Strahlkorper< Frame > Class Template Reference

A star-shaped surface expanded in spherical harmonics. More...

#include <Strahlkorper.hpp>

struct  Center

struct  Lmax

struct  Radius

## Public Types

using options = tmpl::list< Lmax, Radius, Center >

## Public Member Functions

Strahlkorper (size_t l_max, size_t m_max, double radius, std::array< double, 3 > center) noexcept
Construct a sphere of radius radius with a given center.

Strahlkorper (size_t l_max, double radius, std::array< double, 3 > center) noexcept
Construct a sphere of radius radius, setting m_max=l_max.

Strahlkorper (size_t l_max, size_t m_max, const DataVector &radius_at_collocation_points, std::array< double, 3 > center) noexcept
Construct a Strahlkorper from a DataVector containing the radius at the collocation points. More...

Strahlkorper (size_t l_max, size_t m_max, const Strahlkorper &another_strahlkorper) noexcept
Prolong or restrict another surface to the given l_max and m_max.

Strahlkorper (DataVector coefs, const Strahlkorper &another_strahlkorper) noexcept
Construct a Strahlkorper from another Strahlkorper, but explicitly specifying the coefficients. Here coefficients are in the same storage scheme as the coefficients() member function returns.

Strahlkorper (DataVector coefs, Strahlkorper &&another_strahlkorper) noexcept
Move-construct a Strahlkorper from another Strahlkorper, explicitly specifying the coefficients.

void pup (PUP::er &p) noexcept
Serialization for Charm++.

const DataVectorcoefficients () const noexcept

DataVectorcoefficients () noexcept

const std::array< double, 3 > & center () const noexcept
Point about which the spectral basis of the Strahlkorper is expanded. The center is given in the frame in which the Strahlkorper is defined. This center must be somewhere inside the Strahlkorper, but in principle it can be anywhere. See physical_center() for a different measure.

std::array< double, 3 > physical_center () const noexcept
Approximate physical center (determined by $$l=1$$ coefficients) Implementation of Eqs. (38)-(40) in [54].

double average_radius () const noexcept
Average radius of the surface (determined by $$Y_{00}$$ coefficient)

size_t l_max () const noexcept
Maximum $$l$$ in $$Y_{lm}$$ decomposition.

size_t m_max () const noexcept
Maximum $$m$$ in $$Y_{lm}$$ decomposition.

double radius (double theta, double phi) const noexcept
Radius at a particular angle $$(\theta,\phi)$$. This is inefficient if done at multiple points many times. See YlmSpherepack for alternative ways of computing this.

bool point_is_contained (const std::array< double, 3 > &x) const noexcept
Determine if a point x is contained inside the surface. The point must be given in Cartesian coordinates in the frame in which the Strahlkorper is defined. This is inefficient if done at multiple points many times.

const YlmSpherepackylm_spherepack () const noexcept

## Static Public Attributes

static constexpr Options::String help

## Detailed Description

### template<typename Frame> class Strahlkorper< Frame >

A star-shaped surface expanded in spherical harmonics.

## ◆ Strahlkorper()

template<typename Frame >
 Strahlkorper< Frame >::Strahlkorper ( size_t l_max, size_t m_max, const DataVector & radius_at_collocation_points, std::array< double, 3 > center )
noexcept

Construct a Strahlkorper from a DataVector containing the radius at the collocation points.

Note
The collocation points of the constructed Strahlkorper will not be exactly radius_at_collocation_points. Instead, the constructed Strahlkorper will match the shape given by radius_at_collocation_points only to order (l_max,m_max). This is because the YlmSpherepack representation of the Strahlkorper has more collocation points than spectral coefficients. Specifically, radius_at_collocation_points has $$(l_{\rm max} + 1) (2 m_{\rm max} + 1)$$ degrees of freedom, but because there are only $$m_{\rm max}^2+(l_{\rm max}-m_{\rm max})(2m_{\rm max}+1)$$ spectral coefficients, it is not possible to choose spectral coefficients to exactly match all points in radius_at_collocation_points.

## ◆ coefficients()

template<typename Frame >
 const DataVector& Strahlkorper< Frame >::coefficients ( ) const
inlinenoexcept

These coefficients are stored as SPHEREPACK coefficients. Suppose you represent a set of coefficients $$F^{lm}$$ in the expansion

$f(\theta,\phi) = \sum_{l=0}^{l_{max}} \sum_{m=-l}^{l} F^{lm} Y^{lm}(\theta,\phi)$

Here the $$Y^{lm}(\theta,\phi)$$ are the usual complex-valued scalar spherical harmonics, so $$F^{lm}$$ are also complex-valued. But here we assume that $$f(\theta,\phi)$$ is real, so therefore the $$F^{lm}$$ obey $$F^{l-m} = (-1)^m (F^{lm})^\star$$. So one does not need to store both real and imaginary parts for both positive and negative $$m$$, and the stored coefficients can all be real.

So the stored coefficients are:

\begin{align} \text{coefficients()(l,m)} &= (-1)^m \sqrt{\frac{2}{\pi}} \Re(F^{lm}) \quad \text{for} \quad m\ge 0, \\ \text{coefficients()(l,m)} &= (-1)^m \sqrt{\frac{2}{\pi}} \Im(F^{lm}) \quad \text{for} \quad m<0 \end{align}

## ◆ help

template<typename Frame >
 constexpr Options::String Strahlkorper< Frame >::help
staticconstexpr
Initial value:
{
"A star-shaped surface expressed as an expansion in spherical "
"harmonics.\n"
"Currently only a spherical Strahlkorper can be constructed from\n"
"Options. To do this, specify parameters Center, Radius, and Lmax."}

The documentation for this class was generated from the following file: