text: #ifndef BASILISK_HEADER_11
  #define BASILISK_HEADER_11
  #line 1 "/home/michael/basilisk_1/src/fractions.h"
  /**
  # Volume fractions
  
  These functions are used to maintain or define volume and surface
  fractions either from an initial geometric definition or from an
  existing volume fraction field. 
  
  We will use basic geometric functions for square cut cells and the
  "Mixed-Youngs-Centered" (MYC) normal approximation of Ruben
  Scardovelli. */
  
  #include "geometry.h"
  #if dimension == 1
  coord mycs (Point point, scalar c) {
    coord n = {1.};
    return n;
  }
  #elif dimension == 2
  # include "myc2d.h"
  #else // dimension == 3
  # include "myc.h"
  #endif
  
  /**
  By default the interface normal is computed using the MYC
  approximation. This can be overloaded by redefining this macro. */
  
  #ifndef interface_normal
  # define interface_normal(point, c) mycs (point, c)
  #endif
  
  /**
  ## Coarsening and refinement of a volume fraction field 
  
  On trees, we need to define how to coarsen (i.e. "restrict") or
  refine (i.e. "prolongate") interface definitions (see [geometry.h]()
  for a basic explanation of how interfaces are defined). */
  
  #if TREE
  
  void fraction_refine (Point point, scalar c)
  {
    
    /**
    If the parent cell is empty or full, we just use the same value for
    the fine cell. */
  
    double cc = c[];
    if (cc <= 0. || cc >= 1.)
      foreach_child()
        c[] = cc;
    else {
  
      /**
      Otherwise, we reconstruct the interface in the parent cell. */
  
      coord n = mycs (point, c);
      double alpha = plane_alpha (cc, n);
  
      /**
      And compute the volume fraction in the quadrant of the coarse cell
      matching the fine cells. We use symmetries to simplify the
      combinations. */
  
      foreach_child() {
        static const coord a = {0.,0.,0.}, b = {.5,.5,.5};
        coord nc;
        foreach_dimension()
  	nc.x = child.x*n.x;
        c[] = rectangle_fraction (nc, alpha, a, b);
      }
    }
  }
  
  /**
  Finally, we also need to prolongate the reconstructed value of
  $\alpha$. This is done with the simple formula below. We add an
  attribute so that we can access the normal from the refinement
  function. */
  
  attribute {
    vector n;
  }
  
  static void alpha_refine (Point point, scalar alpha)
  {
    vector n = alpha.n;
    double alphac = 2.*alpha[];
    coord m;
    foreach_dimension()
      m.x = n.x[];
    foreach_child() {
      alpha[] = alphac;
      foreach_dimension()
        alpha[] -= child.x*m.x/2.;
    }
  }
  
  #endif // TREE
  
  /**
  ## Computing volume fractions from a "levelset" function 
  
  Initialising a volume fraction field representing an interface is not
  trivial since it involves the numerical evaluation of surface
  integrals.
  
  Here we define a function which allows the approximation of these
  surface integrals in the case of an interface defined by a "levelset"
  function $\Phi$ sampled on the *vertices* of the grid.
  
  By convention the "inside" of the interface corresponds to $\Phi > 0$.
  
  The function takes the vertex scalar field $\Phi$ as input and fills
  `c` with the volume fraction and, optionally if it is given, `s`
  with the surface fractions i.e. the fractions of the faces of the cell
  which are inside the interface.
  
  ![Volume and surface fractions](/src/figures/fractions.svg) */
  
  struct Fractions {
    vertex scalar Phi; // compulsory
    scalar c;          // compulsory
    face vector s;     // optional
    double val;        // optional (default zero)
  };
  
  trace
  void fractions (struct Fractions a)
  {
    vertex scalar Phi = a.Phi;
    scalar c = a.c;
    face vector s = automatic (a.s);
    double val = a.val;
    
    /**
    We store the positions of the intersections of the surface with the
    edges of the cell in vector field `p`. In two dimensions, this field
    is just the transpose of the *line fractions* `s`, in 3D we need to
    allocate a new field. */
    
  #if dimension == 3
    vector p[];
  #else // dimension == 2
    vector p;
    p.x = s.y; p.y = s.x;
  #endif
    
