From bdb54a333144897339f75e8441f9ee2e5a79c2ca Mon Sep 17 00:00:00 2001
From: Stefano Dominici <sfndmn@gmail.com>
Date: Mon, 3 Dec 2018 14:25:35 +0100
Subject: [PATCH] Modified implementation in favor of a constant eccentricity
 manifold Unit tests, comments, and pictures were modified accordingly.

Minor modifications

 - Adjusted style.
 - Used `std_cxx14` instead of `std` for `make_unique()`.
 - Added guard in `pull_back()` to prevent NaNs in `std::acos()`. This
 may happen due to numerical errors: when the argument of acos() falls
 outside the range [-1,1].

Included a special case for pull_back()

If the first coordinate is exactly 0, the center of the manifold is
returned.
---
 doc/doxygen/images/elliptical_hyper_shell.png |  Bin 15192 -> 15498 bytes
 include/deal.II/grid/manifold_lib.h           |   29 +-
 source/grid/manifold_lib.cc                   |   88 +-
 tests/manifold/elliptical_manifold_01.cc      |   73 +-
 tests/manifold/elliptical_manifold_01.output  | 1152 ++++++++---------
 tests/manifold/elliptical_manifold_02.cc      |   36 +-
 tests/manifold/elliptical_manifold_02.output  |   32 +-
 7 files changed, 682 insertions(+), 728 deletions(-)

diff --git a/doc/doxygen/images/elliptical_hyper_shell.png b/doc/doxygen/images/elliptical_hyper_shell.png
index c4e70c62783d021779e22e92624c45f63740fbac..7da426131d36809b098b53b4f3b4112f8e6685a3 100644
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diff --git a/include/deal.II/grid/manifold_lib.h b/include/deal.II/grid/manifold_lib.h
index dfcc581858..6eb5d7e328 100644
--- a/include/deal.II/grid/manifold_lib.h
+++ b/include/deal.II/grid/manifold_lib.h
@@ -468,11 +468,25 @@ private:
  * href="https://en.wikipedia.org/wiki/Elliptic_coordinate_system">Wikipedia
  * </a>.
  *
+ * This is based on the definition of elliptic coordinates $(u,v)$
+ * @f[
+ *  \left\lbrace\begin{align*}
+ *  x &=  x_0 + c \cosh(u) \cos(v) \\
+ *  y &=  y_0 + c \sinh(u) \sin(v)
+ *  \end{align*}\right.
+ * @f]
+ * in which $(x_0,y_0)$ are coordinates of the center of the cartesian system.
+ *
+ * The current implementation uses coordinates $(c,v)$, instead of $(u,v)$, and
+ * fixes $u$ according to a given eccentricity. Therefore, this choice
+ * of coordinates generates an elliptical manifold characterized by a constant
+ * eccentricity: $e=\frac{1}{\cosh(u)}$, with $e\in\left]0,1\right[$.
+ *
  * The constructor of this class will throw an exception if both dim and
  * spacedim are different from two.
  *
  * This manifold can be used to produce hyper_shells with elliptical curvature.
- * As an example, the test elliptical_manifold_01 produces the following
+ * As an example, the test <B>elliptical_manifold_01</B> produces the following
  * triangulation:
  * @image html elliptical_hyper_shell.png
  *
@@ -486,19 +500,19 @@ class EllipticalManifold : public ChartManifold<dim, spacedim, spacedim>
 public:
   /**
    * Constructor that takes the center of the manifold system, the direction of
-   * the major axis, and the eccentricity parameter.
+   * the major axis, and the manifold eccentricity.
    * The default major axis is the <tt>x</tt>-axis. The manifold is rotated in
    * order to align the major axis to the direction specified in input.
    * @param center Center of the manifold.
    * @param major_axis_direction Direction of the major axis of the
    * manifold.
-   * @param c_parameter Parameter controlling the eccentricity of the
-   * manifold.
+   * @param eccentricity Eccentricity of the
+   * manifold $e\in\left]0,1\right[$.
    *
    */
   EllipticalManifold(const Point<spacedim> &    center,
                      const Tensor<1, spacedim> &major_axis_direction,
-                     const double               c_parameter);
+                     const double               eccentricity);
 
