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Package hex (in hex.i) - 3D transport equation solver

Index of documented functions or symbols:

DOCUMENT nlist = bi_dir(tracker, mesh, rays, slimits, c, s)

  Perform hexX_track and track_reduce on a ray that enters
  the problem at the given point on the ray.  This requires
  tracking the ray in both directions from the given point,
  hence this function name indicating bi-directional tracking.
  This is unnecessary when the entry point search was over
  the problem boundary, or when the SLIMITS for the rays
  always lie in one direction relative to the starting point.

  TRACKER is the function used to track the rays, normally
    one of hex5_track, hex_24f_track, or hex24b_track.
  MESH is the problem mesh returned by hex_mesh or hydra_mesh;
    it should be generated using the entry option that finds
    the cell containing the given point on the ray.
  RAYS is the 3-by-nrays-by-2 array of rays, as for hex5_track
  SLIMITS is nil or the ray tracking limits as for track_reduce
  C, S, together with NLIST are the output arrays, as for
    track_reduce

DOCUMENT conv3_rays(rays)

  convert [p,q] representation to or from best_rays representation.
  If the first dimension of RAYS is 3, returns 5-by-raydims array
  of best_rays; if first dimension of RAYS is 5, returns 3-by-raydims-
  by-2 [p,q] for use with hex5_track.

DOCUMENT nlist= cs_adjust(nlist, c, s, ireg)

  adjust NLIST, C, S returned from track_reduce to remove transits
  of cells for which IREG == 0.  Can be called before or after
  c_adjust, depending on layout of IREG.

DOCUMENT c_adjust, c, mesh
      or c_adjust, c, mesh, 1
      or c= c_adjust(c, mesh, how)

  adjust the cell number array C returned by track_reduce to
  allow for a different layout of cell arrays than the one assumed
  by the tracking routines.  Two HOW values are currently
  supported: 0 (or nil) if the cell arrays are the same shape as
  the nodal arrays, but the non-existent cell is at the end of
  each row rather than at the beginning.  And 1 if the cell arrays
  are smaller by one along each dimension than the nodal arrays.

  If you call c_adjust as a subroutine, the input C array
  is modified; if you call it as a function, the input C is
  unchanged and the new values returned.

DOCUMENT c= hex5_track(mesh, rays, s)
         c= hex24f_track(mesh, rays, s)
         c= hex24b_track(mesh, rays, s)

  track 3 x Nrays x 2 RAYS through the 3D MESH.  RAYS(,,1) are
  points on the rays, while RAYS(,,2) are normalized ray directions.

  The c return value and the S parameter are a long and double
  array respectively, with number of elements equal to the total
  number of intersections of all the RAYS with faces of the MESH,
  plus one for any RAY which misses MESH entirely.  The values of
  c are:
    [#hits,cell1,cell2,cell3,..., #hits,cell1,cell2,cell3,..., ...]
  where each #hits is followed by the list of cell indices (assuming
  i=1, j=1, and k=1 are present but meaningless in cell arrays --
  that is, assuming zone centered arrays have the same dimensions
  as XYZ rather than one less in each direction).  Rays which miss
  the mesh entirely have #hits=1, all others have #hits>=2 since they
  must exit.  #hits<0 means a ray reentered the mesh for abs(#hits)
  more face crossings, but this currently cannot happen.  The values
  of S correspond to c:
    [s0,s1,s2,s3,..., s0,s1,s2,s3,..., ...]
  which are the distances along the ray measured from RAYS(,,1) in
  the direction of RAYS(,,2) where the ray pierces a cell face.  For
  rays which miss the mesh, the value of s0 is a diagnostic telling
  why they missed (see compiled code).

  Function hex5_track uses the 5-tet decomposition for hexes,
  which is not unique when the quad faces are non-planar.  You may
  be able to get an idea of this effect by setting hex_triang the
  opposite way and redoing the trace.

  Functions hex24f_track and hex24b_track use the face and body
  centered 24-tet decompositions for hexes.  These are unique;
  however, hex_triang may in rare cases change the trace slightly,
  since the entry search algorithm still involves triangulating
  the surface quads.

