domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge Class Reference

A transition function that falls off linearly from an inner surface of a wedge to an outer surface of a wedge. Meant to be used in domain::CoordinateMaps::Wedge blocks. More...

#include <Wedge.hpp>

Public Types
enum class	Axis : int {   PlusZ = 3 , MinusZ = -3 , PlusY = 2 , MinusY = -2 ,   PlusX = 1 , MinusX = -1 , None = 0 }
	Class to represent the direction of the wedge relative to the outer center.

Public Member Functions
	Wedge (const std::array< double, 3 > &inner_center, double inner_radius, double inner_sphericity, const std::array< double, 3 > &outer_center, double outer_radius, double outer_sphericity, Axis axis, bool reverse=false)
	Construct a Wedge transition for a wedge block in a given direction. More...
double	operator() (const std::array< double, 3 > &source_coords) const override
DataVector	operator() (const std::array< DataVector, 3 > &source_coords) const override
std::optional< double >	original_radius_over_radius (const std::array< double, 3 > &target_coords, double radial_distortion) const override
	The inverse of the transition function. More...
std::array< double, 3 >	gradient (const std::array< double, 3 > &source_coords) const override
std::array< DataVector, 3 >	gradient (const std::array< DataVector, 3 > &source_coords) const override
	WRAPPED_PUPable_decl_template (Wedge)
	Wedge (CkMigrateMessage *msg)
void	pup (PUP::er &p) override
std::unique_ptr< ShapeMapTransitionFunction >	get_clone () const override
bool	operator== (const ShapeMapTransitionFunction &other) const override
bool	operator!= (const ShapeMapTransitionFunction &other) const override
Public Member Functions inherited from domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction
virtual std::optional< double >	original_radius_over_radius (const std::array< double, 3 > &target_coords, double radial_distortion) const =0
	The inverse of the transition function. More...
virtual bool	operator== (const ShapeMapTransitionFunction &other) const =0
virtual bool	operator!= (const ShapeMapTransitionFunction &other) const =0
	WRAPPED_PUPable_abstract (ShapeMapTransitionFunction)
	ShapeMapTransitionFunction (CkMigrateMessage *m)

Friends
std::ostream &	operator<< (std::ostream &os, Axis axis)

Detailed Description

A transition function that falls off linearly from an inner surface of a wedge to an outer surface of a wedge. Meant to be used in domain::CoordinateMaps::Wedge blocks.

Details

The functional form of this transition is

$$\begin{array}{}\text{(1)}& f(r,\theta ,\varphi )=\frac{D_{\text{out}}(r,\theta ,\varphi )-r}{D_{\text{out}}(r,\theta ,\varphi )-D_{\text{in}}(r,\theta ,\varphi )},\end{array}$$

where, in general,

$$\begin{array}{}\text{(2)}& D(r,\theta ,\varphi )=R(\frac{1-s}{\sqrt{3}max(|\sin \theta \cos \varphi |,|\sin \theta \sin \varphi |,|\cos \theta |)}+s).\end{array}$$

Note

If reverse is true, then the functional form of the transition is actually

\begin{equation} f(r, \theta, \phi) = 1 - \frac{D_{\text{out}}(r, \theta, \phi) - r}{D_{\text{out}}(r, \theta, \phi) - D_{\text{in}}(r, \theta, \phi)} = \frac{r - \frac{D_{\text{in}}(r, \theta, \phi)} {D_{\text{out}}(r, \theta, \phi) - D_{\text{in}}(r, \theta, \phi)}. \label{eq:transition_func_reverse} \end{equation}

The function is also defined beyond $D_{\text{in}}(r,\theta ,\varphi )$ and $D_{\text{out}}(r,\theta ,\varphi )$, but slighly differently. Within $D_{\text{in}}(r,\theta ,\varphi )$, $f(r,\theta ,\varphi )=1$ and beyond $D_{\text{out}}(r,\theta ,\varphi )$, $f(r,\theta ,\varphi )=0$. If reverse is true, this logic is flipped.

Here, $s$ is the sphericity of the surface which goes from 0 (flat) to 1 (spherical), $R$ is the radius of the spherical surface, $\text{out}$ is the outer surface, and $\text{in}$ is the inner surface. If the sphericity is 1, then the surface is a sphere so $D=R$. If the sphericity is 0, then the surface is a cube. This cube is circumscribed by a sphere of radius $R$. See domain::CoordinateMaps::Wedge for more of an explanation of these boundary surfaces and their sphericities.