    /**
    ### Line fraction computation
    
    We start by computing the *line fractions* i.e. the (normalised)
    lengths of the edges of the cell within the surface. */
  
    foreach_edge() {
  
      /**
      If the values of $\Phi$ on the vertices of the edge have opposite
      signs, we know that the edge is cut by the interface. */
  
      if ((Phi[] - val)*(Phi[1] - val) < 0.) {
  
        /**
        In that case we can find an approximation of the interface position by
        simple linear interpolation. We also check the sign of one of the
        vertices to orient the interface properly. */
  
        p.x[] = (Phi[] - val)/(Phi[] - Phi[1]);
        if (Phi[] < val)
  	p.x[] = 1. - p.x[];
      }
  
      /**
      If the values of $\Phi$ on the vertices of the edge have the same sign
      (or are zero), then the edge is either entirely outside or entirely
      inside the interface. We check the sign of both vertices to treat
      limit cases properly (when the interface intersects the edge exactly
      on one of the vertices). */
  
      else
        p.x[] = (Phi[] > val || Phi[1] > val);
    }
  
    /**
    ### Surface fraction computation 
  
    We can now compute the surface fractions. In 3D they will be
    computed for each face (in the z, x and y directions) and stored in
    the face field `s`. In 2D the surface fraction in the z-direction is
    the *volume fraction* `c`. The call to `boundary_flux()` defines
    consistent line fractions on trees. */
  
  #if dimension == 3
    scalar s_x = s.x, s_y = s.y, s_z = s.z;
    foreach_face(z,x,y)
  #else // dimension == 2
    boundary_flux ({s});
    scalar s_z = c;
    foreach()
  #endif
    {
  
      /**
      We first compute the normal to the interface. This can be done easily
      using the line fractions. The idea is to compute the circulation of
      the normal along the boundary $\partial\Omega$ of the fraction of the
      cell $\Omega$ inside the interface. Since this is a closed curve, we
      have
      $$
      \oint_{\partial\Omega}\mathbf{n}\;dl = 0
      $$ 
      We can further decompose the integral into its parts along the edges
      of the square and the part along the interface. For the case pictured
      above, we get for one component (and similarly for the other)
      $$
      - s_x[] + \oint_{\Phi=0}n_x\;dl = 0
      $$
      If we now define the *average normal* to the interface as
      $$
      \overline{\mathbf{n}} = \oint_{\Phi=0}\mathbf{n}\;dl
      $$
      We have in the general case
      $$
      \overline{\mathbf{n}}_x = s_x[] - s_x[1,0]
      $$
      and
      $$
      |\overline{\mathbf{n}}| = \oint_{\Phi=0}\;dl
      $$ 
      Note also that this average normal is exact in the case of a linear
      interface. */
  
      coord n;
      double nn = 0.;
      foreach_dimension(2) {
        n.x = p.y[] - p.y[1];
        nn += fabs(n.x);
      }
      
      /**
      If the norm is zero, the cell is full or empty and the surface fraction
      is identical to one of the line fractions. */
  
      if (nn == 0.)
        s_z[] = p.x[];
      else {
      
        /**
        Otherwise we are in a cell containing the interface. We first
        normalise the normal. */
  
        foreach_dimension(2)
  	n.x /= nn;
  
        /**
        To find the intercept $\alpha$, we look for edges which are cut by the
        interface, find the coordinate $a$ of the intersection and use it to
        derive $\alpha$. We take the average of $\alpha$ for all intersections. */
        
        double alpha = 0., ni = 0.;
        for (int i = 0; i <= 1; i++)
  	foreach_dimension(2)
  	  if (p.x[0,i] > 0. && p.x[0,i] < 1.) {
  	    double a = sign(Phi[0,i] - val)*(p.x[0,i] - 0.5);
  	    alpha += n.x*a + n.y*(i - 0.5);
  	    ni++;
  	  }
  
