SpECTRE  v2022.05.05
domain::CoordinateMaps::FocallyLiftedInnerMaps::FlatEndcap Class Reference

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

FlatEndcapoperator= (const FlatEndcap &)=default

FlatEndcapoperator= (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{align} x_0^0 &= R \bar{x} + C^0,\\ x_0^1 &= R \bar{y} + C^1,\\ x_0^2 &= C^2. \end{align}

### sigma

$$\sigma$$ 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{align} \sigma &= \frac{\bar{z}+1}{2}. \end{align}

### deriv_sigma

deriv_sigma returns

\begin{align} \frac{\partial \sigma}{\partial \bar{x}^j} &= (0,0,1/2). \end{align}

### 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{align*} \frac{\partial x_0^0}{\partial \bar{x}} &= R,\\ \frac{\partial x_0^1}{\partial \bar{y}} &= R, \end{align*}

and all other components are zero.

### inverse

inverse takes $$x_0^i$$ and $$\sigma$$ 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 $$\sigma$$ are outside the range of the map. The formula for the inverse is straightforward:

\begin{align} \bar{x} &= \frac{x_0^0-C^0}{R},\\ \bar{y} &= \frac{x_0^1-C^1}{R},\\ \bar{z} &= 2\sigma - 1. \end{align}

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 $$\tilde{\lambda}$$, the solution to

\begin{align} x_0^i = P^i + (x^i - P^i) \tilde{\lambda}.\end{align}

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

\begin{align} \tilde{\lambda} &= \frac{C^3-P^3}{x^3-P^3}.\end{align}

For FocallyLiftedInnerMaps::FlatEndcap, $$x^i$$ is always between $$P^i$$ and $$x_0^i$$, so $$\tilde{\lambda}\ge 1$$. Therefore a default-constructed std::optional<double> is returned if $$\tilde{\lambda}$$ 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 $$\tilde{\lambda}$$, and returns $$\partial \tilde{\lambda}/\partial x^i$$. We have

\begin{align} \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{align}

and other components are zero.

### inv_jacobian

inv_jacobian returns $$\partial \bar{x}^i/\partial x_0^k$$, where $$\sigma$$ 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{align} \frac{\partial \bar{x}}{\partial x_0^0} &= \frac{1}{R},\\ \frac{\partial \bar{y}}{\partial x_0^1} &= \frac{1}{R}. \end{align}

### dxbar_dsigma

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

From Eq. (6) we have

\begin{align} \frac{\partial \bar{x}^i}{\partial \sigma} &= (0,0,2). \end{align}

The documentation for this class was generated from the following file:
• src/Domain/CoordinateMaps/FocallyLiftedFlatEndcap.hpp