Line data    Source code

       1           0 : // Distributed under the MIT License.
       2             : // See LICENSE.txt for details.
       3             : 
       4             : #pragma once
       5             : 
       6             : #include <array>
       7             : #include <cstddef>
       8             : #include <limits>
       9             : #include <optional>
      10             : 
      11             : #include "DataStructures/Tensor/TypeAliases.hpp"
      12             : #include "Utilities/Gsl.hpp"
      13             : #include "Utilities/TypeTraits/RemoveReferenceWrapper.hpp"
      14             : 
      15             : /// \cond
      16             : namespace PUP {
      17             : class er;
      18             : }  // namespace PUP
      19             : /// \endcond
      20             : 
      21             : /// Contains FocallyLiftedInnerMaps
      22             : namespace domain::CoordinateMaps::FocallyLiftedInnerMaps {
      23             : /*!
      24             :  * \brief A FocallyLiftedInnerMap that maps a 3D unit right cylinder
      25             :  *  to a volume that connects a 2D annulus to a spherical
      26             :  *  surface.
      27             :  *
      28             :  * \details The domain of the map is a 3D unit right cylinder with
      29             :  * coordinates \f$(\bar{x},\bar{y},\bar{z})\f$ such that
      30             :  * \f$-1\leq\bar{z}\leq 1\f$ and \f$1\leq \bar{x}^2+\bar{y}^2 \leq
      31             :  * 4\f$.  The range of the map has coordinates \f$(x,y,z)\f$.
      32             :  *
      33             :  * Consider a 2D annulus in 3D space
      34             :  * oriented normal to the \f$z\f$ axis.  The inner and outer radii
      35             :  * of the annulus are \f$R_\mathrm{in}\f$ and \f$R_\mathrm{out}\f$, and the
      36             :  * (3D) center of the annulus is \f$C^i\f$.
      37             :  * `FlatSide` provides the following functions:
      38             :  *
      39             :  * ### forward_map()
      40             :  * `forward_map()` maps \f$(\bar{x},\bar{y},\bar{z}=-1)\f$ to the interior
      41             :  * of the annulus.  The arguments to `forward_map()`
      42             :  * are \f$(\bar{x},\bar{y},\bar{z})\f$, but \f$\bar{z}\f$ is ignored.
      43             :  * `forward_map()` returns \f$x_0^i\f$,
      44             :  * the 3D coordinates on the annulus, which are given by
      45             :  *
      46             :  * \f{align}
      47             :  * x_0^0 &= \left(R_\mathrm{in}+(R_\mathrm{out}-R_\mathrm{in})
      48             :  *    (\bar{\rho}-1)\right)
      49             :  *    \frac{\bar{x}}{\bar{\rho}} + C^0, \label{eq:forward_map_x}\\
      50             :  * x_0^1 &= \left(R_\mathrm{in}+(R_\mathrm{out}-R_\mathrm{in})
      51             :  *    (\bar{\rho}-1)\right)
      52             :  *    \frac{\bar{y}}{\bar{\rho}} + C^1,\\
      53             :  * x_0^2 &= C^2 \label{eq:forward_map_z},
      54             :  * \f}
      55             :  *
      56             :  * where
      57             :  *
      58             :  * \f{align} \bar{\rho} = \sqrt{\bar{x}^2+\bar{y}^2}.\label{eq:rhobar}\f}
      59             :  *
      60             :  * ### sigma
      61             :  *
      62             :  * \f$\sigma\f$ is a function that is zero on the sphere
      63             :  * \f$x^i=x_0^i\f$ and unity at \f$\bar{z}=+1\f$ (corresponding to the
      64             :  * upper surface of the FocallyLiftedMap). We define
      65             :  *
      66             :  * \f{align}
      67             :  *  \sigma &= \frac{\bar{z}+1}{2}.
      68             :  * \f}
      69             :  *
      70             :  * ### deriv_sigma
      71             :  *
      72             :  * `deriv_sigma` returns
      73             :  *
      74             :  * \f{align}
      75             :  *  \frac{\partial \sigma}{\partial \bar{x}^j} &= (0,0,1/2).