   virtual std::unique_ptr<Manifold<dim, spacedim>>
   clone() const override;
@@ -532,9 +546,10 @@ protected:
    */
   const Point<spacedim> center;
   /**
-   * The parameter controlling the eccentricity of the manifold.
+   * Parameters deriving from the eccentricity of the manifold.
    */
-  const double c_parameter;
+  const double cosh_u;
+  const double sinh_u;
 
 private:
   /**
diff --git a/source/grid/manifold_lib.cc b/source/grid/manifold_lib.cc
index e809bcf27e..7387e35f89 100644
--- a/source/grid/manifold_lib.cc
+++ b/source/grid/manifold_lib.cc
@@ -1156,18 +1156,20 @@ template <int dim, int spacedim>
 EllipticalManifold<dim, spacedim>::EllipticalManifold(
   const Point<spacedim> &    center,
   const Tensor<1, spacedim> &major_axis_direction,
-  const double               c_parameter)
+  const double               eccentricity)
   : ChartManifold<dim, spacedim, spacedim>(
       EllipticalManifold<dim, spacedim>::get_periodicity())
   , direction(major_axis_direction)
   , center(center)
-  , c_parameter(c_parameter)
+  , cosh_u(1.0 / eccentricity)
+  , sinh_u(std::sqrt(cosh_u * cosh_u - 1.0))
 {
-  // throws an exception if dim!=2 || spacedim!=2.
+  // Throws an exception if dim!=2 || spacedim!=2.
   Assert(dim == 2 && spacedim == 2, ExcNotImplemented());
-  Assert(this->c_parameter > 0.0,
+  // Throws an exception if eccentricity is not in range.
+  Assert(std::signbit(cosh_u * cosh_u - 1.0) == false,
          ExcMessage(
-           "Invalid ellipsis parameter 'c': It must satisfy c > 0.0."));
+           "Invalid eccentricity: It must satisfy 0 < eccentricity < 1."));
   const double direction_norm = direction.norm();
   Assert(direction_norm != 0.0,
          ExcMessage(
@@ -1181,9 +1183,9 @@ template <int dim, int spacedim>
 std::unique_ptr<Manifold<dim, spacedim>>
 EllipticalManifold<dim, spacedim>::clone() const
 {
-  return std::make_unique<EllipticalManifold<dim, spacedim>>(center,
-                                                             direction,
-                                                             c_parameter);
+  const double eccentricity = 1.0 / cosh_u;
+  return std_cxx14::make_unique<EllipticalManifold<dim, spacedim>>(
+    center, direction, eccentricity);
 }
 
