domain::CoordinateMaps::FocallyLiftedInnerMaps::Side Class Reference

A FocallyLiftedInnerMap that maps a 3D unit right cylindrical shell to a volume that connects portions of two spherical surfaces. More...

#include <FocallyLiftedSide.hpp>

Public Member Functions
	Side (const std::array< double, 3 > &center, const double radius, const double z_lower, const double z_upper)
	Side (Side &&)=default
	Side (const Side &)=default
Side &	operator= (const Side &)=default
Side &	operator= (Side &&)=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 ()

Static Public Attributes
static constexpr size_t	dim = 3

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

Detailed Description

A FocallyLiftedInnerMap that maps a 3D unit right cylindrical shell to a volume that connects portions of two spherical surfaces.

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 sphere with center $C^i$ and radius $R$ that is intersected by two planes normal to the $z$ axis located at $z=z_{\mathrm{L}}$ and $z=z_{\mathrm{U}}$, with $z_{\mathrm{L}}<z_{\mathrm{U}}$. Side provides the following functions:

forward_map()

forward_map() maps $(\overline{x},\overline{y},\overline{z})$ to a point on the inner surface ${\overline{x}}^2+{\overline{y}}^2=1$ by dividing $\overline{x}$ and $\overline{y}$ by $(1+\sigma )$, where $\sigma$ is the function given by Eq. (7) below. Then it maps that point to a point on the portion of the sphere with $z_{\mathrm{L}}\le z\le z_{\mathrm{U}}$. forward_map() returns $x_0^i$, the 3D coordinates on that sphere, which are given by

$$\begin{array}{}\text{(1)}& x_0^0& =R\sin \theta \frac{\overline{x}}{1+\sigma }+C^0,\text{(2)}& x_0^1& =R\sin \theta \frac{\overline{y}}{1+\sigma }+C^1,\text{(3)}& x_0^2& =R\cos \theta +C^2.\end{array}$$

Here

$$\begin{array}{}\text{(4)}& \theta ={\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}+({\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}})\frac{\overline{z}+1}{2},\end{array}$$

where

$$\begin{array}{}\text{(5)}& \cos \left({\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)& =(z_{\mathrm{L}}-C^2)/R,\text{(6)}& \cos \left({\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)& =(z_{\mathrm{U}}-C^2)/R.\end{array}$$

Note that $\theta$ decreases with increasing $\overline{z}$, which is the usual convention for a polar angle but might otherwise cause confusion.

sigma

$\sigma$ is a function that is zero on the sphere $x^i=x_0^i$ and unity at ${\overline{x}}^2+{\overline{y}}^2=4$ (corresponding to the upper surface of the FocallyLiftedMap). We define

$$\begin{array}{}\text{(7)}& \sigma & =\sqrt{{\overline{x}}^2+{\overline{y}}^2}-1.\end{array}$$

deriv_sigma

deriv_sigma returns

$$\begin{array}{}\text{(8)}& \frac{\partial \sigma }{\partial {\overline{x}}^j}& =(\frac{\overline{x}}{1+\sigma },\frac{\overline{y}}{1+\sigma },0).\end{array}$$

jacobian

jacobian returns $\partial x_0^k/\partial {\overline{x}}^j$. The arguments to jacobian are $(\overline{x},\overline{y},\overline{z})$. Differentiating Eqs.(1–4) above yields

$$\begin{array}{rl}\frac{\partial x_0^0}{\partial \overline{x}}& =R\sin \theta \frac{{\overline{y}}^2}{(1+\sigma {)}^3},\\ \frac{\partial x_0^0}{\partial \overline{y}}& =-R\sin \theta \frac{\overline{x}\overline{y}}{(1+\sigma {)}^3},\\ \frac{\partial x_0^0}{\partial \overline{z}}& =R\cos \theta \frac{{\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2(1+\sigma )}\overline{x},\\ \frac{\partial x_0^1}{\partial \overline{x}}& =-R\sin \theta \frac{\overline{x}\overline{y}}{(1+\sigma {)}^3},\\ \frac{\partial x_0^1}{\partial \overline{y}}& =R\sin \theta \frac{{\overline{x}}^2}{(1+\sigma {)}^3},\\ \frac{\partial x_0^1}{\partial \overline{z}}& =R\cos \theta \frac{{\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2(1+\sigma )}\overline{y},\\ \frac{\partial x_0^2}{\partial \overline{x}}& =0,\\ \frac{\partial x_0^2}{\partial \overline{y}}& =0,\\ \frac{\partial x_0^2}{\partial \overline{z}}& =-R\sin \theta \frac{{\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}.\end{array}$$

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.