Note

Because the shape map distorts only radii and does not affect angles, $D$ is not a function of $r$ so we have that $D(r,\theta ,\varphi )=D(\theta ,\varphi )$.

There are several assumptions made for this mapping:

• The coordinates $r,\theta ,\varphi$ are assumed to be from the center of the inner surface, not the center of the computational domain.
• The wedges are concentric. (see the constructor) This is also enforced by the center of the inner surface being within the outer surface
• If the centers of the inner and outer surface are different, then the inner sphericity must be 1 (i.e. $D_{\text{in}}=R$)
• The $max$ in the denominator of $\text{2}$ can be simplified a bit to $max(|x|,|y|,|z|)/r$. It was written the other way in $\text{2}$ to emphasize that $D$ has no radial dependence.

Vector formulation

Useful vectors for Wedge transition

It can be more convenient to work with vectors rather than just distances for the following equations. Therefore, we define the projection center $\overrightarrow{P}$ which points from the center of the outer surface to the center of the inner surface. If the centers of the inner and outer surface are the same, then $\overrightarrow{P}=0$. Then, we define $\overrightarrow{x}$ as the vector from the center of the outer surface to the coordinate we are interested in. We define ${\overrightarrow{x}}_0$ as the vector from the center of the outer surface to the point along ray $\overrightarrow{x}-\overrightarrow{P}$ which intersects the inner surface. Similarly, ${\overrightarrow{x}}_1$ intersects the outer surface along the same ray as $\overrightarrow{x}-\overrightarrow{P}$. Any cube has a side length of $2L$.

In this way, we can reformulate Eq. $\text{1}$ as

$$\begin{array}{}\text{(3)}& f(\overrightarrow{x})=\frac{|{\overrightarrow{x}}_1-\overrightarrow{P}|-r}{|{\overrightarrow{x}}_1-\overrightarrow{P}|-|{\overrightarrow{x}}_0-\overrightarrow{P}|}\end{array}$$

where

$$\begin{array}{}\text{(4)}& r=|\overrightarrow{x}-P|\end{array}$$

Computing vector to inner surface

Similar to Eq. $\text{2}$ we can define ${\overrightarrow{x}}_0$ as

$$\begin{array}{}\text{(5)}& {\overrightarrow{x}}_0=R_0(\frac{r(1-s_0)}{\sqrt{3}max(x_i-P_i)}+s)\hat{r}+\overrightarrow{P}\end{array}$$

with

$$\begin{array}{}\text{(6)}& \hat{r}=\frac{\overrightarrow{x}-\overrightarrow{P}}{|\overrightarrow{x}-\overrightarrow{P}|}.\end{array}$$

and also

$$\begin{array}{}\text{(7)}& |{\overrightarrow{x}}_0-\overrightarrow{P}|=R_0(\frac{r(1-s_0)}{\sqrt{3}max(x_i-P_i)}+s_0)\end{array}$$

Computing vector to outer surface

Then, we define

$$\begin{array}{}\text{(8)}& {\overrightarrow{x}}_1=\lambda (\overrightarrow{x}-\overrightarrow{P})+\overrightarrow{P}\end{array}$$

where $\lambda$ is a scalar to be determined. We can see that this is valid because $\overrightarrow{x}-\overrightarrow{P}$ is centered on $\overrightarrow{P}$ and lies along the ray so $\lambda$ will scale this vector to be the correct length.

