domain::CoordinateMaps::TimeDependent::RotScaleTrans< Dim > Class Template Reference

RotScaleTrans map which applies a combination of rotation, expansion, and translation based on which maps are supplied. More...

#include <RotScaleTrans.hpp>

Public Types
enum class	BlockRegion { Inner , Transition , Outer }

Public Member Functions
	RotScaleTrans (std::optional< std::pair< std::string, std::string > > scale_f_of_t_names, std::optional< std::string > rot_f_of_t_name, std::optional< std::string > trans_f_of_t_name, double inner_radius, double outer_radius, BlockRegion region)
	RotScaleTrans (const RotScaleTrans< Dim > &RotScaleTrans_Map)=default
	RotScaleTrans (RotScaleTrans &&)=default
RotScaleTrans &	operator= (RotScaleTrans &&)=default
RotScaleTrans &	operator= (const RotScaleTrans &RotScaleTrans_Map)=default
template<typename T >
std::array< tt::remove_cvref_wrap_t< T >, Dim >	operator() (const std::array< T, Dim > &source_coords, double time, const std::unordered_map< std::string, std::unique_ptr< domain::FunctionsOfTime::FunctionOfTime > > &functions_of_time) const
std::optional< std::array< double, Dim > >	inverse (const std::array< double, Dim > &target_coords, double time, const std::unordered_map< std::string, std::unique_ptr< domain::FunctionsOfTime::FunctionOfTime > > &functions_of_time) const
	The inverse function is only callable with doubles because the inverse might fail if called for a point out of range, and it is unclear what should happen if the inverse were to succeed for some points in a DataVector but fail for other points.
template<typename T >
std::array< tt::remove_cvref_wrap_t< T >, Dim >	frame_velocity (const std::array< T, Dim > &source_coords, double time, const std::unordered_map< std::string, std::unique_ptr< domain::FunctionsOfTime::FunctionOfTime > > &functions_of_time) const
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, Dim, Frame::NoFrame >	inv_jacobian (const std::array< T, Dim > &source_coords, double time, const std::unordered_map< std::string, std::unique_ptr< domain::FunctionsOfTime::FunctionOfTime > > &functions_of_time) const
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, Dim, Frame::NoFrame >	jacobian (const std::array< T, Dim > &source_coords, double time, const std::unordered_map< std::string, std::unique_ptr< domain::FunctionsOfTime::FunctionOfTime > > &functions_of_time) const
void	pup (PUP::er &p)
const std::unordered_set< std::string > &	function_of_time_names () const

Static Public Member Functions
static bool	is_identity ()

Static Public Attributes
static constexpr size_t	dim = Dim

Friends
template<size_t LocalDim>
bool	operator== (const RotScaleTrans< LocalDim > &lhs, const RotScaleTrans< LocalDim > &rhs)

Detailed Description

template<size_t Dim>
class domain::CoordinateMaps::TimeDependent::RotScaleTrans< Dim >

RotScaleTrans map which applies a combination of rotation, expansion, and translation based on which maps are supplied.

Details

This map adds a rotation, expansion and translation based on what types of maps are needed. Translation and expansion have piecewise functions that map the coordinates \(\vec{\xi}\) based on what region \(|\vec{\xi}|\) is in. Coordinates within the inner radius are translated by the translation function of time \(\vec{T}(t)\) and expanded by the inner expansion function of time \(E_{a}(t)\). Coordinates in between the inner radius and outer radius have a linear radial falloff applied to them. Coordinates at or beyond the outer radius have no translation applied and are expanded by the outer expansion function of time \(E_{b}(t)\). This map assumes that the center of the map is at (0., 0., 0.). There is an enum class to set which region your block is in. Specifying RotScaleTrans::BlockRegion::Inner treats coordinates as if they're inside the inner radius, setting RotScaleTrans::BlockRegion::Transition treats the points as if they're between the inner and outer radius, and setting RotScaleTrans::BlockRegion::Outer treats points as if they're on or beyond the outer boundary.

Note

\(E_{a}(t)\) and \(E_{b}(t)\) are \(a(t)\) and \(b(t)\) from the CubicScale documentation.