      76             :  *  \label{eq:deriv_sigma}
      77             :  * \f}
      78             :  *
      79             :  * ### jacobian
      80             :  *
      81             :  * `jacobian` returns \f$\partial x_0^k/\partial \bar{x}^j\f$.
      82             :  * The arguments to `jacobian`
      83             :  * are \f$(\bar{x},\bar{y},\bar{z})\f$, but \f$\bar{z}\f$ is ignored.
      84             :  *
      85             :  * Differentiating
      86             :  * Eqs. (\f$\ref{eq:forward_map_x}\f$--\f$\ref{eq:forward_map_z}\f$)
      87             :  * above yields
      88             :  *
      89             :  * \f{align*}
      90             :  * \frac{\partial x_0^0}{\partial \bar{x}} &=
      91             :  * R_\mathrm{out}-R_\mathrm{in} + (2 R_\mathrm{in}-R_\mathrm{out})
      92             :  * \frac{\bar{y}^2}{\bar{\rho}^3},\\
      93             :  * \frac{\partial x_0^0}{\partial \bar{y}} &=
      94             :  * -(2 R_\mathrm{in}-R_\mathrm{out})
      95             :  * \frac{\bar{x}\bar{y}}{\bar{\rho}^3},\\
      96             :  * \frac{\partial x_0^1}{\partial \bar{x}} &=
      97             :  * -(2 R_\mathrm{in}-R_\mathrm{out})
      98             :  * \frac{\bar{x}\bar{y}}{\bar{\rho}^3},\\
      99             :  * \frac{\partial x_0^1}{\partial \bar{y}} &=
     100             :  * R_\mathrm{out}-R_\mathrm{in} + (2 R_\mathrm{in}-R_\mathrm{out})
     101             :  * \frac{\bar{x}^2}{\bar{\rho}^3},\\
     102             :  * \f}
     103             :  * and all other components are zero.
     104             :  *
     105             :  * ### inverse
     106             :  *
     107             :  * `inverse` takes \f$x_0^i\f$ and \f$\sigma\f$ as arguments, and
     108             :  * returns \f$(\bar{x},\bar{y},\bar{z})\f$, or a default-constructed
     109             :  * `std::optional<std::array<double, 3>>` if
     110             :  * \f$x_0^i\f$ or \f$\sigma\f$ are outside the range of the map.
     111             :  *
     112             :  * Let
     113             :  * \f{align}
     114             :  *  \rho = \sqrt{(x_0^0-C^0)^2+(x_0^1-C^1)^2}.
     115             :  *  \label{eq:rho}
     116             :  * \f}
     117             :  *
     118             :  * Then
     119             :  * \f{align}
     120             :  *  \bar{x} &= \frac{x_0^0-C^0}{\rho}
     121             :  *  \frac{\rho+R_\mathrm{out}-2 R_\mathrm{in}}{R_\mathrm{out}-R_\mathrm{in}},\\
     122             :  *  \bar{y} &= \frac{x_0^1-C^1}{\rho}
     123             :  *  \frac{\rho+R_\mathrm{out}-2 R_\mathrm{in}}{R_\mathrm{out}-R_\mathrm{in}},\\
     124             :  *  \bar{z} &= 2\sigma - 1.
     125             :  * \f}
     126             :  *
     127             :  * Note that \f$\rho\f$ in Eq. (\f$\ref{eq:rho}\f$) can be written
     128             :  * \f{align}
     129             :  * \rho = R_\mathrm{in}+(R_\mathrm{out}-R_\mathrm{in})(\bar{\rho}-1),
     130             :  * \label{eq:rho_from_rhobar}
     131             :  * \f}
     132             :  * where \f$\bar{\rho}\f$ is given by Eq. (\f$\ref{eq:rhobar}\f$).
     133             :  *
     134             :  * If \f$\bar{z}\f$ is outside the range \f$[-1,1]\f$ or
     135             :  * if \f$\bar{x}^2+\bar{y}^2\f$ is less than 1 or greater than 4
     136             :  * then we return a default-constructed
     137             :  * `std::optional<std::array<double, 3>>`.
