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

TL;DR

This paper introduces a novel finite-difference method for accurately modeling arbitrary free boundaries in 2-D elastodynamics, achieving high precision and stability with minimal computational overhead by using boundary-based fictitious values.

Contribution

The method uniquely calculates fictitious boundary values using boundary and compatibility conditions, enabling high-order stress-free boundary modeling without instability.

Findings

High accuracy with 10 grid nodes per wavelength

Acceptable accuracy with 5 grid nodes per wavelength

Boundary representation avoids staircase effects and spurious diffractions

Abstract

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.