapter 2 General Methodology
2 .1 Governing Equations
Boundary Conditions
The non-reflective boundaries approximate the behavior of the waves that continue to propagate along the edges of the domain. Consequently, the absorbing boundaries allow truncation of the domain with minimal adverse effects on the solution.
Discretization
- Mass Matrix
- Stiffness Matrix
- Damping Matrix
Unfortunately, stiffness-proportional damping significantly reduces the efficiency of the central difference integration scheme, because it contributes off-diagonal terms to the damping matrix. We do this by adapting the formulation of the central-difference scheme as shown by equation (2.16), where [Cm] the.
2 .4 Model of Fault
Slip Degrees of Freedom
We can accomplish these tasks by including the fault plane in the geometry of the finite element model. If we transform to relative and average degrees of freedom, we obtain explicit control of the relative motion over the fault.
Average Slip and Moment Magnitude
Spatial Interpolation
If no such point 2 exists, we choose point 2 as the next closest point in the data set after point 1. If no such point 3 exists, we choose point 3 as the next closest point in the data set after point 2.
Finite-Element Mesh Creation
For example, we want to identify all nodes on the fault plane because we use them to perform fault slip. When forming the absorption limit, we ignore all surfaces that do not coincide with one of the coonliuate CLXe::;.
The Center for Advanced Computing Research Super- computers
State-of-the-art workstations still lack the storage capacity and processing power to effectively handle such problems. Some manufacturers develop their own implementation of MPI to optimize code for supercomputer architectures, such as Intel's MPI on the Intel Paragon.
Domain Partitioning among Processors
We assume that the number of elements in each processor correlates with the computation time, so we partition the domain based solely on the locations of the elements. We consider each element as a point mass with the mass of the element proportional to the computational effort of the degrees of freedom of that element.
Parallel Computation
As in the case of inertial sectioning, we assign a computational effort to each element, and adjust the weights of elements with degrees of slip freedom to account for the additional computational effort they require. As in the case of an inertial bisection, we assign a computational effort to each element, and adjust the freedom-shift elements accordingly to account for the additional computational effort they require.
Parallel Input and Output
The master processor calculates the file position where each processor should start writing based on the current file position (which it stores). The master sends to each processor the file position where it should start writing the8 data.
Global Mesh Refinement
- Element Splitting
During 4x refinement, we do not recursively refine the element using the 2x refinement procedure, because element quality would continue to degrade at the second level of refinement. Instead, we follow a different procedure that adds new nodes to both the edges and the interior of the original element.
Chapter 4 Dynamic Energy Balance for Earthquakes
- Derivation of Dynamic Energy Balance
- Radiated Energy
- Change in Thermal Energy
- Change in Potential Energy
- Topography and Changes in Gravitational Potential Energy
Substituting the stress at slip D(t) into the expression for the incremental change in potential energy and integrating both the slip and fault area yields. On the other hand, when surface topography occurs, the change in gravitational potential energy is of first order.
5 .1 Earthquake Source
Slip Time History
However, the time of the maximum slip velocity must be late enough so that the part we cut off remains negligible. The shape of the slip time history together with the rupture velocity determines the frequency content of the waves generated in the domain.
Slip Distribution
The band-limited features of these three slip time histories match well with our limitation to only simulate long-period ground motion due to the limited resolution of the finite element mesh. Equation (5.1) gives the ultimate slip as a function of location along strike, p, and location along dip, q, where D0 denotes the nominal value of ultimate slip and d denotes a distance controlling the rate of slip decay. We set C 0 when q > q0 and we want to reduce the slip on three edges only, and we set C 1 when we want to reduce the slip along all four edges.
The radius ra indicates the radius at which the height has dropped to 5% of the maximum value.
Rupture Speed
The fault velocity determines when adjacent points on the fault should begin to slide. This means that determining the fault velocity involves setting the time at which each point on the fault begins to slip. The following algorithm describes how to prescribe the time at which the sides of a fault start to move relative to each other at each fault node.
5.3) where Vr is the fracture velocity as a fraction of the local shear wave velocity (/3) and N is the number of elements with edges on the fracture plane containing the hypocenter node and node i.
Chapter 6 Validation of Simulation Software
- Domain Geometry
- Validation of Ground Motion Time Histories
- Validation of Static Displacements
- Discussion
Site A lies directly above the top of the fault, and site D lies at the absorbing boundary. The north-south line passes through the center of the domain, and the east-west line along the top of the fault. This results in a slightly slower decay with distance from the source in the final displacements of the finite element model compared to the analytical solution.
The north-south line passes through the center of the domain and the east-west line along the top of the fault.
