Strahlkorper< Frame > Class Template Reference

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

#include <Strahlkorper.hpp>

Classes
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)
	Construct a sphere of radius radius with a given center.
	Strahlkorper (size_t l_max, double radius, std::array< double, 3 > center)
	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)
	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)
	Prolong or restrict another surface to the given l_max and m_max.
	Strahlkorper (DataVector coefs, const Strahlkorper &another_strahlkorper)
	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)
	Move-construct a Strahlkorper from another Strahlkorper, explicitly specifying the coefficients.
void	pup (PUP::er &p)
	Serialization for Charm++.
const DataVector &	coefficients () const
DataVector &	coefficients ()
const std::array< double, 3 > &	expansion_center () const
	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
	Approximate physical center (determined by \(l=1\) coefficients) Implementation of Eqs. (38)-(40) in [63].
double	average_radius () const
	Average radius of the surface (determined by \(Y_{00}\) coefficient)
size_t	l_max () const
	Maximum \(l\) in \(Y_{lm}\) decomposition.
size_t	m_max () const
	Maximum \(m\) in \(Y_{lm}\) decomposition.
double	radius (double theta, double phi) const
	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
	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 YlmSpherepack &	ylm_spherepack () const

Static Public Attributes
static constexpr Options::String	help

Detailed Description

template<typename Frame>
class Strahlkorper< Frame >

A star-shaped surface expanded in spherical harmonics.

Constructor & Destructor Documentation

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
	)

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.

Member Function Documentation

coefficients()

template<typename Frame >

const DataVector & Strahlkorper< Frame >::coefficients ( ) const

inline

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}

Member Data Documentation

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."}

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

• src/NumericalAlgorithms/SphericalHarmonics/Strahlkorper.hpp