xref: /petsc/src/dm/impls/plex/plexcreate.c (revision 1eb70e550fb8c44dfe55fc2653344863e6ef3627)

#define PETSCDM_DLL
#include <petsc/private/dmpleximpl.h>    /*I   "petscdmplex.h"   I*/
#include <petsc/private/hashseti.h>          /*I   "petscdmplex.h"   I*/
#include <petscsf.h>

PetscLogEvent DMPLEX_CreateFromFile, DMPLEX_BuildFromCellList, DMPLEX_BuildCoordinatesFromCellList;

/*@
  DMPlexCreateDoublet - Creates a mesh of two cells of the specified type, optionally with later refinement.

  Collective

  Input Parameters:
+ comm - The communicator for the DM object
. dim - The spatial dimension
. simplex - Flag for simplicial cells, otherwise they are tensor product cells
. interpolate - Flag to create intermediate mesh pieces (edges, faces)
- refinementLimit - A nonzero number indicates the largest admissible volume for a refined cell

  Output Parameter:
. dm - The DM object

  Level: beginner

.seealso: DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateDoublet(MPI_Comm comm, PetscInt dim, PetscBool simplex, PetscBool interpolate, PetscReal refinementLimit, DM *newdm)
{
  DM             dm;
  PetscMPIInt    rank;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMCreate(comm, &dm);CHKERRQ(ierr);
  ierr = DMSetType(dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(dm, dim);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(comm, &rank);CHKERRMPI(ierr);
  switch (dim) {
  case 2:
    if (simplex) {ierr = PetscObjectSetName((PetscObject) dm, "triangular");CHKERRQ(ierr);}
    else         {ierr = PetscObjectSetName((PetscObject) dm, "quadrilateral");CHKERRQ(ierr);}
    break;
  case 3:
    if (simplex) {ierr = PetscObjectSetName((PetscObject) dm, "tetrahedral");CHKERRQ(ierr);}
    else         {ierr = PetscObjectSetName((PetscObject) dm, "hexahedral");CHKERRQ(ierr);}
    break;
  default:
    SETERRQ1(comm, PETSC_ERR_ARG_OUTOFRANGE, "Cannot make meshes for dimension %D", dim);
  }
  if (rank) {
    PetscInt numPoints[2] = {0, 0};
    ierr = DMPlexCreateFromDAG(dm, 1, numPoints, NULL, NULL, NULL, NULL);CHKERRQ(ierr);
  } else {
    switch (dim) {
    case 2:
      if (simplex) {
        PetscInt    numPoints[2]        = {4, 2};
        PetscInt    coneSize[6]         = {3, 3, 0, 0, 0, 0};
        PetscInt    cones[6]            = {2, 3, 4,  5, 4, 3};
        PetscInt    coneOrientations[6] = {0, 0, 0,  0, 0, 0};
        PetscScalar vertexCoords[8]     = {-0.5, 0.5, 0.0, 0.0, 0.0, 1.0, 0.5, 0.5};

        ierr = DMPlexCreateFromDAG(dm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
      } else {
        PetscInt    numPoints[2]        = {6, 2};
        PetscInt    coneSize[8]         = {4, 4, 0, 0, 0, 0, 0, 0};
        PetscInt    cones[8]            = {2, 3, 4, 5,  3, 6, 7, 4};
        PetscInt    coneOrientations[8] = {0, 0, 0, 0,  0, 0, 0, 0};
        PetscScalar vertexCoords[12]    = {-1.0, -0.5,  0.0, -0.5,  0.0, 0.5,  -1.0, 0.5,  1.0, -0.5,  1.0, 0.5};

        ierr = DMPlexCreateFromDAG(dm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
      }
      break;
    case 3:
      if (simplex) {
        PetscInt    numPoints[2]        = {5, 2};
        PetscInt    coneSize[7]         = {4, 4, 0, 0, 0, 0, 0};
        PetscInt    cones[8]            = {4, 3, 5, 2,  5, 3, 4, 6};
        PetscInt    coneOrientations[8] = {0, 0, 0, 0,  0, 0, 0, 0};
        PetscScalar vertexCoords[15]    = {-1.0, 0.0, 0.0,  0.0, -1.0, 0.0,  0.0, 0.0, 1.0,  0.0, 1.0, 0.0,  1.0, 0.0, 0.0};

        ierr = DMPlexCreateFromDAG(dm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
      } else {
        PetscInt    numPoints[2]         = {12, 2};
        PetscInt    coneSize[14]         = {8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0};
        PetscInt    cones[16]            = {2, 3, 4, 5, 6, 7, 8, 9,  5, 4, 10, 11, 7, 12, 13, 8};
        PetscInt    coneOrientations[16] = {0, 0, 0, 0, 0, 0, 0, 0,  0, 0,  0,  0, 0,  0,  0, 0};
        PetscScalar vertexCoords[36]     = {-1.0, -0.5, -0.5,  -1.0,  0.5, -0.5,  0.0,  0.5, -0.5,   0.0, -0.5, -0.5,
                                            -1.0, -0.5,  0.5,   0.0, -0.5,  0.5,  0.0,  0.5,  0.5,  -1.0,  0.5,  0.5,
                                             1.0,  0.5, -0.5,   1.0, -0.5, -0.5,  1.0, -0.5,  0.5,   1.0,  0.5,  0.5};

        ierr = DMPlexCreateFromDAG(dm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
      }
      break;
    default:
      SETERRQ1(comm, PETSC_ERR_ARG_OUTOFRANGE, "Cannot make meshes for dimension %D", dim);
    }
  }
  *newdm = dm;
  if (refinementLimit > 0.0) {
    DM rdm;
    const char *name;

    ierr = DMPlexSetRefinementUniform(*newdm, PETSC_FALSE);CHKERRQ(ierr);
    ierr = DMPlexSetRefinementLimit(*newdm, refinementLimit);CHKERRQ(ierr);
    ierr = DMRefine(*newdm, comm, &rdm);CHKERRQ(ierr);
    ierr = PetscObjectGetName((PetscObject) *newdm, &name);CHKERRQ(ierr);
    ierr = PetscObjectSetName((PetscObject)    rdm,  name);CHKERRQ(ierr);
    ierr = DMDestroy(newdm);CHKERRQ(ierr);
    *newdm = rdm;
  }
  if (interpolate) {
    DM idm;

    ierr = DMPlexInterpolate(*newdm, &idm);CHKERRQ(ierr);
    ierr = DMDestroy(newdm);CHKERRQ(ierr);
    *newdm = idm;
  }
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateSquareBoundary - Creates a 1D mesh the is the boundary of a square lattice.

  Collective

  Input Parameters:
+ comm  - The communicator for the DM object
. lower - The lower left corner coordinates
. upper - The upper right corner coordinates
- edges - The number of cells in each direction

  Output Parameter:
. dm  - The DM object

  Note: Here is the numbering returned for 2 cells in each direction:
$ 18--5-17--4--16
$  |     |     |
$  6    10     3
$  |     |     |
$ 19-11-20--9--15
$  |     |     |
$  7     8     2
$  |     |     |
$ 12--0-13--1--14

  Level: beginner

.seealso: DMPlexCreateBoxMesh(), DMPlexCreateCubeBoundary(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateSquareBoundary(DM dm, const PetscReal lower[], const PetscReal upper[], const PetscInt edges[])
{
  const PetscInt numVertices    = (edges[0]+1)*(edges[1]+1);
  const PetscInt numEdges       = edges[0]*(edges[1]+1) + (edges[0]+1)*edges[1];
  PetscInt       markerTop      = 1;
  PetscInt       markerBottom   = 1;
  PetscInt       markerRight    = 1;
  PetscInt       markerLeft     = 1;
  PetscBool      markerSeparate = PETSC_FALSE;
  Vec            coordinates;
  PetscSection   coordSection;
  PetscScalar    *coords;
  PetscInt       coordSize;
  PetscMPIInt    rank;
  PetscInt       v, vx, vy;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscOptionsGetBool(((PetscObject) dm)->options,((PetscObject) dm)->prefix, "-dm_plex_separate_marker", &markerSeparate, NULL);CHKERRQ(ierr);
  if (markerSeparate) {
    markerTop    = 3;
    markerBottom = 1;
    markerRight  = 2;
    markerLeft   = 4;
  }
  ierr = MPI_Comm_rank(PetscObjectComm((PetscObject)dm), &rank);CHKERRMPI(ierr);
  if (!rank) {
    PetscInt e, ex, ey;

    ierr = DMPlexSetChart(dm, 0, numEdges+numVertices);CHKERRQ(ierr);
    for (e = 0; e < numEdges; ++e) {
      ierr = DMPlexSetConeSize(dm, e, 2);CHKERRQ(ierr);
    }
    ierr = DMSetUp(dm);CHKERRQ(ierr); /* Allocate space for cones */
    for (vx = 0; vx <= edges[0]; vx++) {
      for (ey = 0; ey < edges[1]; ey++) {
        PetscInt edge   = vx*edges[1] + ey + edges[0]*(edges[1]+1);
        PetscInt vertex = ey*(edges[0]+1) + vx + numEdges;
        PetscInt cone[2];

        cone[0] = vertex; cone[1] = vertex+edges[0]+1;
        ierr    = DMPlexSetCone(dm, edge, cone);CHKERRQ(ierr);
        if (vx == edges[0]) {
          ierr = DMSetLabelValue(dm, "marker", edge,    markerRight);CHKERRQ(ierr);
          ierr = DMSetLabelValue(dm, "marker", cone[0], markerRight);CHKERRQ(ierr);
          if (ey == edges[1]-1) {
            ierr = DMSetLabelValue(dm, "marker", cone[1], markerRight);CHKERRQ(ierr);
            ierr = DMSetLabelValue(dm, "Face Sets", cone[1], markerRight);CHKERRQ(ierr);
          }
        } else if (vx == 0) {
          ierr = DMSetLabelValue(dm, "marker", edge,    markerLeft);CHKERRQ(ierr);
          ierr = DMSetLabelValue(dm, "marker", cone[0], markerLeft);CHKERRQ(ierr);
          if (ey == edges[1]-1) {
            ierr = DMSetLabelValue(dm, "marker", cone[1], markerLeft);CHKERRQ(ierr);
            ierr = DMSetLabelValue(dm, "Face Sets", cone[1], markerLeft);CHKERRQ(ierr);
          }
        }
      }
    }
    for (vy = 0; vy <= edges[1]; vy++) {
      for (ex = 0; ex < edges[0]; ex++) {
        PetscInt edge   = vy*edges[0]     + ex;
        PetscInt vertex = vy*(edges[0]+1) + ex + numEdges;
        PetscInt cone[2];

        cone[0] = vertex; cone[1] = vertex+1;
        ierr    = DMPlexSetCone(dm, edge, cone);CHKERRQ(ierr);
        if (vy == edges[1]) {
          ierr = DMSetLabelValue(dm, "marker", edge,    markerTop);CHKERRQ(ierr);
          ierr = DMSetLabelValue(dm, "marker", cone[0], markerTop);CHKERRQ(ierr);
          if (ex == edges[0]-1) {
            ierr = DMSetLabelValue(dm, "marker", cone[1], markerTop);CHKERRQ(ierr);
            ierr = DMSetLabelValue(dm, "Face Sets", cone[1], markerTop);CHKERRQ(ierr);
          }
        } else if (vy == 0) {
          ierr = DMSetLabelValue(dm, "marker", edge,    markerBottom);CHKERRQ(ierr);
          ierr = DMSetLabelValue(dm, "marker", cone[0], markerBottom);CHKERRQ(ierr);
          if (ex == edges[0]-1) {
            ierr = DMSetLabelValue(dm, "marker", cone[1], markerBottom);CHKERRQ(ierr);
            ierr = DMSetLabelValue(dm, "Face Sets", cone[1], markerBottom);CHKERRQ(ierr);
          }
        }
      }
    }
  }
  ierr = DMPlexSymmetrize(dm);CHKERRQ(ierr);
  ierr = DMPlexStratify(dm);CHKERRQ(ierr);
  /* Build coordinates */
  ierr = DMSetCoordinateDim(dm, 2);CHKERRQ(ierr);
  ierr = DMGetCoordinateSection(dm, &coordSection);CHKERRQ(ierr);
  ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
  ierr = PetscSectionSetChart(coordSection, numEdges, numEdges + numVertices);CHKERRQ(ierr);
  ierr = PetscSectionSetFieldComponents(coordSection, 0, 2);CHKERRQ(ierr);
  for (v = numEdges; v < numEdges+numVertices; ++v) {
    ierr = PetscSectionSetDof(coordSection, v, 2);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldDof(coordSection, v, 0, 2);CHKERRQ(ierr);
  }
  ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
  ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
  ierr = VecCreate(PETSC_COMM_SELF, &coordinates);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
  ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetBlockSize(coordinates, 2);CHKERRQ(ierr);
  ierr = VecSetType(coordinates,VECSTANDARD);CHKERRQ(ierr);
  ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
  for (vy = 0; vy <= edges[1]; ++vy) {
    for (vx = 0; vx <= edges[0]; ++vx) {
      coords[(vy*(edges[0]+1)+vx)*2+0] = lower[0] + ((upper[0] - lower[0])/edges[0])*vx;
      coords[(vy*(edges[0]+1)+vx)*2+1] = lower[1] + ((upper[1] - lower[1])/edges[1])*vy;
    }
  }
  ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(dm, coordinates);CHKERRQ(ierr);
  ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateCubeBoundary - Creates a 2D mesh that is the boundary of a cubic lattice.

  Collective

  Input Parameters:
+ comm  - The communicator for the DM object
. lower - The lower left front corner coordinates
. upper - The upper right back corner coordinates
- faces - The number of faces in each direction (the same as the number of cells)

  Output Parameter:
. dm  - The DM object

  Level: beginner

.seealso: DMPlexCreateBoxMesh(), DMPlexCreateSquareBoundary(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateCubeBoundary(DM dm, const PetscReal lower[], const PetscReal upper[], const PetscInt faces[])
{
  PetscInt       vertices[3], numVertices;
  PetscInt       numFaces    = 2*faces[0]*faces[1] + 2*faces[1]*faces[2] + 2*faces[0]*faces[2];
  Vec            coordinates;
  PetscSection   coordSection;
  PetscScalar    *coords;
  PetscInt       coordSize;
  PetscMPIInt    rank;
  PetscInt       v, vx, vy, vz;
  PetscInt       voffset, iface=0, cone[4];
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if ((faces[0] < 1) || (faces[1] < 1) || (faces[2] < 1)) SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_SUP, "Must have at least 1 face per side");
  ierr = MPI_Comm_rank(PetscObjectComm((PetscObject)dm), &rank);CHKERRMPI(ierr);
  vertices[0] = faces[0]+1; vertices[1] = faces[1]+1; vertices[2] = faces[2]+1;
  numVertices = vertices[0]*vertices[1]*vertices[2];
  if (!rank) {
    PetscInt f;

    ierr = DMPlexSetChart(dm, 0, numFaces+numVertices);CHKERRQ(ierr);
    for (f = 0; f < numFaces; ++f) {
      ierr = DMPlexSetConeSize(dm, f, 4);CHKERRQ(ierr);
    }
    ierr = DMSetUp(dm);CHKERRQ(ierr); /* Allocate space for cones */

    /* Side 0 (Top) */
    for (vy = 0; vy < faces[1]; vy++) {
      for (vx = 0; vx < faces[0]; vx++) {
        voffset = numFaces + vertices[0]*vertices[1]*(vertices[2]-1) + vy*vertices[0] + vx;
        cone[0] = voffset; cone[1] = voffset+1; cone[2] = voffset+vertices[0]+1; cone[3] = voffset+vertices[0];
        ierr    = DMPlexSetCone(dm, iface, cone);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", iface, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+1, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]+1, 1);CHKERRQ(ierr);
        iface++;
      }
    }

    /* Side 1 (Bottom) */
    for (vy = 0; vy < faces[1]; vy++) {
      for (vx = 0; vx < faces[0]; vx++) {
        voffset = numFaces + vy*(faces[0]+1) + vx;
        cone[0] = voffset+1; cone[1] = voffset; cone[2] = voffset+vertices[0]; cone[3] = voffset+vertices[0]+1;
        ierr    = DMPlexSetCone(dm, iface, cone);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", iface, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+1, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]+1, 1);CHKERRQ(ierr);
        iface++;
      }
    }

    /* Side 2 (Front) */
    for (vz = 0; vz < faces[2]; vz++) {
      for (vx = 0; vx < faces[0]; vx++) {
        voffset = numFaces + vz*vertices[0]*vertices[1] + vx;
        cone[0] = voffset; cone[1] = voffset+1; cone[2] = voffset+vertices[0]*vertices[1]+1; cone[3] = voffset+vertices[0]*vertices[1];
        ierr    = DMPlexSetCone(dm, iface, cone);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", iface, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+1, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]*vertices[1]+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]*vertices[1]+1, 1);CHKERRQ(ierr);
        iface++;
      }
    }

    /* Side 3 (Back) */
    for (vz = 0; vz < faces[2]; vz++) {
      for (vx = 0; vx < faces[0]; vx++) {
        voffset = numFaces + vz*vertices[0]*vertices[1] + vertices[0]*(vertices[1]-1) + vx;
        cone[0] = voffset+vertices[0]*vertices[1]; cone[1] = voffset+vertices[0]*vertices[1]+1;
        cone[2] = voffset+1; cone[3] = voffset;
        ierr    = DMPlexSetCone(dm, iface, cone);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", iface, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+1, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]*vertices[1]+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]*vertices[1]+1, 1);CHKERRQ(ierr);
        iface++;
      }
    }