        /**
        Once we have $\mathbf{n}$ and $\alpha$, the (linear) interface
        is fully defined and we can compute the surface fraction using
        our pre-defined function. For marginal cases, the cell is full
        or empty (*ni == 0*) and we look at the line fractions to
        decide. */
  
        if (ni == 0)
  	s_z[] = max (p.x[], p.y[]);
        else if (ni != 4)
  	s_z[] = line_area (n.x, n.y, alpha/ni);
        else {
  #if dimension == 3
  	s_z[] = (p.x[] + p.x[0,1] + p.y[] + p.y[1] > 2.);
  #else
  	s_z[] = 0.;
  #endif
        }
      }
    }
  
    /**
    ### Volume fraction computation
  
    To compute the volume fraction in 3D, we use the same approach. */
    
  #if dimension == 3
    boundary_flux ({s});
    foreach() {
  
      /**
      Estimation of the average normal from the surface fractions. */
         
      coord n;
      double nn = 0.;
      foreach_dimension(3) {
        n.x = s.x[] - s.x[1];
        nn += fabs(n.x);
      }
      if (nn == 0.)
        c[] = s.x[];
      else {
        foreach_dimension(3)
  	n.x /= nn;
  
        /**
        We compute the average value of *alpha* by looking at the
        intersections of the surface with the twelve edges of the
        cube. */
        
        double alpha = 0., ni = 0.;
        for (int i = 0; i <= 1; i++)
  	for (int j = 0; j <= 1; j++)
  	  foreach_dimension(3)
  	    if (p.x[0,i,j] > 0. && p.x[0,i,j] < 1.) {
  	      double a = sign(Phi[0,i,j] - val)*(p.x[0,i,j] - 0.5);
  	      alpha += n.x*a + n.y*(i - 0.5) + n.z*(j - 0.5);
  	      ni++;
  	    }
  
        /**
        Finally we compute the volume fraction. */
  
        if (ni == 0)
  	c[] = s.x[];
        else if (ni < 3 || ni > 6)
  	c[] = 0.; // this is important for robustness of embedded boundaries
        else
  	c[] = plane_volume (n, alpha/ni);
      }
    }
  #endif // dimension == 3
    
    /**
    We apply the boundary conditions. */
  
    boundary ({c});
  }
  
  /**
  The convenience macro below can be used to define a volume fraction
  field directly from a function. */
  
  #define fraction(f,func) do {			\
      vertex scalar phi[];			\
      foreach_vertex()				\
        phi[] = func;				\
      boundary ({phi});				\
      fractions (phi, f);				\
    } while(0)
  
  /**
  ### Boolean operations
  
  Implicit surface representations have the advantage of allowing simple
  [constructive solid
  geometry](https://en.wikipedia.org/wiki/Constructive_solid_geometry)
  operations. */
  
  #define intersection(a,b)   min(a,b)
  #define union(a,b)          max(a,b)
  #define difference(a,b)     min(a,-(b))
  
  /**
  ## Interface reconstruction from volume fractions
  
  The reconstruction of the interface geometry from the volume fraction
  field requires computing an approximation to the interface normal.
  
  ### Youngs normal approximation 
  
  This a simple, but relatively inaccurate way of approximating the
  normal. It is simply a weighted average of centered volume fraction
  gradients. We include it as an example but it is not used. */
  
  coord youngs_normal (Point point, scalar c)
  {
    coord n;
    double nn = 0.;
    assert (dimension == 2);
    foreach_dimension() {
      n.x = (c[-1,1] + 2.*c[-1,0] + c[-1,-1] -
  	   c[+1,1] - 2.*c[+1,0] - c[+1,-1]);
      nn += fabs(n.x);
    }
    // normalize
    if (nn > 0.)
      foreach_dimension()
        n.x /= nn;
    else // this is a small fragment
      n.x = 1.;
    return n;
  }
  
  /**
  ### Normal approximation using MYC or face fractions
  */
  
  coord facet_normal (Point point, scalar c, face vector s)
  {
    if (s.x.i >= 0) { // compute normal from face fractions
      coord n;
      double nn = 0.;
      foreach_dimension() {
        n.x = s.x[] - s.x[1];
        nn += fabs(n.x);
      }
      if (nn > 0.)
        foreach_dimension()
  	n.x /= nn;
      else
        foreach_dimension()
  	n.x = 1./dimension;
      return n;
    }
    return interface_normal (point, c);
  }
  
