Post on 14-Feb-2021
MESHFREE MODELING USING P-REFINED RADIAL-BASIS FINITE DIFFERENCES
Pankaj K Mishra1, Xin Liu1, Leevan Ling2, and Mrinal K Sen1
1Institute for geophysics, The University of Texas at Austin2Department of Mathematics, Hong Kong Baptist University
ABSTRACTRadial basis functions-generated finite difference methods (RBF-FDs) have been gain-ing popularity in the numerical modeling community. In particular, the RBF-FD basedon polyharmonic splines (PHS) augmented with multivariate polynomials (PHS+poly)has been found effective. Many practical problems, in numerical analysis, do not re-quire a uniform node-distribution. Instead, they would be better suited if specific areasof domain, where complicated physics needs to be resolved, have a relatively highernode-density compared to the rest of the domain. In this work, we propose an adap-tive RBF-FD method with a user-defined order of convergence with respect to the totalnumber of (possibly scattered and non-uniform) data points N. Our algorithm outputs asparse differentiation matrix with a desired approximation order. Numerical examplesare provided to show that the proposed adaptive RBF-FD method yields the expectedN-convergence even for highly non-uniform node-distributions. The proposed methodalso reduces the number of non-zero elements in the linear system without sacrificingaccuracy.
1
Meshfree modeling with RBF-FD
0 100 200 300 400 500
X (m)
0
100
200
300
400
500
Z (
m)
Propagating time is 0.1s
-2
0
2
4
6
8
10 -6
0 100 200 300 400 500
X (m)
0
100
200
300
400
500
Z (
m)
Propagating time is 0.2s
-1
-0.5
0
0.5
1
1.5
210 -6
0 100 200 300 400 500
X (m)
0
100
200
300
400
500
Z (
m)
Propagating time is 0.3s
-1
-0.5
0
0.5
1
1.5
2
10 -6
0 100 200 300 400 500
X (m)
0
100
200
300
400
500
Z (
m)
Propagating time is 0.4s
-5
0
5
10
15
10 -7
Figure 1. Snapshots of acoustic wave propagation in a two-layered velocity model, on meshfree nodes.
2