    /* Side 4 (Left) */
    for (vz = 0; vz < faces[2]; vz++) {
      for (vy = 0; vy < faces[1]; vy++) {
        voffset = numFaces + vz*vertices[0]*vertices[1] + vy*vertices[0];
        cone[0] = voffset; cone[1] = voffset+vertices[0]*vertices[1];
        cone[2] = voffset+vertices[0]*vertices[1]+vertices[0]; cone[3] = voffset+vertices[0];
        ierr    = DMPlexSetCone(dm, iface, cone);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", iface, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[1]+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]*vertices[1]+vertices[0], 1);CHKERRQ(ierr);
        iface++;
      }
    }

    /* Side 5 (Right) */
    for (vz = 0; vz < faces[2]; vz++) {
      for (vy = 0; vy < faces[1]; vy++) {
        voffset = numFaces + vz*vertices[0]*vertices[1] + vy*vertices[0] + faces[0];
        cone[0] = voffset+vertices[0]*vertices[1]; cone[1] = voffset;
        cone[2] = voffset+vertices[0]; cone[3] = voffset+vertices[0]*vertices[1]+vertices[0];
        ierr    = DMPlexSetCone(dm, iface, cone);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", iface, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]*vertices[1]+0, 1);CHKERRQ(ierr);
        ierr    = DMSetLabelValue(dm, "marker", voffset+vertices[0]*vertices[1]+vertices[0], 1);CHKERRQ(ierr);
        iface++;
      }
    }
  }
  ierr = DMPlexSymmetrize(dm);CHKERRQ(ierr);
  ierr = DMPlexStratify(dm);CHKERRQ(ierr);
  /* Build coordinates */
  ierr = DMSetCoordinateDim(dm, 3);CHKERRQ(ierr);
  ierr = DMGetCoordinateSection(dm, &coordSection);CHKERRQ(ierr);
  ierr = PetscSectionSetChart(coordSection, numFaces, numFaces + numVertices);CHKERRQ(ierr);
  for (v = numFaces; v < numFaces+numVertices; ++v) {
    ierr = PetscSectionSetDof(coordSection, v, 3);CHKERRQ(ierr);
  }
  ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
  ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
  ierr = VecCreate(PETSC_COMM_SELF, &coordinates);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
  ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetBlockSize(coordinates, 3);CHKERRQ(ierr);
  ierr = VecSetType(coordinates,VECSTANDARD);CHKERRQ(ierr);
  ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
  for (vz = 0; vz <= faces[2]; ++vz) {
    for (vy = 0; vy <= faces[1]; ++vy) {
      for (vx = 0; vx <= faces[0]; ++vx) {
        coords[((vz*(faces[1]+1)+vy)*(faces[0]+1)+vx)*3+0] = lower[0] + ((upper[0] - lower[0])/faces[0])*vx;
        coords[((vz*(faces[1]+1)+vy)*(faces[0]+1)+vx)*3+1] = lower[1] + ((upper[1] - lower[1])/faces[1])*vy;
        coords[((vz*(faces[1]+1)+vy)*(faces[0]+1)+vx)*3+2] = lower[2] + ((upper[2] - lower[2])/faces[2])*vz;
      }
    }
  }
  ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(dm, coordinates);CHKERRQ(ierr);
  ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode DMPlexCreateLineMesh_Internal(MPI_Comm comm,PetscInt segments,PetscReal lower,PetscReal upper,DMBoundaryType bd,DM *dm)
{
  PetscInt       i,fStart,fEnd,numCells = 0,numVerts = 0;
  PetscInt       numPoints[2],*coneSize,*cones,*coneOrientations;
  PetscScalar    *vertexCoords;
  PetscReal      L,maxCell;
  PetscBool      markerSeparate = PETSC_FALSE;
  PetscInt       markerLeft  = 1, faceMarkerLeft  = 1;
  PetscInt       markerRight = 1, faceMarkerRight = 2;
  PetscBool      wrap = (bd == DM_BOUNDARY_PERIODIC || bd == DM_BOUNDARY_TWIST) ? PETSC_TRUE : PETSC_FALSE;
  PetscMPIInt    rank;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidPointer(dm,4);

  ierr = DMCreate(comm,dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm,DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(*dm,1);CHKERRQ(ierr);
  ierr = DMCreateLabel(*dm,"marker");CHKERRQ(ierr);
  ierr = DMCreateLabel(*dm,"Face Sets");CHKERRQ(ierr);

  ierr = MPI_Comm_rank(comm,&rank);CHKERRMPI(ierr);
  if (!rank) numCells = segments;
  if (!rank) numVerts = segments + (wrap ? 0 : 1);

  numPoints[0] = numVerts ; numPoints[1] = numCells;
  ierr = PetscMalloc4(numCells+numVerts,&coneSize,numCells*2,&cones,numCells+numVerts,&coneOrientations,numVerts,&vertexCoords);CHKERRQ(ierr);
  ierr = PetscArrayzero(coneOrientations,numCells+numVerts);CHKERRQ(ierr);
  for (i = 0; i < numCells; ++i) { coneSize[i] = 2; }
  for (i = 0; i < numVerts; ++i) { coneSize[numCells+i] = 0; }
  for (i = 0; i < numCells; ++i) { cones[2*i] = numCells + i%numVerts; cones[2*i+1] = numCells + (i+1)%numVerts; }
  for (i = 0; i < numVerts; ++i) { vertexCoords[i] = lower + (upper-lower)*((PetscReal)i/(PetscReal)numCells); }
  ierr = DMPlexCreateFromDAG(*dm,1,numPoints,coneSize,cones,coneOrientations,vertexCoords);CHKERRQ(ierr);
  ierr = PetscFree4(coneSize,cones,coneOrientations,vertexCoords);CHKERRQ(ierr);

  ierr = PetscOptionsGetBool(((PetscObject)*dm)->options,((PetscObject)*dm)->prefix,"-dm_plex_separate_marker",&markerSeparate,NULL);CHKERRQ(ierr);
  if (markerSeparate) { markerLeft = faceMarkerLeft; markerRight = faceMarkerRight;}
  if (!wrap && !rank) {
    ierr = DMPlexGetHeightStratum(*dm,1,&fStart,&fEnd);CHKERRQ(ierr);
    ierr = DMSetLabelValue(*dm,"marker",fStart,markerLeft);CHKERRQ(ierr);
    ierr = DMSetLabelValue(*dm,"marker",fEnd-1,markerRight);CHKERRQ(ierr);
    ierr = DMSetLabelValue(*dm,"Face Sets",fStart,faceMarkerLeft);CHKERRQ(ierr);
    ierr = DMSetLabelValue(*dm,"Face Sets",fEnd-1,faceMarkerRight);CHKERRQ(ierr);
  }
  if (wrap) {
    L       = upper - lower;
    maxCell = (PetscReal)1.1*(L/(PetscReal)PetscMax(1,segments));
    ierr = DMSetPeriodicity(*dm,PETSC_TRUE,&maxCell,&L,&bd);CHKERRQ(ierr);
  }
  ierr = DMPlexSetRefinementUniform(*dm, PETSC_TRUE);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode DMPlexCreateBoxMesh_Simplex_Internal(MPI_Comm comm, PetscInt dim, const PetscInt faces[], const PetscReal lower[], const PetscReal upper[], const DMBoundaryType periodicity[], PetscBool interpolate, DM *dm)
{
  DM             boundary;
  PetscInt       i;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidPointer(dm, 4);
  for (i = 0; i < dim; ++i) if (periodicity[i] != DM_BOUNDARY_NONE) SETERRQ(comm, PETSC_ERR_SUP, "Periodicity is not supported for simplex meshes");
  ierr = DMCreate(comm, &boundary);CHKERRQ(ierr);
  PetscValidLogicalCollectiveInt(boundary,dim,2);
  ierr = DMSetType(boundary, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(boundary, dim-1);CHKERRQ(ierr);
  ierr = DMSetCoordinateDim(boundary, dim);CHKERRQ(ierr);
  switch (dim) {
  case 2: ierr = DMPlexCreateSquareBoundary(boundary, lower, upper, faces);CHKERRQ(ierr);break;
  case 3: ierr = DMPlexCreateCubeBoundary(boundary, lower, upper, faces);CHKERRQ(ierr);break;
  default: SETERRQ1(comm, PETSC_ERR_SUP, "Dimension not supported: %D", dim);
  }
  ierr = DMPlexGenerate(boundary, NULL, interpolate, dm);CHKERRQ(ierr);
  ierr = DMDestroy(&boundary);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode DMPlexCreateCubeMesh_Internal(DM dm, const PetscReal lower[], const PetscReal upper[], const PetscInt edges[], DMBoundaryType bdX, DMBoundaryType bdY, DMBoundaryType bdZ)
{
  DMLabel        cutLabel = NULL;
  PetscInt       markerTop      = 1, faceMarkerTop      = 1;
  PetscInt       markerBottom   = 1, faceMarkerBottom   = 1;
  PetscInt       markerFront    = 1, faceMarkerFront    = 1;
  PetscInt       markerBack     = 1, faceMarkerBack     = 1;
  PetscInt       markerRight    = 1, faceMarkerRight    = 1;
  PetscInt       markerLeft     = 1, faceMarkerLeft     = 1;
  PetscInt       dim;
  PetscBool      markerSeparate = PETSC_FALSE, cutMarker = PETSC_FALSE;
  PetscMPIInt    rank;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMGetDimension(dm,&dim);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(PetscObjectComm((PetscObject)dm), &rank);CHKERRMPI(ierr);
  ierr = DMCreateLabel(dm,"marker");CHKERRQ(ierr);
  ierr = DMCreateLabel(dm,"Face Sets");CHKERRQ(ierr);
  ierr = PetscOptionsGetBool(((PetscObject) dm)->options,((PetscObject) dm)->prefix, "-dm_plex_periodic_cut", &cutMarker, NULL);CHKERRQ(ierr);
  if (bdX == DM_BOUNDARY_PERIODIC || bdX == DM_BOUNDARY_TWIST ||
      bdY == DM_BOUNDARY_PERIODIC || bdY == DM_BOUNDARY_TWIST ||
      bdZ == DM_BOUNDARY_PERIODIC || bdZ == DM_BOUNDARY_TWIST) {

    if (cutMarker) {ierr = DMCreateLabel(dm, "periodic_cut");CHKERRQ(ierr); ierr = DMGetLabel(dm, "periodic_cut", &cutLabel);CHKERRQ(ierr);}
  }
  switch (dim) {
  case 2:
    faceMarkerTop    = 3;
    faceMarkerBottom = 1;
    faceMarkerRight  = 2;
    faceMarkerLeft   = 4;
    break;
  case 3:
    faceMarkerBottom = 1;
    faceMarkerTop    = 2;
    faceMarkerFront  = 3;
    faceMarkerBack   = 4;
    faceMarkerRight  = 5;
    faceMarkerLeft   = 6;
    break;
  default:
    SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_SUP,"Dimension %D not supported",dim);
  }
  ierr = PetscOptionsGetBool(((PetscObject) dm)->options,((PetscObject) dm)->prefix, "-dm_plex_separate_marker", &markerSeparate, NULL);CHKERRQ(ierr);
  if (markerSeparate) {
    markerBottom = faceMarkerBottom;
    markerTop    = faceMarkerTop;
    markerFront  = faceMarkerFront;
    markerBack   = faceMarkerBack;
    markerRight  = faceMarkerRight;
    markerLeft   = faceMarkerLeft;
  }
  {
    const PetscInt numXEdges    = !rank ? edges[0] : 0;
    const PetscInt numYEdges    = !rank ? edges[1] : 0;
    const PetscInt numZEdges    = !rank ? edges[2] : 0;
    const PetscInt numXVertices = !rank ? (bdX == DM_BOUNDARY_PERIODIC || bdX == DM_BOUNDARY_TWIST ? edges[0] : edges[0]+1) : 0;
    const PetscInt numYVertices = !rank ? (bdY == DM_BOUNDARY_PERIODIC || bdY == DM_BOUNDARY_TWIST ? edges[1] : edges[1]+1) : 0;
    const PetscInt numZVertices = !rank ? (bdZ == DM_BOUNDARY_PERIODIC || bdZ == DM_BOUNDARY_TWIST ? edges[2] : edges[2]+1) : 0;
    const PetscInt numCells     = numXEdges*numYEdges*numZEdges;
    const PetscInt numXFaces    = numYEdges*numZEdges;
    const PetscInt numYFaces    = numXEdges*numZEdges;
    const PetscInt numZFaces    = numXEdges*numYEdges;
    const PetscInt numTotXFaces = numXVertices*numXFaces;
    const PetscInt numTotYFaces = numYVertices*numYFaces;
    const PetscInt numTotZFaces = numZVertices*numZFaces;
    const PetscInt numFaces     = numTotXFaces + numTotYFaces + numTotZFaces;
    const PetscInt numTotXEdges = numXEdges*numYVertices*numZVertices;
    const PetscInt numTotYEdges = numYEdges*numXVertices*numZVertices;
    const PetscInt numTotZEdges = numZEdges*numXVertices*numYVertices;
    const PetscInt numVertices  = numXVertices*numYVertices*numZVertices;
    const PetscInt numEdges     = numTotXEdges + numTotYEdges + numTotZEdges;
    const PetscInt firstVertex  = (dim == 2) ? numFaces : numCells;
    const PetscInt firstXFace   = (dim == 2) ? 0 : numCells + numVertices;
    const PetscInt firstYFace   = firstXFace + numTotXFaces;
    const PetscInt firstZFace   = firstYFace + numTotYFaces;
    const PetscInt firstXEdge   = numCells + numFaces + numVertices;
    const PetscInt firstYEdge   = firstXEdge + numTotXEdges;
    const PetscInt firstZEdge   = firstYEdge + numTotYEdges;
    Vec            coordinates;
    PetscSection   coordSection;
    PetscScalar   *coords;
    PetscInt       coordSize;
    PetscInt       v, vx, vy, vz;
    PetscInt       c, f, fx, fy, fz, e, ex, ey, ez;

    ierr = DMPlexSetChart(dm, 0, numCells+numFaces+numEdges+numVertices);CHKERRQ(ierr);
    for (c = 0; c < numCells; c++) {
      ierr = DMPlexSetConeSize(dm, c, 6);CHKERRQ(ierr);
    }
    for (f = firstXFace; f < firstXFace+numFaces; ++f) {
      ierr = DMPlexSetConeSize(dm, f, 4);CHKERRQ(ierr);
    }
    for (e = firstXEdge; e < firstXEdge+numEdges; ++e) {
      ierr = DMPlexSetConeSize(dm, e, 2);CHKERRQ(ierr);
    }
    ierr = DMSetUp(dm);CHKERRQ(ierr); /* Allocate space for cones */
    /* Build cells */
    for (fz = 0; fz < numZEdges; ++fz) {
      for (fy = 0; fy < numYEdges; ++fy) {
        for (fx = 0; fx < numXEdges; ++fx) {
          PetscInt cell    = (fz*numYEdges + fy)*numXEdges + fx;
          PetscInt faceB   = firstZFace + (fy*numXEdges+fx)*numZVertices +   fz;
          PetscInt faceT   = firstZFace + (fy*numXEdges+fx)*numZVertices + ((fz+1)%numZVertices);
          PetscInt faceF   = firstYFace + (fz*numXEdges+fx)*numYVertices +   fy;
          PetscInt faceK   = firstYFace + (fz*numXEdges+fx)*numYVertices + ((fy+1)%numYVertices);
          PetscInt faceL   = firstXFace + (fz*numYEdges+fy)*numXVertices +   fx;
          PetscInt faceR   = firstXFace + (fz*numYEdges+fy)*numXVertices + ((fx+1)%numXVertices);
                            /* B,  T,  F,  K,  R,  L */
          PetscInt ornt[6] = {-4,  0,  0, -1,  0, -4}; /* ??? */
          PetscInt cone[6];