     138             :  *
     139             :  * ### lambda_tilde
     140             :  *
     141             :  * `lambda_tilde` takes as arguments a point \f$x^i\f$ and a projection point
     142             :  *  \f$P^i\f$, and computes \f$\tilde{\lambda}\f$, the solution to
     143             :  *
     144             :  * \f{align} x_0^i = P^i + (x^i - P^i) \tilde{\lambda}.\f}
     145             :  *
     146             :  * Since \f$x_0^i\f$ must lie on the plane \f$x_0^3=C^3\f$,
     147             :  *
     148             :  * \f{align} \tilde{\lambda} &= \frac{C^3-P^3}{x^3-P^3}.\f}
     149             :  *
     150             :  * If \f$\tilde{\lambda}\f$ is less than unity (indicating that the
     151             :  * supplied point is outside the range of the map), then a
     152             :  * default-constructed `std::optional<double>` is returned.
     153             :  *
     154             :  * ### deriv_lambda_tilde
     155             :  *
     156             :  * `deriv_lambda_tilde` takes as arguments \f$x_0^i\f$, a projection point
     157             :  *  \f$P^i\f$, and \f$\tilde{\lambda}\f$, and
     158             :  *  returns \f$\partial \tilde{\lambda}/\partial x^i\f$. We have
     159             :  *
     160             :  * \f{align}
     161             :  * \frac{\partial\tilde{\lambda}}{\partial x^3} =
     162             :  * -\frac{C^3-P^3}{(x^3-P^3)^2} = -\frac{\tilde{\lambda}^2}{C^3-P^3},
     163             :  * \f}
     164             :  * and other components are zero.
     165             :  *
     166             :  * ### inv_jacobian
     167             :  *
     168             :  * `inv_jacobian` returns \f$\partial \bar{x}^i/\partial x_0^k\f$,
     169             :  *  where \f$\sigma\f$ is held fixed.
     170             :  *  The arguments to `inv_jacobian`
     171             :  *  are \f$(\bar{x},\bar{y},\bar{z})\f$, but \f$\bar{z}\f$ is ignored.
     172             :  *
     173             :  * The nonzero components are
     174             :  * \f{align}
     175             :  * \frac{\partial \bar{x}}{\partial x_0^0} &=
     176             :  *  \frac{1}{R_\mathrm{out}-R_\mathrm{in}} + \frac{\bar{y}^2}{\bar{\rho}^2\rho}
     177             :  *  \frac{R_\mathrm{out}-2 R_\mathrm{in}}{R_\mathrm{out}-R_\mathrm{in}},\\
     178             :  * \frac{\partial \bar{x}}{\partial x_0^1} &=
     179             :  *  - \frac{\bar{x}\bar{y}}{\bar{\rho}^2\rho}
     180             :  *  \frac{R_\mathrm{out}-2 R_\mathrm{in}}{R_\mathrm{out}-R_\mathrm{in}},\\
     181             :  * \frac{\partial \bar{y}}{\partial x_0^1} &=
     182             :  *  \frac{1}{R_\mathrm{out}-R_\mathrm{in}} + \frac{\bar{x}^2}{\bar{\rho}^2\rho}
     183             :  *  \frac{R_\mathrm{out}-2 R_\mathrm{in}}{R_\mathrm{out}-R_\mathrm{in}},\\
     184             :  * \frac{\partial \bar{y}}{\partial x_0^0} &=
     185             :  *  - \frac{\bar{x}\bar{y}}{\bar{\rho}^2\rho}
     186             :  *  \frac{R_\mathrm{out}-2 R_\mathrm{in}}{R_\mathrm{out}-R_\mathrm{in}},
     187             :  * \f}
     188             :  * where \f$\rho\f$ is computed from Eq. (\f$\ref{eq:rho_from_rhobar}\f$).
     189             :  *
     190             :  * ### dxbar_dsigma
     191             :  *
     192             :  * `dxbar_dsigma` returns \f$\partial \bar{x}^i/\partial \sigma\f$,
     193             :  *  where \f$x_0^i\f$ is held fixed.
     194             :  *
     195             :  * From Eq. (\f$\ref{eq:deriv_sigma}\f$) we have
     196             :  *
     197             :  * \f{align}
     198             :  * \frac{\partial \bar{x}^i}{\partial \sigma} &= (0,0,2).
     199             :  * \f}
     200             :  *
     201             :  */
     202           1 : class FlatSide {
     203             :  public:
     204           0 :   FlatSide(const std::array<double, 3>& center, const double inner_radius,
     205             :            const double outer_radius);
     206             : 
     207           0 :   FlatSide() = default;
     208           0 :   ~FlatSide() = default;
     209           0 :   FlatSide(FlatSide&&) = default;
     210           0 :   FlatSide(const FlatSide&) = default;
     211           0 :   FlatSide& operator=(const FlatSide&) = default;
     212           0 :   FlatSide& operator=(FlatSide&&) = default;
     213             : 
     214             :   template <typename T>
     215           0 :   void forward_map(
     216             :       const gsl::not_null<std::array<tt::remove_cvref_wrap_t<T>, 3>*>
     217             :           target_coords,
     218             :       const std::array<T, 3>& source_coords) const;
     219             : 
     220           0 :   std::optional<std::array<double, 3>> inverse(
     221             :       const std::array<double, 3>& target_coords, double sigma_in) const;
     222             : 
     223             :   template <typename T>
     224           0 :   void jacobian(const gsl::not_null<
     225             :                     tnsr::Ij<tt::remove_cvref_wrap_t<T>, 3, Frame::NoFrame>*>
     226             :                     jacobian_out,
     227             :                 const std::array<T, 3>& source_coords) const;
     228             : 
     229             :   template <typename T>
     230           0 :   void inv_jacobian(const gsl::not_null<tnsr::Ij<tt::remove_cvref_wrap_t<T>, 3,
     231             :                                                  Frame::NoFrame>*>
     232             :                         inv_jacobian_out,
     233             :                     const std::array<T, 3>& source_coords) const;
     234             : 
     235             :   template <typename T>
     236           0 :   void sigma(const gsl::not_null<tt::remove_cvref_wrap_t<T>*> sigma_out,
     237             :              const std::array<T, 3>& source_coords) const;
     238             : 
     239             :   template <typename T>
     240           0 :   void deriv_sigma(
     241             :       const gsl::not_null<std::array<tt::remove_cvref_wrap_t<T>, 3>*>
     242             :           deriv_sigma_out,
     243             :       const std::array<T, 3>& source_coords) const;
     244             : 
     245             :   template <typename T>
     246           0 :   void dxbar_dsigma(
     247             :       const gsl::not_null<std::array<tt::remove_cvref_wrap_t<T>, 3>*>
     248             :           dxbar_dsigma_out,
     249             :       const std::array<T, 3>& source_coords) const;
     250             : 
     251           0 :   std::optional<double> lambda_tilde(
     252             :       const std::array<double, 3>& parent_mapped_target_coords,
     253             :       const std::array<double, 3>& projection_point,
     254             :       bool source_is_between_focus_and_target) const;
     255             : 
     256             :   template <typename T>
     257           0 :   void deriv_lambda_tilde(
     258             :       const gsl::not_null<std::array<tt::remove_cvref_wrap_t<T>, 3>*>
     259             :           deriv_lambda_tilde_out,
     260             :       const std::array<T, 3>& target_coords, const T& lambda_tilde,
     261             :       const std::array<double, 3>& projection_point) const;
     262             : 
     263             :   // NOLINTNEXTLINE(google-runtime-references)
     264           0 :   void pup(PUP::er& p);
     265             : 
     266           0 :   static bool is_identity() { return false; }
     267             : 
     268             :  private:
     269           0 :   friend bool operator==(const FlatSide& lhs, const FlatSide& rhs);
     270           0 :   std::array<double, 3> center_{};
     271           0 :   double inner_radius_{std::numeric_limits<double>::signaling_NaN()};
     272           0 :   double outer_radius_{std::numeric_limits<double>::signaling_NaN()};
     273             : };
     274           0 : bool operator!=(const FlatSide& lhs, const FlatSide& rhs);
     275             : }  // namespace domain::CoordinateMaps::FocallyLiftedInnerMaps