Mapped Coordinates

The RotScaleTrans map takes the coordinates \(\vec{\xi}\) to the target coordinates \(\vec{\bar{\xi}}\) through

\begin{equation} \vec{\bar{\xi}} = \left\{\begin{array}{ll}E_{a}(t)R(t)\vec{\xi} + \vec{T}(t), & |\vec{\xi}| \leq R_{in}, \\ (w_{E} + E_{a}(t))R(t)\vec{\xi} + (1 + w_{T})\vec{T}(t), & R_{in} < |\vec{\xi}| \leq 0.5(R_{in} + R_{out}), \\ (w_{E} + E_{b}(t))R(t)\vec{\xi} + w_{T}\vec{T}(t), & 0.5(R_{in} + R_{out}) < |\vec{\xi}| < R_{out}, \\ E_{b}(t)R(t)\vec{\xi}, & |\vec{\xi}| \geq R_{out} \end{array}\right. \end{equation}

Where \(R_{in}\) is the inner radius, \(R_{out}\) is the outer radius, and \(w_{T}\) is the translation falloff factor and \(w_{E}\) is the expansion falloff factor found through

\begin{equation} w_{E} = \left\{\begin{array}{ll}\frac{R_{in}(R_{out} - |\vec{\xi}|)(E_{a}(t) - E_{b}(t))}{|\vec{\xi}|(R_{out} - R_{in})}, & R_{in} < |\vec{\xi}| \leq 0.5(R_{in} + R_{out}), \\ \frac{R_{out}(R_{in} - |\vec{\xi}|)(E_{a}(t) - E_{b}(t))}{|\vec{\xi}|(R_{out} - R_{in})}, & 0.5(R_{in} + R_{out}) < |\vec{\xi}| < R_{out} \end{array}\right. \end{equation}

and

\begin{equation} w_{T} = \left\{\begin{array}{ll}\frac{R_{in} - |\vec{\xi}|}{R_{out} - R_{in}}, & R_{in} < |\vec{\xi}| \leq 0.5(R_{in} + R_{out}), \\ \frac{R_{out} - |\vec{\xi}|}{R_{out} - R_{in}}, & 0.5(R_{in} + R_{out}) < |\vec{\xi}| < R_{out} \end{array}\right. \end{equation}

\(w_{E}\) and \(w_{T}\) are calculated differently based on if you're closer to the inner radius or outer radius to reduce roundoff error.

Inverse

The inverse function maps the coordinates \(\vec{\bar{\xi}}\) back to the original coordinates \(\vec{\xi}\) through different equations based on which maps are supplied.

If Rotation, Expansion, and Translation Maps are supplied then the inverse is given by

\begin{equation} \label{eq:full_inverse} \vec{\xi} = \left\{\begin{array}{ll}R^{T}(t)(\frac{(\vec{\bar{\xi}} - \vec{T}(t))}{E_{a}(t)}), & |\vec{\bar{\xi}}| \leq R_{in}E_{a}(t), \\ R^{T}(t)\frac{\vec{\bar{\xi}} - w_{T}\vec{T}(t)}{w_{E}}, & R_{in}E_{a}(t) < |\vec{\bar{\xi}}| \leq 0.5(R_{in}E_{a}(t) + R_{out}E_{b}(t)), \\ R^{T}(t)\frac{\vec{\bar{\xi}} - (1.0 - w_{T})\vec{T}(t)}{w_{E}}, & 0.5(R_{in}E_{a}(t) + R_{out}E_{b}(t)) < |\vec{\bar{\xi}}| < R_{out}E_{b}(t), \\ R^{T}(t)\frac{\vec{\bar{\xi}}}{E_{b}(t)}, & |\vec{\bar{\xi}}| \geq R_{out}E_{b}(t) \end{array}\right. \end{equation}

Where \(w_{T}\) and \(w_{E}\) are found through different quadratic solves.

When closer to \(R_{in}\) the quadratic has the form

\begin{equation} w_{T}^2((E_{a}(t)R_{in} - E_{b}(t)R_{out})^2 - T(t)^2) + 2w_{T}(E_{a}(t)R_{in}(E_{b}(t)R_{out} - E_{a}(t)R_{in}) + \vec{T}(t) \cdot (\vec{T}(t) - \vec{\bar{\xi}})) + \vec{T}(t) \cdot \vec{\bar{\xi}}) + E_{a}(t)^2 R_{in}^2 - (\vec{\bar{\xi}} - \vec{T}(t))^2 \end{equation}

where \(w_{E} = \frac{(1.0 - w_{T})R_{in}E_{a}(t) + w_{T}R_{out}E_{b}(t)}{(1.0 - w_{T})R_{in} + w_{T}R_{out}}\)