          /* no boundary twisting in 3D */
          cone[0] = faceB; cone[1] = faceT; cone[2] = faceF; cone[3] = faceK; cone[4] = faceR; cone[5] = faceL;
          ierr    = DMPlexSetCone(dm, cell, cone);CHKERRQ(ierr);
          ierr    = DMPlexSetConeOrientation(dm, cell, ornt);CHKERRQ(ierr);
          if (bdX != DM_BOUNDARY_NONE && fx == numXEdges-1 && cutLabel) {ierr = DMLabelSetValue(cutLabel, cell, 2);CHKERRQ(ierr);}
          if (bdY != DM_BOUNDARY_NONE && fy == numYEdges-1 && cutLabel) {ierr = DMLabelSetValue(cutLabel, cell, 2);CHKERRQ(ierr);}
          if (bdZ != DM_BOUNDARY_NONE && fz == numZEdges-1 && cutLabel) {ierr = DMLabelSetValue(cutLabel, cell, 2);CHKERRQ(ierr);}
        }
      }
    }
    /* Build x faces */
    for (fz = 0; fz < numZEdges; ++fz) {
      for (fy = 0; fy < numYEdges; ++fy) {
        for (fx = 0; fx < numXVertices; ++fx) {
          PetscInt face    = firstXFace + (fz*numYEdges+fy)     *numXVertices+fx;
          PetscInt edgeL   = firstZEdge + (fy                   *numXVertices+fx)*numZEdges + fz;
          PetscInt edgeR   = firstZEdge + (((fy+1)%numYVertices)*numXVertices+fx)*numZEdges + fz;
          PetscInt edgeB   = firstYEdge + (fz                   *numXVertices+fx)*numYEdges + fy;
          PetscInt edgeT   = firstYEdge + (((fz+1)%numZVertices)*numXVertices+fx)*numYEdges + fy;
          PetscInt ornt[4] = {0, 0, -2, -2};
          PetscInt cone[4];

          if (dim == 3) {
            /* markers */
            if (bdX != DM_BOUNDARY_PERIODIC) {
              if (fx == numXVertices-1) {
                ierr = DMSetLabelValue(dm, "Face Sets", face, faceMarkerRight);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", face, markerRight);CHKERRQ(ierr);
              }
              else if (fx == 0) {
                ierr = DMSetLabelValue(dm, "Face Sets", face, faceMarkerLeft);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", face, markerLeft);CHKERRQ(ierr);
              }
            }
          }
          cone[0] = edgeB; cone[1] = edgeR; cone[2] = edgeT; cone[3] = edgeL;
          ierr    = DMPlexSetCone(dm, face, cone);CHKERRQ(ierr);
          ierr    = DMPlexSetConeOrientation(dm, face, ornt);CHKERRQ(ierr);
        }
      }
    }
    /* Build y faces */
    for (fz = 0; fz < numZEdges; ++fz) {
      for (fx = 0; fx < numXEdges; ++fx) {
        for (fy = 0; fy < numYVertices; ++fy) {
          PetscInt face    = firstYFace + (fz*numXEdges+fx)*numYVertices + fy;
          PetscInt edgeL   = firstZEdge + (fy*numXVertices+  fx)*numZEdges + fz;
          PetscInt edgeR   = firstZEdge + (fy*numXVertices+((fx+1)%numXVertices))*numZEdges + fz;
          PetscInt edgeB   = firstXEdge + (fz                   *numYVertices+fy)*numXEdges + fx;
          PetscInt edgeT   = firstXEdge + (((fz+1)%numZVertices)*numYVertices+fy)*numXEdges + fx;
          PetscInt ornt[4] = {0, 0, -2, -2};
          PetscInt cone[4];

          if (dim == 3) {
            /* markers */
            if (bdY != DM_BOUNDARY_PERIODIC) {
              if (fy == numYVertices-1) {
                ierr = DMSetLabelValue(dm, "Face Sets", face, faceMarkerBack);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", face, markerBack);CHKERRQ(ierr);
              }
              else if (fy == 0) {
                ierr = DMSetLabelValue(dm, "Face Sets", face, faceMarkerFront);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", face, markerFront);CHKERRQ(ierr);
              }
            }
          }
          cone[0] = edgeB; cone[1] = edgeR; cone[2] = edgeT; cone[3] = edgeL;
          ierr    = DMPlexSetCone(dm, face, cone);CHKERRQ(ierr);
          ierr    = DMPlexSetConeOrientation(dm, face, ornt);CHKERRQ(ierr);
        }
      }
    }
    /* Build z faces */
    for (fy = 0; fy < numYEdges; ++fy) {
      for (fx = 0; fx < numXEdges; ++fx) {
        for (fz = 0; fz < numZVertices; fz++) {
          PetscInt face    = firstZFace + (fy*numXEdges+fx)*numZVertices + fz;
          PetscInt edgeL   = firstYEdge + (fz*numXVertices+  fx)*numYEdges + fy;
          PetscInt edgeR   = firstYEdge + (fz*numXVertices+((fx+1)%numXVertices))*numYEdges + fy;
          PetscInt edgeB   = firstXEdge + (fz*numYVertices+  fy)*numXEdges + fx;
          PetscInt edgeT   = firstXEdge + (fz*numYVertices+((fy+1)%numYVertices))*numXEdges + fx;
          PetscInt ornt[4] = {0, 0, -2, -2};
          PetscInt cone[4];

          if (dim == 2) {
            if (bdX == DM_BOUNDARY_TWIST && fx == numXEdges-1) {edgeR += numYEdges-1-2*fy; ornt[1] = -2;}
            if (bdY == DM_BOUNDARY_TWIST && fy == numYEdges-1) {edgeT += numXEdges-1-2*fx; ornt[2] =  0;}
            if (bdX != DM_BOUNDARY_NONE && fx == numXEdges-1 && cutLabel) {ierr = DMLabelSetValue(cutLabel, face, 2);CHKERRQ(ierr);}
            if (bdY != DM_BOUNDARY_NONE && fy == numYEdges-1 && cutLabel) {ierr = DMLabelSetValue(cutLabel, face, 2);CHKERRQ(ierr);}
          } else {
            /* markers */
            if (bdZ != DM_BOUNDARY_PERIODIC) {
              if (fz == numZVertices-1) {
                ierr = DMSetLabelValue(dm, "Face Sets", face, faceMarkerTop);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", face, markerTop);CHKERRQ(ierr);
              }
              else if (fz == 0) {
                ierr = DMSetLabelValue(dm, "Face Sets", face, faceMarkerBottom);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", face, markerBottom);CHKERRQ(ierr);
              }
            }
          }
          cone[0] = edgeB; cone[1] = edgeR; cone[2] = edgeT; cone[3] = edgeL;
          ierr    = DMPlexSetCone(dm, face, cone);CHKERRQ(ierr);
          ierr    = DMPlexSetConeOrientation(dm, face, ornt);CHKERRQ(ierr);
        }
      }
    }
    /* Build Z edges*/
    for (vy = 0; vy < numYVertices; vy++) {
      for (vx = 0; vx < numXVertices; vx++) {
        for (ez = 0; ez < numZEdges; ez++) {
          const PetscInt edge    = firstZEdge  + (vy*numXVertices+vx)*numZEdges + ez;
          const PetscInt vertexB = firstVertex + (ez                   *numYVertices+vy)*numXVertices + vx;
          const PetscInt vertexT = firstVertex + (((ez+1)%numZVertices)*numYVertices+vy)*numXVertices + vx;
          PetscInt       cone[2];

          if (dim == 3) {
            if (bdX != DM_BOUNDARY_PERIODIC) {
              if (vx == numXVertices-1) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerRight);CHKERRQ(ierr);
              }
              else if (vx == 0) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerLeft);CHKERRQ(ierr);
              }
            }
            if (bdY != DM_BOUNDARY_PERIODIC) {
              if (vy == numYVertices-1) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerBack);CHKERRQ(ierr);
              }
              else if (vy == 0) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerFront);CHKERRQ(ierr);
              }
            }
          }
          cone[0] = vertexB; cone[1] = vertexT;
          ierr = DMPlexSetCone(dm, edge, cone);CHKERRQ(ierr);
        }
      }
    }
    /* Build Y edges*/
    for (vz = 0; vz < numZVertices; vz++) {
      for (vx = 0; vx < numXVertices; vx++) {
        for (ey = 0; ey < numYEdges; ey++) {
          const PetscInt nextv   = (dim == 2 && bdY == DM_BOUNDARY_TWIST && ey == numYEdges-1) ? (numXVertices-vx-1) : (vz*numYVertices+((ey+1)%numYVertices))*numXVertices + vx;
          const PetscInt edge    = firstYEdge  + (vz*numXVertices+vx)*numYEdges + ey;
          const PetscInt vertexF = firstVertex + (vz*numYVertices+ey)*numXVertices + vx;
          const PetscInt vertexK = firstVertex + nextv;
          PetscInt       cone[2];

          cone[0] = vertexF; cone[1] = vertexK;
          ierr = DMPlexSetCone(dm, edge, cone);CHKERRQ(ierr);
          if (dim == 2) {
            if ((bdX != DM_BOUNDARY_PERIODIC) && (bdX != DM_BOUNDARY_TWIST)) {
              if (vx == numXVertices-1) {
                ierr = DMSetLabelValue(dm, "Face Sets", edge, faceMarkerRight);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", edge,    markerRight);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", cone[0], markerRight);CHKERRQ(ierr);
                if (ey == numYEdges-1) {
                  ierr = DMSetLabelValue(dm, "marker", cone[1], markerRight);CHKERRQ(ierr);
                }
              } else if (vx == 0) {
                ierr = DMSetLabelValue(dm, "Face Sets", edge, faceMarkerLeft);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", edge,    markerLeft);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", cone[0], markerLeft);CHKERRQ(ierr);
                if (ey == numYEdges-1) {
                  ierr = DMSetLabelValue(dm, "marker", cone[1], markerLeft);CHKERRQ(ierr);
                }
              }
            } else {
              if (vx == 0 && cutLabel) {
                ierr = DMLabelSetValue(cutLabel, edge,    1);CHKERRQ(ierr);
                ierr = DMLabelSetValue(cutLabel, cone[0], 1);CHKERRQ(ierr);
                if (ey == numYEdges-1) {
                  ierr = DMLabelSetValue(cutLabel, cone[1], 1);CHKERRQ(ierr);
                }
              }
            }
          } else {
            if (bdX != DM_BOUNDARY_PERIODIC) {
              if (vx == numXVertices-1) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerRight);CHKERRQ(ierr);
              } else if (vx == 0) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerLeft);CHKERRQ(ierr);
              }
            }
            if (bdZ != DM_BOUNDARY_PERIODIC) {
              if (vz == numZVertices-1) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerTop);CHKERRQ(ierr);
              } else if (vz == 0) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerBottom);CHKERRQ(ierr);
              }
            }
          }
        }
      }
    }
    /* Build X edges*/
    for (vz = 0; vz < numZVertices; vz++) {
      for (vy = 0; vy < numYVertices; vy++) {
        for (ex = 0; ex < numXEdges; ex++) {
          const PetscInt nextv   = (dim == 2 && bdX == DM_BOUNDARY_TWIST && ex == numXEdges-1) ? (numYVertices-vy-1)*numXVertices : (vz*numYVertices+vy)*numXVertices + (ex+1)%numXVertices;
          const PetscInt edge    = firstXEdge  + (vz*numYVertices+vy)*numXEdges + ex;
          const PetscInt vertexL = firstVertex + (vz*numYVertices+vy)*numXVertices + ex;
          const PetscInt vertexR = firstVertex + nextv;
          PetscInt       cone[2];

          cone[0] = vertexL; cone[1] = vertexR;
          ierr = DMPlexSetCone(dm, edge, cone);CHKERRQ(ierr);
          if (dim == 2) {
            if ((bdY != DM_BOUNDARY_PERIODIC) && (bdY != DM_BOUNDARY_TWIST)) {
              if (vy == numYVertices-1) {
                ierr = DMSetLabelValue(dm, "Face Sets", edge, faceMarkerTop);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", edge,    markerTop);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", cone[0], markerTop);CHKERRQ(ierr);
                if (ex == numXEdges-1) {
                  ierr = DMSetLabelValue(dm, "marker", cone[1], markerTop);CHKERRQ(ierr);
                }
              } else if (vy == 0) {
                ierr = DMSetLabelValue(dm, "Face Sets", edge, faceMarkerBottom);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", edge,    markerBottom);CHKERRQ(ierr);
                ierr = DMSetLabelValue(dm, "marker", cone[0], markerBottom);CHKERRQ(ierr);
                if (ex == numXEdges-1) {
                  ierr = DMSetLabelValue(dm, "marker", cone[1], markerBottom);CHKERRQ(ierr);
                }
              }
            } else {
              if (vy == 0 && cutLabel) {
                ierr = DMLabelSetValue(cutLabel, edge,    1);CHKERRQ(ierr);
                ierr = DMLabelSetValue(cutLabel, cone[0], 1);CHKERRQ(ierr);
                if (ex == numXEdges-1) {
                  ierr = DMLabelSetValue(cutLabel, cone[1], 1);CHKERRQ(ierr);
                }
              }
            }
          } else {
            if (bdY != DM_BOUNDARY_PERIODIC) {
              if (vy == numYVertices-1) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerBack);CHKERRQ(ierr);
              }
              else if (vy == 0) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerFront);CHKERRQ(ierr);
              }
            }
            if (bdZ != DM_BOUNDARY_PERIODIC) {
              if (vz == numZVertices-1) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerTop);CHKERRQ(ierr);
              }
              else if (vz == 0) {
                ierr = DMSetLabelValue(dm, "marker", edge, markerBottom);CHKERRQ(ierr);
              }
            }
          }
        }
      }
    }
    ierr = DMPlexSymmetrize(dm);CHKERRQ(ierr);
    ierr = DMPlexStratify(dm);CHKERRQ(ierr);
    /* Build coordinates */
    ierr = DMGetCoordinateSection(dm, &coordSection);CHKERRQ(ierr);
    ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldComponents(coordSection, 0, dim);CHKERRQ(ierr);
    ierr = PetscSectionSetChart(coordSection, firstVertex, firstVertex+numVertices);CHKERRQ(ierr);
    for (v = firstVertex; v < firstVertex+numVertices; ++v) {
      ierr = PetscSectionSetDof(coordSection, v, dim);CHKERRQ(ierr);
      ierr = PetscSectionSetFieldDof(coordSection, v, 0, dim);CHKERRQ(ierr);
    }
    ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
    ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
    ierr = VecCreate(PETSC_COMM_SELF, &coordinates);CHKERRQ(ierr);
    ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
    ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
    ierr = VecSetBlockSize(coordinates, dim);CHKERRQ(ierr);
    ierr = VecSetType(coordinates,VECSTANDARD);CHKERRQ(ierr);
    ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
    for (vz = 0; vz < numZVertices; ++vz) {
      for (vy = 0; vy < numYVertices; ++vy) {
        for (vx = 0; vx < numXVertices; ++vx) {
          coords[((vz*numYVertices+vy)*numXVertices+vx)*dim+0] = lower[0] + ((upper[0] - lower[0])/numXEdges)*vx;
          coords[((vz*numYVertices+vy)*numXVertices+vx)*dim+1] = lower[1] + ((upper[1] - lower[1])/numYEdges)*vy;
          if (dim == 3) {
            coords[((vz*numYVertices+vy)*numXVertices+vx)*dim+2] = lower[2] + ((upper[2] - lower[2])/numZEdges)*vz;
          }
        }
      }
    }
    ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
    ierr = DMSetCoordinatesLocal(dm, coordinates);CHKERRQ(ierr);
    ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode DMPlexCreateBoxMesh_Tensor_Internal(MPI_Comm comm, PetscInt dim, const PetscInt faces[], const PetscReal lower[], const PetscReal upper[], const DMBoundaryType periodicity[], PetscBool interpolate, DM *dm)
{
  PetscInt       i;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidPointer(dm, 7);
  ierr = DMCreate(comm, dm);CHKERRQ(ierr);
  PetscValidLogicalCollectiveInt(*dm,dim,2);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(*dm, dim);CHKERRQ(ierr);
  switch (dim) {
  case 2: {ierr = DMPlexCreateCubeMesh_Internal(*dm, lower, upper, faces, periodicity[0], periodicity[1], DM_BOUNDARY_NONE);CHKERRQ(ierr);break;}
  case 3: {ierr = DMPlexCreateCubeMesh_Internal(*dm, lower, upper, faces, periodicity[0], periodicity[1], periodicity[2]);CHKERRQ(ierr);break;}
  default: SETERRQ1(comm, PETSC_ERR_SUP, "Dimension not supported: %D", dim);
  }
  if (periodicity[0] == DM_BOUNDARY_PERIODIC || periodicity[0] == DM_BOUNDARY_TWIST ||
      periodicity[1] == DM_BOUNDARY_PERIODIC || periodicity[1] == DM_BOUNDARY_TWIST ||
      (dim > 2 && (periodicity[2] == DM_BOUNDARY_PERIODIC || periodicity[2] == DM_BOUNDARY_TWIST))) {
    PetscReal L[3];
    PetscReal maxCell[3];

    for (i = 0; i < dim; i++) {
      L[i]       = upper[i] - lower[i];
      maxCell[i] = 1.1 * (L[i] / PetscMax(1,faces[i]));
    }
    ierr = DMSetPeriodicity(*dm,PETSC_TRUE,maxCell,L,periodicity);CHKERRQ(ierr);
  }
  if (!interpolate) {
    DM udm;

    ierr = DMPlexUninterpolate(*dm, &udm);CHKERRQ(ierr);
    ierr = DMPlexCopyCoordinates(*dm, udm);CHKERRQ(ierr);
    ierr = DMDestroy(dm);CHKERRQ(ierr);
    *dm  = udm;
  }
  ierr = DMPlexSetRefinementUniform(*dm, PETSC_TRUE);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  DMPlexCreateBoxMesh - Creates a mesh on the tensor product of unit intervals (box) using simplices or tensor cells (hexahedra).

  Collective

  Input Parameters:
+ comm        - The communicator for the DM object
. dim         - The spatial dimension
. simplex     - PETSC_TRUE for simplices, PETSC_FALSE for tensor cells
. faces       - Number of faces per dimension, or NULL for (1,) in 1D and (2, 2) in 2D and (1, 1, 1) in 3D
. lower       - The lower left corner, or NULL for (0, 0, 0)
. upper       - The upper right corner, or NULL for (1, 1, 1)
. periodicity - The boundary type for the X,Y,Z direction, or NULL for DM_BOUNDARY_NONE
- interpolate - Flag to create intermediate mesh pieces (edges, faces)

  Output Parameter:
. dm  - The DM object

  Options Database Keys:
  These options override the hard-wired input values.
+ -dm_plex_box_dim <dim>          - Set the topological dimension
. -dm_plex_box_simplex <bool>     - PETSC_TRUE for simplex elements, PETSC_FALSE for tensor elements
. -dm_plex_box_lower <x,y,z>      - Specify lower-left-bottom coordinates for the box
. -dm_plex_box_upper <x,y,z>      - Specify upper-right-top coordinates for the box
. -dm_plex_box_faces <m,n,p>      - Number of faces in each linear direction
. -dm_plex_box_bd <bx,by,bz>      - Specify the DMBoundaryType for each direction
- -dm_plex_box_interpolate <bool> - PETSC_TRUE turns on topological interpolation (creating edges and faces)

  Notes:
  The options database keys above take lists of length d in d dimensions.

