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

We have recently developed a new lattice-Boltzmann-based approach for model ling compressional wave propagation in heterogeneous media, which we call t he phononic lattice solid by interpolation (PLSI). In this paper, we propos e an absorbing boundary condition for the PLSI method in which the microsco pic reflection coefficients at the boundaries of a model are set to zero an d viscous layers are added to the boundaries. Numerical simulation examples using the PLSI method and comparisons with exact solutions demonstrate tha t artificial boundary reflections can be almost completely eliminated when the incidence angle is less than approximately 70 degrees. Beyond this angl e, remanent artificial boundary reflections become visible. We propose four methods for modelling free-surface reflections in PLSI simu lations. In the first three methods, special collision rules at a free surf ace are specified to take into account the effect of a free surface on quas i-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 s urface 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 l ocated at any position along lattice links and thus term it the backward bo uncing II model. It uses the reflection coefficient at the free surface to calculate the reflected number densities during PLSI simulations. Hence, th e free surface is handled in the same way as an interface within a model. N umerical examples and comparisons with exact solutions show that these four methods used at the microscopic scale are all appropriate for modelling ma croscopic waves reflected from free surfaces.