When closer to \(R_{out}\) the quadratic has the form

\begin{equation} w_{T}^2((E_{a}(t)R_{in} - E_{b}(t)R_{out})^2 - T(t)^2) + 2w_{T}(E_{b}(t)R_{out}(E_{a}(t)R_{in} - E_{b}(t)R_{out}) + \vec{T}(t) \cdot \vec{\bar{\xi}}) + E_{b}(t)^2 R_{out}^2 - \vec{\bar{\xi}}^2 \end{equation}

where \(w_{E} = \frac{w_{T}R_{in}E_{a}(t) + (1.0 - w_{T})R_{out}E_{b}(t)}{w_{T}R_{in} + (1.0 - w_{T})R_{out}}\)

If Rotation and Expansion are supplied then the inverse is given by

\begin{equation} \vec{\xi} = \left\{\begin{array}{ll}R^{T}(t)(\frac{\vec{\bar{\xi}}}{E_{a}(t)}), & |\vec{\bar{\xi}}| \leq R_{in}E_{a}(t), \\ R^{T}(t)\frac{\vec{\bar{\xi}}}{w_{E}}, & R_{in}E_{a}(t) < |\vec{\bar{\xi}}| \leq 0.5(R_{in}E_{a}(t) + R_{out}E_{b}(t)), \\ R^{T}(t)\frac{\vec{\bar{\xi}}}{w_{E}}, & 0.5(R_{in}E_{a}(t) + R_{out}E_{b}(t)) < |\vec{\bar{\xi}}| < R_{out}E_{b}(t), \\ R^{T}(t)\frac{\vec{\bar{\xi}}}{E_{b}(t)}, & |\vec{\bar{\xi}}| \geq R_{out}E_{b}(t) \end{array}\right. \end{equation}

Where \(w_{E}\) is found through different quadratic solves.

When closer to \(R_{in}\) the quadratic has the form

\begin{equation} w^2(E_{a}(t)R_{in} - E_{b}(t)R_{out})^2 + 2wE_{a}(t)R_{in}(E_{b}(t)R_{out} - E_{a}(t)R_{in}) + (E_{a}(t) R_{in})^2 - \bar{\xi}^2 \end{equation}

with \(w_{E} = \frac{E_{a}(t)R_{in}(1.0 - w) + wE_{b}(t)R_{out}}{R_{in}(1.0 - w) + wR_{out}}\)

When closer to \(R_{out}\) the quadratic has the form

\begin{equation} w^2(E_{a}(t)R_{in} - E_{b}(t)R_{out})^2 + 2wE_{b}(t)R_{out}(E_{a}(t)R_{in} - E_{b}(t)R_{out}) + (E_{b}(t) R_{out})^2 - \bar{\xi}^2 \end{equation}

with \(w_{E} = \frac{wE_{a}(t)R_{in} + E_{b}(t)R_{out}(1.0 - w)}{wR_{in} + R_{out}(1.0 - w)}\)

If Rotation and Translation are supplied, then the inverse is given by

\begin{equation} \vec{\xi} = \left\{\begin{array}{ll}R^{T}(t)(\vec{\bar{\xi}} - \vec{T}(t)), & |\vec{\bar{\xi}}| \leq R_{in}, \\ R^{T}(t)(\vec{\bar{\xi}} - w_{T}\vec{T}(t)), & R_{in} < |\vec{\bar{\xi}}| \leq 0.5(R_{in} + R_{out}), \\ R^{T}(t)(\vec{\bar{\xi}} - (1.0 - w_{T})\vec{T}(t)), & 0.5(R_{in} + R_{out}) < |\vec{\bar{\xi}}| < R_{out}, \\ R^{T}(t)\vec{\bar{\xi}}, & |\vec{\bar{\xi}}| \geq R_{out} \end{array}\right. \end{equation}

Where \(w_{T}\) is found through different quadratic solves.

When closer to \(R_{in}\) the quadratic has the form

\begin{equation} w_{T}^2(T(t)^2 - (R_{out} - R_{in})^2) - 2w_{T}(\vec{T}(t) \cdot (\vec{T}(t) - \vec{\bar{\xi}}) + R_{in}(R_{out} - R_{in}) + (\vec{T}(t) - \vec{\bar{\xi}})^2 - R_{in}^2 \end{equation}

When closer to \(R_{out}\) the quadratic has the form

\begin{equation} w_{T}^2(T(t)^2 - (R_{out} - R_{in})^2) + 2w_{T}(R_{out}(R_{out} - R_{in}) - \vec{T}(t) \cdot \vec{\bar{\xi}}) + \vec{\bar{\xi}}^2 - R_{out}^2 \end{equation}

If Expansion and Translation are supplied, then the inverse is given by Eq. \(\ref{eq:full_inverse}\), with no transpose of rotation applied.

