# Embedded boundaries on adaptive trees

This file defines the restriction/prolongation functions which are necessary to implement embedded boundaries on adaptive meshes.

## Volume fraction field cs

For the embedded fraction field cs, the function below is modelled closely on the volume fraction refinement function fraction_refine().

static void embed_fraction_refine (Point point, scalar cs)
{
double cc = cs[];

If the cell is empty or full, simple injection from the coarse cell value is used.

  if (cc <= 0. || cc >= 1.) {
foreach_child()
cs[] = cc;
}
else {

If the cell contains the embedded boundary, we reconstruct the boundary using VOF linear reconstruction and a normal estimated from the surface fractions.

    coord n = facet_normal (point, cs, fs);
double alpha = plane_alpha (cc, n);

foreach_child() {
static const coord a = {0.,0.,0.}, b = {.5,.5,.5};
coord nc;
foreach_dimension()
nc.x = child.x*n.x;
cs[] = rectangle_fraction (nc, alpha, a, b);
}
}
}

## Surface fractions field fs

The embedded surface fractions fs are reconstructed using this function.

foreach_dimension()
static void embed_face_fraction_refine_x (Point point, scalar s)
{
vector fs = s.v;

If the cell is empty or full, simple injection from the coarse cell value is used.

  if (cs[] <= 0. || cs[] >= 1.) {

We need to make sure that the fine cells face fractions match those of their neighbours.

    for (int j = 0; j <= 1; j++)
for (int k = 0; k <= 1; k++)
fine(fs.x,1,j,k) = cs[];
for (int i = 0; i <= 1; i++)
if (!is_refined(neighbor(2*i-1)) && neighbor(2*i-1).neighbors &&
(is_local(cell) || is_local(neighbor(2*i-1))))
for (int j = 0; j <= 1; j++)
for (int k = 0; k <= 1; k++)
fine(fs.x,2*i,j,k) = fs.x[i];
}
else {

If the cell contains the embedded boundary, we reconstruct the boundary using VOF linear reconstruction and a normal estimated from the surface fractions.

    coord n = facet_normal (point, cs, fs);
double alpha = plane_alpha (cs[], n);

We need to reconstruct the face fractions fs for the fine cells.

For the fine face fractions contained within the coarse cell, we compute the intersections directly using the VOF reconstruction.

#if dimension == 2

In 2D, we obtain the face fractions by taking into account the orientation of the normal.

    if (2.*fabs(alpha) < fabs(n.y)) {
double yc = alpha/n.y;
int i = yc > 0.;
fine(fs.x,1,1 - i) = n.y < 0. ? 1. - i : i;
fine(fs.x,1,i) = n.y < 0. ? i - 2.*yc : 1. - i + 2.*yc;
}
else
fine(fs.x,1,0) = fine(fs.x,1,1) = alpha > 0.;

#else // dimension == 3

in 3D, we use the 2D projection of the reconstruction.

    for (int j = 0; j <= 1; j++)
for (int k = 0; k <= 1; k++)
if (!fine(cs,0,j,k) || !fine(cs,1,j,k))
fine(fs.x,1,j,k) = 0.;
else {
static const coord a = {0.,0.,0.}, b = {.5,.5,.5};
coord nc;
nc.x = 0., nc.y = (2.*j - 1.)*n.y, nc.z = (2.*k - 1.)*n.z;
fine(fs.x,1,j,k) = rectangle_fraction (nc, alpha, a, b);
}

#endif // dimension == 3

For the fine face fractions coincident with the faces of the coarse cell, we obtain the intersection position from the coarse cell face fraction.

    for (int i = 0; i <= 1; i++)
if (neighbor(2*i-1).neighbors &&
(is_local(cell) || is_local(neighbor(2*i-1)))) {
if (!is_refined(neighbor(2*i-1))) {
if (fs.x[i] <= 0. || fs.x[i] >= 1.)
for (int j = 0; j <= 1; j++)
for (int k = 0; k <= 1; k++)
fine(fs.x,2*i,j,k) = fs.x[i];
else {
#if dimension == 2

In 2D the orientation is obtained by looking at the values of face fractions in the transverse direction.

	    double a = fs.y[0,1] <= 0. || fs.y[2*i-1,1] <= 0. ||
fs.y[] >= 1. || fs.y[2*i-1] >= 1.;
if ((2.*a - 1)*(fs.x[i] - 0.5) > 0.) {
fine(fs.x,2*i,0) = a;
fine(fs.x,2*i,1) = 2.*fs.x[i] - a;
}
else {
fine(fs.x,2*i,0) = 2.*fs.x[i] + a - 1.;
fine(fs.x,2*i,1) = 1. - a;
}

#else  // dimension == 3

In 3D we reconstruct the face fraction from the projection of the cell interface reconstruction, as above.

