domain::CoordinateMaps::FocallyLiftedInnerMaps::FlatSide Class Reference

A FocallyLiftedInnerMap that maps a 3D unit right cylinder to a volume that connects a 2D annulus to a spherical surface. More...

#include <FocallyLiftedFlatSide.hpp>

Public Member Functions
	FlatSide (const std::array< double, 3 > &center, const double inner_radius, const double outer_radius)
	FlatSide (FlatSide &&)=default
	FlatSide (const FlatSide &)=default
FlatSide &	operator= (const FlatSide &)=default
FlatSide &	operator= (FlatSide &&)=default
template<typename T >
void	forward_map (const gsl::not_null< std::array< tt::remove_cvref_wrap_t< T >, 3 > * > target_coords, const std::array< T, 3 > &source_coords) const
std::optional< std::array< double, 3 > >	inverse (const std::array< double, 3 > &target_coords, double sigma_in) const
template<typename T >
void	jacobian (const gsl::not_null< tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFrame > * > jacobian_out, const std::array< T, 3 > &source_coords) const
template<typename T >
void	inv_jacobian (const gsl::not_null< tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFrame > * > inv_jacobian_out, const std::array< T, 3 > &source_coords) const
template<typename T >
void	sigma (const gsl::not_null< tt::remove_cvref_wrap_t< T > * > sigma_out, const std::array< T, 3 > &source_coords) const
template<typename T >
void	deriv_sigma (const gsl::not_null< std::array< tt::remove_cvref_wrap_t< T >, 3 > * > deriv_sigma_out, const std::array< T, 3 > &source_coords) const
template<typename T >
void	dxbar_dsigma (const gsl::not_null< std::array< tt::remove_cvref_wrap_t< T >, 3 > * > dxbar_dsigma_out, const std::array< T, 3 > &source_coords) const
std::optional< double >	lambda_tilde (const std::array< double, 3 > &parent_mapped_target_coords, const std::array< double, 3 > &projection_point, bool source_is_between_focus_and_target) const
template<typename T >
void	deriv_lambda_tilde (const gsl::not_null< std::array< tt::remove_cvref_wrap_t< T >, 3 > * > deriv_lambda_tilde_out, const std::array< T, 3 > &target_coords, const T &lambda_tilde, const std::array< double, 3 > &projection_point) const
void	pup (PUP::er &p)

Static Public Member Functions
static bool	is_identity ()

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

Detailed Description

A FocallyLiftedInnerMap that maps a 3D unit right cylinder to a volume that connects a 2D annulus to a spherical surface.

Details

The domain of the map is a 3D unit right cylinder with coordinates $(\overline{x},\overline{y},\overline{z})$ such that $-1\le \overline{z}\le 1$ and $1\le {\overline{x}}^2+{\overline{y}}^2\le 4$. The range of the map has coordinates $(x,y,z)$.

Consider a 2D annulus in 3D space oriented normal to the $z$ axis. The inner and outer radii of the annulus are $R_{\mathrm{i}\mathrm{n}}$ and $R_{\mathrm{o}\mathrm{u}\mathrm{t}}$, and the (3D) center of the annulus is $C^i$. FlatSide provides the following functions:

forward_map()

forward_map() maps $(\overline{x},\overline{y},\overline{z}=-1)$ to the interior of the annulus. The arguments to forward_map() are $(\overline{x},\overline{y},\overline{z})$, but $\overline{z}$ is ignored. forward_map() returns $x_0^i$, the 3D coordinates on the annulus, which are given by

$$\begin{array}{}\text{(1)}& x_0^0& =(R_{\mathrm{i}\mathrm{n}}+(R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}})(\overline{\rho }-1))\frac{\overline{x}}{\overline{\rho }}+C^0,\text{(2)}& x_0^1& =(R_{\mathrm{i}\mathrm{n}}+(R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}})(\overline{\rho }-1))\frac{\overline{y}}{\overline{\rho }}+C^1,\text{(3)}& x_0^2& =C^2,\end{array}$$

where

$$\begin{array}{}\text{(4)}& \overline{\rho }=\sqrt{{\overline{x}}^2+{\overline{y}}^2}.\end{array}$$

sigma

$\sigma$ is a function that is zero on the sphere $x^i=x_0^i$ and unity at $\overline{z}=+1$ (corresponding to the upper surface of the FocallyLiftedMap). We define

$$\begin{array}{}\text{(5)}& \sigma & =\frac{\overline{z}+1}{2}.\end{array}$$

deriv_sigma

deriv_sigma returns

$$\begin{array}{}\text{(6)}& \frac{\partial \sigma }{\partial {\overline{x}}^j}& =(0,0,1/2).\end{array}$$

jacobian

jacobian returns $\partial x_0^k/\partial {\overline{x}}^j$. The arguments to jacobian are $(\overline{x},\overline{y},\overline{z})$, but $\overline{z}$ is ignored.