 
@@ -1215,14 +1217,12 @@ template <>
 Point<2>
 EllipticalManifold<2, 2>::push_forward(const Point<2> &chart_point) const
 {
-  const double ch = std::cosh(chart_point[0]);
-  const double sh = std::sinh(chart_point[0]);
   const double cs = std::cos(chart_point[1]);
   const double sn = std::sin(chart_point[1]);
   // Coordinates in the reference frame (i.e. major axis direction is
   // x-axis)
-  const double x = c_parameter * ch * cs;
-  const double y = c_parameter * sh * sn;
+  const double x = chart_point[0] * cosh_u * cs;
+  const double y = chart_point[0] * sinh_u * sn;
   // Rotate them according to the major axis direction
   const Point<2> p(direction[0] * x - direction[1] * y,
                    direction[1] * x + direction[0] * y);
@@ -1246,45 +1246,29 @@ Point<2>
 EllipticalManifold<2, 2>::pull_back(const Point<2> &space_point) const
 {
   // Moving space_point in the reference coordinate system.
-  const double c2 = c_parameter * c_parameter;
   const double x0 = space_point[0] - center[0];
   const double y0 = space_point[1] - center[1];
   const double x  = direction[0] * x0 + direction[1] * y0;
   const double y  = -direction[1] * x0 + direction[0] * y0;
-  // From here we try to find solutions of two equations:
-  // x^2/(1-q)-y^2/q = c^2 for q, given q = -sinh^2(pt[0]),
-  // and
-  // x^2/(1-p)-y^2/p = c^2 for p, given p = sin^2(pt[1]).
-  // Note that, in the end, p and q are solutions to the same quadratic
-  // equation.
-  const double x2 = x * x;
-  const double y2 = y * y;
-  //
-  const double b       = x2 + y2 - c2;
-  const double srdelta = std::sqrt(b * b + 4.0 * c2 * y2);
-  double       p       = (-b + srdelta) / (2.0 * c2);
-  // Since p=sin^2(pt[1]) is bounded in [0,1],
-  // guard against numerical overflows.
-  if (p < 0.0)
-    p = 0.0;
-  if (p > 1.0)
-    p = 1.0;
-  const double q             = (-b - srdelta) / (2.0 * c2);
-  const bool   x_is_positive = !std::signbit(x);
-  const bool   y_is_positive = !std::signbit(y);
-  const double eta0          = std::asin(std::sqrt(p));
-  // Given q = -sinh^2(pt[0]),
-  // pt[0] is calculated by straight inversion since q <= 0 for any
-  // pt[0].
-  Point<2> pt(std::log(std::sqrt(-q) + std::sqrt(1.0 - q)), eta0);
-  // Unfolding pt[1] according to the quadrant.
-  if (x_is_positive && !y_is_positive)
-    pt[1] = 2.0 * numbers::PI - eta0;
-  else if (!x_is_positive && y_is_positive)
-    pt[1] = numbers::PI - eta0;
-  else if (!x_is_positive && !y_is_positive)
-    pt[1] = numbers::PI + eta0;
-  return pt;
+  const double pt0 =
+    std::sqrt((x * x) / (cosh_u * cosh_u) + (y * y) / (sinh_u * sinh_u));
+  // If the radius is exactly zero, the point coincides with the origin.
+  if (pt0 == 0.0)
+    {
+      return center;
+    }
+  double cos_eta = x / (pt0 * cosh_u);
+  if (cos_eta < -1.0)
+    {
+      cos_eta = -1.0;
+    }
+  if (cos_eta > 1.0)
+    {
+      cos_eta = 1.0;
+    }
+  const double eta = std::acos(cos_eta);
+  const double pt1 = (std::signbit(y) ? 2.0 * numbers::PI - eta : eta);
+  return Point<2>(pt0, pt1);
 }
 
 
@@ -1305,15 +1289,13 @@ DerivativeForm<1, 2, 2>
 EllipticalManifold<2, 2>::push_forward_gradient(
   const Point<2> &chart_point) const
 {
-  const double            ch = std::cosh(chart_point[0]);
-  const double            sh = std::sinh(chart_point[0]);
   const double            cs = std::cos(chart_point[1]);
   const double            sn = std::sin(chart_point[1]);
   DerivativeForm<1, 2, 2> dX;
-  dX[0][0] = c_parameter * sh * cs;
-  dX[0][1] = -c_parameter * ch * sn;
-  dX[1][0] = c_parameter * ch * sn;
-  dX[1][1] = c_parameter * sh * cs;
+  dX[0][0] = cosh_u * cs;
+  dX[0][1] = -chart_point[0] * cosh_u * sn;
+  dX[1][0] = sinh_u * sn;
+  dX[1][1] = chart_point[1] * sinh_u * cs;
   return dX;
 }
 
