functions in hex.i -

             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  

             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.  

             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.  

             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.  

 hex24b_track  
 

 hex24f_track  
 

             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.  

             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  

             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  

             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  

             old_flag= hex_startflag(new_flag)  
 
     possibly set flag to NEW_FLAG, always return OLD_FLAG, where  
     flag value is 0 (default) to begin search for new entry point  
     at previous entry point, 1 to begin search for new entry point  
     from mesh start face for every ray.  Any other value of NEW_FLAG  
     returns OLD_FLAG without changing it.  

             old_flag= hex_triang(new_flag)  
 
     possibly set flag to NEW_FLAG, always return OLD_FLAG, where  
     flag value is 0 for default mesh triangulation, 1 for opposite  
     triangulation, and 2 on input to signal not to change the  
     current value.  The triangulation value can affect the result  
     of hex5_track if the quad faces of the mesh are not planar.  

 HX_blkbnd  
 
struct HX_blkbnd {  /* must match hex.h */  
  long block;  
  long cell;  
  int orient;  
}  

 HX_block  
 
struct HX_block {   /* must match hex.h */  
  long stride(3);  
  long length(3);  
  long first;  
  long final;  
}  

             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.  

             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.  

             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.  

             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).  

             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).  

             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.  

             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.  

             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.  

             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);