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

Abstract

accepted and to be published in Geophys. J. Int.International audienceA 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

A hybrid FDTD-Rayleigh integral computational method for the simulation of the ultrasound measurement of proximal femur

“…The grid step ∆x should be sufficiently small to model the geometry accurately. Note that alternative FDTD schemes which have been proposed to model explicitly curved boundaries [29,30] could be used advantageously to model QUS experiments. Finally, the method of coupling between the Rayleigh integral formulation and the FDTD assumes that the diffracting object inside the FDTD box is in the far field.…”

A Fast 3-D Free-surface Topography Method for Acoustic Full-waveform Inversion

“…Wavefields are extended beyond the free surface such that regular highorder discretization stencils may be used to approximate derivatives near the free surface. The method described in Lombard et al (2008) is not stable for standard staggered grids. It is implemented using a high-order, single-grid ADER method.…”

“…Bohlen and Saenger, 2006) or the approach of Robertsson (1996) require a relatively large number of grid points per wavelength (c.f. Lombard et al, 2008).…”

“…For immersed boundary methods (Lombard et al, 2008), the free surface does not have to coincide with the discretization grid. Wavefields are extended beyond the free surface such that regular highorder discretization stencils may be used to approximate derivatives near the free surface.…”

Accurate seabed modeling using finite difference methods

“…In addition, it is relatively difficult to derive and implement, especially for the accuracy of order higher than four. The similar methods called "immersed boundary method" are also used to deal with an irregular topography of the free surface [19,29].…”