diff --git a/tests/manifold/elliptical_manifold_01.cc b/tests/manifold/elliptical_manifold_01.cc
index a9a6b0affd..cc1bf38599 100644
--- a/tests/manifold/elliptical_manifold_01.cc
+++ b/tests/manifold/elliptical_manifold_01.cc
@@ -41,64 +41,27 @@
 // hyper_shell made of 8 elements.
 namespace
 {
-  /* The following system of equations
-  // /
-  // |x = ellPar*cosh(u)*cos(v)
-  // |y = ellPar*sinh(u)*sin(v)
-  // \
-  // allows to transform a point (u,v) in the chart space to a
-  // point (x,y) in the cartesian space.
-  //
-  // Setting v=0, the system of equations becomes
-  // /
-  // |x = x_0 = ellPar*cosh(u)
-  // |y = 0
-  // \
-  // which represents the crossing point of an ellipsis, non-rotated and
-  // centered at the origin of the cartesian system, with its major axis.
-  //
-  // For the sake of completeness, by setting v=pi/2, one finds y_0 =
-  // ellPar*sinh(u) as the crossing point on the minor axis.
-  //
-  // This function takes in input a point pt={x_0,v}
-  // and transforms it into (x,y) in accordance with the first system of
-  // equations above.
-  //
-  // On a final note, the condition x_0 >= ellPar has to be satisfied. This is
-  // required because min(cosh(u)) = 1 is obtained for u = 0, and imposing v = 0
-  // the minimum value that x can assume is x = ellPar.
-  // For the sake of completeness, it's worth to notice that there would be no
-  // restrictions on the input if we passed y_0, instead of x_0, as pt[0].
-  */
-  Point<2>
-  chart_to_cartesian(const Point<2> &pt, const double ellPar)
-  {
-    Point<2>     c;
-    const double ch = pt[0] / ellPar;
-    const double sh = std::sqrt(ch * ch - 1.0);
-    c[0]            = ellPar * ch * std::cos(pt[1]);
-    c[1]            = ellPar * sh * std::sin(pt[1]);
-    return c;
-  }
-
-
-
   std::vector<Point<2>>
   generate_shell_points(const Point<2> &center,
                         const double    radius0,
                         const double    radius1,
-                        const double    ellPar)
+                        const double    eccentricity)
   {
-    const std::array<double, 2> radii{{radius0, radius1}};
+    const std::array<double, 2> radii{
+      {radius0 * eccentricity, radius1 * eccentricity}};
     const std::array<double, 5> angle{
       {0, numbers::PI_4, numbers::PI_2, 3.0 * numbers::PI_4, numbers::PI}};
     std::vector<Point<2>> points;
+    // Create an elliptical manifold to use push_forward()
+    Tensor<1, 2> axis;
+    axis[0] = 1.0;
+    axis[1] = 0.0;
+    EllipticalManifold<2, 2> mani(center, axis, eccentricity);
     for (auto &i : radii)
       {
         for (auto &j : angle)
           {
-            points.emplace_back(chart_to_cartesian(Point<2>(i, j), ellPar) +
-                                center);
+            points.emplace_back(mani.push_forward(Point<2>(i, j)) + center);
           }
       }
     std::array<int, 6> dup{{1, 2, 3, 6, 7, 8}};
@@ -117,19 +80,15 @@ namespace
   // center, and the major axis oriented in the direction of the x-axis.
   //
   // inner_radius and outer_radius parameters correspond to the
-  // distances from the center of the manifold of the
-  // crossing points of two concentric ellipsis with their major axis.
+  // semi-major axis length for the inner and outer ellipsis.
   //
-  // Input parameters must respect the following constrains:
-  // c_parameter < inner_radius < outer_radius.
+  // Eccentricity must be in range ]0,1[
   //
-  // To understand the constrain on the c_parameter refer to the
-  // documentation of function chart_to_cartesian() whithin this namespace.
   void build_simple_hyper_shell(Triangulation<2, 2> &grid,
                                 const Point<2> &     center,
                                 const double         inner_radius,
                                 const double         outer_radius,
-                                const double         c_param)
+                                const double         eccentricity)
   {
     unsigned int          cell[][4] = {{5, 6, 0, 1},
                               {6, 7, 1, 2},
@@ -140,7 +99,7 @@ namespace
                               {15, 14, 12, 11},
                               {9, 15, 4, 12}};
     std::vector<Point<2>> points =
-      generate_shell_points(center, inner_radius, outer_radius, c_param);
+      generate_shell_points(center, inner_radius, outer_radius, eccentricity);
     std::vector<CellData<2>> cells(8, CellData<2>());
     for (int i = 0; i < 8; ++i)
       for (int j = 0; j < 4; ++j)
@@ -151,7 +110,7 @@ namespace
     Tensor<1, 2> axis;
     axis[0] = 1.0;
     axis[1] = 0.0;
-    grid.set_manifold(0, EllipticalManifold<2, 2>(center, axis, c_param));
+    grid.set_manifold(0, EllipticalManifold<2, 2>(center, axis, eccentricity));
     grid.set_all_manifold_ids(0);
   }
 
