Keywords: acoustics, time-domain, local, multi-quadric radial basis function.

In the last decade, meshless methods have found widespread application in different fields of engineering and science. Beyond novelty, their mathematical and modeling simplicity and their accuracy have been the key to their rapid dissemination. Among meshless techniques, radial basis function (RBF) based methods can be simple and general, allowing the solution of problems related to multiple areas of applied physics and engineering. In the specific field of acoustics, there are usually two possible approaches for the solution of a problem: time- and frequency-domain.

In this work, the authors propose a local RBF-based interpolation scheme for the solution of the acoustic wave equation in the time-domain. The chosen interpolation function is the multi-quadric (MQ) function which is used as interpolation function, and a very limited number of points around each node is used for interpolation. Thus, the interpolation becomes local, and is performed in overlapping regions. As the solution is pursued in the time-domain, the proposed approach is used to reconstruct the relevant spatial partial derivatives, and an explicit time-marching scheme based on central differences is used.

First, a quick overview of the mathematical formulation of the method is presented. Then, the method is verified using closed-form solutions, known for a simple geometrical configuration, also testing the effect of different choices for the interpolation domain. The computed responses revealed a good fit between the numerical and analytical solutions, in particular when the interpolation domain is not too small.

A second and a third test case are also presented, where the propagation of a Ricker pulse in a heterogeneous acoustic media is simulated, and is used to assess the applicability of the method to more realistic situations. For those cases, the results were, then, compared with those provided using a finite-difference implementation. Those two test cases allowed observing that the arrival times predicted by the RBF and by the finite-difference model match very well, although small amplitude mismatches can be detected, even for the final part of the first arriving pulse. However, in general, the results computed using both approaches followed the same trend, with the signal exhibiting the same patterns of pulse propagation. Similar conclusions were obtained when observing the first time derivative of the computed pressure, although with slightly more pronounced differences between the two models.

All the presented examples revealed an accurate behavior of the proposed strategy, and indicated that the method may be further developed and used to accurately solve large scale applications in several fields, such as ocean acoustics or geophysics.

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