A FocallyLiftedInnerMap that maps a 3D unit right cylinder to a volume that connects a portion of a plane and a spherical surface. More...

#include <FocallyLiftedFlatEndcap.hpp>

Public Member Functions
	FlatEndcap (const std::array< double, 3 > &center, double radius)

	FlatEndcap (FlatEndcap &&)=default

	FlatEndcap (const FlatEndcap &)=default

FlatEndcap &	operator= (const FlatEndcap &)=default

FlatEndcap &	operator= (FlatEndcap &&)=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 FlatEndcap &lhs, const FlatEndcap &rhs)

Detailed Description

A FocallyLiftedInnerMap that maps a 3D unit right cylinder to a volume that connects a portion of a plane and a spherical surface.

Details

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

Consider a 2D circle in 3D space that is normal to the $z$ axis and has (3D) center $C^{i}$ and radius $R$ . FlatEndcap provides the following functions:

forward_map()

forward_map() maps $(\bar{x}, \bar{y}, \bar{z} = - 1)$ to the interior of the circle. The arguments to forward_map() are $(\bar{x}, \bar{y}, \bar{z})$ , but $\bar{z}$ is ignored. forward_map() returns $x_{0}^{i}$ , the 3D coordinates on the circle, which are given by

$\begin{aligned} (1) & x_{0}^{0} & = R \bar{x} + C^{0}, \\ (2) & x_{0}^{1} & = R \bar{y} + C^{1}, \\ (3) & x_{0}^{2} & = C^{2} . \end{aligned}$

sigma

$σ$ is a function that is zero on the plane $x^{i} = x_{0}^{i}$ and unity at $\bar{z} = + 1$ (corresponding to the upper surface of the FocallyLiftedMap). We define

$\begin{aligned} (4) & σ & = \frac{\bar{z} + 1}{2} . \end{aligned}$

deriv_sigma

deriv_sigma returns

$\begin{aligned} (5) & \frac{\partial σ}{\partial {\bar{x}}^{j}} & = (0, 0, 1 / 2) . \end{aligned}$

jacobian

jacobian returns $\partial x_{0}^{k} / \partial {\bar{x}}^{j}$ . The arguments to jacobian are $(\bar{x}, \bar{y}, \bar{z})$ , but $\bar{z}$ is ignored.

Differentiating Eqs.(1–3) above yields

$\begin{aligned} \frac{\partial x_{0}^{0}}{\partial \bar{x}} & = R, \\ \frac{\partial x_{0}^{1}}{\partial \bar{y}} & = R, \end{aligned}$

and all other components are zero.

inverse

inverse takes $x_{0}^{i}$ and $σ$ as arguments, and returns $(\bar{x}, \bar{y}, \bar{z})$ , or a default-constructed std::optional<std::array<double, 3>> if $x_{0}^{i}$ or $σ$ are outside the range of the map. The formula for the inverse is straightforward:

$\begin{aligned} (6) & \bar{x} & = \frac{x_{0}^{0} - C^{0}}{R}, \\ (7) & \bar{y} & = \frac{x_{0}^{1} - C^{1}}{R}, \\ (8) & \bar{z} & = 2 σ - 1. \end{aligned}$

If $\bar{z}$ is outside the range $[- 1, 1]$ or if ${\bar{x}}^{2} + {\bar{y}}^{2} > 1$ 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 $\tilde{λ}$ , the solution to

$\begin{aligned} (9) & x_{0}^{i} = P^{i} + (x^{i} - P^{i}) \tilde{λ} . \end{aligned}$

Since $x_{0}^{i}$ must lie on the plane $x_{0}^{3} = C^{3}$ ,

$\begin{aligned} (10) & \tilde{λ} & = \frac{C^{3} - P^{3}}{x^{3} - P^{3}} . \end{aligned}$

For FocallyLiftedInnerMaps::FlatEndcap, $x^{i}$ is always between $P^{i}$ and $x_{0}^{i}$ , so $\tilde{λ} \geq 1$ . Therefore a default-constructed std::optional<double> is returned if $\tilde{λ}$ is less than unity (meaning that $x^{i}$ is outside the range of the map).

deriv_lambda_tilde

deriv_lambda_tilde takes as arguments $x_{0}^{i}$ , a projection point $P^{i}$ , and $\tilde{λ}$ , and returns $\partial \tilde{λ} / \partial x^{i}$ . We have

$\begin{aligned} (11) & \frac{\partial \tilde{λ}}{\partial x^{3}} = - \frac{C^{3} - P^{3}}{(x^{3} - P^{3})^{2}} = - \frac{{\tilde{λ}}^{2}}{C^{3} - P^{3}}, \end{aligned}$

and other components are zero.

inv_jacobian

inv_jacobian returns $\partial {\bar{x}}^{i} / \partial x_{0}^{k}$ , where $σ$ is held fixed. The arguments to inv_jacobian are $(\bar{x}, \bar{y}, \bar{z})$ , but $\bar{z}$ is ignored.

The nonzero components are

$\begin{aligned} (12) & \frac{\partial \bar{x}}{\partial x_{0}^{0}} & = \frac{1}{R}, \\ (13) & \frac{\partial \bar{y}}{\partial x_{0}^{1}} & = \frac{1}{R} . \end{aligned}$

dxbar_dsigma

dxbar_dsigma returns $\partial {\bar{x}}^{i} / \partial σ$ , where $x_{0}^{i}$ is held fixed.

From Eq. (6) we have

$\begin{aligned} (14) & \frac{\partial {\bar{x}}^{i}}{\partial σ} & = (0, 0, 2) . \end{aligned}$

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

src/Domain/CoordinateMaps/FocallyLiftedFlatEndcap.hpp

Public Member Functions

Static Public Member Functions

Friends