An adaptive node-distribution method for radial-basis-function finite-difference modeling with optimal shape parameter

The radial-basis-function finite-difference (RBF-FD) method has been proven successful in modeling seismic-wave propagation. Node distribution is typically the first and most critical step in RBF-FD. Regarding the difficulties in seismic modeling, such as node distribution of complex geologic structures, we have designed an adaptive node-distribution method that can generate nodes automatically and flexibly as the computation proceeds with the adaptive grain-radius satisfied dispersion relation and stability condition of seismic-wave propagation. Our method consists of two novel points. The first one is that we adopt an adaptive grain-radius generation method, which can automatically provide a wider scope of grain radius in seismic modeling while satisfying the dispersion relation and stability condition; the second one is that the node-generation algorithm is built by a smoothed model, which significantly improves the modeling stability at a reduced computational cost. Excessive or undesirable shape parameters will create a very ill-conditioned problem. A set of optimal shape parameters for different numbers of neighbor nodes is found quantitatively by minimizing root-mean-square error functions. This optimization method enables us to achieve an improved meshfree modeling process with higher accuracy and practicability and fewer spurious diffractions caused by the transition of different sampling areas. Several numerical results verify the feasibility of our adaptive node-distribution method and the optimal shape parameters.