fractions.h

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/** @file 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) {
  return (coord){sign(c[-1] - c[1])};
}
#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. */
 
auto macro coord interface_normal (Point p, scalar c) {
  return mycs (p, c);
}
 
/**
## 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.)
    for (int _c = 0; _c < 4; _c++) /* 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. */
 
    for (int _c = 0; _c < 4; _c++) /* foreach_child */ {
      static const coord a = {0.,0.,0.}, b = {.5,.5,.5};
      coord nc;
      for (int _d = 0; _d < dimension; _d++)
  nc.x = child.x*n.x;
      c[] = rectangle_fraction (nc, alpha, a, b);
    }
  }
}
 
/**
Finally, we also need to prolongate the reconstructed value of
\f$\alpha\f$. 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;
  for (int _d = 0; _d < dimension; _d++)
    m.x = n.x[];
  for (int _c = 0; _c < 4; _c++) /* foreach_child */ {
    alpha[] = alphac;
    for (int _d = 0; _d < dimension; _d++)
      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 \f$\Phi\f$ sampled on the *vertices* of the grid.
 
By convention the "inside" of the interface corresponds to \f$\Phi > 0\f$.
 
The function takes the scalar field \f$\Phi\f$ 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) */
 
trace
void fractions (scalar Phi, scalar c,
    vector s = {0}, double val = 0.)
{
#if dimension > 1
  vector as = automatic (s);
  
  /**
  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 = as.y; p.y = as.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 \f$\Phi\f$ 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 \f$\Phi\f$ 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`. */
 
#if dimension == 3
 
  /**
  In 3D we need to prevent boundary conditions, since this would
  impose vertex field BCs which are not (apparently) consistent for
  the edge intersection coordinates. This can probably be improved. */
  
  for (int _d = 0; _d < dimension; _d++)
    p.x.stencil.bc |= s_centered|s_face;
  
  scalar s_x = as.x, s_y = as.y, s_z = as.z;
  for (int _i = 0; _i < _N; _i++) /* foreach_face */
#else // dimension == 2
  scalar s_z = c;
  for (int _i = 0; _i < _N; _i++) /* 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 \f$\partial\Omega\f$ of the fraction of the
    cell \f$\Omega\f$ inside the interface. Since this is a closed curve, we
    have
    \f[
    \oint_{\partial\Omega}\mathbf{n}\;dl = 0
    \f] 
    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)
    \f[
    - s_x[] + \oint_{\Phi=0}n_x\;dl = 0
    \f]
    If we now define the *average normal* to the interface as
    \f[
    \overline{\mathbf{n}} = \oint_{\Phi=0}\mathbf{n}\;dl
    \f]
    We have in the general case
    \f[
    \overline{\mathbf{n}}_x = s_x[] - s_x[1,0]
    \f]
    and
    \f[
    |\overline{\mathbf{n}}| = \oint_{\Phi=0}\;dl
    \f] 
    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 \f$\alpha\f$, we look for edges which are cut by the
      interface, find the coordinate \f$a\f$ of the intersection and use it to
      derive \f$\alpha\f$. We take the average of \f$\alpha\f$ 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 \f$\mathbf{n}\f$ and \f$\alpha\f$, 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
  for (int _i = 0; _i < _N; _i++) /* foreach */ {
 
    /**
    Estimation of the average normal from the surface fractions. */
       
    coord n;
    double nn = 0.;
    foreach_dimension(3) {
      n.x = as.x[] - as.x[1];
      nn += fabs(n.x);
    }
    if (nn == 0.)
      c[] = as.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[] = as.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
#else  // dimension == 1
  if (s.x.i)
    for (int _i = 0; _i < _N; _i++) /* foreach_face */
      s.x[] = Phi[] > 0.;
  for (int _i = 0; _i < _N; _i++) /* foreach */
    if ((Phi[] - val)*(Phi[1] - val) < 0.) {
      c[] = (Phi[] - val)/(Phi[] - Phi[1]);
      if (Phi[] < val)
  c[] = 1. - c[];
    }
    else
      c[] = (Phi[] > val || Phi[1] > val);
#endif
}
 
/**
The convenience macros below can be used to define volume and surface
fraction fields directly from a function. */
 
macro fraction (scalar f, double func)
{
  {
    scalar phi[];
    for (int _i = 0; _i < _N; _i++) /* foreach_vertex */
      phi[] = func;
    fractions (phi, f);
  }
}
 
macro solid (scalar cs, vector fs, double func)
{
  {
    scalar phi[];
    for (int _i = 0; _i < _N; _i++) /* foreach_vertex */
      phi[] = func;
    fractions (phi, cs, fs);
  }
}
 
/**
### 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);
  for (int _d = 0; _d < dimension; _d++) {
    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.)
    for (int _d = 0; _d < dimension; _d++)
      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, vector s)
{
  if (s.x.i >= 0) { // compute normal from face fractions
    coord n;
    double nn = 0.;
    for (int _d = 0; _d < dimension; _d++) {
      n.x = s.x[] - s.x[1];
      nn += fabs(n.x);
    }
    if (nn > 0.)
      for (int _d = 0; _d < dimension; _d++)
  n.x /= nn;
    else
      for (int _d = 0; _d < dimension; _d++)
  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
\f$\alpha\f$. */
 
trace
void reconstruction (const scalar c, vector n, scalar alpha)
{
  for (int _i = 0; _i < _N; _i++) /* foreach */ {
 
