Internal wave reflections and transmissions arising from a non‐uniform mesh. Part I: An analysis for the Crank–Nicolson linear finite element scheme

Cathers, B.; Bates, Susan M.; O’Connor, Brian A.

doi:10.1002/fld.1650090704

Cited by 4 publications

(3 citation statements)

References 9 publications

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“…First, outside the layers of small cells, the maximum frequency of seismic waves that can be modelled is reduced by a factor of two. Second, there is the possibility of seismic wave reflection from the interface between small and large cells (Vichnevetsky 1987;Cathers et al 1989). Our testing indicates that grid-doubling yields satisfactory results despite these effects, although they may contribute to increased spurious oscillations of the model.…”

Section: G R I D -D O U B L I N G M E S H E Smentioning

confidence: 99%

A grid-doubling finite-element technique for calculating dynamic three-dimensional spontaneous rupture on an earthquake fault

Barall¹

2009

Geophysical Journal International

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S U M M A R YWe present a new finite-element technique for calculating dynamic 3-D spontaneous rupture on an earthquake fault, which can reduce the required computational resources by a factor of six or more, without loss of accuracy. The grid-doubling technique employs small cells in a thin layer surrounding the fault. The remainder of the modelling volume is filled with larger cells, typically two or four times as large as the small cells. In the resulting non-conforming mesh, an interpolation method is used to join the thin layer of smaller cells to the volume of larger cells. Grid-doubling is effective because spontaneous rupture calculations typically require higher spatial resolution on and near the fault than elsewhere in the model volume. The technique can be applied to non-planar faults by morphing, or smoothly distorting, the entire mesh to produce the desired 3-D fault geometry. Using our FaultMod finite-element software, we have tested grid-doubling with both slip-weakening and rate-and-state friction laws, by running the SCEC/USGS 3-D dynamic rupture benchmark problems. We have also applied it to a model of the Hayward fault, Northern California, which uses realistic fault geometry and rock properties. FaultMod implements fault slip using common nodes, which represent motion common to both sides of the fault, and differential nodes, which represent motion of one side of the fault relative to the other side. We describe how to modify the traction-at-split-nodes method to work with common and differential nodes, using an implicit time stepping algorithm.The paper presents a new grid-doubling technique for dynamic 3-D spontaneous rupture calculations. This technique substantially reduces the required computational resources compared to standard finite-element techniques, without loss of accuracy.A dynamic 3-D spontaneous rupture calculation is a threedimensional numerical simulation of the physical processes that occur during an earthquake (see, e.g. Harris 2004). It is constructed by specifying the relevant physical laws, which include the elastodynamic wave equation, the constitutive equations which describe the rock properties, and the frictional equations which describe the properties of the fault. It is also necessary to specify a set of initial conditions, including the initial stresses acting on the fault. Then, the computer calculates the resulting behaviour of the system, which includes the nucleation and propagation of the earthquake rupture, and the resulting seismic waves.Such calculations are extremely computationally intensive. In finite-element approaches, space is discretized into a set of elements or cells. The standard technique is to use the same sized elements throughout all or most of the modelling volume. However, accurate modelling of fault frictional behaviour requires high spatial resolution, typically requiring an element size in the neighbourhood of 100 m or less. So, even relatively modest models may need many millions of elements. Models of that size consume so much computer...

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Section: G R I D -D O U B L I N G M E S H E Smentioning

confidence: 99%

A grid-doubling finite-element technique for calculating dynamic three-dimensional spontaneous rupture on an earthquake fault

Barall¹

2009

Geophysical Journal International

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“…The y-direction is treated similarly. The discretizations of the convective terms are second order accurate on Cartesian grids with constant step sizes h x and h y in the x-and y-directions with (4) and (5). The stencils for the convective term w x on such grids are Methods for systems of conservation laws governing fluid flow are found in [20].…”

Section: Finite Volume Discretizationmentioning

confidence: 99%

“…Maxwell's equations. One dimensional problems are treated in [5] and interfaces in two dimensions in [4].…”

Section: Introductionmentioning

confidence: 99%

Accurate and stable grid interfaces for finite volume methods

Ferm

Lötstedt

2004

Applied Numerical Mathematics

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A convection-diffusion equation is discretized by a finite volume method in two space dimensions. The grid is partitioned into blocks with jumps in the grid size at the block interfaces. Interpolation in the cells adjacent to the interfaces is necessary to be able to apply the difference stencils. Second order accuracy is achieved and the stability of the discretizations is investigated. The interface treatment is tested in the solution of the compressible Navier-Stokes equations. The conclusions from the scalar equation are valid also for these equations.

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Internal wave reflections and transmissions arising from a non‐uniform mesh. Part II: A generalized analysis for the Crank–Nicolson linear finite element scheme

Cathers¹,

Bates²,

Penoyre³

et al. 1989

Numerical Methods in Fluids

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SUMMARYInternal wave reflections and transmissions are examined for the Crank-Nicolson linear finite element scheme applied to the linear shallow water equations in a 1D domain containing an abrupt change in nodal spacing. In Part I of this series the reflection/transmission analysis was verified by some 'hot-start' numerical experiments. Here in Part 11, however, that analysis is found wanting when it comes to providing a description of the pseudo-steady state wave configuration which develops with some 'cold-start' experiments. It is shown that the analysis of Part I can be extended to take in both the 'hot-' and 'cold-start' experimental results such that four essentially different wave configurations can be identified. The four configurations are discernible on the basis of group velocity. In order to be sustained, two of the configurations require one energy source whereas the other two require two energy sburces. Numerical experiments confirmed the analysis.

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Internal wave reflections and transmissions arising from a non‐uniform mesh. Part I: An analysis for the Crank–Nicolson linear finite element scheme

Cited by 4 publications

References 9 publications

A grid-doubling finite-element technique for calculating dynamic three-dimensional spontaneous rupture on an earthquake fault

A grid-doubling finite-element technique for calculating dynamic three-dimensional spontaneous rupture on an earthquake fault

Accurate and stable grid interfaces for finite volume methods

Internal wave reflections and transmissions arising from a non‐uniform mesh. Part II: A generalized analysis for the Crank–Nicolson linear finite element scheme

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