Modeling dynamic rupture in a 3D earthquake fault model

1998 ◽  
Vol 88 (5) ◽  
pp. 1182-1197 ◽  
Author(s):  
R. Madariaga ◽  
K. Olsen ◽  
R. Archuleta
Keyword(s):  
Boundary Conditions ◽  
Fault Zone ◽  
Spontaneous Rupture ◽  
Grid Method ◽  
Three Dimensions ◽  
Initial Stresses ◽  
Staggered Grid ◽  
Dynamic Rupture ◽  
Plane Shear ◽  
Frictional Instability

Abstract We propose a fourth-order staggered-grid finite-difference method to study dynamic faulting in three dimensions. The method uses an implementation of the boundary conditions on the fault that allows the use of general friction models including slip weakening and rate dependence. Because the staggered-grid method defines stresses and particle velocities at different grid points, we preserve symmetry by implementing a two-grid-row “thick” fault zone. Slip is computed between points located at the borders of the fault zone, while the two components of shear traction on the fault are forced to be symmetric inside the fault zone. We study the properties of the numerical method comparing our simulations with well-known properties of seismic ruptures in 3D. Among the properties that are well modeled by our method are full elastic-wave interactions, frictional instability, rupture initiation from a finite initial patch, spontaneous rupture growth at subsonic and supersonic speeds, as well as healing by either stopping phases or rate-dependent friction. We use this method for simulating spontaneous rupture propagation along an arbitrarily loaded planar fault starting from a localized asperity on circular and rectangular faults. The shape of the rupture front is close to elliptical and is systematically elongated in the inplane direction of traction drop. This elongation is due to the presence of a strong shear stress peak that moves ahead of the rupture in the in-plane direction. At high initial stresses the rupture front becomes unstable and jumps to super-shear speeds in the direction of in-plane shear. Another interesting effect is the development of relatively narrow rupture fronts due to the presence of rate-weakening friction. The solutions for the “thick fault” boundary conditions scale with the slip-weakening distance (D0) and are stable and reproducible for D0 greater than about 4 in terms of 2Tu/μ × Δx. Finally, a comparison of scalar and vector boundary conditions for the friction shows that slip is dominant along the direction of the prestress, with the largest deviations in slip-rate direction occurring near the rupture front and the edges of the fault.

scholarly journals An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li-Bin Liu ◽  
Ying Liang ◽  
Xiaobing Bao ◽  
Honglin Fang
Keyword(s):  
Boundary Conditions ◽  
Grid Method ◽  
Posteriori Error ◽  
Adaptive Grid ◽  
Singularly Perturbed ◽  
Robin Boundary Conditions ◽  
Euler Formula ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Robin Boundary

AbstractA system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval $[0,1]$ [ 0 , 1 ] . It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder–Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

scholarly journals Surface Wave Speed of Functionally Graded Magneto-Electro-Elastic Materials with Initial Stresses

2014 ◽  
Vol 44 (3) ◽  
pp. 49-64 ◽  
Author(s):  
Li Li ◽  
P. J. Wei
Keyword(s):  
Surface Wave ◽  
Boundary Conditions ◽  
Elastic Material ◽  
Short Circuit ◽  
Equations Of Motion ◽  
Open Circuit ◽  
Functionally Graded ◽  
Initial Stresses ◽  
Shear Surface ◽  
Traction Boundary Conditions

Abstract The shear surface wave at the free traction surface of half- infinite functionally graded magneto-electro-elastic material with initial stress is investigated. The material parameters are assumed to vary ex- ponentially along the thickness direction, only. The velocity equations of shear surface wave are derived on the electrically or magnetically open circuit and short circuit boundary conditions, based on the equations of motion of the graded magneto-electro-elastic material with the initial stresses and the free traction boundary conditions. The dispersive curves are obtained numerically and the influences of the initial stresses and the material gradient index on the dispersive curves are discussed. The investigation provides a basis for the development of new functionally graded magneto-electro-elastic surface wave devices.

Influence of Boundary Conditions on Response of Pipelines Crossing Reverse Fault Zone

2021 ◽  
Author(s):  
Hamid Rostami ◽  
Abdolreza Osouli ◽  
Brent Vaughn ◽  
Hamed Gholizadeh Touchaei
Keyword(s):  
Boundary Conditions ◽  
Fault Zone ◽  
Reverse Fault

scholarly journals Eigenvalues of the negative Laplacian for arbitrary multiply connected domains

1996 ◽  
Vol 19 (3) ◽  
pp. 581-586
Author(s):  
E. M. E. Zayed
Keyword(s):  
Boundary Conditions ◽  
Spectral Function ◽  
Three Dimensions ◽  
Bounded Domains ◽  
Multiply Connected Domains ◽  
Multiply Connected ◽  
Asymptotic Formulae ◽  
Negative Laplacian

The purpose of this paper is to derive some interesting asymptotic formulae for spectra of arbitrary multiply connected bounded domains in two or three dimensions, linked with variation of positive distinct functions entering the boundary conditions, using the spectral function∑k=1∞{μk(σ1,…,σn)+P}−2asP→∞. Further results may be obtained.