Note

For all the maps with rotation, the inverse of rotation is the transpose of the original rotation. For maps with translation, the inverse map also assumes that if \(\vec{\bar{\xi}} - \vec{T}(t) \leq R_{in}\) then the translated point originally came from within the inner radius so it'll be translated back without a quadratic solve.

Frame Velocity

The Frame Velocity is found through different equations based on which maps are supplied.

If Rotation, Expansion, and Translation are supplied then the frame velocity is found through

\begin{equation} \vec{v} = \left\{\begin{array}{ll}(E_{a}(t)dR(t) + dE_{a}(t)R(t))\vec{\xi} + d \vec{T}(t), & |\vec{\xi}| \leq R_{in}, \\ ((E_{a}(t) + w_{E})dR(t) + (dE_{a}(t) + dw_{E})R(t))\vec{\xi} + (1 + w_{T})d \vec{T}(t), & R_{in} < |\vec{\xi}| \leq 0.5(R_{in} + R_{out}), \\ ((E_{b}(t) + w_{E})dR(t) + (dE_{b}(t) + dw_{E})R(t))\vec{\xi} + w_{T}d \vec{T}(t), & 0.5(R_{in} + R_{out}) < |\vec{\xi}| < R_{out}, \\ (E_{b}(t)dR(t) + dE_{b}(t)R(t))\vec{\xi}, & |\vec{\xi}| \geq R_{out} \end{array}\right. \end{equation}

where \(dw_{E}\) is the derivative of the \(w_{E}\) given by

\begin{equation} dw_{E} = \left\{\begin{array}{ll}\frac{R_{out}(R_{in} - |\vec{\xi}|)(dE_{a}(t) - dE_{b}(t))}{|\vec{\xi}|(R_{out} - R_{in})}, & R_{in} < |\vec{\xi}| \leq 0.5(R_{in} + R_{out}), \\ \frac{R_{in}(R_{out} - |\vec{\xi}|)(dE_{a}(t) - dE_{b}(t))}{|\vec{\xi}|(R_{out} - R_{in})}, & 0.5(R_{in} + R_{out}) < |\vec{\xi}| < R_{out} \end{array}\right. \end{equation}

Jacobian

The jacobian is also found through different equations based on which maps are supplied.

If Rotation, Expansion and Translation maps are supplied then the jacobian is found through

\begin{equation} {J^{i}}_{j} = \left\{\begin{array}{ll}E_{a}(t){R^{i}}_{j}(t), & |\vec{\xi}| \leq R_{in}, \\ {R^{i}}_{j}(t)E_{a}(t) + \frac{\alpha {R^{i}}_{l}(t)\vec{\xi}^{l}\vec{\xi}_{j}(E_{A}(t) - E_{B}(t))}{|\vec{\xi}|} + w_{E}{R^{i}}_{j}(t) + \frac{dw_{T}T^{i}\xi_{j}}{|\vec{\xi}|}, & R_{in} < |\vec{\xi}| \leq 0.5(R_{in} + R_{out}), \\ {R^{i}}_{j}(t)E_{b}(t) + \frac{\alpha {R^{i}}_{l}(t)\vec{\xi}^{l}\vec{\xi}_{j}(E_{A}(t) - E_{B}(t))}{|\vec{\xi}|} + w_{E}{R^{i}}_{j}(t) + \frac{dw_{T}T^{i}\xi_{j}}{|\vec{\xi}|}, & 0.5(R_{in} + R_{out}) < |\vec{\xi}| < R_{out}, \\ E_{b}(t){R^{i}}_{j}(t), & |\vec{\xi}| \geq R_{out} \end{array}\right. \end{equation}

where \(\alpha = \frac{R_{in}R_{out}}{\vec{\xi}^2(R_{in} - R_{out})}\) and \(dw_{T} = \frac{-1.0}{R_{out} - R_{in}}\)

Note

For the translation map, the map returns the identity for all regions except between \(R_{in}\) and \(R_{out}\)

Inverse Jacobian

The inverse jacobian is computed numerically by inverting the jacobian.

Member Enumeration Documentation

BlockRegion

template<size_t Dim>

enum class domain::CoordinateMaps::TimeDependent::RotScaleTrans::BlockRegion

strong

Enumerator
Inner	Within the inner radius.
Transition	Between inner and outer radius.
Outer	At or beyond outer boundary.

---

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

• src/Domain/CoordinateMaps/TimeDependent/RotScaleTrans.hpp