Function reference¶

Throughout this document the sum \(\sum_\tau\) implies the periodic sum \(\sum_{p \in \mathbb Z^3}\) with \(\tau = (p_1L_1, p_2L_2, p_3L_3)\).

SE¶

Ewald summation for the 3-periodic electrostatic potential

\[u(x) = \sum_{\tau} \sum_{n=1}^N \frac{q_n}{|x - x_n + \tau|} .\]

SE.se3p_fourier_space(x, f, opt)¶: initialize time array

SE.se3p_precomp(x, opt)¶: SPECTRAL EWALD 3P, pre-computation of FGG vectors Fast Ewald method for electrostatic potential calculation, k-space part.

SE.spectral_ewald(eval_idx, x, q, xi, opt)¶

SPECTRAL EWALD Fast Ewald method for electrostatic potential calculation, k-space part.

phi = spectral_ewald(idx, x, q, xi, box, opt)

Returns:

phi – k-space part of periodic potential

Parameters:

idx – evaluation indices (particle indices)
x – particle positions (N-by-3)
q – particle charge (N-by-1)
xi – Ewald parameter
opt – Options:
opt.M – grid size, eg. [31 31 31] (required)
opt.P – Gaussian support (required)
opt.box – Periodic cell, eg. [1 1 1]
opt.m – Gaussian shape function (default: m=m(P))
opt.w – Gaussian width (default: w=h*P/2)

SE2P¶

Ewald summation for the 2-periodic electrostatic potential.

SE1P¶

Ewald summation for the 1-periodic electrostatic potential.

SE0P¶

Ewald summation for the free-space electrostatic (stokeslet, stresslet, rotlet) potential.

SE_kaiser¶

Ewald summation for the electrostatic potential with arbitrary periodicity using Barnett-Magland kernel.

SE2P_Stokes¶

Ewald summation for the 2-periodic stokeslet potential.

SE2P_Stokes.SE2P_Stokes(eval_idx, x, f, xi, opt)¶

SPECTRAL EWALD 2DP Fast Ewald method for electrostatic potential calculation, k-space part.

phi = spectral_ewald_2dp(idx, x, q, xi, opt)

Dag Lindbo, dag@kth.se, Jul. 2011

SE2P_Stokes.SE2P_Stokes_pre(x, xi, opt)¶

SPECTRAL EWALD 2DP, pre-computation of FGG vectors

Dag Lindbo, dag@kth.se, Jul. 2011

SE2P_Stokes.parse_params(opt)¶: check that we have all mandatory options

SE_fast_gridding¶

Routines for fast gaussian gridding (FGG) and fast Kaiser gridding (FKG), which is central to Spectral Ewald.

SE_fast_gridding.SE_FGG_precomp(x, xi, opt)¶: SPECTRAL EWALD 3P, pre-computation of FGG vectors Fast Ewald method for electrostatic potential calculation, k-space part.

SE_fast_gridding.SE_fg_extend_fcn(F, opt)¶: parameters and constants

SE_fast_gridding.SE_fg_int(x, U, opt)¶

u = SE_fg_int(x, U, opt): Return integration at x of on-grid values U
u = SE_fg_int(x, {U1, …, UN}, opt): Return integration at x of on-grid values U1,…,UN u has N columns

SE_fast_gridding.SE_fg_int_extended(x, U, opt)¶: parameters and constants

SE_fast_gridding.parse_params(opt)¶

Parse parameter struct for fast Gaussian gridding

Mandatory parameters:

Parameters:	opt.M – grid size (1 x 3) opt.P – Gaussian width opt.box – box size (1 x 3)

SE_Rotlet¶

Ewald summation for the 3-periodic rotlet potential

\[u(x) = \sum_{\tau} \sum_{n=1}^N R(x - x_n + \tau) t_n\]

where

\[R_{ij}(r) = \epsilon_{ijk} \frac{r_k}{|r|^3} .\]