  Here is the numbering returned for 2 faces in each direction for tensor cells:
$ 10---17---11---18----12
$  |         |         |
$  |         |         |
$ 20    2   22    3    24
$  |         |         |
$  |         |         |
$  7---15----8---16----9
$  |         |         |
$  |         |         |
$ 19    0   21    1   23
$  |         |         |
$  |         |         |
$  4---13----5---14----6

and for simplicial cells

$ 14----8---15----9----16
$  |\     5  |\      7 |
$  | \       | \       |
$ 13   2    14    3    15
$  | 4   \   | 6   \   |
$  |       \ |       \ |
$ 11----6---12----7----13
$  |\        |\        |
$  | \    1  | \     3 |
$ 10   0    11    1    12
$  | 0   \   | 2   \   |
$  |       \ |       \ |
$  8----4----9----5----10

  Level: beginner

.seealso: DMPlexCreateFromFile(), DMPlexCreateHexCylinderMesh(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateBoxMesh(MPI_Comm comm, PetscInt dim, PetscBool simplex, const PetscInt faces[], const PetscReal lower[], const PetscReal upper[], const DMBoundaryType periodicity[], PetscBool interpolate, DM *dm)
{
  PetscInt       fac[3] = {0, 0, 0};
  PetscReal      low[3] = {0, 0, 0};
  PetscReal      upp[3] = {1, 1, 1};
  DMBoundaryType bdt[3] = {DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE};
  PetscInt       i, n;
  PetscBool      flg;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscOptionsGetInt(NULL, NULL, "-dm_plex_box_dim", &dim, &flg);CHKERRQ(ierr);
  if ((dim < 0) || (dim > 3)) SETERRQ1(comm, PETSC_ERR_ARG_OUTOFRANGE, "Dimension %D should be in [1, 3]", dim);
  ierr = PetscOptionsGetBool(NULL, NULL, "-dm_plex_box_simplex", &simplex, &flg);CHKERRQ(ierr);
  n    = dim;
  ierr = PetscOptionsGetIntArray(NULL, NULL, "-dm_plex_box_faces", fac, &n, &flg);CHKERRQ(ierr);
  for (i = 0; i < dim; ++i) fac[i] = faces ? faces[i] : (flg && i < n ? fac[i] : (dim == 1 ? 1 : 4-dim));
  if (lower) for (i = 0; i < dim; ++i) low[i] = lower[i];
  if (upper) for (i = 0; i < dim; ++i) upp[i] = upper[i];
  if (periodicity) for (i = 0; i < dim; ++i) bdt[i] = periodicity[i];
  /* Allow bounds to be specified from the command line */
  n    = 3;
  ierr = PetscOptionsGetRealArray(NULL, NULL, "-dm_plex_box_lower", low, &n, &flg);CHKERRQ(ierr);
  if (flg && (n != dim)) SETERRQ2(comm, PETSC_ERR_ARG_SIZ, "Lower box point had %D values, should have been %D", n, dim);
  n    = 3;
  ierr = PetscOptionsGetRealArray(NULL, NULL, "-dm_plex_box_upper", upp, &n, &flg);CHKERRQ(ierr);
  if (flg && (n != dim)) SETERRQ2(comm, PETSC_ERR_ARG_SIZ, "Upper box point had %D values, should have been %D", n, dim);
  n    = 3;
  ierr = PetscOptionsGetEnumArray(NULL, NULL, "-dm_plex_box_bd", DMBoundaryTypes, (PetscEnum *) bdt, &n, &flg);CHKERRQ(ierr);
  if (flg && (n != dim)) SETERRQ2(comm, PETSC_ERR_ARG_SIZ, "Box boundary types had %D values, should have been %D", n, dim);
  ierr = PetscOptionsGetBool(NULL, NULL, "-dm_plex_box_interpolate", &interpolate, &flg);CHKERRQ(ierr);

  if (dim == 1)      {ierr = DMPlexCreateLineMesh_Internal(comm, fac[0], low[0], upp[0], bdt[0], dm);CHKERRQ(ierr);}
  else if (simplex)  {ierr = DMPlexCreateBoxMesh_Simplex_Internal(comm, dim, fac, low, upp, bdt, interpolate, dm);CHKERRQ(ierr);}
  else               {ierr = DMPlexCreateBoxMesh_Tensor_Internal(comm, dim, fac, low, upp, bdt, interpolate, dm);CHKERRQ(ierr);}
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateWedgeBoxMesh - Creates a 3-D mesh tesselating the (x,y) plane and extruding in the third direction using wedge cells.

  Collective

  Input Parameters:
+ comm        - The communicator for the DM object
. faces       - Number of faces per dimension, or NULL for (1, 1, 1)
. lower       - The lower left corner, or NULL for (0, 0, 0)
. upper       - The upper right corner, or NULL for (1, 1, 1)
. periodicity - The boundary type for the X,Y,Z direction, or NULL for DM_BOUNDARY_NONE
. orderHeight - If PETSC_TRUE, orders the extruded cells in the height first. Otherwise, orders the cell on the layers first
- interpolate - Flag to create intermediate mesh pieces (edges, faces)

  Output Parameter:
. dm  - The DM object

  Level: beginner

.seealso: DMPlexCreateHexCylinderMesh(), DMPlexCreateWedgeCylinderMesh(), DMPlexExtrude(), DMPlexCreateBoxMesh(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateWedgeBoxMesh(MPI_Comm comm, const PetscInt faces[], const PetscReal lower[], const PetscReal upper[], const DMBoundaryType periodicity[], PetscBool orderHeight, PetscBool interpolate, DM *dm)
{
  DM             bdm, botdm;
  PetscInt       i;
  PetscInt       fac[3] = {0, 0, 0};
  PetscReal      low[3] = {0, 0, 0};
  PetscReal      upp[3] = {1, 1, 1};
  DMBoundaryType bdt[3] = {DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE};
  PetscErrorCode ierr;

  PetscFunctionBegin;
  for (i = 0; i < 3; ++i) fac[i] = faces ? (faces[i] > 0 ? faces[i] : 1) : 1;
  if (lower) for (i = 0; i < 3; ++i) low[i] = lower[i];
  if (upper) for (i = 0; i < 3; ++i) upp[i] = upper[i];
  if (periodicity) for (i = 0; i < 3; ++i) bdt[i] = periodicity[i];
  for (i = 0; i < 3; ++i) if (bdt[i] != DM_BOUNDARY_NONE) SETERRQ(comm, PETSC_ERR_SUP, "Periodicity not yet supported");

  ierr = DMCreate(comm, &bdm);CHKERRQ(ierr);
  ierr = DMSetType(bdm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(bdm, 1);CHKERRQ(ierr);
  ierr = DMSetCoordinateDim(bdm, 2);CHKERRQ(ierr);
  ierr = DMPlexCreateSquareBoundary(bdm, low, upp, fac);CHKERRQ(ierr);
  ierr = DMPlexGenerate(bdm, NULL, PETSC_FALSE, &botdm);CHKERRQ(ierr);
  ierr = DMDestroy(&bdm);CHKERRQ(ierr);
  ierr = DMPlexExtrude(botdm, fac[2], upp[2] - low[2], orderHeight, NULL, interpolate, dm);CHKERRQ(ierr);
  if (low[2] != 0.0) {
    Vec         v;
    PetscScalar *x;
    PetscInt    cDim, n;

    ierr = DMGetCoordinatesLocal(*dm, &v);CHKERRQ(ierr);
    ierr = VecGetBlockSize(v, &cDim);CHKERRQ(ierr);
    ierr = VecGetLocalSize(v, &n);CHKERRQ(ierr);
    ierr = VecGetArray(v, &x);CHKERRQ(ierr);
    x   += cDim;
    for (i=0; i<n; i+=cDim) x[i] += low[2];
    ierr = VecRestoreArray(v,&x);CHKERRQ(ierr);
    ierr = DMSetCoordinatesLocal(*dm, v);CHKERRQ(ierr);
  }
  ierr = DMDestroy(&botdm);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  DMPlexExtrude - Creates a (d+1)-D mesh by extruding a d-D mesh in the normal direction using prismatic cells.

  Collective on idm

  Input Parameters:
+ idm         - The mesh to be extruded
. layers      - The number of layers, or PETSC_DETERMINE to use the default
. height      - The total height of the extrusion, or PETSC_DETERMINE to use the default
. orderHeight - If PETSC_TRUE, orders the extruded cells in the height first. Otherwise, orders the cell on the layers first
. extNormal   - The normal direction in which the mesh should be extruded, or NULL to extrude using the surface normal
- interpolate - Flag to create intermediate mesh pieces (edges, faces)

  Output Parameter:
. dm  - The DM object

  Notes:
  The mesh created has prismatic cells, and the vertex ordering in the cone of the cell is that of the tensor prismatic cells. Not currently supported in Fortran.

  Options Database Keys:
+   -dm_plex_extrude_layers <k> - Sets the number of layers k
.   -dm_plex_extrude_height <h> - Sets the total height of the extrusion
.   -dm_plex_extrude_heights <h0,h1,...> - Sets the height of each layer
.   -dm_plex_extrude_order_height - If true, order cells by height first
-   -dm_plex_extrude_normal <n0,...,nd> - Sets the normal vector along which to extrude

  Level: advanced

.seealso: DMPlexCreateWedgeCylinderMesh(), DMPlexCreateWedgeBoxMesh(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexExtrude(DM idm, PetscInt layers, PetscReal height, PetscBool orderHeight, const PetscReal extNormal[], PetscBool interpolate, DM* dm)
{
  PetscScalar       *coordsB;
  const PetscScalar *coordsA;
  PetscReal         *normals = NULL, *heights = NULL;
  PetscReal         clNormal[3];
  Vec               coordinatesA, coordinatesB;
  PetscSection      coordSectionA, coordSectionB;
  PetscInt          dim, cDim, cDimB, c, l, v, coordSize, *newCone, nl;
  PetscInt          cStart, cEnd, vStart, vEnd, cellV, numCells, numVertices;
  const char       *prefix;
  PetscBool         haveCLNormal, flg;
  PetscErrorCode    ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(idm, DM_CLASSID, 1);
  PetscValidLogicalCollectiveInt(idm, layers, 2);
  PetscValidLogicalCollectiveReal(idm, height, 3);
  PetscValidLogicalCollectiveBool(idm, interpolate, 4);
  ierr = DMGetDimension(idm, &dim);CHKERRQ(ierr);
  ierr = DMGetCoordinateDim(idm, &cDim);CHKERRQ(ierr);
  cDimB = cDim == dim ? cDim+1 : cDim;
  if (dim < 1 || dim > 3) SETERRQ1(PetscObjectComm((PetscObject)idm), PETSC_ERR_SUP, "Support for dimension %D not coded", dim);

  ierr = PetscObjectGetOptionsPrefix((PetscObject) idm, &prefix);CHKERRQ(ierr);
  if (layers < 0) layers = 1;
  ierr = PetscOptionsGetInt(NULL, prefix, "-dm_plex_extrude_layers", &layers, NULL);CHKERRQ(ierr);
  if (layers <= 0) SETERRQ1(PetscObjectComm((PetscObject) idm), PETSC_ERR_ARG_OUTOFRANGE, "Number of layers %D must be positive", layers);
  if (height < 0.) height = 1.;
  ierr = PetscOptionsGetReal(NULL, prefix, "-dm_plex_extrude_height", &height, NULL);CHKERRQ(ierr);
  if (height <= 0.) SETERRQ1(PetscObjectComm((PetscObject) idm), PETSC_ERR_ARG_OUTOFRANGE, "Height of layers %g must be positive", (double) height);
  ierr = PetscMalloc1(layers, &heights);CHKERRQ(ierr);
  nl   = layers;
  ierr = PetscOptionsGetRealArray(NULL, prefix, "-dm_plex_extrude_heights", heights, &nl, &flg);CHKERRQ(ierr);
  if (flg) {
    if (!nl) SETERRQ(PetscObjectComm((PetscObject) idm), PETSC_ERR_ARG_OUTOFRANGE, "Must give at least one height for -dm_plex_extrude_heights");
    for (l = nl; l < layers; ++l) heights[l] = heights[l-1];
    for (l = 0; l < layers; ++l) if (heights[l] <= 0.) SETERRQ2(PetscObjectComm((PetscObject) idm), PETSC_ERR_ARG_OUTOFRANGE, "Height %g of layers %D must be positive", (double) heights[l], l);
  } else {
    for (l = 0; l < layers; ++l) heights[l] = height/layers;
  }
  ierr = PetscOptionsGetBool(NULL, prefix, "-dm_plex_extrude_order_height", &orderHeight, NULL);CHKERRQ(ierr);
  c = 3;
  ierr = PetscOptionsGetRealArray(NULL, prefix, "-dm_plex_extrude_normal", clNormal, &c, &haveCLNormal);CHKERRQ(ierr);
  if (haveCLNormal && c != cDimB) SETERRQ2(PetscObjectComm((PetscObject)idm), PETSC_ERR_ARG_SIZ, "Input normal has size %D != %D extruded coordinate dimension", c, cDimB);

  ierr = DMPlexGetHeightStratum(idm, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetDepthStratum(idm, 0, &vStart, &vEnd);CHKERRQ(ierr);
  numCells = (cEnd - cStart)*layers;
  numVertices = (vEnd - vStart)*(layers+1);
  ierr = DMCreate(PetscObjectComm((PetscObject)idm), dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(*dm, dim+1);CHKERRQ(ierr);
  ierr = DMPlexSetChart(*dm, 0, numCells+numVertices);CHKERRQ(ierr);
  /* Must create the celltype label here so that we do not automatically try to compute the types */
  ierr = DMCreateLabel(*dm, "celltype");CHKERRQ(ierr);
  for (c = cStart, cellV = 0; c < cEnd; ++c) {
    DMPolytopeType ct, nct;
    PetscInt      *closure = NULL;
    PetscInt       closureSize, numCorners = 0;

    ierr = DMPlexGetCellType(idm, c, &ct);CHKERRQ(ierr);
    switch (ct) {
      case DM_POLYTOPE_SEGMENT:       nct = DM_POLYTOPE_SEG_PRISM_TENSOR;break;
      case DM_POLYTOPE_TRIANGLE:      nct = DM_POLYTOPE_TRI_PRISM_TENSOR;break;
      case DM_POLYTOPE_QUADRILATERAL: nct = DM_POLYTOPE_QUAD_PRISM_TENSOR;break;
      default: nct = DM_POLYTOPE_UNKNOWN;
    }
    ierr = DMPlexGetTransitiveClosure(idm, c, PETSC_TRUE, &closureSize, &closure);CHKERRQ(ierr);
    for (v = 0; v < closureSize*2; v += 2) if ((closure[v] >= vStart) && (closure[v] < vEnd)) numCorners++;
    ierr = DMPlexRestoreTransitiveClosure(idm, c, PETSC_TRUE, &closureSize, &closure);CHKERRQ(ierr);
    for (l = 0; l < layers; ++l) {
      const PetscInt cell = orderHeight ? layers*(c - cStart) + l : l*(cEnd - cStart) + c - cStart;

      ierr = DMPlexSetConeSize(*dm, cell, 2*numCorners);CHKERRQ(ierr);
      ierr = DMPlexSetCellType(*dm, cell, nct);CHKERRQ(ierr);
    }
    cellV = PetscMax(numCorners,cellV);
  }
  ierr = DMSetUp(*dm);CHKERRQ(ierr);

  if (dim != cDim && !(extNormal || haveCLNormal)) {ierr = PetscCalloc1(cDim*(vEnd - vStart), &normals);CHKERRQ(ierr);}
  ierr = PetscMalloc1(3*cellV,&newCone);CHKERRQ(ierr);
  for (c = cStart; c < cEnd; ++c) {
    PetscInt *closure = NULL;
    PetscInt closureSize, numCorners = 0, l;
    PetscReal normal[3] = {0, 0, 0};

    if (normals) {ierr = DMPlexComputeCellGeometryFVM(idm, c, NULL, NULL, normal);CHKERRQ(ierr);}
    ierr = DMPlexGetTransitiveClosure(idm, c, PETSC_TRUE, &closureSize, &closure);CHKERRQ(ierr);
    for (v = 0; v < closureSize*2; v += 2) {
      if ((closure[v] >= vStart) && (closure[v] < vEnd)) {
        PetscInt d;

        newCone[numCorners++] = closure[v] - vStart;
        if (normals) {for (d = 0; d < cDim; ++d) normals[cDim*(closure[v]-vStart)+d] += normal[d];}
      }
    }
    ierr = DMPlexRestoreTransitiveClosure(idm, c, PETSC_TRUE, &closureSize, &closure);CHKERRQ(ierr);
    for (l = 0; l < layers; ++l) {
      PetscInt i;

      for (i = 0; i < numCorners; ++i) {
        newCone[  numCorners + i] = orderHeight ? (layers+1)*newCone[i] + l     + numCells :     l*(vEnd - vStart) + newCone[i] + numCells;
        newCone[2*numCorners + i] = orderHeight ? (layers+1)*newCone[i] + l + 1 + numCells : (l+1)*(vEnd - vStart) + newCone[i] + numCells;
      }
      ierr = DMPlexSetCone(*dm, orderHeight ? layers*(c - cStart) + l : l*(cEnd - cStart) + c - cStart, newCone + numCorners);CHKERRQ(ierr);
    }
  }
  ierr = DMPlexSymmetrize(*dm);CHKERRQ(ierr);
  ierr = DMPlexStratify(*dm);CHKERRQ(ierr);
  ierr = PetscFree(newCone);CHKERRQ(ierr);

  ierr = DMGetCoordinateSection(*dm, &coordSectionB);CHKERRQ(ierr);
  ierr = PetscSectionSetNumFields(coordSectionB, 1);CHKERRQ(ierr);
  ierr = PetscSectionSetFieldComponents(coordSectionB, 0, cDimB);CHKERRQ(ierr);
  ierr = PetscSectionSetChart(coordSectionB, numCells, numCells+numVertices);CHKERRQ(ierr);
  for (v = numCells; v < numCells+numVertices; ++v) {
    ierr = PetscSectionSetDof(coordSectionB, v, cDimB);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldDof(coordSectionB, v, 0, cDimB);CHKERRQ(ierr);
    ierr = DMPlexSetCellType(*dm, v, DM_POLYTOPE_POINT);CHKERRQ(ierr);
  }
  ierr = PetscSectionSetUp(coordSectionB);CHKERRQ(ierr);
  ierr = PetscSectionGetStorageSize(coordSectionB, &coordSize);CHKERRQ(ierr);
  ierr = VecCreate(PETSC_COMM_SELF, &coordinatesB);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) coordinatesB, "coordinates");CHKERRQ(ierr);
  ierr = VecSetSizes(coordinatesB, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetBlockSize(coordinatesB, cDimB);CHKERRQ(ierr);
  ierr = VecSetType(coordinatesB,VECSTANDARD);CHKERRQ(ierr);