Computing $\lambda$ is a bit complicated because the outer surface can either be a cube, a sphere, or something in between and the way to solve for $\lambda$ in the cube case is different than in the sphere case. Therefore we define ${\lambda }_c$ and ${\lambda }_s$ as the scalar factors for the cube and sphere case, respectively. This allows us to define $\lambda$ as

$$\begin{array}{}\text{(9)}& \lambda =(1-s_1){\lambda }_c+s_1{\lambda }_s\end{array}$$

Computing cube lambda

To compute ${\lambda }_c$ for the cube case, we notice that (in the image)

$$\begin{array}{}\text{(10)}& L& ={\overrightarrow{x}}_1\cdot \hat{z}\text{(11)}& & =({\lambda }_c^{+z}(\overrightarrow{x}-\overrightarrow{P})+\overrightarrow{P})\cdot \hat{z}\text{(12)}& & ={\lambda }_c^{+z}(x_z-P_z)+P_z\end{array}$$

If we generalize this to all six unit vectors and solve for ${\lambda }_c^{\pm k}$ we get

$$\begin{array}{}\text{(13)}& {\lambda }_c^{\pm k}=\frac{\pm L-\overrightarrow{P}\cdot {\hat{x}}_k}{(\overrightarrow{x}-\overrightarrow{P})\cdot {\hat{x}}_k}\end{array}$$

for $k\in \{x,y,z\}$. This quantity ${\lambda }_c^{\pm k}$ represents the factor needed to scale the vector $\overrightarrow{x}-\overrightarrow{P}$ to intersect with the infinite planes that contain the six faces of the cube.

Some ${\lambda }_c^{\pm k}$ will be negative as they point to the opposite planes. We can discard those. Then for the ${\lambda }_c^{\pm k}$ that are positive, we want the one that is smallest in magnitude. This can be written as

$$\begin{array}{}\text{(14)}& {\lambda }_c=\underset{\begin{array}{c}{\lambda }_c^{\pm k}>0\end{array}}{min}({\lambda }_c^{\pm k})\end{array}$$

Computing sphere lambda

To compute ${\lambda }_s$ for the sphere case, again we notice that (in the image)

$$\begin{array}{}\text{(15)}& R_1^2& =|{\overrightarrow{x}}_1{|}^2\text{(16)}& & =|{\lambda }_s(\overrightarrow{x}-\overrightarrow{P})+\overrightarrow{P}{|}^2\text{(17)}& & ={\lambda }_s^2|\overrightarrow{x}-\overrightarrow{P}{|}^2+2{\lambda }_s(\overrightarrow{x}-\overrightarrow{P})\cdot \overrightarrow{P}+|\overrightarrow{P}{|}^2\end{array}$$

This is simply a quadratic equation for ${\lambda }_s$:

$$\begin{array}{}\text{(18)}& 0={\lambda }_s^2|\overrightarrow{x}-\overrightarrow{P}{|}^2+2{\lambda }_s(\overrightarrow{x}-\overrightarrow{P})\cdot \overrightarrow{P}+|\overrightarrow{P}{|}^2-R_1^2\end{array}$$

We explicitly don't write the typical form of the solution of a quadratic equation because that form behaviors poorly numerically if the two terms in the numerator are close to each other (bad numerical roundoff). Therefore, we just resolve to compute the solution numerically, and say that we have ${\lambda }_s$.

We have now solved for ${\overrightarrow{x}}_1$.

$$\begin{array}{}\text{(19)}& {\overrightarrow{x}}_1=((1-s_1){\lambda }_c+s_1{\lambda }_s)(\overrightarrow{x}-\overrightarrow{P})+\overrightarrow{P}\end{array}$$

given the solutions for ${\lambda }_c$ in Eqn. $\text{14}$ and ${\lambda }_s$ in Eqn. $\text{18}$.

Original radius divided by mapped radius

Given an already mapped point $\tilde{\overrightarrow{x}}$ and the radial distortion $\mathrm{\Sigma }(\theta ,\varphi )=\sum _{\ell ,m}{\lambda }_{\ell m}{Y}_{\ell m}(\theta ,\varphi )$, we can figure out the ratio of the original radius $r$ to the mapped $\tilde{r}=|\tilde{\overrightarrow{x}}|$ by solving

$$\begin{array}{}\text{(20)}& \frac{r}{\tilde{r}}=\frac{1}{1-\frac{f(r,\theta ,\varphi )}{r}\mathrm{\Sigma }(\theta ,\varphi )}\end{array}$$

After plugging in the transition and solving, we get

$$\begin{array}{}\text{(21)}& \frac{r}{\tilde{r}}=\frac{1+\frac{|{\overrightarrow{x}}_1-\overrightarrow{P}|\mathrm{\Sigma }(\theta ,\varphi )}{\tilde{r}(|{\overrightarrow{x}}_1-\overrightarrow{P}|-|{\overrightarrow{x}}_0-\overrightarrow{P}|)}}{1+\frac{\mathrm{\Sigma }(\theta ,\varphi )}{|{\overrightarrow{x}}_1-\overrightarrow{P}|-|{\overrightarrow{x}}_0-\overrightarrow{P}|}}\end{array}$$

Note

Since ${\overrightarrow{x}}_0-\overrightarrow{P}$ and ${\overrightarrow{x}}_1-\overrightarrow{P}$ are not functions of the radius, we can use $\tilde{\overrightarrow{x}}$ in Eq. $\text{5}$ instead of $\overrightarrow{x}$.

