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.