Non-planar dynamic rupture modelling across diffuse, deforming fault zones using a spectral finite element method with a non-mesh aligned embedded diffuse discontinuity

2021 ◽  
Author(s):  
Jorge Nicolas Hayek Valencia ◽  
Dave A. May ◽  
Alice-Agnes Gabriel
Keyword(s):  
Finite Element ◽  
Galerkin Method ◽  
Fault Zone ◽  
Fault Zones ◽  
Diffuse Interface ◽  
Earthquake Rupture ◽  
Dynamic Rupture ◽  
Failure Patterns ◽  
Continuous Galerkin ◽  
Spectral Finite Element

<p>Faults in earthquake rupture dynamic simulations are typically treated as infinitesimally thin planes with distinct on- versus off-fault rheologies. These faults are prescribed and can be explicitly accounted for with hexahedral or unstructured tetrahedral meshing approaches.  <br>We present a diffuse interface alternative to dynamic rupture modelling on non-mesh aligned faults and, by design, permits modelling of non-planar faults and time-dependent fault geometries. We use se2dr, a spectral finite element (continuous Galerkin) method with a non-mesh aligned embedded diffuse discontinuity for dynamic rupture simulations.</p><p>Natural fault systems are characterised by fault zone complexity, e.g. the frictional strength and spatio-temporal slip localisation may change drastically from the outer damage zone to the fault core. Complex volumetric failure patterns are observed in well-recorded large complex earthquakes (e.g., the 2016 Mw7.8 Kaikōura event, Klinger et al. 2018), small events (e.g.,  in the San Jacinto Fault Zone, Cheng et al. 2018), and laboratory-scale experiments (e.g., in high-velocity friction experiments, Passelègue et al., 2016).</p><p>We develop a diffuse description of fault slip to better understand complex volumetric failure patterns and the mechanics of slip in diffuse fault zones. The fault is defined via a signed distance function (s(x)), which is in turn used to define a fault indicator function with compact support H. If s(x) > H the material behaves as a pure elastic solid - otherwise the tangential stress is governed by a frictional sliding law.<br>Our approach is implemented on a structured hexahedral mesh using a spectral finite element (continuous Galerkin) method for wave propagation using PETSc. Our diffuse fault SEM method is inspired by the stress-glut method of Andrews, 1999.  A non-mesh aligned embedded diffusive discontinuity allows for complex dynamic rupture simulations. We present 2D numerical experiments of kinematically driven rupture and spontaneous dynamic rupture on non-planar and non-mesh aligned complex fault geometries. The method can be used to model earthquake rupture dynamics on specifically complex and evolving fault faults such as the San Jacinto, CA, fault, or shallowly dipping megathrusts and splay faulting structures in subduction zones.</p>

scholarly journals A unified first order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones

2021 ◽  
Author(s):  
Alice-Agnes Gabriel ◽  
Duo Li ◽  
Simone Chiocchetti ◽  
Maurizio Tavelli ◽  
Ilya Peshkov ◽  
...  
Keyword(s):  
Fault Zone ◽  
Scalar Fields ◽  
Fault Zones ◽  
Diffuse Interface ◽  
Material Damage ◽  
Earthquake Rupture ◽  
Shear Cracks ◽  
Dynamic Rupture ◽  
First Order ◽  
Cartesian Meshes

