domain::CoordinateMaps::Interval Class Reference

Maps $\xi$ in the 1D interval $[A,B]$ to $x$ in the interval $[a,b]$ according to a domain::CoordinateMaps::Distribution. More...

#include <Interval.hpp>

Public Member Functions
	Interval (double A, double B, double a, double b, Distribution distribution, std::optional< double > singularity_pos=std::nullopt)
template<typename T >
std::array< tt::remove_cvref_wrap_t< T >, 1 >	operator() (const std::array< T, 1 > &source_coords) const
std::optional< std::array< double, 1 > >	inverse (const std::array< double, 1 > &target_coords) const
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 1, Frame::NoFrame >	jacobian (const std::array< T, 1 > &source_coords) const
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 1, Frame::NoFrame >	inv_jacobian (const std::array< T, 1 > &source_coords) const
void	pup (PUP::er &p)
bool	is_identity () const

Static Public Attributes
static constexpr size_t	dim = 1

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

Detailed Description

Maps $\xi$ in the 1D interval $[A,B]$ to $x$ in the interval $[a,b]$ according to a domain::CoordinateMaps::Distribution.

Details

The mapping takes a domain::CoordinateMaps::Distribution and distributes the grid points accordingly.

The formula for the mapping is, in case of a Linear distribution

$$\begin{array}{}\text{(1)}& x& =\frac{b}{B-A}(\xi -A)+\frac{a}{B-A}(B-\xi )\text{(2)}& \xi & =\frac{B}{b-a}(x-a)+\frac{A}{b-a}(b-x)\end{array}$$

For every other distribution we use this linear mapping to map onto an interval $[-1,1]$, so we define:

$$\begin{array}{}\text{(3)}& f(\xi )& :=\frac{A+B-2\xi }{A-B}\in [-1,1]\text{(4)}& g(x)& :=\frac{a+b-2x}{a-b}\in [-1,1]\end{array}$$

With this an Equiangular distribution is described by

$$\begin{array}{}\text{(5)}& x& =\frac{a}{2}(1-\mathrm{t}\mathrm{a}\mathrm{n}(\frac{\pi }{4}f(\xi )))+\frac{b}{2}(1+\mathrm{t}\mathrm{a}\mathrm{n}(\frac{\pi }{4}f(\xi )))\text{(6)}& \xi & =\frac{A}{2}(1-\frac{4}{\pi }\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(g(x)))+\frac{B}{2}(1+\frac{4}{\pi }\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}(g(x)))\end{array}$$

Note

The equiangular distribution is intended to be used with the Wedge map when equiangular coordinates are chosen for those maps. For more information on this choice of coordinates, see the documentation for Wedge.

For both the Logarithmic and Inverse distribution, we first specify a position for the singularity $c:=$singularity_pos outside the target interval $[a,b]$.

The Logarithmic distribution further requires the introduction of a variable $\sigma$ dependent on whether $c$ is left ( $\sigma =1$) or right ( $\sigma =-1$) of the target interval. With this, the Logarithmic distribution is described by

$$\begin{array}{}\text{(7)}& x& =\sigma \mathrm{e}\mathrm{x}\mathrm{p}(\frac{\ln (b-c)+\ln (a-c)}{2}+f(\xi )\frac{\ln (b-c)-\ln (a-c)}{2})+c\text{(8)}& \xi & =\frac{B-A}{2}\frac{2\ln (\sigma [x-c])-\ln (b-c)-\ln (a-c)}{\ln (b-c)-\ln (a-c)}+\frac{B+A}{2}\end{array}$$

and the Inverse distribution by

$$\begin{array}{}\text{(9)}& x& =\frac{2(a-c)(b-c)}{a+b-2c+(a-b)f(\xi )}+c\text{(10)}& \xi & =-\frac{A-B}{2(a-b)}(\frac{2(a-c)(b-c)}{x-c}-(a-c)-(b-c))+\frac{A+B}{2}\end{array}$$

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

• src/Domain/CoordinateMaps/Interval.hpp