SE_Rotlet.SE_Rotlet(xe, x, f, xi, opt)¶

u = SE_Rotlet(x_targets, x_sources, f_sources, xi, opt): Returns the rotlet potential
[U1, U2, U3] = SE_Rotlet(x_targets, x_sources, f_sources, xi, opt): Returns the grid with Fourier coefficients, \(U_i\in\mathbb{C}^{M_1\times M_2 \times M_3}\)

To compute u from U1,U2,U3:

F{1} = real( ifftn( ifftshift( U1 )));
F{2} = real( ifftn( ifftshift( U2 )));
F{3} = real( ifftn( ifftshift( U3 )));
u = SE_fg_int(xe, F, opt);

opt must be identical to that passed to SE_Rotlet

TODO: Add possibility of passing precomputed structures

SE_Rotlet.rotlet_direct_fd(xe, x, f, xi, L, kmax)¶

Direct summation of rotlet k-space part.

Parameters:	xe – target positions, M-by-3 x – source positions, N-by-3 f – source strengths, N-by-3 xi – Ewald parameter, L – box size, 1-by-3 kmax – k-space truncation

SE_Rotlet.rotlet_direct_real(idx, x, f, xi, L, varargin)¶

Direct summation of rotlet real space part

phi = rotlet_direct_real(idx, x, t, xi, box, ‘layers’, M, ‘tol’, TOL);: Compute interactions from all particles within M layers, stop at tolerance TOL.
phi = rotlet_direct_real(idx, x, t, xi, box, ‘mode’,’cutoff’, ‘rc’, rc);: Compute interactions from all particles within cutoff radius rc.

SE_Rotlet.rotlet_direct_rsrc(xe, x, t, opt)¶

Rotlet real space direct summation with cutoff radius, MEX implementation.

Parameters:	xe – target locations x – source locaitons t – source strengths opt – Ewald options opt.box – box size opt.xi – Ewald parameter \(\xi\) opt.rc – Cutoff radius

SE_Rotlet.rotlet_ewald_sum(xe, x, t, opt)¶

Ewald summation for 3P rotlet.

Parameters:	xe – target locations x – source locaitons t – source strengths opt – Ewald options opt.box – box size opt.xi – Ewald parameter \(\xi\) opt.M – Fourier space grid size opt.P – Gaussian width opt.rc – Real space cutoff radius

SE_Stokes¶

Ewald summation for the 3-periodic stokeslet potential

\[u(x) = \sum_{\tau} \sum_{n=1}^N S(x - x_n + \tau) f_n,\]

where

\[S(r) = \frac{I}{|r|} + \frac{r \otimes r}{|r|^3}.\]

SE_Stokes.SE_Stokes(eval_idx, x, f, xi, opt)¶

Compute Fourier space part of Ewald sum for periodic stokeslet potential.

u = SE_Stokes(eval_idx,x,f,xi,opt): Return potential
[U1, U2, U3] = SE_Stokes(eval_idx,x,f,xi,opt): Return Fourier coefficients

Parameters:	eval_idx – index of source locations where potential should be evaluated x – source locations (Nx3) f – source strengths (Nx3) xi – Ewald paramter opt – Ewald options opt.M – grid size (M1, M2, M3) opt.P – Gaussian width opt.box – Box size (L1, L2, L3)
Returns:	phi – Fourier space potential
Returns:	U1,U2,U3 – Fourier space potential

SE_Stokes.SE_Stokes_ext(xe, xs, f, xi, opt)¶: Evaluate Stokeslet at external points

SE_Stokes.SE_Stokes_par(eval_idx, x, f, xi, opt)¶: parfor:ed version

SE_Stresslet¶

Ewald summation for the 3-periodic stresslet potential

\[u(x) = \sum_{\tau} \sum_{m=1}^N q_m T(x - x_m + \tau) \hat n_m,\]

where

\[T_{ijk}(r) = -6\frac{r_i r_j r_k}{|r|^5} .\]