<p>Earthquake fault zones are more complex, both geometrically and rheologically, than an idealised infinitely thin plane embedded in linear elastic material.  Field and laboratory measurements reveal complex fault zone structure involving tensile and shear fractures spanning a wide spectrum of length scales (e.g., Mitchell & Faulkner, 2009), dense seismic and geodetic recording of small and large earthquakes show hierarchical volumetric faulting patterns (e.g., Cheng et al., 2018, Ross et al., 2019) and 2D numerical models explicitly accounting for off-fault fractures demonstrate important feedback with rupture dynamics and ground motions (e.g., Thomas & Bhat 2018, Okubo et al., 2019).</p><p>Here (Gabriel et al., 2021) we adopt a diffuse crack representation to incorporate finite strain nonlinear material behaviour, natural complexities and multi-physics coupling within and outside of fault zones into dynamic earthquake rupture modeling. We use a first-order hyperbolic and thermodynamically compatible mathematical model, namely the GPR model (Godunov & Romenski, 1972; Romenski, 1988),  to describe a continuum in a gravitational field which provides a unified description of nonlinear elasto-plasticity, material damage and of viscous Newtonian flows with phase transition between solid and liquid phases.</p><p>The model shares common features with phase-field approaches but substantially extends them. Pre-damaged faults as well as dynamically induced secondary cracks are therein described via a scalar function indicating the local level of material damage (Tavelli et al., 2020); arbitrarily complex geometries are represented via a diffuse interface approach based on a solid volume fraction function (Tavelli et al., 2019). Neither of the two scalar fields needs to be mesh-aligned, allowing thus faults and cracks with complex topology and the use of adaptive Cartesian meshes (AMR). High-order accuracy and adaptive Cartesian meshes are enabled in 2D and 3D by using the extreme scale hyperbolic PDE solver ExaHyPE (Reinarz et al., 2019).</p><p>We show a wide range of numerical applications that are relevant for dynamic earthquake rupture in fault zones, including the co-seismic generation of secondary off-fault shear cracks, tensile rock fracture in the Brazilian disc test, as well as a natural convection problem in molten rock-like material. We compare diffuse interface fault models of kinematic cracks, spontaneous dynamic rupture and dynamically generated off-fault shear cracks to sharp interface reference models. To this end, we calibrate the GPR model to resemble empirical tensile and shear crack formation and friction laws. We find that the continuum model can resemble and extend classical solutions, while introducing dynamic differences (i) on the scale of pre-damaged/low-rigidity fault zone, such as out-of- plane rupture rotation; and (ii) on the scale of the intact host rock, such as conjugate shear cracking in tensile lobes. </p><p>Our approach is part of the TEAR ERC project (www.tear-erc.eu) and will potentially allow to fully model volumetric fault zone shearing during earthquake rupture, which includes spontaneous partition of fault slip into intensely localized shear deformation within weaker (possibly cohesionless/ultracataclastic) fault-core gouge and more distributed damage within fault rocks and foliated gouges.</p>

Modeling earthquake rupture dynamics across diffuse deforming fault zones

2020 ◽  
Author(s):  
Jorge Nicolas Hayek Valencia ◽  
Duo Li ◽  
Dave A. May ◽  
Alice-Agnes Gabriel
Keyword(s):  
Solid Earth ◽  
Fault Zone ◽  
Fault Zones ◽  
Displacement Discontinuity ◽  
Diffuse Interface ◽  
Dynamic Rupture ◽  
San Jacinto Fault ◽  
Rupture Dynamics ◽  
San Jacinto Fault Zone ◽  
Plastic Materials

<p>Earthquakes are a multi-scale, multi-physics problem. For the last decades, earthquakes have been modeled as a sudden displacement discontinuity across a simplified (potentially heterogeneous) surface of infinitesimal thickness in the framework of linear elastodynamics. Thus, earthquake models are commonly forced to distinguish artificially between on-fault frictional failure and the off-fault response of rock.<span> </span></p><p>While complex volumetric failure patterns of fault networks are observed from well-recorded large earthquakes (e.g., the 2016 M<sub>w</sub>7.8 Kaikōura event, <em>Klinger et al. 2018</em>) and small earthquakes (e.g., events in the San Jacinto Fault Zone, <em>Cheng et al. 2018</em>) as well as in laboratory experiments (e.g., in high-velocity friction experiments,<em> Passelègue et al., 2016</em>) inelastic deformation within a larger volume around the fault is generally neglected when studying kinematics, dynamics and the energy budget of earthquakes. Fault behaviour is then dominantly controlled by lab-derived friction on a surface. Recent 2D collapsing of material properties, stresses, geometry, and strength conditions from seismo-thermo-mechanical models to elastodynamic frictional interfaces illustrated resulting earthquake complexity and modeling challenges (<em>van Zelst et al., 2019</em>).</p><p>To understand the mechanics of slip in extended fault zones the ERC project <strong>TEAR</strong> (https://www.tear-erc.eu) aims to solve the governing equations of earthquake sources based on the conservation of mass, momentum and energy and rheological models for generalized visco-elasto-plastic materials. We here present (i) 2D numerical experiments of rupture dynamics and displacement decoupling under loading for varying fault zone properties resembling observations from the San Jacinto Fault Zone in a weak discontinuity approach<span>  </span>sing a diffuse fault representation (adapted stress-glut approach, Madariaga et al., 1998) within a <em>PETSc</em> spectral element discretisation of the seismic wave equation; (ii) Verification of modeling rupture dynamics using a novel diffuse interface approach using<em> ExaHyPE</em> (www.exahype.eu, <em>Reinarz et al. 2019</em>) that allows spontaneous, finite crack formation (<em>Tavelli et al.,</em> in prep.) and adaptive mesh refinement (AMR) zooming into the process zone at the rupture tip.</p><p>By this means, we start exploring scalable software for modelling shear rupture across extended, spontaneously developing fault systems for testing the hypothesis, that earthquake dynamics in fault zones can be jointly captured based on the theory of generalized visco-elasto-plastic materials.</p><p>References:</p><ul><li>Cheng, Y. et al. Diverse volumetric faulting patterns in the San Jacinto fault zone. JGR: Solid Earth, 123.6, 5068-5081 (2018). https://doi.org/10.1029/2017JB015408</li> <li>Klinger, Y. et al. Earthquake damage patterns resolve complex rupture processes. GRL, 45, 10,279– 10,287 (2018). https://doi.org/10.1029/2018GL078842</li> <li>Madariaga, R. et al. Modeling dynamic rupture in a 3D earthquake fault model. BSSA, 88.5 (1998): 1182-1197.</li> <li>Passelègue, F. X. et al. Frictional evolution, acoustic emissions activity, and off‐fault damage in simulated faults sheared at seismic slip rates. JGR: Solid Earth, 121(10), 7490-7513 (2016). doi:10.1002/2016JB012988</li> <li>Reinarz, A. et al. ExaHyPE: An Engine for Parallel Dynamically Adaptive Simulations of Wave Problems. arXiv preprint (2019), arXiv:1905.07987.</li> <li>Tavelli, M. et al. Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure, in prep.</li> <li>Van Zelst, I. et al. Modeling Megathrust Earthquakes Across Scales: One-way Coupling From Geodynamics and Seismic Cycles to Dynamic Rupture. JGR: Solid Earth, <span>124</span>, <span>11414</span>–<span>11446</span> (2019). https://doi.org/10.1029/2019JB017539</li> </ul>