Numerical simulation of the Late Jurassic closure of the Vardar Tethys

2021 ◽  
Author(s):  
Nikola Stanković ◽  
Vesna Cvetkov ◽  
Vladica Cvetković
Keyword(s):  
Boundary Conditions ◽  
Numerical Simulations ◽  
Stokes Equations ◽  
Numerical Study ◽  
Phase Evolution ◽  
Staggered Grid ◽  
Second Phase ◽  
Weak Zone ◽  
Ongoing Research ◽  
Magmatic Complex

<p>We report updated results of our ongoing research on constraining geodynamic conditions associated with the final closure of the Vardar branch of the Tethys Ocean by means of application of numerical simulations (previous interim results reported in EGU2020-5919).</p><p>The aim of our numerical study is to test the hypothesis that a single eastward subduction in the Jurassic is a valid explanation for the occurrence of three major, presently observed geological entities that are left behind after the closure of the Vardar Tethys. These include: ophiolite-like igneous rocks of the Sava-Vardar zone and presumably subduction related Timok Magmatic Complex, both Late Cretaceous in age as well as Jurassic ophiolites obducted onto the Adriatic margin. In our simulations we initiate an intraoceanic subduction in the Early/Middle Jurassic, which eventually transitions into an oceanic closure and subsequent continental collision processes.</p><p>In the scope of our study numerical simulations are performed by solving a set of partial differential equations: the continuity equation, the Navier-Stokes equations and the temperature equation. To this end we used I2VIS thermo-mechanical code which utilizes marker in cell approach with finite difference discretization of equations on a staggered grid [Gerya et al., 2000; Gerya&Yuen, 2003].</p><p>The 2D model consists of two continental plates separated by two oceanic slabs connected at a mid-oceanic ridge. Intraoceanic subduction is initiated along the ridge by assigning a weak zone beneath the ridge. Time-dependent boundary conditions for velocity are imposed on the simulation in order to model a transient spreading period. The change of sign in plate velocities is found to be useful for both obtaining obduction / ophiolite emplacement [Duretz et al., 2016] and causing back-arc extension. Changes in velocities are linear in time. Simulations follow a three-phase evolution of velocity boundary conditions consisting of two convergent phases separated by a single divergent phase where spreading regime is dominant. Effect of duration and magnitude of the second phase on model evolution is also explored.</p><p>Our so far obtained simulations were able to reproduce the westward obduction and certain extension processes along the active (European) margin, which match the existing geological relationships. However, the simulations involve an unreasonably short geodynamic event (cca 15-20 My) and we are working on solving this problem with new simulations. </p>

On the impact of alternative seismic time imaging methods on subsurface fault mapping in the northernmost Molasse Basin, Switzerland

2020 ◽  
Vol 8 (4) ◽  
pp. SQ1-SQ13
Author(s):  
Christoph G. Eichkitz ◽  
Sarah Schneider ◽  
Andreas B. Hölker ◽  
Philip Birkhäuser ◽  
Herfried Madritsch
Keyword(s):  
Fault Zone ◽  
Three Dimensions ◽  
Small Scale ◽  
Zone Model ◽  
Molasse Basin ◽  
Data Set ◽  
Volume Comparison ◽  
Imaging Methods ◽  
Seismic Amplitude ◽  
The Impact

The identification and characterization of tectonic faults in the subsurface represent key aspects of geologic exploration activities across the world. We have evaluated the impact of alternative seismic time imaging methods on initial subsurface fault mapping in three dimensions in the form of a case study situated in the most external foreland of the European Central Alps (the northernmost Molasse Basin). Four different seismic amplitude volumes of one and the same 3D seismic data set, differing in imaging technologies and parameterizations applied, were considered for the interpretation of a fault zone dissecting a Mesozoic sedimentary sequence that is characterized by a pronounced mechanical stratigraphy and has witnessed a multiphase tectonic evolution. For this purpose, we interpreted each seismic amplitude volume separately. In addition, we computed a series of seismic attributes individually for each volume. Comparison of the different data interpretations revealed consistent results concerning the mapping of the seismic marker horizons and main fault segments. Deviations concern the apparent degree of vertical and lateral fault zone segmentation and the occurrence of small-scale fault strands that may be regarded as important fault kinematic indicators. The compilation of all fault interpretations in map form allows the critical assessment of the robustness of the initial seismic fault mapping, highlighting well-constrained from poorly defined fault zone elements. We conclude that the consideration of multiple seismic processing products for subsurface fault mapping is advisable to evaluate general imaging uncertainties and potentially guide the development of fault zone model variants to tackle previously discussed aspects of conceptual interpretation uncertainties.

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