  ierr = DMGetCoordinateSection(idm, &coordSectionA);CHKERRQ(ierr);
  ierr = DMGetCoordinatesLocal(idm, &coordinatesA);CHKERRQ(ierr);
  ierr = VecGetArray(coordinatesB, &coordsB);CHKERRQ(ierr);
  ierr = VecGetArrayRead(coordinatesA, &coordsA);CHKERRQ(ierr);
  for (v = vStart; v < vEnd; ++v) {
    const PetscScalar *cptr;
    PetscReal         ones2[2] = { 0., 1.}, ones3[3] = { 0., 0., 1.};
    PetscReal         normal[3];
    PetscReal         norm;
    PetscInt          offA, d, cDimA = cDim;

    if (normals)           {for (d = 0; d < cDimB; ++d) normal[d] = normals[cDimB*(v - vStart)+d];}
    else if (haveCLNormal) {for (d = 0; d < cDimB; ++d) normal[d] = clNormal[d];}
    else if (extNormal)    {for (d = 0; d < cDimB; ++d) normal[d] = extNormal[d];}
    else if (cDimB == 2)   {for (d = 0; d < cDimB; ++d) normal[d] = ones2[d];}
    else if (cDimB == 3)   {for (d = 0; d < cDimB; ++d) normal[d] = ones3[d];}
    else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Unable to determine normal for extrusion");
    for (d = 0, norm = 0.0; d < cDimB; ++d) norm += normal[d]*normal[d];
    for (d = 0; d < cDimB; ++d) normal[d] *= 1./PetscSqrtReal(norm);

    ierr = PetscSectionGetOffset(coordSectionA, v, &offA);CHKERRQ(ierr);
    cptr = coordsA + offA;
    for (l = 0; l <= layers; ++l) {
      PetscInt offB, d, newV;

      newV = orderHeight ? (layers+1)*(v -vStart) + l + numCells : (vEnd -vStart)*l + (v -vStart) + numCells;
      ierr = PetscSectionGetOffset(coordSectionB, newV, &offB);CHKERRQ(ierr);
      for (d = 0; d < cDimA; ++d) { coordsB[offB+d]  = cptr[d]; }
      for (d = 0; d < cDimB; ++d) { coordsB[offB+d] += l ? normal[d]*heights[l-1] : 0.0; }
      cptr    = coordsB + offB;
      cDimA   = cDimB;
    }
  }
  ierr = VecRestoreArrayRead(coordinatesA, &coordsA);CHKERRQ(ierr);
  ierr = VecRestoreArray(coordinatesB, &coordsB);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(*dm, coordinatesB);CHKERRQ(ierr);
  ierr = VecDestroy(&coordinatesB);CHKERRQ(ierr);
  ierr = PetscFree(normals);CHKERRQ(ierr);
  ierr = PetscFree(heights);CHKERRQ(ierr);
  if (interpolate) {
    DM idm;

    ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
    ierr = DMPlexCopyCoordinates(*dm, idm);CHKERRQ(ierr);
    ierr = DMDestroy(dm);CHKERRQ(ierr);
    *dm  = idm;
  }
  PetscFunctionReturn(0);
}

/*@C
  DMPlexSetOptionsPrefix - Sets the prefix used for searching for all DM options in the database.

  Logically Collective on dm

  Input Parameters:
+ dm - the DM context
- prefix - the prefix to prepend to all option names

  Notes:
  A hyphen (-) must NOT be given at the beginning of the prefix name.
  The first character of all runtime options is AUTOMATICALLY the hyphen.

  Level: advanced

.seealso: SNESSetFromOptions()
@*/
PetscErrorCode DMPlexSetOptionsPrefix(DM dm, const char prefix[])
{
  DM_Plex       *mesh = (DM_Plex *) dm->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  ierr = PetscObjectSetOptionsPrefix((PetscObject) dm, prefix);CHKERRQ(ierr);
  ierr = PetscObjectSetOptionsPrefix((PetscObject) mesh->partitioner, prefix);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateHexCylinderMesh - Creates a mesh on the tensor product of the unit interval with the circle (cylinder) using hexahedra.

  Collective

  Input Parameters:
+ comm      - The communicator for the DM object
. numRefine - The number of regular refinements to the basic 5 cell structure
- periodicZ - The boundary type for the Z direction

  Output Parameter:
. dm  - The DM object

  Options Database Keys:
These options override the hard-wired input values.
+ -dm_plex_hex_cyl_refine <r> - Refine the sylinder r times
- -dm_plex_hex_cyl_bd <bz>    - Specify the DMBoundaryType in the z-direction

  Note:
  Here is the output numbering looking from the bottom of the cylinder:
$       17-----14
$        |     |
$        |  2  |
$        |     |
$ 17-----8-----7-----14
$  |     |     |     |
$  |  3  |  0  |  1  |
$  |     |     |     |
$ 19-----5-----6-----13
$        |     |
$        |  4  |
$        |     |
$       19-----13
$
$ and up through the top
$
$       18-----16
$        |     |
$        |  2  |
$        |     |
$ 18----10----11-----16
$  |     |     |     |
$  |  3  |  0  |  1  |
$  |     |     |     |
$ 20-----9----12-----15
$        |     |
$        |  4  |
$        |     |
$       20-----15

  Level: beginner

.seealso: DMPlexCreateBoxMesh(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateHexCylinderMesh(MPI_Comm comm, PetscInt numRefine, DMBoundaryType periodicZ, DM *dm)
{
  const PetscInt dim = 3;
  PetscInt       numCells, numVertices, r;
  PetscMPIInt    rank;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidPointer(dm, 4);
  ierr = MPI_Comm_rank(comm, &rank);CHKERRMPI(ierr);
  ierr = PetscOptionsGetInt(NULL, NULL, "-dm_plex_hex_cyl_refine", &numRefine, NULL);CHKERRQ(ierr);
  if (numRefine < 0) SETERRQ1(comm, PETSC_ERR_ARG_OUTOFRANGE, "Number of refinements %D cannot be negative", numRefine);
  ierr = PetscOptionsGetEnum(NULL, NULL, "-dm_plex_box_bd", DMBoundaryTypes, (PetscEnum *) &periodicZ, NULL);CHKERRQ(ierr);
  ierr = DMCreate(comm, dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(*dm, dim);CHKERRQ(ierr);
  /* Create topology */
  {
    PetscInt cone[8], c;

    numCells    = !rank ?  5 : 0;
    numVertices = !rank ? 16 : 0;
    if (periodicZ == DM_BOUNDARY_PERIODIC) {
      numCells   *= 3;
      numVertices = !rank ? 24 : 0;
    }
    ierr = DMPlexSetChart(*dm, 0, numCells+numVertices);CHKERRQ(ierr);
    for (c = 0; c < numCells; c++) {ierr = DMPlexSetConeSize(*dm, c, 8);CHKERRQ(ierr);}
    ierr = DMSetUp(*dm);CHKERRQ(ierr);
    if (!rank) {
      if (periodicZ == DM_BOUNDARY_PERIODIC) {
        cone[0] = 15; cone[1] = 18; cone[2] = 17; cone[3] = 16;
        cone[4] = 31; cone[5] = 32; cone[6] = 33; cone[7] = 34;
        ierr = DMPlexSetCone(*dm, 0, cone);CHKERRQ(ierr);
        cone[0] = 16; cone[1] = 17; cone[2] = 24; cone[3] = 23;
        cone[4] = 32; cone[5] = 36; cone[6] = 37; cone[7] = 33; /* 22 25 26 21 */
        ierr = DMPlexSetCone(*dm, 1, cone);CHKERRQ(ierr);
        cone[0] = 18; cone[1] = 27; cone[2] = 24; cone[3] = 17;
        cone[4] = 34; cone[5] = 33; cone[6] = 37; cone[7] = 38;
        ierr = DMPlexSetCone(*dm, 2, cone);CHKERRQ(ierr);
        cone[0] = 29; cone[1] = 27; cone[2] = 18; cone[3] = 15;
        cone[4] = 35; cone[5] = 31; cone[6] = 34; cone[7] = 38;
        ierr = DMPlexSetCone(*dm, 3, cone);CHKERRQ(ierr);
        cone[0] = 29; cone[1] = 15; cone[2] = 16; cone[3] = 23;
        cone[4] = 35; cone[5] = 36; cone[6] = 32; cone[7] = 31;
        ierr = DMPlexSetCone(*dm, 4, cone);CHKERRQ(ierr);

        cone[0] = 31; cone[1] = 34; cone[2] = 33; cone[3] = 32;
        cone[4] = 19; cone[5] = 22; cone[6] = 21; cone[7] = 20;
        ierr = DMPlexSetCone(*dm, 5, cone);CHKERRQ(ierr);
        cone[0] = 32; cone[1] = 33; cone[2] = 37; cone[3] = 36;
        cone[4] = 22; cone[5] = 25; cone[6] = 26; cone[7] = 21;
        ierr = DMPlexSetCone(*dm, 6, cone);CHKERRQ(ierr);
        cone[0] = 34; cone[1] = 38; cone[2] = 37; cone[3] = 33;
        cone[4] = 20; cone[5] = 21; cone[6] = 26; cone[7] = 28;
        ierr = DMPlexSetCone(*dm, 7, cone);CHKERRQ(ierr);
        cone[0] = 35; cone[1] = 38; cone[2] = 34; cone[3] = 31;
        cone[4] = 30; cone[5] = 19; cone[6] = 20; cone[7] = 28;
        ierr = DMPlexSetCone(*dm, 8, cone);CHKERRQ(ierr);
        cone[0] = 35; cone[1] = 31; cone[2] = 32; cone[3] = 36;
        cone[4] = 30; cone[5] = 25; cone[6] = 22; cone[7] = 19;
        ierr = DMPlexSetCone(*dm, 9, cone);CHKERRQ(ierr);

        cone[0] = 19; cone[1] = 20; cone[2] = 21; cone[3] = 22;
        cone[4] = 15; cone[5] = 16; cone[6] = 17; cone[7] = 18;
        ierr = DMPlexSetCone(*dm, 10, cone);CHKERRQ(ierr);
        cone[0] = 22; cone[1] = 21; cone[2] = 26; cone[3] = 25;
        cone[4] = 16; cone[5] = 23; cone[6] = 24; cone[7] = 17;
        ierr = DMPlexSetCone(*dm, 11, cone);CHKERRQ(ierr);
        cone[0] = 20; cone[1] = 28; cone[2] = 26; cone[3] = 21;
        cone[4] = 18; cone[5] = 17; cone[6] = 24; cone[7] = 27;
        ierr = DMPlexSetCone(*dm, 12, cone);CHKERRQ(ierr);
        cone[0] = 30; cone[1] = 28; cone[2] = 20; cone[3] = 19;
        cone[4] = 29; cone[5] = 15; cone[6] = 18; cone[7] = 27;
        ierr = DMPlexSetCone(*dm, 13, cone);CHKERRQ(ierr);
        cone[0] = 30; cone[1] = 19; cone[2] = 22; cone[3] = 25;
        cone[4] = 29; cone[5] = 23; cone[6] = 16; cone[7] = 15;
        ierr = DMPlexSetCone(*dm, 14, cone);CHKERRQ(ierr);
      } else {
        cone[0] =  5; cone[1] =  8; cone[2] =  7; cone[3] =  6;
        cone[4] =  9; cone[5] = 12; cone[6] = 11; cone[7] = 10;
        ierr = DMPlexSetCone(*dm, 0, cone);CHKERRQ(ierr);
        cone[0] =  6; cone[1] =  7; cone[2] = 14; cone[3] = 13;
        cone[4] = 12; cone[5] = 15; cone[6] = 16; cone[7] = 11;
        ierr = DMPlexSetCone(*dm, 1, cone);CHKERRQ(ierr);
        cone[0] =  8; cone[1] = 17; cone[2] = 14; cone[3] =  7;
        cone[4] = 10; cone[5] = 11; cone[6] = 16; cone[7] = 18;
        ierr = DMPlexSetCone(*dm, 2, cone);CHKERRQ(ierr);
        cone[0] = 19; cone[1] = 17; cone[2] =  8; cone[3] =  5;
        cone[4] = 20; cone[5] =  9; cone[6] = 10; cone[7] = 18;
        ierr = DMPlexSetCone(*dm, 3, cone);CHKERRQ(ierr);
        cone[0] = 19; cone[1] =  5; cone[2] =  6; cone[3] = 13;
        cone[4] = 20; cone[5] = 15; cone[6] = 12; cone[7] =  9;
        ierr = DMPlexSetCone(*dm, 4, cone);CHKERRQ(ierr);
      }
    }
    ierr = DMPlexSymmetrize(*dm);CHKERRQ(ierr);
    ierr = DMPlexStratify(*dm);CHKERRQ(ierr);
  }
  /* Interpolate */
  {
    DM idm;

    ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
    ierr = DMDestroy(dm);CHKERRQ(ierr);
    *dm  = idm;
  }
  /* Create cube geometry */
  {
    Vec             coordinates;
    PetscSection    coordSection;
    PetscScalar    *coords;
    PetscInt        coordSize, v;
    const PetscReal dis = 1.0/PetscSqrtReal(2.0);
    const PetscReal ds2 = dis/2.0;

    /* Build coordinates */
    ierr = DMGetCoordinateSection(*dm, &coordSection);CHKERRQ(ierr);
    ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldComponents(coordSection, 0, dim);CHKERRQ(ierr);
    ierr = PetscSectionSetChart(coordSection, numCells, numCells+numVertices);CHKERRQ(ierr);
    for (v = numCells; v < numCells+numVertices; ++v) {
      ierr = PetscSectionSetDof(coordSection, v, dim);CHKERRQ(ierr);
      ierr = PetscSectionSetFieldDof(coordSection, v, 0, dim);CHKERRQ(ierr);
    }
    ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
    ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
    ierr = VecCreate(PETSC_COMM_SELF, &coordinates);CHKERRQ(ierr);
    ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
    ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
    ierr = VecSetBlockSize(coordinates, dim);CHKERRQ(ierr);
    ierr = VecSetType(coordinates,VECSTANDARD);CHKERRQ(ierr);
    ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
    if (!rank) {
      coords[0*dim+0] = -ds2; coords[0*dim+1] = -ds2; coords[0*dim+2] = 0.0;
      coords[1*dim+0] =  ds2; coords[1*dim+1] = -ds2; coords[1*dim+2] = 0.0;
      coords[2*dim+0] =  ds2; coords[2*dim+1] =  ds2; coords[2*dim+2] = 0.0;
      coords[3*dim+0] = -ds2; coords[3*dim+1] =  ds2; coords[3*dim+2] = 0.0;
      coords[4*dim+0] = -ds2; coords[4*dim+1] = -ds2; coords[4*dim+2] = 1.0;
      coords[5*dim+0] = -ds2; coords[5*dim+1] =  ds2; coords[5*dim+2] = 1.0;
      coords[6*dim+0] =  ds2; coords[6*dim+1] =  ds2; coords[6*dim+2] = 1.0;
      coords[7*dim+0] =  ds2; coords[7*dim+1] = -ds2; coords[7*dim+2] = 1.0;
      coords[ 8*dim+0] =  dis; coords[ 8*dim+1] = -dis; coords[ 8*dim+2] = 0.0;
      coords[ 9*dim+0] =  dis; coords[ 9*dim+1] =  dis; coords[ 9*dim+2] = 0.0;
      coords[10*dim+0] =  dis; coords[10*dim+1] = -dis; coords[10*dim+2] = 1.0;
      coords[11*dim+0] =  dis; coords[11*dim+1] =  dis; coords[11*dim+2] = 1.0;
      coords[12*dim+0] = -dis; coords[12*dim+1] =  dis; coords[12*dim+2] = 0.0;
      coords[13*dim+0] = -dis; coords[13*dim+1] =  dis; coords[13*dim+2] = 1.0;
      coords[14*dim+0] = -dis; coords[14*dim+1] = -dis; coords[14*dim+2] = 0.0;
      coords[15*dim+0] = -dis; coords[15*dim+1] = -dis; coords[15*dim+2] = 1.0;
      if (periodicZ == DM_BOUNDARY_PERIODIC) {
        /* 15 31 19 */ coords[16*dim+0] = -ds2; coords[16*dim+1] = -ds2; coords[16*dim+2] = 0.5;
        /* 16 32 22 */ coords[17*dim+0] =  ds2; coords[17*dim+1] = -ds2; coords[17*dim+2] = 0.5;
        /* 17 33 21 */ coords[18*dim+0] =  ds2; coords[18*dim+1] =  ds2; coords[18*dim+2] = 0.5;
        /* 18 34 20 */ coords[19*dim+0] = -ds2; coords[19*dim+1] =  ds2; coords[19*dim+2] = 0.5;
        /* 29 35 30 */ coords[20*dim+0] = -dis; coords[20*dim+1] = -dis; coords[20*dim+2] = 0.5;
        /* 23 36 25 */ coords[21*dim+0] =  dis; coords[21*dim+1] = -dis; coords[21*dim+2] = 0.5;
        /* 24 37 26 */ coords[22*dim+0] =  dis; coords[22*dim+1] =  dis; coords[22*dim+2] = 0.5;
        /* 27 38 28 */ coords[23*dim+0] = -dis; coords[23*dim+1] =  dis; coords[23*dim+2] = 0.5;
      }
    }
    ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
    ierr = DMSetCoordinatesLocal(*dm, coordinates);CHKERRQ(ierr);
    ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  }
  /* Create periodicity */
  if (periodicZ == DM_BOUNDARY_PERIODIC || periodicZ == DM_BOUNDARY_TWIST) {
    PetscReal      L[3];
    PetscReal      maxCell[3];
    DMBoundaryType bdType[3];
    PetscReal      lower[3] = {0.0, 0.0, 0.0};
    PetscReal      upper[3] = {1.0, 1.0, 1.5};
    PetscInt       i, numZCells = 3;

    bdType[0] = DM_BOUNDARY_NONE;
    bdType[1] = DM_BOUNDARY_NONE;
    bdType[2] = periodicZ;
    for (i = 0; i < dim; i++) {
      L[i]       = upper[i] - lower[i];
      maxCell[i] = 1.1 * (L[i] / numZCells);
    }
    ierr = DMSetPeriodicity(*dm, PETSC_TRUE, maxCell, L, bdType);CHKERRQ(ierr);
  }
  /* Refine topology */
  for (r = 0; r < numRefine; ++r) {
    DM rdm = NULL;

    ierr = DMRefine(*dm, comm, &rdm);CHKERRQ(ierr);
    ierr = DMDestroy(dm);CHKERRQ(ierr);
    *dm  = rdm;
  }
  /* Remap geometry to cylinder
       Interior square: Linear interpolation is correct
       The other cells all have vertices on rays from the origin. We want to uniformly expand the spacing
       such that the last vertex is on the unit circle. So the closest and farthest vertices are at distance

         phi     = arctan(y/x)
         d_close = sqrt(1/8 + 1/4 sin^2(phi))
         d_far   = sqrt(1/2 + sin^2(phi))

       so we remap them using

         x_new = x_close + (x - x_close) (1 - d_close) / (d_far - d_close)
         y_new = y_close + (y - y_close) (1 - d_close) / (d_far - d_close)

