Divergence Free Interpolation

Divergence free vector field interpolant for 2D and 3D cases. Described in [1], relies on radial basis functions [2]. The current implementation works as expected, but does not scale well, an improvement would be to implement a multilevel approach [3], but implementing a thinning algorithm [4] proved to be challenging.

Installation

  pip install Divergence-Free-Interpolant

Test case dependecies

  pip install pyvista matplotlib

Basic functionality

Import

  import numpy as np
  import Divergence_Free_Interpolant as dfi

Initialize

  initialized_interpolant = dfi.interpolant(nu = 5, k = 3, dim = 3)

nu - Radial basis function parameter: int, default value 5, in most cases does not have to be changed

k - Radial basis function parameter: int, default value 3, in most cases does not have to be changed

dim - Dimensionality of space to interpolate: int, default value 3, currently only supports 2 and 3, can be expanded indefinitely.

Condition the interpolant

  positions = np.random(3, 10)
  vectorfield = np.random(3, 10)
  initialized_interpolant.condition(positions, vectorfield, support_radius = 0.2, method = 'linsolve')

positions - vector field coordinates: np.ndarray, shape = (dim, N)

vectorfield - vector field values: np.ndarray, shape = (dim, N)

support_radius - kernel radius: float, default value 1

method - method to use for solving the linear system: str, default value linsolve, accepts SVD, penrose, linsolve, lstsq

Interpolate

  x, y, z = 0.3, 0.4, 0.6
  vector = initialized_interpolant(x, y, z)

x - x coordinates at which to interpolate: array_like

y - y coordinates at which to interpolate: array_like

z - z coordinates at which to interpolate: array_like

vector - interoplated vector values at the given points: np.ndarray, shape = (..., dim)

__call__ is vectorized

if dim == 2 will not accept the z component

See tests/test_case_2D.py and tests/test_case_3D.py for more detailed examples.

References

[1] Fuselier, Edward J. “Sobolev-Type Approximation Rates for Divergence-Free and Curl-Free RBF Interpolants.” Mathematics of Computation, vol. 77, no. 263, 2008, pp. 1407–23. http://www.jstor.org/stable/40234564

[2] Wendland, H. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4, 389–396 (1995). https://doi.org/10.1007/BF02123482

[3] Patricio Farrell, Kathryn Gillow, Holger Wendland, Multilevel interpolation of divergence-free vector fields, IMA Journal of Numerical Analysis, Volume 37, Issue 1, January 2017, Pages 332–353, https://doi.org/10.1093/imanum/drw006

[4] Floater M. S. Iske A. Thinning algorithms for scattered data interpolation . BIT , 38 , 705 –720 . (1998)