Differentiating Eqs. ( $\text{1}$– $\text{3}$) above yields

$$\begin{array}{rl}\frac{\partial x_0^0}{\partial \overline{x}}& =R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}+(2R_{\mathrm{i}\mathrm{n}}-R_{\mathrm{o}\mathrm{u}\mathrm{t}})\frac{{\overline{y}}^2}{{\overline{\rho }}^3},\\ \frac{\partial x_0^0}{\partial \overline{y}}& =-(2R_{\mathrm{i}\mathrm{n}}-R_{\mathrm{o}\mathrm{u}\mathrm{t}})\frac{\overline{x}\overline{y}}{{\overline{\rho }}^3},\\ \frac{\partial x_0^1}{\partial \overline{x}}& =-(2R_{\mathrm{i}\mathrm{n}}-R_{\mathrm{o}\mathrm{u}\mathrm{t}})\frac{\overline{x}\overline{y}}{{\overline{\rho }}^3},\\ \frac{\partial x_0^1}{\partial \overline{y}}& =R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}+(2R_{\mathrm{i}\mathrm{n}}-R_{\mathrm{o}\mathrm{u}\mathrm{t}})\frac{{\overline{x}}^2}{{\overline{\rho }}^3},\end{array}$$

and all other components are zero.

inverse

inverse takes $x_0^i$ and $\sigma$ as arguments, and returns $(\overline{x},\overline{y},\overline{z})$, or a default-constructed std::optional<std::array<double, 3>> if $x_0^i$ or $\sigma$ are outside the range of the map.

Let

$$\begin{array}{}\text{(7)}& \rho =\sqrt{(x_0^0-C^0{)}^2+(x_0^1-C^1{)}^2}.\end{array}$$

Then

$$\begin{array}{}\text{(8)}& \overline{x}& =\frac{x_0^0-C^0}{\rho }\frac{\rho +R_{\mathrm{o}\mathrm{u}\mathrm{t}}-2R_{\mathrm{i}\mathrm{n}}}{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}},\text{(9)}& \overline{y}& =\frac{x_0^1-C^1}{\rho }\frac{\rho +R_{\mathrm{o}\mathrm{u}\mathrm{t}}-2R_{\mathrm{i}\mathrm{n}}}{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}},\text{(10)}& \overline{z}& =2\sigma -1.\end{array}$$

Note that $\rho$ in Eq. ( $\text{7}$) can be written

$$\begin{array}{}\text{(11)}& \rho =R_{\mathrm{i}\mathrm{n}}+(R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}})(\overline{\rho }-1),\end{array}$$

where $\overline{\rho }$ is given by Eq. ( $\text{4}$).

If $\overline{z}$ is outside the range $[-1,1]$ or if ${\overline{x}}^2+{\overline{y}}^2$ is less than 1 or greater than 4 then we return a default-constructed std::optional<std::array<double, 3>>.

lambda_tilde

lambda_tilde takes as arguments a point $x^i$ and a projection point $P^i$, and computes $\tilde{\lambda }$, the solution to

$$\begin{array}{}\text{(12)}& x_0^i=P^i+(x^i-P^i)\tilde{\lambda }.\end{array}$$

Since $x_0^i$ must lie on the plane $x_0^3=C^3$,

$$\begin{array}{}\text{(13)}& \tilde{\lambda }& =\frac{C^3-P^3}{x^3-P^3}.\end{array}$$

If $\tilde{\lambda }$ is less than unity (indicating that the supplied point is outside the range of the map), then a default-constructed std::optional<double> is returned.

deriv_lambda_tilde

deriv_lambda_tilde takes as arguments $x_0^i$, a projection point $P^i$, and $\tilde{\lambda }$, and returns $\partial \tilde{\lambda }/\partial x^i$. We have

$$\begin{array}{}\text{(14)}& \frac{\partial \tilde{\lambda }}{\partial x^3}=-\frac{C^3-P^3}{(x^3-P^3{)}^2}=-\frac{{\tilde{\lambda }}^2}{C^3-P^3},\end{array}$$

and other components are zero.

inv_jacobian

inv_jacobian returns $\partial {\overline{x}}^i/\partial x_0^k$, where $\sigma$ is held fixed. The arguments to inv_jacobian are $(\overline{x},\overline{y},\overline{z})$, but $\overline{z}$ is ignored.

The nonzero components are

$$\begin{array}{}\text{(15)}& \frac{\partial \overline{x}}{\partial x_0^0}& =\frac{1}{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}}+\frac{{\overline{y}}^2}{{\overline{\rho }}^2\rho }\frac{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-2R_{\mathrm{i}\mathrm{n}}}{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}},\text{(16)}& \frac{\partial \overline{x}}{\partial x_0^1}& =-\frac{\overline{x}\overline{y}}{{\overline{\rho }}^2\rho }\frac{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-2R_{\mathrm{i}\mathrm{n}}}{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}},\text{(17)}& \frac{\partial \overline{y}}{\partial x_0^1}& =\frac{1}{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}}+\frac{{\overline{x}}^2}{{\overline{\rho }}^2\rho }\frac{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-2R_{\mathrm{i}\mathrm{n}}}{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}},\text{(18)}& \frac{\partial \overline{y}}{\partial x_0^0}& =-\frac{\overline{x}\overline{y}}{{\overline{\rho }}^2\rho }\frac{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-2R_{\mathrm{i}\mathrm{n}}}{R_{\mathrm{o}\mathrm{u}\mathrm{t}}-R_{\mathrm{i}\mathrm{n}}},\end{array}$$

where $\rho$ is computed from Eq. ( $\text{11}$).

dxbar_dsigma

dxbar_dsigma returns $\partial {\overline{x}}^i/\partial \sigma$, where $x_0^i$ is held fixed.

From Eq. ( $\text{6}$) we have

$$\begin{array}{}\text{(19)}& \frac{\partial {\overline{x}}^i}{\partial \sigma }& =(0,0,2).\end{array}$$

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

• src/Domain/CoordinateMaps/FocallyLiftedFlatSide.hpp