DOCUMENT mesh= hex_mesh(xyz, bound, nbnds, &mbnds, nblk, &blks, start)

  create a 3D mesh object from the multiblock mesh parameters
  XYZ   is NBLK 3 x Ni x Nj x Nk coordinate arrays packed together
  BOUND is NBLK 3 x Ni x Nj x Nk face boundary markers packed
  NBNDS is length of MBNDS
  MBNDS is HX_blkbnd describing each internal block boundary face
  NBLK  is number of blocks
  BLKS  is NBLK HX_block objects describing the block structure
  START is 0-origin 6*cell+face index of first boundary face/cell
         or -1-cell to trace from centroid of that cell to point
         p on ray to begin tracking

DOCUMENT mesh= hex_mesh2(xyz, bounds)
  old interface for hex_mesh

  create a 3D mesh object from the 3 x Ni x Nj x Nk coordinate
  array XYZ and the list of 6 BOUNDS:
    BOUNDS(1), BOUNDS(2)  for the i=1,Ni boundaries
    BOUNDS(3), BOUNDS(4)  for the j=1,Nj boundaries
    BOUNDS(5), BOUNDS(6)  for the k=1,Nk boundaries
  The BOUNDS values are:
    1   if this is a problem boundary
    2   if this is a reflecting boundary
    3   if this is a periodic boundary

DOCUMENT start= hex_query(mesh, xyz, bound, mbnds, blks)

  query a mesh created by hex_mesh, returning the arrays
  passed to that function (these are not copies -- be careful
  not to clobber them)
  function return value is the start index

DOCUMENT mesh= hydra_mesh(f)
      or mesh= hydra_mesh(f, ublk, i0, j0, k0, face)
      or mesh= hydra_mesh(f, ublk, i0, j0, k0)

  read a 3D mesh object from the hydra PDB/Silo file F.

  Note that the boundary arrays are adjusted to the hex convention
  that cells with i=1, j=1, k=1 are missing, rather than the hydra
  convention that i=imax, j=jmax, k=kmax are missing.

  In the first form, the ray entry search will start on the
  first open boundary face in the mesh.  If the actual problem
  boundary is not convex, you need to identify a surface of
  constant i, j, or k in the problem which is convex, and which
  all the rays you intend to trace intersect.
  UBLK is the user block number (starting from 0),
  I0, J0, K0 are the (1-origin) logical coordinates of a
    hydra *cell*.  Note that unlike hex cells, the hydra
    cell bounded by nodes (1,1,1) and (2,2,2) is numbered (1,1,1).
    (Hex numbers it (2,2,2).)
  FACE is the face number on cell (I0,J0,K0) which you want a
    ray to enter.  0 means the -I face, 1 the +I face, 2 the -J
    face, 3 the +J face, 4 the -K face, and 5 the +K face.
    As you step from this cell to its neighbors, then to their
    neighbors, and so on, this face must trace out a convex
    surface for the ray entry search.  Rays not intersecting
    this surface will not enter the problem; the ray trace
    will begin at this surface, not at -infinity.

  If FACE==-1 or is omitted (as in the third form), then the
  given points on the rays are assumed to lie inside the mesh,
  and a pseudo ray from the centroid of cell (I0, J0, K0) will be
  tracked to the given point on each ray; the ray will be launched
  into the cell containing that point.

DOCUMENT hydra_start, mesh, start

  change the starting cell of the hydra MESH (returned by hydra_mesh)
  to START.  If called as a function, returns old start value.

DOCUMENT make_sphere(radius, [imax,jmax,kmax],
                     [phi1, phi2], [theta1, theta2])

  return a mesh (see hex_mesh) representing the given section
  of the sphere of given RADIUS.  IMAX, JMAX, and KMAX are the
  number of nodes (cells+1) in the radial, longitude (phi), and
  colatitude (theta) directions, respectively.  Note that for
  a right handed coordinate system, phi1theta2.

DOCUMENT rays= pic3_rays(xpict, ypict, ray)
      or rays= pic3_rays(xpict, ypict, ray, q_up)

  Like picture_rays, but returns rays in the [p,q] representation
  appropriate for hex5_track.

  (XPICT,YPICT) are 2D arrays of pixel corners in the image plane;
  RAY is the central ray (0,0) in (XPICT,YPICT) coordinates, given
  in [p,q] representation (i.e. RAY is a 3-by-2 array).  The
  optional Q_UP is a 3-vector specifying the orientation of the
  y-axis in the picture plane (see theta_up, phi_up in picture_rays
  for a description of default orientation).  Q_UP must not be
  parallel to RAY(,2).

DOCUMENT c= reg_track(x, y, z, rays, s)

  track RAYS through regular mesh defined by the 1D coordinate
  arrays X, Y, and Z.  Return values S and C are as for
  hex5_track, where the mesh is numberof(X) by numberof(Y) by
  numberof(Z).

DOCUMENT nlist = track_combine(nm,cm,sm, np,cp,sp, c, s)

  combine two track_reduce results NM,CM,SM, and NP,CP,SP,
  which represent the first and second halves of a set of
  rays.  See bi_dir for a typical application.  The returned
  NLIST is NM+NP, or NM+NP-1 for those rays where the
  final CM is identical to the initial CP.