       If pi/4 < phi < 3pi/4 or -3pi/4 < phi < -pi/4, then we switch x and y.
  */
  {
    Vec           coordinates;
    PetscSection  coordSection;
    PetscScalar  *coords;
    PetscInt      vStart, vEnd, v;
    const PetscReal dis = 1.0/PetscSqrtReal(2.0);
    const PetscReal ds2 = 0.5*dis;

    ierr = DMPlexGetDepthStratum(*dm, 0, &vStart, &vEnd);CHKERRQ(ierr);
    ierr = DMGetCoordinateSection(*dm, &coordSection);CHKERRQ(ierr);
    ierr = DMGetCoordinatesLocal(*dm, &coordinates);CHKERRQ(ierr);
    ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
    for (v = vStart; v < vEnd; ++v) {
      PetscReal phi, sinp, cosp, dc, df, x, y, xc, yc;
      PetscInt  off;

      ierr = PetscSectionGetOffset(coordSection, v, &off);CHKERRQ(ierr);
      if ((PetscAbsScalar(coords[off+0]) <= ds2) && (PetscAbsScalar(coords[off+1]) <= ds2)) continue;
      x    = PetscRealPart(coords[off]);
      y    = PetscRealPart(coords[off+1]);
      phi  = PetscAtan2Real(y, x);
      sinp = PetscSinReal(phi);
      cosp = PetscCosReal(phi);
      if ((PetscAbsReal(phi) > PETSC_PI/4.0) && (PetscAbsReal(phi) < 3.0*PETSC_PI/4.0)) {
        dc = PetscAbsReal(ds2/sinp);
        df = PetscAbsReal(dis/sinp);
        xc = ds2*x/PetscAbsReal(y);
        yc = ds2*PetscSignReal(y);
      } else {
        dc = PetscAbsReal(ds2/cosp);
        df = PetscAbsReal(dis/cosp);
        xc = ds2*PetscSignReal(x);
        yc = ds2*y/PetscAbsReal(x);
      }
      coords[off+0] = xc + (coords[off+0] - xc)*(1.0 - dc)/(df - dc);
      coords[off+1] = yc + (coords[off+1] - yc)*(1.0 - dc)/(df - dc);
    }
    ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
    if (periodicZ == DM_BOUNDARY_PERIODIC || periodicZ == DM_BOUNDARY_TWIST) {
      ierr = DMLocalizeCoordinates(*dm);CHKERRQ(ierr);
    }
  }
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateWedgeCylinderMesh - Creates a mesh on the tensor product of the unit interval with the circle (cylinder) using wedges.

  Collective

  Input Parameters:
+ comm - The communicator for the DM object
. n    - The number of wedges around the origin
- interpolate - Create edges and faces

  Output Parameter:
. dm  - The DM object

  Level: beginner

.seealso: DMPlexCreateHexCylinderMesh(), DMPlexCreateBoxMesh(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateWedgeCylinderMesh(MPI_Comm comm, PetscInt n, PetscBool interpolate, DM *dm)
{
  const PetscInt dim = 3;
  PetscInt       numCells, numVertices, v;
  PetscMPIInt    rank;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidPointer(dm, 3);
  ierr = MPI_Comm_rank(comm, &rank);CHKERRMPI(ierr);
  if (n < 0) SETERRQ1(comm, PETSC_ERR_ARG_OUTOFRANGE, "Number of wedges %D cannot be negative", n);
  ierr = DMCreate(comm, dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(*dm, dim);CHKERRQ(ierr);
  /* Must create the celltype label here so that we do not automatically try to compute the types */
  ierr = DMCreateLabel(*dm, "celltype");CHKERRQ(ierr);
  /* Create topology */
  {
    PetscInt cone[6], c;

    numCells    = !rank ?        n : 0;
    numVertices = !rank ?  2*(n+1) : 0;
    ierr = DMPlexSetChart(*dm, 0, numCells+numVertices);CHKERRQ(ierr);
    for (c = 0; c < numCells; c++) {ierr = DMPlexSetConeSize(*dm, c, 6);CHKERRQ(ierr);}
    ierr = DMSetUp(*dm);CHKERRQ(ierr);
    for (c = 0; c < numCells; c++) {
      cone[0] =  c+n*1; cone[1] = (c+1)%n+n*1; cone[2] = 0+3*n;
      cone[3] =  c+n*2; cone[4] = (c+1)%n+n*2; cone[5] = 1+3*n;
      ierr = DMPlexSetCone(*dm, c, cone);CHKERRQ(ierr);
      ierr = DMPlexSetCellType(*dm, c, DM_POLYTOPE_TRI_PRISM_TENSOR);CHKERRQ(ierr);
    }
    ierr = DMPlexSymmetrize(*dm);CHKERRQ(ierr);
    ierr = DMPlexStratify(*dm);CHKERRQ(ierr);
  }
  for (v = numCells; v < numCells+numVertices; ++v) {
    ierr = DMPlexSetCellType(*dm, v, DM_POLYTOPE_POINT);CHKERRQ(ierr);
  }
  /* Interpolate */
  if (interpolate) {
    DM idm;

    ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
    ierr = DMDestroy(dm);CHKERRQ(ierr);
    *dm  = idm;
  }
  /* Create cylinder geometry */
  {
    Vec          coordinates;
    PetscSection coordSection;
    PetscScalar *coords;
    PetscInt     coordSize, c;

    /* Build coordinates */
    ierr = DMGetCoordinateSection(*dm, &coordSection);CHKERRQ(ierr);
    ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldComponents(coordSection, 0, dim);CHKERRQ(ierr);
    ierr = PetscSectionSetChart(coordSection, numCells, numCells+numVertices);CHKERRQ(ierr);
    for (v = numCells; v < numCells+numVertices; ++v) {
      ierr = PetscSectionSetDof(coordSection, v, dim);CHKERRQ(ierr);
      ierr = PetscSectionSetFieldDof(coordSection, v, 0, dim);CHKERRQ(ierr);
    }
    ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
    ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
    ierr = VecCreate(PETSC_COMM_SELF, &coordinates);CHKERRQ(ierr);
    ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
    ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
    ierr = VecSetBlockSize(coordinates, dim);CHKERRQ(ierr);
    ierr = VecSetType(coordinates,VECSTANDARD);CHKERRQ(ierr);
    ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
    for (c = 0; c < numCells; c++) {
      coords[(c+0*n)*dim+0] = PetscCosReal(2.0*c*PETSC_PI/n); coords[(c+0*n)*dim+1] = PetscSinReal(2.0*c*PETSC_PI/n); coords[(c+0*n)*dim+2] = 1.0;
      coords[(c+1*n)*dim+0] = PetscCosReal(2.0*c*PETSC_PI/n); coords[(c+1*n)*dim+1] = PetscSinReal(2.0*c*PETSC_PI/n); coords[(c+1*n)*dim+2] = 0.0;
    }
    if (!rank) {
      coords[(2*n+0)*dim+0] = 0.0; coords[(2*n+0)*dim+1] = 0.0; coords[(2*n+0)*dim+2] = 1.0;
      coords[(2*n+1)*dim+0] = 0.0; coords[(2*n+1)*dim+1] = 0.0; coords[(2*n+1)*dim+2] = 0.0;
    }
    ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
    ierr = DMSetCoordinatesLocal(*dm, coordinates);CHKERRQ(ierr);
    ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

PETSC_STATIC_INLINE PetscReal DiffNormReal(PetscInt dim, const PetscReal x[], const PetscReal y[])
{
  PetscReal prod = 0.0;
  PetscInt  i;
  for (i = 0; i < dim; ++i) prod += PetscSqr(x[i] - y[i]);
  return PetscSqrtReal(prod);
}
PETSC_STATIC_INLINE PetscReal DotReal(PetscInt dim, const PetscReal x[], const PetscReal y[])
{
  PetscReal prod = 0.0;
  PetscInt  i;
  for (i = 0; i < dim; ++i) prod += x[i]*y[i];
  return prod;
}

/* The first constant is the sphere radius */
static void snapToSphere(PetscInt dim, PetscInt Nf, PetscInt NfAux,
                         const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[],
                         const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[],
                         PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  PetscReal r = PetscRealPart(constants[0]);
  PetscReal norm2 = 0.0, fac;
  PetscInt  n = uOff[1] - uOff[0], d;

  for (d = 0; d < n; ++d) norm2 += PetscSqr(PetscRealPart(u[d]));
  fac = r/PetscSqrtReal(norm2);
  for (d = 0; d < n; ++d) f0[d] = u[d]*fac;
}

/*@
  DMPlexCreateSphereMesh - Creates a mesh on the d-dimensional sphere, S^d.

  Collective

  Input Parameters:
+ comm    - The communicator for the DM object
. dim     - The dimension
. simplex - Use simplices, or tensor product cells
- R       - The radius

  Output Parameter:
. dm  - The DM object

  Level: beginner

.seealso: DMPlexCreateBallMesh(), DMPlexCreateBoxMesh(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateSphereMesh(MPI_Comm comm, PetscInt dim, PetscBool simplex, PetscReal R, DM *dm)
{
  const PetscInt  embedDim = dim+1;
  PetscSection    coordSection;
  Vec             coordinates;
  PetscScalar    *coords;
  PetscReal      *coordsIn;
  PetscInt        numCells, numEdges, numVerts, firstVertex, v, firstEdge, coordSize, d, c, e;
  PetscMPIInt     rank;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  PetscValidPointer(dm, 4);
  ierr = DMCreate(comm, dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(*dm, dim);CHKERRQ(ierr);
  ierr = DMSetCoordinateDim(*dm, dim+1);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(PetscObjectComm((PetscObject) *dm), &rank);CHKERRMPI(ierr);
  switch (dim) {
  case 2:
    if (simplex) {
      DM              idm;
      const PetscReal radius    = PetscSqrtReal(1 + PETSC_PHI*PETSC_PHI)/(1.0 + PETSC_PHI);
      const PetscReal edgeLen   = 2.0/(1.0 + PETSC_PHI) * (R/radius);
      const PetscInt  degree    = 5;
      PetscReal       vertex[3] = {0.0, 1.0/(1.0 + PETSC_PHI), PETSC_PHI/(1.0 + PETSC_PHI)};
      PetscInt        s[3]      = {1, 1, 1};
      PetscInt        cone[3];
      PetscInt       *graph, p, i, j, k;

      vertex[0] *= R/radius; vertex[1] *= R/radius; vertex[2] *= R/radius;
      numCells    = !rank ? 20 : 0;
      numVerts    = !rank ? 12 : 0;
      firstVertex = numCells;
      /* Use icosahedron, which for a R-sphere has coordinates which are all cyclic permutations of

           (0, \pm 1/\phi+1, \pm \phi/\phi+1)

         where \phi^2 - \phi - 1 = 0, meaning \phi is the golden ratio \frac{1 + \sqrt{5}}{2}. The edge
         length is then given by 2/(1+\phi) = 2 * 0.38197 = 0.76393.
      */
      /* Construct vertices */
      ierr = PetscCalloc1(numVerts * embedDim, &coordsIn);CHKERRQ(ierr);
      if (!rank) {
        for (p = 0, i = 0; p < embedDim; ++p) {
          for (s[1] = -1; s[1] < 2; s[1] += 2) {
            for (s[2] = -1; s[2] < 2; s[2] += 2) {
              for (d = 0; d < embedDim; ++d) coordsIn[i*embedDim+d] = s[(d+p)%embedDim]*vertex[(d+p)%embedDim];
              ++i;
            }
          }
        }
      }
      /* Construct graph */
      ierr = PetscCalloc1(numVerts * numVerts, &graph);CHKERRQ(ierr);
      for (i = 0; i < numVerts; ++i) {
        for (j = 0, k = 0; j < numVerts; ++j) {
          if (PetscAbsReal(DiffNormReal(embedDim, &coordsIn[i*embedDim], &coordsIn[j*embedDim]) - edgeLen) < PETSC_SMALL) {graph[i*numVerts+j] = 1; ++k;}
        }
        if (k != degree) SETERRQ3(comm, PETSC_ERR_PLIB, "Invalid icosahedron, vertex %D degree %D != %D", i, k, degree);
      }
      /* Build Topology */
      ierr = DMPlexSetChart(*dm, 0, numCells+numVerts);CHKERRQ(ierr);
      for (c = 0; c < numCells; c++) {
        ierr = DMPlexSetConeSize(*dm, c, embedDim);CHKERRQ(ierr);
      }
      ierr = DMSetUp(*dm);CHKERRQ(ierr); /* Allocate space for cones */
      /* Cells */
      for (i = 0, c = 0; i < numVerts; ++i) {
        for (j = 0; j < i; ++j) {
          for (k = 0; k < j; ++k) {
            if (graph[i*numVerts+j] && graph[j*numVerts+k] && graph[k*numVerts+i]) {
              cone[0] = firstVertex+i; cone[1] = firstVertex+j; cone[2] = firstVertex+k;
              /* Check orientation */
              {
                const PetscInt epsilon[3][3][3] = {{{0, 0, 0}, {0, 0, 1}, {0, -1, 0}}, {{0, 0, -1}, {0, 0, 0}, {1, 0, 0}}, {{0, 1, 0}, {-1, 0, 0}, {0, 0, 0}}};
                PetscReal normal[3];
                PetscInt  e, f;

                for (d = 0; d < embedDim; ++d) {
                  normal[d] = 0.0;
                  for (e = 0; e < embedDim; ++e) {
                    for (f = 0; f < embedDim; ++f) {
                      normal[d] += epsilon[d][e][f]*(coordsIn[j*embedDim+e] - coordsIn[i*embedDim+e])*(coordsIn[k*embedDim+f] - coordsIn[i*embedDim+f]);
                    }
                  }
                }
                if (DotReal(embedDim, normal, &coordsIn[i*embedDim]) < 0) {PetscInt tmp = cone[1]; cone[1] = cone[2]; cone[2] = tmp;}
              }
              ierr = DMPlexSetCone(*dm, c++, cone);CHKERRQ(ierr);
            }
          }
        }
      }
      ierr = DMPlexSymmetrize(*dm);CHKERRQ(ierr);
      ierr = DMPlexStratify(*dm);CHKERRQ(ierr);
      ierr = PetscFree(graph);CHKERRQ(ierr);
      /* Interpolate mesh */
      ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
      ierr = DMDestroy(dm);CHKERRQ(ierr);
      *dm  = idm;
    } else {
      /*
        12-21--13
         |     |
        25  4  24
         |     |
  12-25--9-16--8-24--13
   |     |     |     |
  23  5 17  0 15  3  22
   |     |     |     |
  10-20--6-14--7-19--11
         |     |
        20  1  19
         |     |
        10-18--11
         |     |
        23  2  22
         |     |
        12-21--13
       */
      PetscInt cone[4], ornt[4];

