parkipy.ewald.stokes_comb

parkipy.ewald.stokes_comb(trg, src, dens, normal, options)

Compute the combined Stokes single and double layer potential.

\[\boldsymbol{u}(\boldsymbol{x}_i) = \sum_{j=1}^{N} \sum_{p \in P} \left(\left( \frac{\boldsymbol{q}_j}{\lVert \boldsymbol{r}_{ij} \rVert} + \frac{\boldsymbol{r}_{ij}}{\lVert \boldsymbol{r}_{ij} \rVert^3} (\boldsymbol{r}_{ij} \cdot \boldsymbol{q}_j) \right) + \left( \epsilon_{jlm} \frac{\boldsymbol{r}_m}{\|\boldsymbol{r}_{ij}\|^3} \boldsymbol{q}_l \boldsymbol{n}_m \right) \right),\]

where \(\boldsymbol{r}_{ij} = \boldsymbol{x}_i - \boldsymbol{y}_j\) and \(P\) is the specified periodicity.

Warning

Currently only supported for \(P=1,3\).

Parameters:

• trg (ndarray) – Array of target values \(\boldsymbol{x}_i\) to compute the Stokes potential. Array should be shape (3,nt) with the default ordering (i.e., 'C' ordering for cupy and 'F' ordering for numpy.

• src (ndarray) – Array of source values \(\boldsymbol{y}_j\) of shape (3, ns) with default ordering.

• dens (ndarray) – Array of density values of the source points of shape (6,ns) with default ordering where dens[:3] represent the single layer density \(\boldsymbol{q}_j\) and dens[3:] the double layer density \(\boldsymbol{q}_l\).

• normal (ndarray) – Array of normal vectors for the source points of shape (3,ns) with default ordering.

• options (EwaldOptions) – Dataclass specifying Ewald parameters.

Returns:

• potential (ndarray) – Array containing the Ewald approximation for the Stokes potential at the target points \(\boldsymbol{u}(\boldsymbol{x}_i)\).

• walltimes (dict, optional) – Dictionary containing the wall-time for each stage of the Ewald summation. Only returned if options.return_walltime == True.

• params (SEParams, optional) – SEParams dataclass containing derived Ewald parameters for the Ewald summation run. Only returned if options.return_params == True.

Raises:

NotImplementedError – If options.periodicity is not 1 or 3.

Notes

Parameter selection based off [1].

References

[1]

Bagge, J., & Tornberg, A.-K. (2023). Fast Ewald summation for Stokes flow with arbitrary periodicity. Journal of Computational Physics, 493, 112473. https://doi.org/10.1016/j.jcp.2023.112473

Examples

>>> import parkipy
>>> import numpy as np
>>> trg = src = np.random.rand(3, 100000)
>>> dens_sl = np.random.randn(3, 100000)
>>> dens_dl = np.random.randn(3, 100000)
>>> norms = np.random.randn(3, 100000)
>>> dens = np.vstack((dens_sl, dens_dl))  # stack densities for ewald call
>>> options = parkipy.ewald.EwaldOptions(
... box=[1,1,1], periodicity=1, tolerance=1e-4,
... execution_space="openmp", cell_size=224, return_walltime=True
... )
>>> pot, walltime = parkipy.ewald.stokes_comb(trg, src, dens, norms, options)
>>> walltime
{'p2p': {'args': 3.0994415283203125e-06, 'sort': 0.004508018493652344, 'kernel': 0.08536601066589355, 'tot': 0.08987712860107422}, 'p2g': {'args': 8.106231689453125e-06, 'sort': 0.002873659133911133, 'kernel': 0.06609272956848145, 'tot': 0.06897449493408203}, 'fft': {'tot': 0.05228924751281738}, 'cnv': {'tot': 0.025938749313354492}, 'ifft': {'tot': 0.010168075561523438}, 'g2p': {'args': 2.4318695068359375e-05, 'sort': 0.0019330978393554688, 'kernel': 0.0029044151306152344, 'adj': 0.0005638599395751953, 'tot': 0.005425691604614258}}