Software for Efficient Static Dislocation-Traction Calculations in Fault Simulators

Abstract

Online Material: Software packages with documentation.Quasistatic or quasidynamic rate-state friction (QRSF) simulators are used to study the mechanics of faults (e.g., Shibazaki and Shimamoto, 2007). The displacement discontinuity method (DDM;Crouch and Starfield, 1983) meshes the fault or faults into N elements and constructs a matrix of Green's functions (GFs) relating slip to stress. The simulator evolves strength and slip in time. Usually, the most expensive part of a simulation time step is the matrix-vector product (MVP) of the slip distribution with the DDM matrix; the straightforward implementation performs ON 2 operations. A simulator performs thousands to millions of MVPs with the same DDM matrix.My free and open-source software (FOSS) package hmmvp speeds up simulations by an asymptotic factor of a little less than N faster than the straightforward implementation for both forming an approximation to and performing MVP with the GF matrix.In QRSF simulations, rupture tip length scales as f r ∝ μ ′ d c =bσ for the aging evolution law and f r multiplied by a factor that depends on slip speed and background values for the slip law. In this case, μ ′ μ=1 − ν, μ is the shear modulus, ν is Poisson's ratio, b is the constant multiplying the state term in rate-state friction, σ is the effective normal stress, and d c is the characteristic slip distance for friction evolution (Rubin and Ampuero, 2009). Rupture tips must be well resolved in simulations. If f r is nonuniform on the fault, discretization also can be nonuniform for efficiency. However, not all DDM operators are accurate on nonuniform meshes. My FOSS package dc3dm implements a method, IGA, that is as accurate on a nonuniform mesh as the standard method is on a uniform mesh. dc3dm uses hmmvp to efficiently approximate the IGA operator. H-MATRIXES AND hmmvpA variety of methods can speed up the MVP. Let a rectangular fault be discretized uniformly into N N s N d elements, in which N s is the number of along-strike elements and N d is the number along dip. The straightforward implementation is ON 2 , in which O is big-O notation. In a homogeneous elastic (HE) full-space, the fast Fourier transform (FFT) can be used to calculate stress from slip in ON log N work. In an HE halfspace, the FFTcan be used in the along-strike direction to implement the convolution, resulting in ON 2 d N s log N s time. In general, the FFTcannot be applied to a nonuniformly discretized or nonplanar fault. A class of methods, including fast multipole methods (Greengard and Rokhlin, 1987;Ying et al., 2004), Barnes-Hut (Barnes and Hut, 1986), and hierarchical matrixes (H-matrixes;Hackbusch, 1999), are instead applicable. These methods take advantage of off-diagonal block (for an appropriate permutation) approximate low-rank structure. A block B ≈ UV T , with columnsU ≪ minrowsB; columnsB. For a mesh having N elements, work is close to ON . Among these methods, the H-matrix is best suited to QRSF simulators because it is fastest when the operator is formed infrequently relative to ...

The 2012 Mw 8.6 Wharton Basin sequence: A cascade of great earthquakes generated by near‐orthogonal, young, oceanic mantle faults

“…We use a uniform fault discretization, so the stress singularity at the termination of each fault patch has a minimal impact on this calculation. Nevertheless, we perform the stress change calculation using dc3dm , the numerical method of Bradley [], that produces results that are comparably accurate for both uniform and nonuniform meshes. We then express the average stress drop using the formulation of Noda et al [], where s represents coseismic slip and the integration is performed over the entire fault area ∂ Ω.…”

“…We use a uniform fault discretization, so the stress singularity at the termination of each fault patch has a minimal impact on this calculation. Nevertheless, we perform the stress change calculation using dc3dm, the numerical method of Bradley [2014], that produces results that are comparably accurate for both Figure S12 shows the same results plotted with a color bar saturated at 30 m. The chi-square model to data fits for these slip distributions are shown in Figure 10. uniform and nonuniform meshes.…”

Small interseismic asperities and widespread aseismic creep on the northern Japan subduction interface

“…In both models, slip is computed from boundary element calculations using efficient hierarchical matrix techniques [Bradley, 2014;Johnson et al, 2013]. In both models, slip is computed from boundary element calculations using efficient hierarchical matrix techniques [Bradley, 2014;Johnson et al, 2013].…”

“…The interseismic strain accumulation model assumes that seismic asperities are fully coupled during the interseismic period, and creep occurs on the surrounding areas of the interface at constant shear stress. In both models, slip is computed from boundary element calculations using efficient hierarchical matrix techniques [Bradley, 2014;Johnson et al, 2013].…”

Fully Coupled Hydromechanical Simulation of Hydraulic Fracturing in 3D Discrete-Fracture Networks

“…Most often, DFN codes have been used to describe hydraulic stimulation in enhanced geothermal systems (Lanyon et al 1993;Bruel 1995;Kohl and Hopkirk 1995;Willis-Richards et al 1996;Jing et al 2000;Rahman et al 2002;Rachez and Gentier 2010;Zhou and Ghassemi 2011;Riahi and Damjanac 2013). In some cases, shear stimulation has been simulated with continuum models (Cladouhos et al 2011;Dempsey et al 2013).…”