      numCells    = !rank ?  6 : 0;
      numEdges    = !rank ? 12 : 0;
      numVerts    = !rank ?  8 : 0;
      firstVertex = numCells;
      firstEdge   = numCells + numVerts;
      /* Build Topology */
      ierr = DMPlexSetChart(*dm, 0, numCells+numEdges+numVerts);CHKERRQ(ierr);
      for (c = 0; c < numCells; c++) {
        ierr = DMPlexSetConeSize(*dm, c, 4);CHKERRQ(ierr);
      }
      for (e = firstEdge; e < firstEdge+numEdges; ++e) {
        ierr = DMPlexSetConeSize(*dm, e, 2);CHKERRQ(ierr);
      }
      ierr = DMSetUp(*dm);CHKERRQ(ierr); /* Allocate space for cones */
      if (!rank) {
        /* Cell 0 */
        cone[0] = 14; cone[1] = 15; cone[2] = 16; cone[3] = 17;
        ierr = DMPlexSetCone(*dm, 0, cone);CHKERRQ(ierr);
        ornt[0] = 0; ornt[1] = 0; ornt[2] = 0; ornt[3] = 0;
        ierr = DMPlexSetConeOrientation(*dm, 0, ornt);CHKERRQ(ierr);
        /* Cell 1 */
        cone[0] = 18; cone[1] = 19; cone[2] = 14; cone[3] = 20;
        ierr = DMPlexSetCone(*dm, 1, cone);CHKERRQ(ierr);
        ornt[0] = 0; ornt[1] = 0; ornt[2] = -2; ornt[3] = 0;
        ierr = DMPlexSetConeOrientation(*dm, 1, ornt);CHKERRQ(ierr);
        /* Cell 2 */
        cone[0] = 21; cone[1] = 22; cone[2] = 18; cone[3] = 23;
        ierr = DMPlexSetCone(*dm, 2, cone);CHKERRQ(ierr);
        ornt[0] = 0; ornt[1] = 0; ornt[2] = -2; ornt[3] = 0;
        ierr = DMPlexSetConeOrientation(*dm, 2, ornt);CHKERRQ(ierr);
        /* Cell 3 */
        cone[0] = 19; cone[1] = 22; cone[2] = 24; cone[3] = 15;
        ierr = DMPlexSetCone(*dm, 3, cone);CHKERRQ(ierr);
        ornt[0] = -2; ornt[1] = -2; ornt[2] = 0; ornt[3] = -2;
        ierr = DMPlexSetConeOrientation(*dm, 3, ornt);CHKERRQ(ierr);
        /* Cell 4 */
        cone[0] = 16; cone[1] = 24; cone[2] = 21; cone[3] = 25;
        ierr = DMPlexSetCone(*dm, 4, cone);CHKERRQ(ierr);
        ornt[0] = -2; ornt[1] = -2; ornt[2] = -2; ornt[3] = 0;
        ierr = DMPlexSetConeOrientation(*dm, 4, ornt);CHKERRQ(ierr);
        /* Cell 5 */
        cone[0] = 20; cone[1] = 17; cone[2] = 25; cone[3] = 23;
        ierr = DMPlexSetCone(*dm, 5, cone);CHKERRQ(ierr);
        ornt[0] = -2; ornt[1] = -2; ornt[2] = -2; ornt[3] = -2;
        ierr = DMPlexSetConeOrientation(*dm, 5, ornt);CHKERRQ(ierr);
        /* Edges */
        cone[0] =  6; cone[1] =  7;
        ierr = DMPlexSetCone(*dm, 14, cone);CHKERRQ(ierr);
        cone[0] =  7; cone[1] =  8;
        ierr = DMPlexSetCone(*dm, 15, cone);CHKERRQ(ierr);
        cone[0] =  8; cone[1] =  9;
        ierr = DMPlexSetCone(*dm, 16, cone);CHKERRQ(ierr);
        cone[0] =  9; cone[1] =  6;
        ierr = DMPlexSetCone(*dm, 17, cone);CHKERRQ(ierr);
        cone[0] = 10; cone[1] = 11;
        ierr = DMPlexSetCone(*dm, 18, cone);CHKERRQ(ierr);
        cone[0] = 11; cone[1] =  7;
        ierr = DMPlexSetCone(*dm, 19, cone);CHKERRQ(ierr);
        cone[0] =  6; cone[1] = 10;
        ierr = DMPlexSetCone(*dm, 20, cone);CHKERRQ(ierr);
        cone[0] = 12; cone[1] = 13;
        ierr = DMPlexSetCone(*dm, 21, cone);CHKERRQ(ierr);
        cone[0] = 13; cone[1] = 11;
        ierr = DMPlexSetCone(*dm, 22, cone);CHKERRQ(ierr);
        cone[0] = 10; cone[1] = 12;
        ierr = DMPlexSetCone(*dm, 23, cone);CHKERRQ(ierr);
        cone[0] = 13; cone[1] =  8;
        ierr = DMPlexSetCone(*dm, 24, cone);CHKERRQ(ierr);
        cone[0] = 12; cone[1] =  9;
        ierr = DMPlexSetCone(*dm, 25, cone);CHKERRQ(ierr);
      }
      ierr = DMPlexSymmetrize(*dm);CHKERRQ(ierr);
      ierr = DMPlexStratify(*dm);CHKERRQ(ierr);
      /* Build coordinates */
      ierr = PetscCalloc1(numVerts * embedDim, &coordsIn);CHKERRQ(ierr);
      if (!rank) {
        coordsIn[0*embedDim+0] = -R; coordsIn[0*embedDim+1] =  R; coordsIn[0*embedDim+2] = -R;
        coordsIn[1*embedDim+0] =  R; coordsIn[1*embedDim+1] =  R; coordsIn[1*embedDim+2] = -R;
        coordsIn[2*embedDim+0] =  R; coordsIn[2*embedDim+1] = -R; coordsIn[2*embedDim+2] = -R;
        coordsIn[3*embedDim+0] = -R; coordsIn[3*embedDim+1] = -R; coordsIn[3*embedDim+2] = -R;
        coordsIn[4*embedDim+0] = -R; coordsIn[4*embedDim+1] =  R; coordsIn[4*embedDim+2] =  R;
        coordsIn[5*embedDim+0] =  R; coordsIn[5*embedDim+1] =  R; coordsIn[5*embedDim+2] =  R;
        coordsIn[6*embedDim+0] = -R; coordsIn[6*embedDim+1] = -R; coordsIn[6*embedDim+2] =  R;
        coordsIn[7*embedDim+0] =  R; coordsIn[7*embedDim+1] = -R; coordsIn[7*embedDim+2] =  R;
      }
    }
    break;
  case 3:
    if (simplex) {
      DM              idm;
      const PetscReal edgeLen         = 1.0/PETSC_PHI;
      PetscReal       vertexA[4]      = {0.5, 0.5, 0.5, 0.5};
      PetscReal       vertexB[4]      = {1.0, 0.0, 0.0, 0.0};
      PetscReal       vertexC[4]      = {0.5, 0.5*PETSC_PHI, 0.5/PETSC_PHI, 0.0};
      const PetscInt  degree          = 12;
      PetscInt        s[4]            = {1, 1, 1};
      PetscInt        evenPerm[12][4] = {{0, 1, 2, 3}, {0, 2, 3, 1}, {0, 3, 1, 2}, {1, 0, 3, 2}, {1, 2, 0, 3}, {1, 3, 2, 0},
                                         {2, 0, 1, 3}, {2, 1, 3, 0}, {2, 3, 0, 1}, {3, 0, 2, 1}, {3, 1, 0, 2}, {3, 2, 1, 0}};
      PetscInt        cone[4];
      PetscInt       *graph, p, i, j, k, l;

      vertexA[0] *= R; vertexA[1] *= R; vertexA[2] *= R; vertexA[3] *= R;
      vertexB[0] *= R; vertexB[1] *= R; vertexB[2] *= R; vertexB[3] *= R;
      vertexC[0] *= R; vertexC[1] *= R; vertexC[2] *= R; vertexC[3] *= R;
      numCells    = !rank ? 600 : 0;
      numVerts    = !rank ? 120 : 0;
      firstVertex = numCells;
      /* Use the 600-cell, which for a unit sphere has coordinates which are

           1/2 (\pm 1, \pm 1,    \pm 1, \pm 1)                          16
               (\pm 1,    0,       0,      0)  all cyclic permutations   8
           1/2 (\pm 1, \pm phi, \pm 1/phi, 0)  all even permutations    96

         where \phi^2 - \phi - 1 = 0, meaning \phi is the golden ratio \frac{1 + \sqrt{5}}{2}. The edge
         length is then given by 1/\phi = 0.61803.

         http://buzzard.pugetsound.edu/sage-practice/ch03s03.html
         http://mathworld.wolfram.com/600-Cell.html
      */
      /* Construct vertices */
      ierr = PetscCalloc1(numVerts * embedDim, &coordsIn);CHKERRQ(ierr);
      i    = 0;
      if (!rank) {
        for (s[0] = -1; s[0] < 2; s[0] += 2) {
          for (s[1] = -1; s[1] < 2; s[1] += 2) {
            for (s[2] = -1; s[2] < 2; s[2] += 2) {
              for (s[3] = -1; s[3] < 2; s[3] += 2) {
                for (d = 0; d < embedDim; ++d) coordsIn[i*embedDim+d] = s[d]*vertexA[d];
                ++i;
              }
            }
          }
        }
        for (p = 0; p < embedDim; ++p) {
          s[1] = s[2] = s[3] = 1;
          for (s[0] = -1; s[0] < 2; s[0] += 2) {
            for (d = 0; d < embedDim; ++d) coordsIn[i*embedDim+d] = s[(d+p)%embedDim]*vertexB[(d+p)%embedDim];
            ++i;
          }
        }
        for (p = 0; p < 12; ++p) {
          s[3] = 1;
          for (s[0] = -1; s[0] < 2; s[0] += 2) {
            for (s[1] = -1; s[1] < 2; s[1] += 2) {
              for (s[2] = -1; s[2] < 2; s[2] += 2) {
                for (d = 0; d < embedDim; ++d) coordsIn[i*embedDim+d] = s[evenPerm[p][d]]*vertexC[evenPerm[p][d]];
                ++i;
              }
            }
          }
        }
      }
      if (i != numVerts) SETERRQ2(comm, PETSC_ERR_PLIB, "Invalid 600-cell, vertices %D != %D", i, numVerts);
      /* Construct graph */
      ierr = PetscCalloc1(numVerts * numVerts, &graph);CHKERRQ(ierr);
      for (i = 0; i < numVerts; ++i) {
        for (j = 0, k = 0; j < numVerts; ++j) {
          if (PetscAbsReal(DiffNormReal(embedDim, &coordsIn[i*embedDim], &coordsIn[j*embedDim]) - edgeLen) < PETSC_SMALL) {graph[i*numVerts+j] = 1; ++k;}
        }
        if (k != degree) SETERRQ3(comm, PETSC_ERR_PLIB, "Invalid 600-cell, vertex %D degree %D != %D", i, k, degree);
      }
      /* Build Topology */
      ierr = DMPlexSetChart(*dm, 0, numCells+numVerts);CHKERRQ(ierr);
      for (c = 0; c < numCells; c++) {
        ierr = DMPlexSetConeSize(*dm, c, embedDim);CHKERRQ(ierr);
      }
      ierr = DMSetUp(*dm);CHKERRQ(ierr); /* Allocate space for cones */
      /* Cells */
      if (!rank) {
        for (i = 0, c = 0; i < numVerts; ++i) {
          for (j = 0; j < i; ++j) {
            for (k = 0; k < j; ++k) {
              for (l = 0; l < k; ++l) {
                if (graph[i*numVerts+j] && graph[j*numVerts+k] && graph[k*numVerts+i] &&
                    graph[l*numVerts+i] && graph[l*numVerts+j] && graph[l*numVerts+k]) {
                  cone[0] = firstVertex+i; cone[1] = firstVertex+j; cone[2] = firstVertex+k; cone[3] = firstVertex+l;
                  /* Check orientation: https://ef.gy/linear-algebra:normal-vectors-in-higher-dimensional-spaces */
                  {
                    const PetscInt epsilon[4][4][4][4] = {{{{0,  0,  0,  0}, { 0, 0,  0,  0}, { 0,  0, 0,  0}, { 0,  0,  0, 0}},
                                                           {{0,  0,  0,  0}, { 0, 0,  0,  0}, { 0,  0, 0,  1}, { 0,  0, -1, 0}},
                                                           {{0,  0,  0,  0}, { 0, 0,  0, -1}, { 0,  0, 0,  0}, { 0,  1,  0, 0}},
                                                           {{0,  0,  0,  0}, { 0, 0,  1,  0}, { 0, -1, 0,  0}, { 0,  0,  0, 0}}},

                                                          {{{0,  0,  0,  0}, { 0, 0,  0,  0}, { 0,  0, 0, -1}, { 0,  0,  1, 0}},
                                                           {{0,  0,  0,  0}, { 0, 0,  0,  0}, { 0,  0, 0,  0}, { 0,  0,  0, 0}},
                                                           {{0,  0,  0,  1}, { 0, 0,  0,  0}, { 0,  0, 0,  0}, {-1,  0,  0, 0}},
                                                           {{0,  0, -1,  0}, { 0, 0,  0,  0}, { 1,  0, 0,  0}, { 0,  0,  0, 0}}},

                                                          {{{0,  0,  0,  0}, { 0, 0,  0,  1}, { 0,  0, 0,  0}, { 0, -1,  0, 0}},
                                                           {{0,  0,  0, -1}, { 0, 0,  0,  0}, { 0,  0, 0,  0}, { 1,  0,  0, 0}},
                                                           {{0,  0,  0,  0}, { 0, 0,  0,  0}, { 0,  0, 0,  0}, { 0,  0,  0, 0}},
                                                           {{0,  1,  0,  0}, {-1, 0,  0,  0}, { 0,  0, 0,  0}, { 0,  0,  0, 0}}},

                                                          {{{0,  0,  0,  0}, { 0, 0, -1,  0}, { 0,  1, 0,  0}, { 0,  0,  0, 0}},
                                                           {{0,  0,  1,  0}, { 0, 0,  0,  0}, {-1,  0, 0,  0}, { 0,  0,  0, 0}},
                                                           {{0, -1,  0,  0}, { 1, 0,  0,  0}, { 0,  0, 0,  0}, { 0,  0,  0, 0}},
                                                           {{0,  0,  0,  0}, { 0, 0,  0,  0}, { 0,  0, 0,  0}, { 0,  0,  0, 0}}}};
                    PetscReal normal[4];
                    PetscInt  e, f, g;

                    for (d = 0; d < embedDim; ++d) {
                      normal[d] = 0.0;
                      for (e = 0; e < embedDim; ++e) {
                        for (f = 0; f < embedDim; ++f) {
                          for (g = 0; g < embedDim; ++g) {
                            normal[d] += epsilon[d][e][f][g]*(coordsIn[j*embedDim+e] - coordsIn[i*embedDim+e])*(coordsIn[k*embedDim+f] - coordsIn[i*embedDim+f])*(coordsIn[l*embedDim+f] - coordsIn[i*embedDim+f]);
                          }
                        }
                      }
                    }
                    if (DotReal(embedDim, normal, &coordsIn[i*embedDim]) < 0) {PetscInt tmp = cone[1]; cone[1] = cone[2]; cone[2] = tmp;}
                  }
                  ierr = DMPlexSetCone(*dm, c++, cone);CHKERRQ(ierr);
                }
              }
            }
          }
        }
      }
      ierr = DMPlexSymmetrize(*dm);CHKERRQ(ierr);
      ierr = DMPlexStratify(*dm);CHKERRQ(ierr);
      ierr = PetscFree(graph);CHKERRQ(ierr);
      /* Interpolate mesh */
      ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
      ierr = DMDestroy(dm);CHKERRQ(ierr);
      *dm  = idm;
      break;
    }
  default: SETERRQ1(comm, PETSC_ERR_SUP, "Unsupported dimension for sphere: %D", dim);
  }
  /* Create coordinates */
  ierr = DMGetCoordinateSection(*dm, &coordSection);CHKERRQ(ierr);
  ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
  ierr = PetscSectionSetFieldComponents(coordSection, 0, embedDim);CHKERRQ(ierr);
  ierr = PetscSectionSetChart(coordSection, firstVertex, firstVertex+numVerts);CHKERRQ(ierr);
  for (v = firstVertex; v < firstVertex+numVerts; ++v) {
    ierr = PetscSectionSetDof(coordSection, v, embedDim);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldDof(coordSection, v, 0, embedDim);CHKERRQ(ierr);
  }
  ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
  ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
  ierr = VecCreate(PETSC_COMM_SELF, &coordinates);CHKERRQ(ierr);
  ierr = VecSetBlockSize(coordinates, embedDim);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
  ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetType(coordinates,VECSTANDARD);CHKERRQ(ierr);
  ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
  for (v = 0; v < numVerts; ++v) for (d = 0; d < embedDim; ++d) {coords[v*embedDim+d] = coordsIn[v*embedDim+d];}
  ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(*dm, coordinates);CHKERRQ(ierr);
  ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  ierr = PetscFree(coordsIn);CHKERRQ(ierr);
  /* Create coordinate function space */
  {
    DM          cdm;
    PetscDS     cds;
    PetscFE     fe;
    PetscScalar radius = R;
    PetscInt    dT, dE;

    ierr = DMGetCoordinateDM(*dm, &cdm);CHKERRQ(ierr);
    ierr = DMGetDimension(*dm, &dT);CHKERRQ(ierr);
    ierr = DMGetCoordinateDim(*dm, &dE);CHKERRQ(ierr);
    ierr = PetscFECreateLagrange(PETSC_COMM_SELF, dT, dE, simplex, 1, -1, &fe);CHKERRQ(ierr);
    ierr = DMSetField(cdm, 0, NULL, (PetscObject) fe);CHKERRQ(ierr);
    ierr = PetscFEDestroy(&fe);CHKERRQ(ierr);
    ierr = DMCreateDS(cdm);CHKERRQ(ierr);

    ierr = DMGetDS(cdm, &cds);CHKERRQ(ierr);
    ierr = PetscDSSetConstants(cds, 1, &radius);CHKERRQ(ierr);
  }
  ((DM_Plex *) (*dm)->data)->coordFunc = snapToSphere;
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateBallMesh - Creates a simplex mesh on the d-dimensional ball, B^d.