Chapter 7 Sensitivity Study of Near-Source Ground Motion with Prescribed Ruptures
- Strike-Slip Fault
- Finite-Element Model
- Simulation Results
- layered I -0.8
- Thrust Fault
Source inversions of the Landers earthquake (Wald and Heaton 1994) and the Kobe earthquake (Wald 1996) resulted in slip durations of more than four seconds for the shallow slip, but eyewitnesses reported a slip duration of one second or less for the Luzon earthquake in 1990. Philippines (Yomogida and Nakata Hl94) and the Borah Peak earthquake in 1983 (Wallace 1984). The directivity of the fault causes large variations in the ground motions with changes in the azimuth (angle between the projection of the slip vector on the ground surface and the vector from the epicenter to the site). Site S1 is 10 km north of the northern end of the fault (0 degree azimuth) and site S2 is 10 km east of the fault center (90 degree azimuth).
The directivity of the fault causes large Vdriations in the ground motions with changes in the azimuth (angle between the projection of the slip vector on the ground surface and the vector from the epicenter to the site). Site S1 is 10 km north of the northern end of the fault (0 degree azimuth) and site S2 is 10 km east of the fault center (90 degree azimuth). The ground motions at location Sl show small differences with changes in the location of the.
7 .3 Discussion
Ground Motion Characteristics
Sensitivity of Grmmcl Motions to Variations in Parameters
On both the strike-slip fault and the thrust fault, ground motions show a strong sensitivity to the depth of the fault. We consider the distribution of ground motions on the ground surface for the base cases in the sensitivity study (scenario baseII for the strike-slip fault and scenario base for the thrust fault). The magnitude of the moment thus provides a poor measure of the severity of the ground motion with variations in the depth of the fault.
We now consider a change in the formulation of the near-source factor curve to account for blind thrust errors.
Earthquake Source
- Governing Equations with Friction
- Forces on Slip Degrees of Freedom
- Initiation of Sliding
Thus, to calculate the friction force at time t, we use the slip rate at time t - At instead of the slip rate at time t. The force of friction does not require any transformation; the product of the friction coefficient and the force acting on the relative normal degree of freedom gives the magnitude of the friction force vector acting on the sliding degree of freedom. To initiate sliding at a point on a fault, the frictional force must be less than the sum of the other forces acting on that point.
In more mathematical terms, the sliding stops when the frictional force becomes greater than the sum of all the other forces.
Initial Tractions on Fault
- Effective Normal Tractions
- Shear Traction
- Effect of Gravity
- Spatial Distribution of Parameters for Dynamic Failure
If there is little or no water in the interstices, the pore pressures are negligible and the effective normal stresses are equal to the normal stresses. For homogeneous material properties and much larger lithostatic stresses than tectonic normal stresses, the effective normal stresses increase approximately linearly with depth. We also expect that the normal stresses on the fault from gravity are much greater than the stresses from plate tectonics, so that the effective normal stresses are similar to the normal stresses from gravity.
Madariaga et al. 1998) without recognizing that assuming uniform effective normal stresses with depth implies very large confining pressures.
Overview of Rupture Dynamics
- Stress and Rupture Dynamics
Time Figure 8.4: Diagrams of shear stress concentration near the crack front as a function of space (left diagram) and as a function of time (right diagram). The dynamic stress drop and the distance from failure (the difference between the failure stress and the initial shear stress) determine the nature of the stress concentration ahead of the leading edge of rupture. The increase in shear stress in the fault before rupture means the storage of deformation energy in the surrounding region.
However, accurately capturing the stress concentration in shear stress near the leading edge of the fracture requires much smaller elements than those needed to model the wave propagation ( Madariaga et al. 1998 ).
Friction Models
- Constant
- Two-Phase
- Slip-Weakening I
- Slip-Weakening II
The instantaneous drop in the coefficient of friction means that no energy is required for breakage. We modify the two-phase friction model by decreasing the coefficient of friction over a slip distance of D0 • This includes the latent heat (fracture energy) generated by fracture in the friction model. We assume immediate recovery of the coefficient of friction upon cessation of sliding, so that as soon as sliding stops, the coefficient of friction increases to µ,max.
In this case, instead of a linear decrease in the friction coefficient with the slip distance, we decrease the friction coefficient inversely proportional to the slip distance, as shown in Figure 8.7.
DID 0
- Rate-Weakening II
- Finite-Element Models
- l Strike-Slip Fault
- Thrust Fault
- General Features of Dynamic Rupture
- Role of Fracture Energy
We also consider various variations in the effective normal stresses with depth and their implications on the parameters of the friction model. Likewise, when we bury the fault, the largest slips occur near the center of the fault. We can increase this slow fracture rate to a more realistic value by decreasing the fracture energy without changing the fundamental behavior of the fracture.
We can increase this slow fracture speed to a more realistic value by reducing the fracture energy without changing the fundamental behavior of the fracture.