If $\sigma$ is outside the range $[0,1]$ then we return a default-constructed std::optional<std::array<double, 3>>.

To get $\overline{z}$ we invert Eq. (4):

$$\begin{array}{}\text{(9)}& \overline{z}& =2\frac{\mathrm{acos}((x_0^2-C^2)/R)-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}}-1.\end{array}$$

If $\overline{z}$ is outside the range $[-1,1]$ then we return a default-constructed std::optional<std::array<double, 3>>.

To compute $\overline{x}$ and $\overline{y}$, we invert Eqs. (1–3) and use $\sigma$:

$$\begin{array}{}\text{(10)}& \overline{x}& =\frac{(x_0^0-C^0)(1+\sigma )}{\rho },\text{(11)}& \overline{y}& =\frac{(x_0^1-C^1)(1+\sigma )}{\rho },\end{array}$$

where

$$\begin{array}{}\text{(12)}& \rho =\sqrt{(x_0^0-C^0{)}^2+(x_0^1-C^1{)}^2}.\end{array}$$

lambda_tilde

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

$$\begin{array}{}\text{(13)}& x_0^i=P^i+(x^i-P^i)\tilde{\lambda }.\end{array}$$

Since $x_0^i$ must lie on the sphere, $\tilde{\lambda }$ is the solution of the quadratic equation

$$\begin{array}{}\text{(14)}& |P^i+(x^i-P^i)\tilde{\lambda }-C^i{|}^2-R^2=0.\end{array}$$

In solving the quadratic, we choose the larger root if $x^2>z_{\mathrm{P}}$ and the smaller root otherwise. We demand that the root is greater than unity. If there is no such root, this means that the point $x^i$ is not in the range of the map so we return a default-constructed std::optional<double>.

deriv_lambda_tilde

deriv_lambda_tilde takes as arguments $x_0^i$, a projection point $P^i$, and $\tilde{\lambda }$, and returns $\partial \tilde{\lambda }/\partial x^i$. By differentiating Eq. (14), we find

$$\begin{array}{rl}\frac{\partial \tilde{\lambda }}{\partial x^j}& ={\tilde{\lambda }}^2\frac{C^j-x_0^j}{(x_0^i-P^i)(x_{0i}-C_i)}\\ \text{(15)}& & ={\tilde{\lambda }}^2\frac{C^j-x_0^j}{|x_0^i-P^i{|}^2+(x_0^i-P^i)(P_i-C_i)}.\end{array}$$

inv_jacobian

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

Note from Eqs. (9–12) that $\overline{x}$ and $\overline{y}$ depend only on $x_0^0$ and $x_0^1$ but not on $x_0^2$.

By differentiating Eqs. (9–12), we find

$$\begin{array}{rl}\frac{\partial \overline{x}}{\partial x_0^0}& =\frac{{\overline{y}}^2}{(1+\sigma )\rho },\\ \frac{\partial \overline{x}}{\partial x_0^1}& =-\frac{\overline{x}\overline{y}}{(1+\sigma )\rho },\\ \frac{\partial \overline{x}}{\partial x_0^2}& =0,\\ \frac{\partial \overline{y}}{\partial x_0^0}& =-\frac{\overline{x}\overline{y}}{(1+\sigma )\rho },\\ \frac{\partial \overline{y}}{\partial x_0^1}& =\frac{{\overline{x}}^2}{(1+\sigma )\rho },\\ \frac{\partial \overline{y}}{\partial x_0^2}& =0,\\ \frac{\partial \overline{z}}{\partial x_0^0}& =0,\\ \frac{\partial \overline{z}}{\partial x_0^1}& =0,\\ \frac{\partial \overline{z}}{\partial x_0^2}& =-\frac{2}{\rho ({\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}})},\end{array}$$

where

$$\rho =R\sin \theta =R\sin ({\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}}+({\theta }_{\mathrm{m}\mathrm{i}\mathrm{n}}-{\theta }_{\mathrm{m}\mathrm{a}\mathrm{x}})\frac{\overline{z}+1}{2}),$$

which is also equal to the quantity in Eq. (12).

dxbar_dsigma

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

From Eqs. (10) and (11) we have

$$\begin{array}{}\text{(16)}& \frac{\partial {\overline{x}}^i}{\partial \sigma }& =(\frac{\overline{x}}{\sqrt{{\overline{x}}^2+{\overline{y}}^2}},\frac{\overline{y}}{\sqrt{{\overline{x}}^2+{\overline{y}}^2}},0).\end{array}$$

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

• src/Domain/CoordinateMaps/FocallyLiftedSide.hpp