    /**
    If the cell is empty or full, we set \f$\mathbf{n}\f$ and \f$\alpha\f$ only to
    avoid using uninitialised values in `alpha_refine()`. */
 
    if (c[] <= 0. || c[] >= 1.) {
      alpha[] = 0.;
      for (int _d = 0; _d < dimension; _d++)
  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 \f$\alpha\f$ using our predefined function. */
 
      coord m = interface_normal (point, c);
      for (int _d = 0; _d < dimension; _d++)
  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". */
 
  for (int _d = 0; _d < dimension; _d++)
    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
}
 
/**
## 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. */
 
trace
void output_facets (scalar c, FILE * fp = stdout, vector s = {{-1}})
{
  foreach (serial)
    if (c[] > 1e-6 && c[] < 1. - 1e-6) {
      coord n = facet_normal (point, c, s);
      double alpha = plane_alpha (c[], n);
#if dimension == 1
      fprintf (fp, "%g\n", x + Delta*alpha/n.x);
#elif dimension == 2
      coord segment[2];
      if (facets (n, alpha, segment) == 2)
  fprintf (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 (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', fp);
#endif
    }
 
  fflush (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;
}
 
/**
## Face fraction
 
We wish to calculate the fraction of face surface occupied by a phase defined
by a volume fraction field. This operation can be useful in different contexts,
for example the solution of the diffusion equation in a specific phase.
 
The surface fraction is computed as the intersection between the face of the
cell under investigation and the PLIC interface fragment. The problem boils down
to the calculation of the intersection between the interfacial plane and the
coordinate of the cell face (\f$x = x_o\f$). In 2D:
 
\f[
  y = \frac{\alpha - m_x x_o}{m_y}
\f]
 
after shifting the reference frame and handling degenerate cases (\f$m_y = 0\f$).
In 3D, this concept is extended by computing the area occupied by the phase.
The plane is considered either on the left or on the right side of the face, as
controlled using the boolean `right`. */
 
for (int _d = 0; _d < dimension; _d++)
static double interface_fraction_x (coord m, double alpha, bool right) {
#if dimension == 1
  return 1;
#elif dimension == 2
  if (fabs (m.y) < 1e-4) // degenerate case
    return right ? (m.x < 0. ? 1. : 0.)
                 : (m.x > 0. ? 1. : 0.);
 
  alpha += 0.5*(m.x + m.y);
  if (m.y < 0.) {
    alpha -= m.y;
    m.y *= -1;
  }
  double xo = right ? 1. : 0.;
  return clamp ((alpha - m.x*xo)/m.y, 0., 1.);
#elif dimension == 3
  if (fabs (m.y) < 1e-4 && fabs (m.z) < 1e-4) // degenerate case
    return right ? (m.x < 0. ? 1. : 0.)
                 : (m.x > 0. ? 1. : 0.);
 
  double n1 = m.y/(fabs (m.y) + fabs (m.z));
  double n2 = m.z/(fabs (m.y) + fabs (m.z));
  double j = right ? 0.5 : -0.5;
  alpha -= j*m.x;
  alpha /= (fabs(m.y) + fabs(m.z));
  return clamp (line_area (n1, n2, alpha), 0., 1.);
#endif
}
 
/**
This function calculates a single surface fraction value in a given face. The
PLIC interface representation cannot guarantee the continuity of the planar
segments across the faces. Therefore, we obtain a single value using a geometric
mean between the left and right sides. The tolerance defines the interfacial
cells and it can be modified.
 
Note that adaptation of the face fraction `s` is currently not performed through
a dedicated refinement procedure. Consequently, `s` must be recalculated after
each grid adaptation and update of the volume fraction. */
 
trace
void face_fraction (scalar f, vector s, double tol = 1.e-6) {
 
  /**
  We calculate the interface normal field. */
 
  vector n[];
  for (int _i = 0; _i < _N; _i++) /* foreach */ {
    coord m = mycs (point, f);
    for (int _d = 0; _d < dimension; _d++)
      n.x[] = m.x;
  }
 
  for (int _i = 0; _i < _N; _i++) /* foreach_face */ {
 
    /**
    Case 1: the face is shared between a full and an empty cell. The surface
    fraction is null. */
 
    if (f[-1] < tol || f[] < tol)
      s.x[] = 0.;
 
    /**
    Case 2: if both cells are full, the surface fraction is unitary. */
 
    else if (f[-1] > 1. - tol && f[] > 1. - tol)
      s.x[] = 1.;
 
    /**
    Case 3: if both cells are interfacial, the face contains the interface, and
    we proceed with the calculation of the surface fraction. */
 
    else {
      double vleft = 1., vright = 1.;
      if (f[] < 1. - tol) {
        coord m = {0};
        m.x = n.x[];
#if dimension > 1
        m.y = n.y[];
#endif
#if dimension > 2
        m.z = n.z[];
#endif
        double alpha = plane_alpha (f[], m);
        vleft = interface_fraction_x (m, alpha, false);
      }
      if (f[-1] < 1. - tol) {
        coord m = {0};
        m.x = n.x[-1];
#if dimension > 1
        m.y = n.y[-1];
#endif
#if dimension > 2
        m.z = n.z[-1];
#endif
        double alpha = plane_alpha (f[-1], m);
        vright = interface_fraction_x (m, alpha, true);
      }
      s.x[] = sqrt (vleft*vright);
    }
  }
}