SE_Stresslet.SE_Stresslet(eval_idx, x, f, n, xi, opt, varargin)¶

Compute Fourier space part of Ewald sum for periodic stresslet potential.

phi = SE_Stresslet(eval_idx,x,f,n,xi,opt): Return potential
[U1, U2, U3] = SE_Stresslet(eval_idx,x,f,n,xi,opt): Return Fourier coefficients

Parameters:	eval_idx – index of source locations where potential should be evaluated x – source locations (Nx3) f – source strengths (Nx3) n – source normals (Nx3) xi – Ewald paramter opt – Ewald options: opt.M – grid size (M1, M2, M3) opt.P – Gaussian width opt.box – Box size (L1, L2, L3) varargin=SE_static – Precomputations from SE_Stresslet_pre() to use SIMD FGG code.
Returns:	phi – Fourier space potential
Returns:	U1,U2,U3 – Fourier space coefficients

SE_Stresslet.SE_Stresslet_pre(x, xi, opt)¶

Precompute data for fast Gaussian gridding to enable optimized kernels

Returns:	SE_static

SE_Stresslet.generate_state(N, L, varargin)¶: Returning x: Nx3 matrix with coords inside box L, f: Nx3 matrix with point forces. nvec: Nx3 matrix with associated normal vectors.

SE_Stresslet.stresslet_direct_fd(idx, x, f, nvec, xi, L, kmax)¶

Fourier space Ewald sum for stresslet, using (very slow) direct summation.

Parameters:	kmax – Fourier space truncation

SE_Stresslet.stresslet_direct_fd_zero(idx, xvec, fvec, nvec, L)¶

Component for \(k=0\) of stresslet Fourier sum.

Parameters:	idx – index of target positions (Nx1) xvec – source positions (Nx3) fvec – source strengths (Nx3) nvec – normal vectors (Nx3) L – box size (1x3)
Returns:	phi_k0 – (Nx3)

SE_Stresslet.stresslet_direct_real(idx, x, f, nvec, xi, L, nbox, TOL, varargin)¶

Real space Ewald sum for stresslet, using (very slow) direct summation.

Parameters:	nbox – Number of periodic replications in each direction. TOL – Break when truncation error < TOL. varargin=eval_x – Evaluate at eval_x instead of x(idx,:)

SE_Stresslet.stresslet_direct_real_fast(idx, x, f, nvec, xi, L, nbox, rc, varargin)¶

Ewald summation for the stresslet – Real space part. Fast in the sense that it saves interactions as matrix for subsequent iterations Sums over layers

[phi A] = stresslet_direct_real_fast( idx, x, f, nvec, xi, L, nbox, rc, [A, sing_sub, ns_quad_mtrx])

Parameters:	x – positions :param (N-by-3) f – source strengths (N-by-3) nvec – normal vectors (N-by-3) xi – Ewald parameter L – Box size (3) nbox – periodic replications rc – cutoff radius A – Precomputed matrix sing_sub – Use singularity subtraction ns_quad_mtrx – Correction matrices for nearly singular quadrature

SE_Stresslet.stresslet_op_fd(k, xi)¶: Returns imaginary part of Bi(k, xi) ie B = 1i*Bi;

SE_Stresslet.stresslet_real_rc(x, q, nvec, xi, box, rc, varargin)¶

Stresslet real space summation with truncation radius rc

Usage:

[res] = stresslet_real_rc( x, f, nvec, xi, box, rc);
[res AMAT] = stresslet_real_rc( x, f, nvec, xi, box, rc, [], sing_sub, ns_quad_mtrx);
[res AMAT] = stresslet_real_rc( x, f, nvec, xi, box, rc, AMAT, sing_sub, ns_quad_mtrx);
[res AMAT R C V PER] = stresslet_real_rc( x, f, nvec, xi, box, rc);

Parameters:	sing_sub – singularity subtraction, default 0. Only for matrix comp. rc – cutoff radius, must be <= min(box)/2