meshfree modeling using p-refined radial-basis finite … · 2020. 3. 19. · meshfree modeling...

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