@@ -168,13 +127,11 @@ template <int dim, int spacedim>
 void
 test(unsigned int ref = 1)
 {
-  // deallog << "Testing dim " << dim << ", spacedim " << spacedim << std::endl;
-
   Triangulation<dim, spacedim> tria;
   switch (dim)
     {
       case 2:
-        build_simple_hyper_shell(tria, Point<spacedim>(), 1.5, 1.8, 1.4);
+        build_simple_hyper_shell(tria, Point<spacedim>(), 1.0, 2.0, 0.8);
         break;
       default:
         AssertThrow(0, ExcNotImplemented());
diff --git a/tests/manifold/elliptical_manifold_01.output b/tests/manifold/elliptical_manifold_01.output
index d380473c18..edd50e266f 100644
--- a/tests/manifold/elliptical_manifold_01.output
+++ b/tests/manifold/elliptical_manifold_01.output
@@ -1,582 +1,582 @@
 
 $NOD
 576
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-2  1.06066 0.380789 0
-3  9.18485e-17 0.538516 0
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 $ELM
 512
diff --git a/tests/manifold/elliptical_manifold_02.cc b/tests/manifold/elliptical_manifold_02.cc
index 17b8966091..e6b99121e5 100644
--- a/tests/manifold/elliptical_manifold_02.cc
+++ b/tests/manifold/elliptical_manifold_02.cc
@@ -28,7 +28,7 @@
 #include <deal.II/grid/tria_accessor.h>
 #include <deal.II/grid/tria_iterator.h>
 
-#define TOLERANCE 1e-12
+#define TOLERANCE 1e-14
 
 
 
@@ -36,13 +36,13 @@
 template <int dim, int spacedim>
 Point<spacedim>
 test_push(const Tensor<1, spacedim> &axis,
-          const double               c_parameter,
+          const double               eccentricity,
           const Point<spacedim> &    space_point,
           unsigned int               ref = 1)
 {
   EllipticalManifold<dim, spacedim> manifold(Point<spacedim>(),
                                              axis,
-                                             c_parameter);
+                                             eccentricity);
   const Point<spacedim> chart_point(manifold.push_forward(space_point));
   deallog << space_point << " -> ";
   deallog << chart_point << std::endl;
@@ -54,13 +54,13 @@ test_push(const Tensor<1, spacedim> &axis,
 template <int dim, int spacedim>
 Point<spacedim>
 test_pull(const Tensor<1, spacedim> &axis,
-          const double               c_parameter,
+          const double               eccentricity,
           const Point<spacedim> &    chart_point,
           unsigned int               ref = 1)
 {
   EllipticalManifold<dim, spacedim> manifold(Point<spacedim>(),
                                              axis,
-                                             c_parameter);
+                                             eccentricity);
   const Point<spacedim> space_point(manifold.pull_back(chart_point));
   deallog << space_point << " <- ";
   deallog << chart_point << std::endl;
@@ -72,11 +72,11 @@ test_pull(const Tensor<1, spacedim> &axis,
 // Function that tests pull_back() and push_forward().
 void
 local_test(const Tensor<1, 2> &axis,
-           const double        c_parameter,
+           const double        eccentricity,
            const Point<2> &    space_point)
 {
-  const Point<2> pt1 = test_push<2, 2>(axis, c_parameter, space_point);
-  const Point<2> pt2 = test_pull<2, 2>(axis, c_parameter, pt1);
+  const Point<2> pt1 = test_push<2, 2>(axis, eccentricity, space_point);
+  const Point<2> pt2 = test_pull<2, 2>(axis, eccentricity, pt1);
   if ((space_point - pt2).norm() < TOLERANCE)
     {
       deallog << "OK" << std::endl;
@@ -95,20 +95,20 @@ test()
   {
     // test on a default manifold
     const Tensor<1, 2> axis({1.0, 0.0});
-    const double       c_parameter(0.1);
-    local_test(axis, c_parameter, Point<2>(1.1, 0.0));
-    local_test(axis, c_parameter, Point<2>(2.0, 0.0));
-    local_test(axis, c_parameter, Point<2>(3.0, 0.0));
-    local_test(axis, c_parameter, Point<2>(4.0, 2.0));
+    const double       eccentricity(0.1);
+    local_test(axis, eccentricity, Point<2>(1.1, 0.0));
+    local_test(axis, eccentricity, Point<2>(2.0, 0.0));
+    local_test(axis, eccentricity, Point<2>(3.0, 0.0));
+    local_test(axis, eccentricity, Point<2>(4.0, 2.0));
   }
   {
     // test on a rotated manifold
     const Tensor<1, 2> axis({1.0, -1.0});
-    const double       c_parameter(0.1);
-    local_test(axis, c_parameter, Point<2>(1.1, 0.0));
-    local_test(axis, c_parameter, Point<2>(2.0, 0.0));
-    local_test(axis, c_parameter, Point<2>(3.0, 0.0));
-    local_test(axis, c_parameter, Point<2>(4.0, 2.0));
+    const double       eccentricity(0.1);
+    local_test(axis, eccentricity, Point<2>(1.1, 0.0));
+    local_test(axis, eccentricity, Point<2>(2.0, 0.0));
+    local_test(axis, eccentricity, Point<2>(3.0, 0.0));
+    local_test(axis, eccentricity, Point<2>(4.0, 2.0));
   }
 }
 