  /**
  ### Interface reconstruction 
  
  The reconstruction function takes a volume fraction field `c` and
  returns the corresponding normal vector field `n` and intercept field
  $\alpha$. */
  
  trace
  void reconstruction (const scalar c, vector n, scalar alpha)
  {
    foreach() {
  
      /**
      If the cell is empty or full, we set $\mathbf{n}$ and $\alpha$ only to
      avoid using uninitialised values in `alpha_refine()`. */
  
      if (c[] <= 0. || c[] >= 1.) {
        alpha[] = 0.;
        foreach_dimension()
  	n.x[] = 0.;
      }
      else {
  
        /**
        Otherwise, we compute the interface normal using the
        Mixed-Youngs-Centered scheme, copy the result into the normal field
        and compute the intercept $\alpha$ using our predefined function. */
  
        coord m = interface_normal (point, c);
        foreach_dimension()
  	n.x[] = m.x;
        alpha[] = plane_alpha (c[], m);
      }
    }
  
  #if TREE
  
    /**
    On a tree grid, for the normal to the interface, we don't use
    any interpolation from coarse to fine i.e. we use straight
    "injection". */
  
    foreach_dimension()
      n.x.refine = n.x.prolongation = refine_injection;
  
    /**
    We set our refinement function for *alpha*. */
  
    alpha.n = n;
    alpha.refine = alpha.prolongation = alpha_refine;
  #endif
  
    /**
    Finally we apply the boundary conditions to define $\mathbf{n}$ and
    $\alpha$ everywhere (using the prolongation functions when necessary
    on tree grids). */
  
    boundary ({n, alpha});
  }
  
  /**
  ## Interface output
  
  This function "draws" interface facets in a file. The segment
  endpoints are defined by pairs of coordinates. Each pair of endpoints
  is separated from the next pair by a newline, so that the resulting
  file is directly visualisable with gnuplot.
  
  The input parameters are a volume fraction field `c`, an optional file
  pointer `fp` (which defaults to stdout) and an optional face
  vector field `s` containing the surface fractions.
  
  If `s` is specified, the surface fractions are used to compute the
  interface normals which leads to a continuous interface representation
  in most cases. Otherwise the interface normals are approximated from
  the volume fraction field, which results in a piecewise continuous
  (i.e. geometric VOF) interface representation. */
  
  struct OutputFacets {
    scalar c;
    FILE * fp;     // optional: default is stdout
    face vector s; // optional: default is none
  };
  
  trace
  void output_facets (struct OutputFacets p)
  {
    scalar c = p.c;
    face vector s = p.s;
    if (!p.fp) p.fp = stdout;
    if (!s.x.i) s.x.i = -1;
  
    foreach()
      if (c[] > 1e-6 && c[] < 1. - 1e-6) {
        coord n = facet_normal (point, c, s);
        double alpha = plane_alpha (c[], n);
  #if dimension == 2      
        coord segment[2];
        if (facets (n, alpha, segment) == 2)
  	fprintf (p.fp, "%g %g\n%g %g\n\n", 
  		 x + segment[0].x*Delta, y + segment[0].y*Delta, 
  		 x + segment[1].x*Delta, y + segment[1].y*Delta);
  #else // dimension == 3
        coord v[12];
        int m = facets (n, alpha, v, 1.);
        for (int i = 0; i < m; i++)
  	fprintf (p.fp, "%g %g %g\n",
  		 x + v[i].x*Delta, y + v[i].y*Delta, z + v[i].z*Delta);
        if (m > 0)
  	fputc ('\n', p.fp);
  #endif
      }
  
    fflush (p.fp);
  }
  
  /**
  ## Interfacial area
  
  This function returns the surface area of the interface as estimated
  using its VOF reconstruction. */
  
  trace
  double interface_area (scalar c)
  {
    double area = 0.;
    foreach (reduction(+:area))
      if (c[] > 1e-6 && c[] < 1. - 1e-6) {
        coord n = interface_normal (point, c), p;
        double alpha = plane_alpha (c[], n);
        area += pow(Delta, dimension - 1)*plane_area_center (n, alpha, &p);
      }
    return area;
  }
  
  #endif
  