Note

If reverse is true, then the value multiplying $\mathrm{\Sigma }$ in the numerator is now $-|{\overrightarrow{x}}_0-\overrightarrow{P}|$ and in the denomintor $\mathrm{\Sigma }$ picks up a minus sign factor.

Note

If we are inside the inner surface, then this simplifies to

$$\begin{array}{}\text{(22)}& \frac{r}{\tilde{r}}=1+\frac{\mathrm{\Sigma }(\theta ,\varphi )}{\tilde{r}}\end{array}$$

because $f(r,\theta ,\varphi )=1$. If we are outside the outer surface, then $r/\tilde{r}=1$ because $f(r,\theta ,\varphi )=0$. If reverse is true, this logic is reversed.

Gradient

The cartesian gradient of the transition function is

$$\begin{array}{}\text{(23)}& \frac{\partial f}{\partial x_i}=\frac{\frac{\partial |{\overrightarrow{x}}_1-\overrightarrow{P}|}{\partial x_i}-\frac{x_i-P_i}{r}}{|{\overrightarrow{x}}_1-\overrightarrow{P}|-|{\overrightarrow{x}}_0-\overrightarrow{P}|}-\frac{(|{\overrightarrow{x}}_1-\overrightarrow{P}|-r)(\frac{\partial |{\overrightarrow{x}}_1-\overrightarrow{P}|}{\partial x_i}-\frac{\partial |{\overrightarrow{x}}_0-\overrightarrow{P}|}{\partial x_i})}{{(|{\overrightarrow{x}}_1-\overrightarrow{P}|-|{\overrightarrow{x}}_0-\overrightarrow{P}|)}^2}.\end{array}$$

Note

If reverse is true, the gradient picks up an overall factor of -1.0.

Note

The gradient is not supported if points lie beyond the inner or outer surface.

Therefore, we need to compute the gradients of ${\overrightarrow{x}}_0$ and ${\overrightarrow{x}}_1$.

Gradient of vector to inner surface

If $\overrightarrow{P}\ne 0$, then we require the inner surface to be a sphere, which means the gradient is trivially zero. Therefore the following is only for when $\overrightarrow{P}=0$.

The gradient of $|{\overrightarrow{x}}_0-\overrightarrow{P}|$ depends on the result of the $max$ in the denominator of $\text{7}$. To simplify the expression, let $i\in \{0,1,2\}$ correspond to the $max(x_i-P_i)$. Then we can write the gradient of as

$$\begin{array}{}\text{(24)}& \frac{\partial |{\overrightarrow{x}}_0-\overrightarrow{P}|}{\partial x_i}& =-\text{sgn}(x_i-P_i)\frac{R(1-s)(r^2-(x_i-P_i{)}^2)}{r(x_i-P_i{)}^2\sqrt{3}}\text{(25)}& \frac{\partial |{\overrightarrow{x}}_0-\overrightarrow{P}|}{\partial x_{i+1}}& =\frac{R(1-s)(x_{i+1}-P_{i+1})}{r|x_i-P_i|\sqrt{3}}\text{(26)}& \frac{\partial |{\overrightarrow{x}}_0-\overrightarrow{P}|}{\partial x_{i+2}}& =\frac{R(1-s)(x_{i+2}-P_{i+2})}{r|x_i-P_i|\sqrt{3}}\end{array}$$

where $i+1$ and $i+2$ are understood to be $\mathrm{mod}3$. Also since we don't allow points at the center of the inner surfaces, we can be assured that for whichever $i$ is max, that $x_i-P_i$ won't be zero.