	    for (int j = 0; j <= 1; j++)
for (int k = 0; k <= 1; k++) {
static const coord a = {0.,0.,0.}, b = {.5,.5,.5};
coord nc;
nc.x = 0., nc.y = (2.*j - 1.)*n.y, nc.z = (2.*k - 1.)*n.z;
fine(fs.x,2*i,j,k) =
rectangle_fraction (nc, alpha - n.x*(2.*i - 1.)/2., a, b);
}

#endif // dimension == 3
}
}

The face fractions of empty children cells must be zero.

	for (int j = 0; j <= 1; j++)
#if dimension > 2
for (int k = 0; k <= 1; k++)
#endif
if (fine(fs.x,2*i,j,k) && !fine(cs,i,j,k))
fine(fs.x,2*i,j,k) = 0.;
}
}
}

## Restriction of cell-centered fields

We now define restriction and prolongation functions for cell-centered fields. The goal is to define second-order operators which do not use any values from cells entirely contained within the embedded boundary (for which cs = 0).

When restricting it is unfortunately not always possible to obtain a second-order interpolation. This happens when the parent cell does not contain enough child cells not entirely contained within the embedded boundary. In these cases, some external information (i.e. a boundary gradient condition) is required to be able to maintain second-order accuracy. This information can be passed by defining the embed_gradient() function of the field being restricted.

attribute {
void (* embed_gradient) (Point, scalar, coord *);
}

static inline void restriction_embed_linear (Point point, scalar s)
{
// 0 children
if (!cs[]) {
s[] = 0.;
return;
}

We first try to interpolate “diagonally”. If enough child cells are defined (i.e. have non-zero embedded fractions), we return the corresponding value.

  double val = 0., nv = 0.;
for (int i = 0; i <= 1; i++)
#if dimension > 2
for (int j = 0; j <= 1; j++)
#endif
if (fine(cs,0,i,j) && fine(cs,1,!i,!j))
val += (fine(s,0,i,j) + fine(s,1,!i,!j))/2., nv++;
if (nv > 0.) {
s[] = val/nv;
return;
}

Otherwise, we use the average of the child cells which are defined (there is at least one).

  coord p = {0.,0.,0.};
foreach_child()
if (cs[])
p.x += x, p.y += y, p.z += z, val += s[], nv++;
assert (nv > 0.);
s[] = val/nv;

If the gradient is defined and if the variable is not using homogeneous boundary conditions, we improve the interpolation using this information.

  if (s.embed_gradient && s.boundary != s.boundary_homogeneous) {
coord o = {x,y,z}, g;
foreach_dimension()
s[] += (o.x - p.x/nv)*g.x;
}
}

## Refinement/prolongation of cell-centered fields

For refinement, we use either bilinear interpolation, if the required four coarse cell values are defined or trilinear interpolation if only three coarse cell values are defined. If less that three coarse cell values are defined (“pathological cases” below), we try to estimate gradients in each direction and add the corresponding correction.

