influence_function.hpp File Reference

#include "config/config.hpp"
#include "p3m/common.hpp"
#include "p3m/for_each_3d.hpp"
#include "p3m/math.hpp"
#include <utils/Vector.hpp>
#include <utils/index.hpp>
#include <utils/math/int_pow.hpp>
#include <utils/math/sqr.hpp>
#include <cmath>
#include <cstddef>
#include <functional>
#include <numbers>
#include <utility>
#include <vector>

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Functions
template<std::size_t S, std::size_t m>
double	G_opt (P3MParameters const &params, Utils::Vector3d const &k)
	Calculate the aliasing sums for the optimal influence function.
template<typename FloatType , std::size_t S, std::size_t m>
std::vector< FloatType >	grid_influence_function (P3MParameters const &params, Utils::Vector3i const &n_start, Utils::Vector3i const &n_stop, Utils::Vector3d const &inv_box_l)
	Map influence function over a grid.

Function Documentation

G_opt()

template<std::size_t S, std::size_t m>

double G_opt	(	P3MParameters const &	params,
		Utils::Vector3d const &	k
	)

Calculate the aliasing sums for the optimal influence function.

This implements Eq. 30 of [10], which can be used for monopole and dipole P3M by choosing the appropriate S factor.

$$\tilde{G}_{\text{opt}} ( k ) = \frac{\sum_{m \in \mathbb{Z}^{3}} [ [ \tilde{ \left ( D \right ) } ( k ) \cdot i k_{m} ]^{S} ( \hat{U} ( k_{m} ) )^{2} \hat{\phi} ( k_{m} ) ]}{[ \tilde{ \left ( D \right ) } ( k ) ]^{2 S} [ \sum_{m \in \mathbb{Z}^{3}} ( \hat{U} ( k_{m} ) )^{2} ]^{2}}$$

Eq 8.29 in Hockney:

$$G_{\text{opt}}(\mathbf{k}) = \frac{\hat{\mathbf{D}}\sum_n \hat{\mathbf{R}}^\ast \hat{U}^2}{|\hat{\mathbf{D}}|^2\sum_n \hat{U}^2\sum_{n'}^{'} \hat{U}^2}$$

The full equation is:

$$\begin{align*} G_{\text{opt}}(\vec{n}, S, \text{cao}) &= \displaystyle\frac{4\pi}{\sum_\vec{m} U^2} \sum_\vec{m}U^2 \displaystyle\frac{\left(\vec{k}\odot\vec{k}_m\right)^S} {|\vec{k}_m|^2} e^{-1/(2\alpha)^2|\vec{k}_m|^2} \\ U &= \operatorname{det}\left[I_3\cdot\operatorname{sinc}\left( \frac{\vec{k}_m\odot\vec{a}}{2\pi}\right)\right]^{\text{cao}} \\ \vec{k} &= \frac{2\pi}{\vec{N}\odot\vec{a}}\vec{s}[\vec{n}] \\ \vec{k}_m &= \frac{2\pi}{\vec{N}\odot\vec{a}} \left(\vec{s}[\vec{n}]+\vec{m}\odot\vec{N}\right) \end{align*}$$

with $I_3$ the 3x3 identity matrix, $\vec{N}$ the global mesh size in k-space coordinates, $\vec{a}$ the mesh size, $\vec{m}$ the Brillouin zone coordinates, $\odot$ the Hadamard product.

Template Parameters

S	Order of the differential operator (0 for potential, 1 for E-field)
m	Number of Brillouin zones that contribute to the aliasing sums

Parameters

params	P3M parameters
k	k-vector to evaluate the function for.

Returns

The optimal influence function for a specific k-vector.

Definition at line 91 of file influence_function.hpp.

References Utils::Vector< T, N >::broadcast(), for_each_3d(), Utils::Vector< T, N >::norm2(), params, Utils::product(), math::sinc(), and Utils::sqr().

grid_influence_function()

template<typename FloatType , std::size_t S, std::size_t m>

std::vector< FloatType > grid_influence_function	(	P3MParameters const &	params,
		Utils::Vector3i const &	n_start,
		Utils::Vector3i const &	n_stop,
		Utils::Vector3d const &	inv_box_l
	)

Map influence function over a grid.

This evaluates the optimal influence function G_opt over a regular grid of k vectors, and returns the values as a vector.

Template Parameters

S	Order of the differential operator (0 for potential, 1 for E-field)
m	Number of Brillouin zones that contribute to the aliasing sums

Parameters

params	P3M parameters.
n_start	Lower left corner of the grid in k-space.
n_stop	Upper right corner of the grid in k-space.
inv_box_l	Inverse box length.

Returns

Values of G_opt at regular grid points.

Definition at line 151 of file influence_function.hpp.

References calc_p3m_mesh_shift(), Utils::COLUMN_MAJOR, for_each_3d(), Utils::get_linear_index(), params, and Utils::product().