diff --git a/tests/manifold/elliptical_manifold_02.output b/tests/manifold/elliptical_manifold_02.output
index 755935b8f6..0cf687080f 100644
--- a/tests/manifold/elliptical_manifold_02.output
+++ b/tests/manifold/elliptical_manifold_02.output
@@ -1,25 +1,25 @@
 
-DEAL::1.10000 0.00000 -> 0.166852 0.00000
-DEAL::1.10000 0.00000 <- 0.166852 0.00000
+DEAL::1.10000 0.00000 -> 11.0000 0.00000
+DEAL::1.10000 0.00000 <- 11.0000 0.00000
 DEAL::OK
-DEAL::2.00000 0.00000 -> 0.376220 0.00000
-DEAL::2.00000 0.00000 <- 0.376220 0.00000
+DEAL::2.00000 0.00000 -> 20.0000 0.00000
+DEAL::2.00000 0.00000 <- 20.0000 0.00000
 DEAL::OK
-DEAL::3.00000 0.00000 -> 1.00677 0.00000
-DEAL::3.00000 0.00000 <- 1.00677 0.00000
+DEAL::3.00000 0.00000 -> 30.0000 0.00000
+DEAL::3.00000 0.00000 <- 30.0000 0.00000
 DEAL::OK
-DEAL::4.00000 2.00000 -> -1.13642 2.48147
-DEAL::4.00000 2.00000 <- -1.13642 2.48147
+DEAL::4.00000 2.00000 -> -16.6459 36.1896
+DEAL::4.00000 2.00000 <- -16.6459 36.1896
 DEAL::OK
-DEAL::1.10000 0.00000 -> 0.117982 -0.117982
-DEAL::1.10000 0.00000 <- 0.117982 -0.117982
+DEAL::1.10000 0.00000 -> 7.77817 -7.77817
+DEAL::1.10000 0.00000 <- 7.77817 -7.77817
 DEAL::OK
-DEAL::2.00000 0.00000 -> 0.266027 -0.266027
-DEAL::2.00000 0.00000 <- 0.266027 -0.266027
+DEAL::2.00000 0.00000 -> 14.1421 -14.1421
+DEAL::2.00000 0.00000 <- 14.1421 -14.1421
 DEAL::OK
-DEAL::3.00000 0.00000 -> 0.711891 -0.711891
-DEAL::3.00000 0.00000 <- 0.711891 -0.711891
+DEAL::3.00000 0.00000 -> 21.2132 -21.2132
+DEAL::3.00000 0.00000 <- 21.2132 -21.2132
 DEAL::OK
-DEAL::4.00000 2.00000 -> 0.951088 2.55823
-DEAL::4.00000 2.00000 <- 0.951088 2.55823
+DEAL::4.00000 2.00000 -> 13.8195 37.3603
+DEAL::4.00000 2.00000 <- 13.8195 37.3603
 DEAL::OK
-- 
2.39.5