Gradient of vector to outer surface

This gradient is much more complicated because of how we have to define $\lambda$. To start off with, we need to compute

$$\begin{array}{}\text{(27)}& \frac{\partial }{\partial x_i}|{\overrightarrow{x}}_1-\overrightarrow{P}|& =\frac{\partial }{\partial x_i}(\lambda |\overrightarrow{x}-\overrightarrow{P}|)\text{(28)}& & =\frac{\lambda (x_i-P_i)}{|\overrightarrow{x}-\overrightarrow{P}|}+\frac{\partial \lambda }{\partial x_i}|\overrightarrow{x}-\overrightarrow{P}|\end{array}$$

Since we know $\lambda$ and $\overrightarrow{x}$, that just leaves

$$\begin{array}{}\text{(29)}& \frac{\partial \lambda }{\partial x_i}=\frac{\partial {\lambda }_c}{\partial x_i}(1-s_1)+s_1\frac{\partial {\lambda }_s}{\partial x_i}\end{array}$$

Gradient of cube lambda

Taking the gradient of Eqn. $\text{13}$ we notice that $L$, $\overrightarrow{P}$, and ${\hat{x}}_k$ are constant, so we only have to worry about $\overrightarrow{x}$

$$\begin{array}{}\text{(30)}& \frac{\partial }{\partial x_i}{\lambda }_c^{\pm k}=-\frac{(\pm L-\overrightarrow{P}\cdot {\hat{x}}_k)(\partial /\partial x_i(\overrightarrow{x}-\overrightarrow{P})\cdot {\hat{x}}_k)}{((\overrightarrow{x}-\overrightarrow{P})\cdot {\hat{x}}_k{)}^2}\end{array}$$

The gradient in the numerator is simple to take so we end up with

$$\begin{array}{}\text{(31)}& \frac{\partial }{\partial x_i}{\lambda }_c^{\pm k}=-\frac{(\pm L-\overrightarrow{P}\cdot {\hat{x}}_k)}{(x_k-P_k{)}^2}{\delta }_{ik}\end{array}$$

Gradient of sphere lambda

Taking the gradient of Eqn. $\text{18}$ (and using the fact that $\overrightarrow{P}$ and $R$ are constant), we get

$$\begin{array}{}\text{(32)}& 0={\lambda }_s\frac{\partial {\lambda }_s}{\partial x_i}|\overrightarrow{x}-\overrightarrow{P}{|}^2+{\lambda }_s^2(\overrightarrow{x}-\overrightarrow{P})\cdot \frac{\partial }{\partial x_i}(\overrightarrow{x}-\overrightarrow{P})+\frac{\partial {\lambda }_s}{\partial x_i}(\overrightarrow{x}-\overrightarrow{P})\cdot \overrightarrow{P}+{\lambda }_s\overrightarrow{P}\cdot \frac{\partial }{\partial x_i}(\overrightarrow{x}-\overrightarrow{P})\end{array}$$

Taking the simple gradients and moving things around, we get

$$\begin{array}{}\text{(33)}& \frac{\partial {\lambda }_s}{\partial x_i}=-\frac{{\lambda }_s^2(x_i-P_i)+{\lambda }_s{P}_i}{{\lambda }_s|\overrightarrow{x}-\overrightarrow{P}{|}^2+(\overrightarrow{x}-\overrightarrow{P})\cdot \overrightarrow{P}}\end{array}$$

where ${\lambda }_s$ can be computed from Eqn. $\text{18}$.

So now we can put it all together and we get

$$\begin{array}{}\text{(34)}& \frac{\partial \lambda }{\partial x_i}=-\frac{(\pm L-\overrightarrow{P}\cdot {\hat{x}}_k)}{(x_k-P_k{)}^2}{\delta }_{ik}(1-s_1)+s_1\frac{{\lambda }_s^2(x_i-P_i)+{\lambda }_s{P}_i}{{\lambda }_s|\overrightarrow{x}-\overrightarrow{P}{|}^2+(\overrightarrow{x}-\overrightarrow{P})\cdot \overrightarrow{P}}\end{array}$$

which can then be substituted into Eqn. $\text{27}$ to get the gradient of $|{\overrightarrow{x}}_1-\overrightarrow{P}|$.