  C, S, together with NLIST are the output arrays, as for
  track_reduce.

DOCUMENT result= track_integ(nlist, transp, selfem, last)

  integrates a transport equation by doing the sums:

     transparency(i) = transparency(i-1) * TRANSP(i)
     emissivity(i) = emissivity(i-1) * TRANSP(i) + SELFEM(i)

  returning only the final values transparency(n) and emissivity(n).
  The NLIST is a list of n values, so that many transport integrals
  can be performed simultaneously; sum(NLIST) = numberof(TRANSP) =
  numberof(SELFEM).  The result is 2-by-dimsof(NLIST).

  If TRANSP is nil, result is dimsof(NLIST) sums of SELFEM.
  If SELFEM is nil, result is dimsof(NLIST) products of TRANSP.

  TRANSP and SELFEM may by 2D to do multigroup integrations
  simultaneously.  By default, the group dimension is first, but
  if LAST is non-nil and non-zero, the group dimension is second.
  In either case, the result will be ngroup-by-2-by-dimsof(NLIST).

  track_solve is the higher-level interface.

DOCUMENT nlist= track_reduce(c, s)
      or nlist= track_reduce(c, s, rays, slimits)

  compresses the C and S returns from the tracking routines (see
  hex5_track) to the following form:
    [cell1,cell2,cell3,..., cell1,cell2,cell3,..., ...]
    [s1-s0,s2-s1,s3-s2,..., s1-s0,s2-s1,s3-s2,..., ...]
  returning nlist as
    [#hits, #hits, ...]

  In this form, any negative #hits are combined with the preceding
  positive values, and #hits=1 (indicating a miss) appear as #hits=0
  in nlist.  Hence, nlist always has exactly Nrays elements.

  If RAYS is supplied, it is used to force the dimensions of the
  returned nlist to match the dimensions of RAYS (the value of RAYS
  is never used).  The RAYS argument need not have the trailing 2
  dimension, so if you specified RAYS as [P,Q] if the call to
  hex5_track, you can use just P or Q as the RAYS argument to
  track_reduce.

  If SLIMITS is supplied, it should be [smin,smax] or [smin,smax]-
  by-dimsof(nlist) in order to reject input S values outside the
  specified limits.  The C list will be culled appropriately, and
  the first and last returned ds values adjusted.

  With a non-zero flip= keyword, the order of the elements of
  C and S within each group of #hits is reversed, so that a
  subsequent track_solve will track the ray backwards.  If you
  use this, both the ray direction input to the tracking routine
  and any SLIMITS argument here should refer to the reverse of
  the ray you intend to track.

DOCUMENT result= track_solve(nlist, c, s, akap, ekap, last)

  integrates a transport equation for NLIST, C, and S returned
  by track_reduce (and optionally c_adjust).  The RAYS argument
  is used only to set the dimensions of the result.  AKAP and
  EKAP are mesh-sized arrays of opacity and emissivity, respectively.
  They may have an additional group dimension, as well.  The
  units of AKAP are 1/length (where length is the unit of S),
  while EKAP is (spectral) power per unit area (length^2), where
  the power is what ever units you want the result in.  The
  emission per unit volume of material is EKAP*AKAP; an optically
  thick block of material emits EKAP per unit surface.

  The NLIST is a list of n values, so that many transport integrals
  can be performed simultaneously; sum(NLIST) = numberof(AKAP) =
  numberof(EKAP).  The result is 2-by-dimsof(NLIST), where the
  first element of the first index is the transmission fraction
  through the entire ray path, and the second element of the
  result is the self-emission along the ray, which has the same
  units as EKAP.

  If EKAP is nil, result is dimsof(NLIST) -- exactly the same as
  the transparency (1st element of result) when both EKAP and AKAP
  are specified.

  If AKAP is nil, result is dimsof(NLIST).  In this case, EKAP
  must have units of emission per unit volume instead of per unit
  area; the result will be the sum of EKAP*S along each ray.

  AKAP and EKAP may by 2D to do multigroup integrations
  simultaneously.  By default, the group dimension is first, but
  if LAST is non-nil and non-zero, the group dimension is last.
  In either case, the result will be ngroup-by-2-by-dimsof(NLIST).

  To use in conjuction with hex5_track, one might do this:

     c= hex5_track(mesh, rays, s);
     nlist= track_reduce(c, s, rays);
     c_adjust, c, mesh;  // if necessary
     result= track_solve(nlist, c, s, akap, ekap);

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