scholarly journals Seismic velocity structure at the southern termination of the 2016 Kumamoto Earthquake rupture, Japan

2020 ◽  
Vol 72 (1) ◽  
Author(s):  
Yasuhira Aoyagi ◽  
Haruo Kimura ◽  
Kazuo Mizoguchi
Keyword(s):  
Fault Zone ◽  
Velocity Structure ◽  
Seismic Velocity ◽  
Earthquake Rupture ◽  
2016 Kumamoto Earthquake ◽  
Seismic Velocity Structure ◽  
Kumamoto Earthquake ◽  
Seismogenic Layer ◽  
The 2016 Kumamoto Earthquake ◽  
Tectonic Line

Abstract The earthquake rupture termination mechanism and size of the ruptured area are crucial parameters for earthquake magnitude estimations and seismic hazard assessments. The 2016 Mw 7.0 Kumamoto Earthquake, central Kyushu, Japan, ruptured a 34-km-long area along previously recognized active faults, eastern part of the Futagawa fault zone and northernmost part of the Hinagu fault zone. Many researchers have suggested that a magma chamber under Aso Volcano terminated the eastward rupture. However, the termination mechanism of the southward rupture has remained unclear. Here, we conduct a local seismic tomographic inversion using a dense temporary seismic network to detail the seismic velocity structure around the southern termination of the rupture. The compressional-wave velocity (Vp) results and compressional- to shear-wave velocity (Vp/Vs) structure indicate several E–W- and ENE–WSW-trending zonal anomalies in the upper to middle crust. These zonal anomalies may reflect regional geological structures that follow the same trends as the Oita–Kumamoto Tectonic Line and Usuki–Yatsushiro Tectonic Line. While the 2016 Kumamoto Earthquake rupture mainly propagated through a low-Vp/Vs area (1.62–1.74) along the Hinagu fault zone, the southern termination of the earthquake at the focal depth of the mainshock is adjacent to a 3-km-diameter high-Vp/Vs body. There is a rapid 5-km step in the depth of the seismogenic layer across the E–W-trending velocity boundary between the low- and high-Vp/Vs areas that corresponds well with the Rokkoku Tectonic Line; this geological boundary is the likely cause of the dislocation of the seismogenic layer because it is intruded by serpentinite veins. A possible factor in the southern rupture termination of the 2016 Kumamoto Earthquake is the existence of a high-Vp/Vs body in the direction of southern rupture propagation. The provided details of this inhomogeneous barrier, which are inferred from the seismic velocity structures, may improve future seismic hazard assessments for a complex fault system composed of multiple segments.

scholarly journals Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

2020 ◽  
Vol 85 (2) ◽  
Author(s):  
R. Abgrall ◽  
J. Nordström ◽  
P. Öffner ◽  
S. Tokareva
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Discontinuous Galerkin ◽  
Finite Element Methods ◽  
Main Idea ◽  
Galerkin Methods ◽  
Exact Integration ◽  
Galerkin Scheme ◽  
Continuous Galerkin ◽  
Continuous Galerkin Methods

AbstractIn the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

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