domain::CoordinateMaps::CylindricalSphericalShell Class Reference

Maps a cylindrical shell block to a region bounded by an inner right cylinder and an outer spherical surface. More...

#include <CylindricalSphericalShell.hpp>

Public Member Functions
	CylindricalSphericalShell (double x_inner_lower, double x_inner_upper, double x_outer_lower, double x_outer_upper, double r_inner, double r_sphere)
	Construct the map.
	CylindricalSphericalShell (CylindricalSphericalShell &&)=default
	CylindricalSphericalShell (const CylindricalSphericalShell &)=default
CylindricalSphericalShell &	operator= (const CylindricalSphericalShell &)=default
CylindricalSphericalShell &	operator= (CylindricalSphericalShell &&)=default
template<typename T>
std::array< tt::remove_cvref_wrap_t< T >, 3 >	operator() (const std::array< T, 3 > &source_coords) const
std::optional< std::array< double, 3 > >	inverse (const std::array< double, 3 > &target_coords) const
template<typename T>
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFrame >	jacobian (const std::array< T, 3 > &source_coords) const
template<typename T>
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFrame >	inv_jacobian (const std::array< T, 3 > &source_coords) const
void	pup (PUP::er &p)

Static Public Member Functions
static bool	is_identity ()

Static Public Attributes
static constexpr size_t	dim = 3
static constexpr bool	supports_hessian {false}

Friends
bool	operator== (const CylindricalSphericalShell &lhs, const CylindricalSphericalShell &rhs)

Detailed Description

Maps a cylindrical shell block to a region bounded by an inner right cylinder and an outer spherical surface.

The shaded region is the image.

Details

The logical coordinates are:

• radial \(\xi \in [-1,1]\): interpolates between the inner right cylinder and the outer sphere.
• angular \(\eta \in (-\pi, \pi]\): azimuthal angle \(\phi\) (S1 periodic direction). The inverse map returns \(\eta = \mathrm{atan2}(z, y)\), which lies in \((-\pi, \pi]\).
• axial \(\zeta \in [-1,1]\): blends between the lower and upper ends of the shell.

The physical coordinates \((x, y, z)\) are computed as follows. Let \(\alpha = (\xi+1)/2\) and \(\beta = (\zeta+1)/2\). Define

\begin{align} x_\mathrm{inner}(\beta) &= x^\mathrm{inner}_\mathrm{lower} + \beta\,(x^\mathrm{inner}_\mathrm{upper} - x^\mathrm{inner}_\mathrm{lower}), \\ x_\mathrm{outer}(\beta) &= x^\mathrm{outer}_\mathrm{lower} + \beta\,(x^\mathrm{outer}_\mathrm{upper} - x^\mathrm{outer}_\mathrm{lower}), \\ r_\mathrm{outer}(\beta) &= \sqrt{r_\mathrm{sphere}^2 - x_\mathrm{outer}(\beta)^2}. \end{align}

Then

\begin{align} x &= (1-\alpha)\,x_\mathrm{inner}(\beta) + \alpha\,x_\mathrm{outer}(\beta), \\ r &= (1-\alpha)\,r_\mathrm{inner} + \alpha\,r_\mathrm{outer}(\beta), \\ y &= r\cos\eta, \quad z = r\sin\eta. \end{align}

The six block faces have the following geometry:

• \(\xi = -1\) ( \(\alpha=0\)): the inner right cylinder at \(r = r_\mathrm{inner}\), extending in \(x\) from \(x^\mathrm{inner}_\mathrm{lower}\) to \(x^\mathrm{inner}_\mathrm{upper}\).
• \(\xi = +1\) ( \(\alpha=1\)): a portion of the sphere \(r^2 + x^2 = r_\mathrm{sphere}^2\).
• \(\zeta = \pm 1\): generically curved surfaces in Cartesian coordinates (ruled surfaces that blend linearly in \((x,r)\) between the inner cylinder edge and the outer sphere edge at the corresponding axial end).
• \(\eta\) direction: periodic (connected by the cylindrical_shell topology).

Requirements

• \(x^\mathrm{inner}_\mathrm{lower} < x^\mathrm{inner}_\mathrm{upper}\)
• \(x^\mathrm{outer}_\mathrm{lower} < x^\mathrm{outer}_\mathrm{upper}\)
• \(r_\mathrm{inner} > 0\)
• \(|x^\mathrm{outer}_\mathrm{lower}|, |x^\mathrm{outer}_\mathrm{upper}| < r_\mathrm{sphere}\)
• \(r_\mathrm{inner} < r_\mathrm{outer}(\beta)\) for all \(\beta \in [0,1]\) (equivalently, \(r_\mathrm{inner} < \min(r_\mathrm{outer}(0), r_\mathrm{outer}(1))\)).

Constructor & Destructor Documentation

CylindricalSphericalShell()

domain::CoordinateMaps::CylindricalSphericalShell::CylindricalSphericalShell	(	double	x_inner_lower,
		double	x_inner_upper,
		double	x_outer_lower,
		double	x_outer_upper,
		double	r_inner,
		double	r_sphere )

Construct the map.

Parameters

x_inner_lower	Axial coordinate \(x\) at the lower end of the inner cylinder ( \(\xi=-1, \zeta=-1\)).
x_inner_upper	Axial coordinate \(x\) at the upper end of the inner cylinder ( \(\xi=-1, \zeta=+1\)).
x_outer_lower	Axial coordinate \(x\) at the lower end of the outer spherical face ( \(\xi=+1, \zeta=-1\)).
x_outer_upper	Axial coordinate \(x\) at the upper end of the outer spherical face ( \(\xi=+1, \zeta=+1\)).
r_inner	Radius of the inner right cylinder.
r_sphere	Radius of the outer bounding sphere (centered at the origin).

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

• src/Domain/CoordinateMaps/CylindricalSphericalShell.hpp