Constructor & Destructor Documentation

Wedge()

domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::Wedge	(	const std::array< double, 3 > &	inner_center,
		double	inner_radius,
		double	inner_sphericity,
		const std::array< double, 3 > &	outer_center,
		double	outer_radius,
		double	outer_sphericity,
		Axis	axis,
		bool	reverse = false
	)

Construct a Wedge transition for a wedge block in a given direction.

Details

Many concentric wedges can be part of the same falloff from 1 at the inner boundary of the innermost wedge, to 0 at the outer boundary of the outermost wedge.

Note

If inner_center and outer_center are different, then inner_sphericity must be 1.0.

Parameters

inner_center	Center of the inner surface
inner_radius	Inner radius of innermost wedge
inner_sphericity	Sphericity of innermost surface of innermost wedge
outer_center	Center of the outer surface
outer_radius	Outermost radius of outermost wedge
outer_sphericity	Sphericity of outermost surface of outermost wedge
axis	The direction that this wedge is in.
reverse	If true, the transition function will be 0 at the inner boundary and 1 at the outer boundary (useful for deforming star surfaces). Otherwise, it will be 1 at the inner boundary and 0 at the outer boundary (useful for deforming black hole excision surfaces).

Member Function Documentation

get_clone()

std::unique_ptr< ShapeMapTransitionFunction > domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::get_clone ( ) const

inlineoverridevirtual

Evaluate the gradient of the transition function with respect to the Cartesian coordinates x, y and z at the Cartesian coordinates source_coords.

Implements domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction.

gradient() [1/2]

std::array< DataVector, 3 > domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::gradient ( const std::array< DataVector, 3 > &  source_coords ) const

overridevirtual

Evaluate the gradient of the transition function with respect to the Cartesian coordinates x, y and z at the Cartesian coordinates source_coords.

Implements domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction.

gradient() [2/2]

std::array< double, 3 > domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::gradient ( const std::array< double, 3 > &  source_coords ) const

overridevirtual

Evaluate the gradient of the transition function with respect to the Cartesian coordinates x, y and z at the Cartesian coordinates source_coords.

Implements domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction.

operator!=()

bool domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::operator!= ( const ShapeMapTransitionFunction &  other ) const

overridevirtual

Implements domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction.

operator()() [1/2]

DataVector domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::operator() ( const std::array< DataVector, 3 > &  source_coords ) const

overridevirtual

Evaluate the transition function $f(r,\theta ,\varphi )\in [0,1]$ at the Cartesian coordinates source_coords.

Implements domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction.

operator()() [2/2]

double domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::operator() ( const std::array< double, 3 > &  source_coords ) const

overridevirtual

Evaluate the transition function $f(r,\theta ,\varphi )\in [0,1]$ at the Cartesian coordinates source_coords.

Implements domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction.

operator==()

bool domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::operator== ( const ShapeMapTransitionFunction &  other ) const

overridevirtual

Implements domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction.

original_radius_over_radius()

std::optional< double > domain::CoordinateMaps::ShapeMapTransitionFunctions::Wedge::original_radius_over_radius ( const std::array< double, 3 > &  target_coords, double  radial_distortion  ) const

overridevirtual

The inverse of the transition function.

This method returns $r/\tilde{r}$ given the mapped coordinates ${\tilde{x}}^i$ (target_coords) and the spherical harmonic expansion $\mathrm{\Sigma }(t,\theta ,\varphi )=\sum _{lm}{\lambda }_{lm}(t)Y_{lm}(\theta ,\varphi )$ (radial_distortion). See domain::CoordinateMaps::TimeDependent::Shape for details on how this quantity is used to compute the inverse of the Shape map.

To derive the expression for this inverse, solve Eq. ( $\text{???}$) for $r$ after substituting $f(r,\theta ,\varphi )$.

Parameters

target_coords	The mapped Cartesian coordinates ${\tilde{x}}^i$.
radial_distortion	The spherical harmonic expansion $\mathrm{\Sigma }(t,\theta ,\varphi )$.

Returns: The quantity $r/\tilde{r}$.

Implements domain::CoordinateMaps::ShapeMapTransitionFunctions::ShapeMapTransitionFunction.

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

• src/Domain/CoordinateMaps/TimeDependent/ShapeMapTransitionFunctions/Wedge.hpp