static inline void refine_embed_linear (Point point, scalar s)
{
foreach_child() {
if (!cs[])
s[] = 0.;
else {
assert (coarse(cs));
int i = (child.x + 1)/2, j = (child.y + 1)/2;
#if dimension == 2
if (coarse(fs.x,i) && coarse(fs.y,0,j) &&
(coarse(cs) == 1. || coarse(cs,child.x) == 1. ||
coarse(cs,0,child.y) == 1. || coarse(cs,child.x,child.y) == 1.)) {
assert (coarse(cs,child.x) && coarse(cs,0,child.y));
if (coarse(fs.x,i,child.y) && coarse(fs.y,child.x,j)) {
// bilinear interpolation
assert (coarse(cs,child.x,child.y));
s[] = (9.*coarse(s) +
3.*(coarse(s,child.x) + coarse(s,0,child.y)) +
coarse(s,child.x,child.y))/16.;
}
else
// triangular interpolation
s[] = (2.*coarse(s) + coarse(s,child.x) + coarse(s,0,child.y))/4.;
}
else if (coarse(cs,child.x,child.y) &&
((coarse(fs.x,i) && coarse(fs.y,child.x,j)) ||
(coarse(fs.y,0,j) && coarse(fs.x,i,child.y)))) {
// diagonal interpolation
s[] = (3.*coarse(s) + coarse(s,child.x,child.y))/4.;
}
#else // dimension == 3
int k = (child.z + 1)/2;
if (coarse(fs.x,i) > 0.25 && coarse(fs.y,0,j) > 0.25 &&
coarse(fs.z,0,0,k) > 0.25 &&
(coarse(cs) == 1. || coarse(cs,child.x) == 1. ||
coarse(cs,0,child.y) == 1. || coarse(cs,child.x,child.y) == 1. ||
coarse(cs,0,0,child.z) == 1. || coarse(cs,child.x,0,child.z) == 1. ||
coarse(cs,0,child.y,child.z) == 1. ||
coarse(cs,child.x,child.y,child.z) == 1.)) {
assert (coarse(cs,child.x) && coarse(cs,0,child.y) &&
coarse(cs,0,0,child.z));
if (coarse(fs.x,i,child.y) && coarse(fs.y,child.x,j) &&
coarse(fs.z,child.x,child.y,k) &&
coarse(fs.z,child.x,0,k) && coarse(fs.z,0,child.y,k)) {
assert (coarse(cs,child.x,child.y) && coarse(cs,child.x,0,child.z) &&
coarse(cs,0,child.y,child.z) &&
coarse(cs,child.x,child.y,child.z));
// bilinear interpolation
s[] = (27.*coarse(s) +
9.*(coarse(s,child.x) + coarse(s,0,child.y) +
coarse(s,0,0,child.z)) +
3.*(coarse(s,child.x,child.y) + coarse(s,child.x,0,child.z) +
coarse(s,0,child.y,child.z)) +
coarse(s,child.x,child.y,child.z))/64.;
}
else
// tetrahedral interpolation
s[] = (coarse(s) + coarse(s,child.x) + coarse(s,0,child.y) +
coarse(s,0,0,child.z))/4.;
}
else if (coarse(cs,child.x,child.y,child.z) &&
((coarse(fs.z,child.x,child.y,k) &&
((coarse(fs.x,i) && coarse(fs.y,child.x,j)) ||
(coarse(fs.y,0,j) && coarse(fs.x,i,child.y))))
||
(coarse(fs.z,0,0,k) &&
((coarse(fs.x,i,0,child.z) && coarse(fs.y,child.x,j,child.z)) ||
(coarse(fs.y,0,j,child.z) && coarse(fs.x,i,child.y,child.z))))
||
(coarse(fs.z,child.x,0,k) &&
coarse(fs.x,i) && coarse(fs.y,child.x,j,child.z))
||
(coarse(fs.z,0,child.y,k) &&
coarse(fs.y,0,j) && coarse(fs.x,i,child.y,child.z))
))
// diagonal interpolation
s[] = (3.*coarse(s) + coarse(s,child.x,child.y,child.z))/4.;
#endif // dimension == 3
else {
// Pathological cases, use 1D gradients.
s[] = coarse(s);
foreach_dimension() {
if (coarse(fs.x,(child.x + 1)/2) && coarse(cs,child.x))
s[] += (coarse(s,child.x) - coarse(s))/4.;
else if (coarse(fs.x,(- child.x + 1)/2) && coarse(cs,- child.x))
s[] -= (coarse(s,- child.x) - coarse(s))/4.;
}
}
}
}
}

## Refinement/prolongation of face-centered velocity

This function is modelled on refine_face_x() and is typically used to refine the values of the face-centered velocity field uf. It uses linear interpolation, taking into account the weighting by the embedded fractions fs.

foreach_dimension()
void refine_embed_face_x (Point point, scalar s)
{
vector v = s.v;
for (int i = 0; i <= 1; i++)
if (neighbor(2*i - 1).neighbors &&
(is_local(cell) || is_local(neighbor(2*i - 1)))) {
double g1 = fs.x[i] >= 1. && fs.x[i,+1] && fs.x[i,-1] ?
(v.x[i,+1]/fs.x[i,+1] - v.x[i,-1]/fs.x[i,-1])/8. : 0.;
double g2 = fs.x[i] >= 1. && fs.x[i,0,+1] && fs.x[i,0,-1] ?
(v.x[i,0,+1]/fs.x[i,0,+1] - v.x[i,0,-1]/fs.x[i,0,-1])/8. : 0.;
for (int j = 0; j <= 1; j++)
for (int k = 0; k <= 1; k++)
fine(v.x,2*i,j,k) = fs.x[i] ?
fine(fs.x,2*i,j,k)*(v.x[i]/fs.x[i] +
(2*j - 1)*g1 + (2*k - 1)*g2) : 0.;
}
if (is_local(cell)) {
double g1 = (fs.x[0,+1] + fs.x[1,+1]) && (fs.x[0,-1] + fs.x[1,-1]) ?
((v.x[0,+1] + v.x[1,+1])/(fs.x[0,+1] + fs.x[1,+1]) -
(v.x[0,-1] + v.x[1,-1])/(fs.x[0,-1] + fs.x[1,-1]))/8. : 0.;
double g2 = (fs.x[1,0,+1] + fs.x[0,0,+1]) && (fs.x[1,0,-1] + fs.x[0,0,-1]) ?
((v.x[0,0,+1] + v.x[1,0,+1])/(fs.x[1,0,+1] + fs.x[0,0,+1]) -
(v.x[0,0,-1] + v.x[1,0,-1])/(fs.x[1,0,-1] + fs.x[0,0,-1]))/8. : 0.;
for (int j = 0; j <= 1; j++)
for (int k = 0; k <= 1; k++)
fine(v.x,1,j,k) = fs.x[] + fs.x ?
fine(fs.x,1,j,k)*((v.x[] + v.x)/(fs.x[] + fs.x) +
(2*j - 1)*g1 + (2*k - 1)*g2) : 0.;
}
}