Absorbing boundary and free-surface conditions in the phononic lattice solid by interpolation

We have recently developed a new lattice-Boltzmann-based approach for modelling compressional wave propagation in heterogeneous media, which we call the phononic lattice solid by interpolation (PLSI). In this paper, we propose an absorbing boundary condition for the PLSI method in which the microscopic reflection coefficients at the boundaries of a model are set to zero and viscous layers are added to the boundaries. Numerical simulation examples using the PLSI method and comparisons with exact solutions demonstrate that artificial boundary reflections can be almost completely eliminated when the incidence angle is less than approximately 70°. Beyond this angle, remanent artificial boundary reflections become visible. We propose four methods for modelling free-surface reflections in PLSI simulations. In the first three methods, special collision rules at a free surface are specified to take into account the effect of a free surface on quasi-particle movements (i.e. wave propagation). They are termed the specular bouncing, backward bouncing I, and combined bouncing methods. They involve quasi-particle reflections with a coefficient of - 1 and require the free surface to be located exactly along lattice nodes. For the fourth method, we modify the backward bouncing I model for the case when a free surface is located at any position along lattice links and thus term it the backward bouncing II model. It uses the reflection coefficient at the free surface to calculate the reflected number densities during PLSI simulations. Hence, the free surface is handled in the same way as an interface within a model. Numerical examples and comparisons with exact solutions show that these four methods used at the microscopic scale are all appropriate for modelling macroscopic waves reflected from free surfaces

The earth’s core: an approach from first principles

The Earth’s core is largely composed of iron (Fe), alloyed with less dense elements such as sulphur, silicon and/or oxygen. The phase relations and physical properties of both solid and liquid Fe-alloys are therefore of great geophysical importance. As a result, over the past fifty years the properties of Fe and its alloys have been extensively studied experimentally. However, achieving the extreme pressures (up to 360 GPa) and temperatures (~6000K) found in the core provide a major experimental challenge, and it is not surprising that there are still considerable discrepancies in the results obtained by using different experimental techniques. In the past fifteen years quantum mechanical techniques have been applied to predict the properties of Fe. Here we review the progress that has been made in the use of first principles methods to study Fe and its alloys, and as a result of these studies we conclude: (i) that pure Fe adopts an hexagonal close packed structure under core conditions and melts at ~6200 K at 360 GPa, (ii) that thermodynamic equilibrium and observed seismic data are satisfied by a liquid Fe alloy outer core with a composition of ~10 mole% S (or Si) and 8 mole% O crystallising at ~ 5500 K to give an Fe alloy inner core with ~8 mole% S (or Si) and 0.2 mole % O, and (iii) that with such concentrations of S (or Si), an Fe alloy might adopt a body centred cubic structure in all or part of the inner core. In the future the roles of Ni, C, H and K in the core need to be studied, and techniques to predict the transport and rheological properties of Fe alloys need to be developed

Free and smooth boundaries in 2-D finite-difference schemes for transient elastic waves

A method is proposed for accurately describing arbitrary-shaped free boundaries in single-grid finite-difference schemes for elastodynamics, in a time-domain velocity-stress framework. The basic idea is as follows: fictitious values of the solution are built in vacuum, and injected into the numerical integration scheme near boundaries. The most original feature of this method is the way in which these fictitious values are calculated. They are based on boundary conditions and compatibility conditions satisfied by the successive spatial derivatives of the solution, up to a given order that depends on the spatial accuracy of the integration scheme adopted. Since the work is mostly done during the preprocessing step, the extra computational cost is negligible. Stress-free conditions can be designed at any arbitrary order without any numerical instability, as numerically checked. Using 10 grid nodes per minimal S-wavelength with a propagation distance of 50 wavelengths yields highly accurate results. With 5 grid nodes per minimal S-wavelength, the solution is less accurate but still acceptable. A subcell resolution of the boundary inside the Cartesian meshing is obtained, and the spurious diffractions induced by staircase descriptions of boundaries are avoided. Contrary to what occurs with the vacuum method, the quality of the numerical solution obtained with this method is almost independent of the angle between the free boundary and the Cartesian meshing.Comment: accepted and to be published in Geophys. J. In

Rupture by damage accumulation in rocks

The deformation of rocks is associated with microcracks nucleation and propagation, i.e. damage. The accumulation of damage and its spatial localization lead to the creation of a macroscale discontinuity, so-called "fault" in geological terms, and to the failure of the material, i.e. a dramatic decrease of the mechanical properties as strength and modulus. The damage process can be studied both statically by direct observation of thin sections and dynamically by recording acoustic waves emitted by crack propagation (acoustic emission). Here we first review such observations concerning geological objects over scales ranging from the laboratory sample scale (dm) to seismically active faults (km), including cliffs and rock masses (Dm, hm). These observations reveal complex patterns in both space (fractal properties of damage structures as roughness and gouge), time (clustering, particular trends when the failure approaches) and energy domains (power-law distributions of energy release bursts). We use a numerical model based on progressive damage within an elastic interaction framework which allows us to simulate these observations. This study shows that the failure in rocks can be the result of damage accumulation

Seismic Anisotropy of Temperate Ice in Polar Ice Sheets

We present a series of simple shear numerical simulations of dynamic recrystallization of two‐phase nonlinear viscous materials that represent temperate ice. First, we investigate the effect of the presence of water on the resulting microstructures and, second, how water influences on P wave (Vp) and fast S wave (Vs) velocities. Regardless the water percentage, all simulations evolve from a random fabric to a vertical single maximum. For a purely solid aggregate, the highest Vp quickly aligns with the maximum c‐axis orientation. At the same time, the maximum c‐axis development reduces Vs in this orientation. When water is present, the developed maximum c‐axis orientation is less intense, which results in lower Vp and Vs. At high percentage of water, Vp does not align with the maximum c‐axis orientation. If the bulk modulus of ice is assumed for the water phase (i.e., implying that water is at high pressure), we find a remarkable decrease of Vs while Vp remains close to the value for purely solid ice. These results suggest that the decrease in Vs observed at the base of the ice sheets could be explained by the presence of water at elevated pressure, which would reside in isolated pockets at grain triple junctions. Under these conditions water would not favor sliding between ice grains. However, if we consider that deformation dominates over recrystallization, water pockets get continuously stretched, allowing water films to be located at grain boundaries. This configuration would modify and even overprint the maximum c‐axis‐dependent orientation and the magnitude of seismic anisotropy