  Collective

  Input Parameters:
+ comm  - The communicator for the DM object
. dim   - The dimension
- R     - The radius

  Output Parameter:
. dm  - The DM object

  Options Database Keys:
- bd_dm_refine - This will refine the surface mesh preserving the sphere geometry

  Level: beginner

.seealso: DMPlexCreateSphereMesh(), DMPlexCreateBoxMesh(), DMSetType(), DMCreate()
@*/
PetscErrorCode DMPlexCreateBallMesh(MPI_Comm comm, PetscInt dim, PetscReal R, DM *dm)
{
  DM             sdm;
  DMLabel        bdlabel;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMPlexCreateSphereMesh(comm, dim-1, PETSC_TRUE, R, &sdm);CHKERRQ(ierr);
  ierr = PetscObjectSetOptionsPrefix((PetscObject) sdm, "bd_");CHKERRQ(ierr);
  ierr = DMSetFromOptions(sdm);CHKERRQ(ierr);
  ierr = DMPlexGenerate(sdm, NULL, PETSC_TRUE, dm);CHKERRQ(ierr);
  ierr = DMDestroy(&sdm);CHKERRQ(ierr);
  ierr = DMCreateLabel(*dm, "marker");CHKERRQ(ierr);
  ierr = DMGetLabel(*dm, "marker", &bdlabel);CHKERRQ(ierr);
  ierr = DMPlexMarkBoundaryFaces(*dm, PETSC_DETERMINE, bdlabel);CHKERRQ(ierr);
  ierr = DMPlexLabelComplete(*dm, bdlabel);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* External function declarations here */
extern PetscErrorCode DMCreateInterpolation_Plex(DM dmCoarse, DM dmFine, Mat *interpolation, Vec *scaling);
extern PetscErrorCode DMCreateInjection_Plex(DM dmCoarse, DM dmFine, Mat *mat);
extern PetscErrorCode DMCreateMassMatrix_Plex(DM dmCoarse, DM dmFine, Mat *mat);
extern PetscErrorCode DMCreateLocalSection_Plex(DM dm);
extern PetscErrorCode DMCreateDefaultConstraints_Plex(DM dm);
extern PetscErrorCode DMCreateMatrix_Plex(DM dm,  Mat *J);
extern PetscErrorCode DMCreateCoordinateDM_Plex(DM dm, DM *cdm);
extern PetscErrorCode DMCreateCoordinateField_Plex(DM dm, DMField *field);
PETSC_INTERN PetscErrorCode DMClone_Plex(DM dm, DM *newdm);
extern PetscErrorCode DMSetUp_Plex(DM dm);
extern PetscErrorCode DMDestroy_Plex(DM dm);
extern PetscErrorCode DMView_Plex(DM dm, PetscViewer viewer);
extern PetscErrorCode DMLoad_Plex(DM dm, PetscViewer viewer);
extern PetscErrorCode DMCreateSubDM_Plex(DM dm, PetscInt numFields, const PetscInt fields[], IS *is, DM *subdm);
extern PetscErrorCode DMCreateSuperDM_Plex(DM dms[], PetscInt len, IS **is, DM *superdm);
static PetscErrorCode DMInitialize_Plex(DM dm);

/* Replace dm with the contents of dmNew
   - Share the DM_Plex structure
   - Share the coordinates
   - Share the SF
*/
static PetscErrorCode DMPlexReplace_Static(DM dm, DM dmNew)
{
  PetscSF               sf;
  DM                    coordDM, coarseDM;
  DMField               coordField;
  Vec                   coords;
  PetscBool             isper;
  const PetscReal      *maxCell, *L;
  const DMBoundaryType *bd;
  PetscErrorCode        ierr;

  PetscFunctionBegin;
  ierr = DMGetPointSF(dmNew, &sf);CHKERRQ(ierr);
  ierr = DMSetPointSF(dm, sf);CHKERRQ(ierr);
  ierr = DMGetCoordinateDM(dmNew, &coordDM);CHKERRQ(ierr);
  ierr = DMGetCoordinatesLocal(dmNew, &coords);CHKERRQ(ierr);
  ierr = DMGetCoordinateField(dmNew, &coordField);CHKERRQ(ierr);
  ierr = DMSetCoordinateDM(dm, coordDM);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(dm, coords);CHKERRQ(ierr);
  ierr = DMSetCoordinateField(dm, coordField);CHKERRQ(ierr);
  ierr = DMGetPeriodicity(dm, &isper, &maxCell, &L, &bd);CHKERRQ(ierr);
  ierr = DMSetPeriodicity(dmNew, isper, maxCell, L, bd);CHKERRQ(ierr);
  ierr = DMDestroy_Plex(dm);CHKERRQ(ierr);
  ierr = DMInitialize_Plex(dm);CHKERRQ(ierr);
  dm->data = dmNew->data;
  ((DM_Plex *) dmNew->data)->refct++;
  ierr = DMDestroyLabelLinkList_Internal(dm);CHKERRQ(ierr);
  ierr = DMCopyLabels(dmNew, dm, PETSC_OWN_POINTER, PETSC_TRUE);CHKERRQ(ierr);
  ierr = DMGetCoarseDM(dmNew,&coarseDM);CHKERRQ(ierr);
  ierr = DMSetCoarseDM(dm,coarseDM);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Swap dm with the contents of dmNew
   - Swap the DM_Plex structure
   - Swap the coordinates
   - Swap the point PetscSF
*/
static PetscErrorCode DMPlexSwap_Static(DM dmA, DM dmB)
{
  DM              coordDMA, coordDMB;
  Vec             coordsA,  coordsB;
  PetscSF         sfA,      sfB;
  void            *tmp;
  DMLabelLink     listTmp;
  DMLabel         depthTmp;
  PetscInt        tmpI;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = DMGetPointSF(dmA, &sfA);CHKERRQ(ierr);
  ierr = DMGetPointSF(dmB, &sfB);CHKERRQ(ierr);
  ierr = PetscObjectReference((PetscObject) sfA);CHKERRQ(ierr);
  ierr = DMSetPointSF(dmA, sfB);CHKERRQ(ierr);
  ierr = DMSetPointSF(dmB, sfA);CHKERRQ(ierr);
  ierr = PetscObjectDereference((PetscObject) sfA);CHKERRQ(ierr);

  ierr = DMGetCoordinateDM(dmA, &coordDMA);CHKERRQ(ierr);
  ierr = DMGetCoordinateDM(dmB, &coordDMB);CHKERRQ(ierr);
  ierr = PetscObjectReference((PetscObject) coordDMA);CHKERRQ(ierr);
  ierr = DMSetCoordinateDM(dmA, coordDMB);CHKERRQ(ierr);
  ierr = DMSetCoordinateDM(dmB, coordDMA);CHKERRQ(ierr);
  ierr = PetscObjectDereference((PetscObject) coordDMA);CHKERRQ(ierr);

  ierr = DMGetCoordinatesLocal(dmA, &coordsA);CHKERRQ(ierr);
  ierr = DMGetCoordinatesLocal(dmB, &coordsB);CHKERRQ(ierr);
  ierr = PetscObjectReference((PetscObject) coordsA);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(dmA, coordsB);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(dmB, coordsA);CHKERRQ(ierr);
  ierr = PetscObjectDereference((PetscObject) coordsA);CHKERRQ(ierr);

  tmp       = dmA->data;
  dmA->data = dmB->data;
  dmB->data = tmp;
  listTmp   = dmA->labels;
  dmA->labels = dmB->labels;
  dmB->labels = listTmp;
  depthTmp  = dmA->depthLabel;
  dmA->depthLabel = dmB->depthLabel;
  dmB->depthLabel = depthTmp;
  depthTmp  = dmA->celltypeLabel;
  dmA->celltypeLabel = dmB->celltypeLabel;
  dmB->celltypeLabel = depthTmp;
  tmpI         = dmA->levelup;
  dmA->levelup = dmB->levelup;
  dmB->levelup = tmpI;
  PetscFunctionReturn(0);
}

PetscErrorCode DMSetFromOptions_NonRefinement_Plex(PetscOptionItems *PetscOptionsObject,DM dm)
{
  DM_Plex       *mesh = (DM_Plex*) dm->data;
  PetscBool      flg;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  /* Handle viewing */
  ierr = PetscOptionsBool("-dm_plex_print_set_values", "Output all set values info", "DMPlexMatSetClosure", PETSC_FALSE, &mesh->printSetValues, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBoundedInt("-dm_plex_print_fem", "Debug output level all fem computations", "DMPlexSNESComputeResidualFEM", 0, &mesh->printFEM, NULL,0);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-dm_plex_print_tol", "Tolerance for FEM output", "DMPlexSNESComputeResidualFEM", mesh->printTol, &mesh->printTol, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBoundedInt("-dm_plex_print_l2", "Debug output level all L2 diff computations", "DMComputeL2Diff", 0, &mesh->printL2, NULL,0);CHKERRQ(ierr);
  ierr = DMMonitorSetFromOptions(dm, "-dm_plex_monitor_throughput", "Monitor the simulation throughput", "DMPlexMonitorThroughput", DMPlexMonitorThroughput, NULL, &flg);CHKERRQ(ierr);
  if (flg) {ierr = PetscLogDefaultBegin();CHKERRQ(ierr);}
  /* Point Location */
  ierr = PetscOptionsBool("-dm_plex_hash_location", "Use grid hashing for point location", "DMInterpolate", PETSC_FALSE, &mesh->useHashLocation, NULL);CHKERRQ(ierr);
  /* Partitioning and distribution */
  ierr = PetscOptionsBool("-dm_plex_partition_balance", "Attempt to evenly divide points on partition boundary between processes", "DMPlexSetPartitionBalance", PETSC_FALSE, &mesh->partitionBalance, NULL);CHKERRQ(ierr);
  /* Generation and remeshing */
  ierr = PetscOptionsBool("-dm_plex_remesh_bd", "Allow changes to the boundary on remeshing", "DMAdapt", PETSC_FALSE, &mesh->remeshBd, NULL);CHKERRQ(ierr);
  /* Projection behavior */
  ierr = PetscOptionsBoundedInt("-dm_plex_max_projection_height", "Maxmimum mesh point height used to project locally", "DMPlexSetMaxProjectionHeight", 0, &mesh->maxProjectionHeight, NULL,0);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-dm_plex_regular_refinement", "Use special nested projection algorithm for regular refinement", "DMPlexSetRegularRefinement", mesh->regularRefinement, &mesh->regularRefinement, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnum("-dm_plex_cell_refiner", "Strategy for cell refinment", "ex40.c", DMPlexCellRefinerTypes, (PetscEnum) mesh->cellRefiner, (PetscEnum *) &mesh->cellRefiner, NULL);CHKERRQ(ierr);
  /* Checking structure */
  {
    PetscBool   flg = PETSC_FALSE, flg2 = PETSC_FALSE, all = PETSC_FALSE;

    ierr = PetscOptionsBool("-dm_plex_check_all", "Perform all checks", NULL, PETSC_FALSE, &all, &flg2);CHKERRQ(ierr);
    ierr = PetscOptionsBool("-dm_plex_check_symmetry", "Check that the adjacency information in the mesh is symmetric", "DMPlexCheckSymmetry", PETSC_FALSE, &flg, &flg2);CHKERRQ(ierr);
    if (all || (flg && flg2)) {ierr = DMPlexCheckSymmetry(dm);CHKERRQ(ierr);}
    ierr = PetscOptionsBool("-dm_plex_check_skeleton", "Check that each cell has the correct number of vertices (only for homogeneous simplex or tensor meshes)", "DMPlexCheckSkeleton", PETSC_FALSE, &flg, &flg2);CHKERRQ(ierr);
    if (all || (flg && flg2)) {ierr = DMPlexCheckSkeleton(dm, 0);CHKERRQ(ierr);}
    ierr = PetscOptionsBool("-dm_plex_check_faces", "Check that the faces of each cell give a vertex order this is consistent with what we expect from the cell type", "DMPlexCheckFaces", PETSC_FALSE, &flg, &flg2);CHKERRQ(ierr);
    if (all || (flg && flg2)) {ierr = DMPlexCheckFaces(dm, 0);CHKERRQ(ierr);}
    ierr = PetscOptionsBool("-dm_plex_check_geometry", "Check that cells have positive volume", "DMPlexCheckGeometry", PETSC_FALSE, &flg, &flg2);CHKERRQ(ierr);
    if (all || (flg && flg2)) {ierr = DMPlexCheckGeometry(dm);CHKERRQ(ierr);}
    ierr = PetscOptionsBool("-dm_plex_check_pointsf", "Check some necessary conditions for PointSF", "DMPlexCheckPointSF", PETSC_FALSE, &flg, &flg2);CHKERRQ(ierr);
    if (all || (flg && flg2)) {ierr = DMPlexCheckPointSF(dm);CHKERRQ(ierr);}
    ierr = PetscOptionsBool("-dm_plex_check_interface_cones", "Check points on inter-partition interfaces have conforming order of cone points", "DMPlexCheckInterfaceCones", PETSC_FALSE, &flg, &flg2);CHKERRQ(ierr);
    if (all || (flg && flg2)) {ierr = DMPlexCheckInterfaceCones(dm);CHKERRQ(ierr);}
    ierr = PetscOptionsBool("-dm_plex_check_cell_shape", "Check cell shape", "DMPlexCheckCellShape", PETSC_FALSE, &flg, &flg2);CHKERRQ(ierr);
    if (flg && flg2) {ierr = DMPlexCheckCellShape(dm, PETSC_TRUE, PETSC_DETERMINE);CHKERRQ(ierr);}
  }

  ierr = PetscPartitionerSetFromOptions(mesh->partitioner);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode DMSetFromOptions_Plex(PetscOptionItems *PetscOptionsObject,DM dm)
{
  PetscReal      volume = -1.0;
  PetscInt       prerefine = 0, refine = 0, r, coarsen = 0, overlap = 0;
  PetscBool      uniformOrig, uniform = PETSC_TRUE, distribute = PETSC_FALSE, isHierarchy, flg;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  ierr = PetscOptionsHead(PetscOptionsObject,"DMPlex Options");CHKERRQ(ierr);
  /* Handle DMPlex refinement before distribution */
  ierr = DMPlexGetRefinementUniform(dm, &uniformOrig);CHKERRQ(ierr);
  ierr = PetscOptionsBool("-dm_refine_uniform_pre", "Flag for uniform refinement before distribution", "DMCreate", uniform, &uniform, &flg);CHKERRQ(ierr);
  if (flg) {ierr = DMPlexSetRefinementUniform(dm, uniform);CHKERRQ(ierr);}
  ierr = PetscOptionsReal("-dm_refine_volume_limit_pre", "The maximum cell volume after refinement before distribution", "DMCreate", volume, &volume, &flg);CHKERRQ(ierr);
  if (flg) {ierr = DMPlexSetRefinementLimit(dm, volume);CHKERRQ(ierr);}
  ierr = PetscOptionsBoundedInt("-dm_refine_pre", "The number of refinements before distribution", "DMCreate", prerefine, &prerefine, NULL,0);CHKERRQ(ierr);
  for (r = 0; r < prerefine; ++r) {
    DM             rdm;
    PetscPointFunc coordFunc = ((DM_Plex*) dm->data)->coordFunc;

    ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dm);CHKERRQ(ierr);
    ierr = DMRefine(dm, PetscObjectComm((PetscObject) dm), &rdm);CHKERRQ(ierr);
    /* Total hack since we do not pass in a pointer */
    ierr = DMPlexReplace_Static(dm, rdm);CHKERRQ(ierr);
    ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dm);CHKERRQ(ierr);
    if (coordFunc) {
      ierr = DMPlexRemapGeometry(dm, 0.0, coordFunc);CHKERRQ(ierr);
      ((DM_Plex*) dm->data)->coordFunc = coordFunc;
    }
    ierr = DMDestroy(&rdm);CHKERRQ(ierr);
  }
  ierr = DMPlexSetRefinementUniform(dm, uniformOrig);CHKERRQ(ierr);
  /* Handle DMPlex distribution */
  ierr = PetscOptionsBool("-dm_distribute", "Flag to redistribute a mesh among processes", "DMCreate", distribute, &distribute, NULL);CHKERRQ(ierr);
  ierr = PetscOptionsBoundedInt("-dm_distribute_overlap", "The size of the overlap halo", "DMCreate", overlap, &overlap, NULL, 0);CHKERRQ(ierr);
  if (distribute) {
    DM               pdm = NULL;
    PetscPartitioner part;

    ierr = DMPlexGetPartitioner(dm, &part);CHKERRQ(ierr);
    ierr = PetscPartitionerSetFromOptions(part);CHKERRQ(ierr);
    ierr = DMPlexDistribute(dm, overlap, NULL, &pdm);CHKERRQ(ierr);
    if (pdm) {
      ierr = DMPlexReplace_Static(dm, pdm);CHKERRQ(ierr);
      ierr = DMDestroy(&pdm);CHKERRQ(ierr);
    }
  }
  /* Handle DMPlex refinement */
  ierr = PetscOptionsBoundedInt("-dm_refine", "The number of uniform refinements", "DMCreate", refine, &refine, NULL,0);CHKERRQ(ierr);
  ierr = PetscOptionsBoundedInt("-dm_refine_hierarchy", "The number of uniform refinements", "DMCreate", refine, &refine, &isHierarchy,0);CHKERRQ(ierr);
  if (refine) {ierr = DMPlexSetRefinementUniform(dm, PETSC_TRUE);CHKERRQ(ierr);}
  if (refine && isHierarchy) {
    DM *dms, coarseDM;

    ierr = DMGetCoarseDM(dm, &coarseDM);CHKERRQ(ierr);
    ierr = PetscObjectReference((PetscObject)coarseDM);CHKERRQ(ierr);
    ierr = PetscMalloc1(refine,&dms);CHKERRQ(ierr);
    ierr = DMRefineHierarchy(dm, refine, dms);CHKERRQ(ierr);
    /* Total hack since we do not pass in a pointer */
    ierr = DMPlexSwap_Static(dm, dms[refine-1]);CHKERRQ(ierr);
    if (refine == 1) {
      ierr = DMSetCoarseDM(dm, dms[0]);CHKERRQ(ierr);
      ierr = DMPlexSetRegularRefinement(dm, PETSC_TRUE);CHKERRQ(ierr);
    } else {
      ierr = DMSetCoarseDM(dm, dms[refine-2]);CHKERRQ(ierr);
      ierr = DMPlexSetRegularRefinement(dm, PETSC_TRUE);CHKERRQ(ierr);
      ierr = DMSetCoarseDM(dms[0], dms[refine-1]);CHKERRQ(ierr);
      ierr = DMPlexSetRegularRefinement(dms[0], PETSC_TRUE);CHKERRQ(ierr);
    }
    ierr = DMSetCoarseDM(dms[refine-1], coarseDM);CHKERRQ(ierr);
    ierr = PetscObjectDereference((PetscObject)coarseDM);CHKERRQ(ierr);
    /* Free DMs */
    for (r = 0; r < refine; ++r) {
      ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dms[r]);CHKERRQ(ierr);
      ierr = DMDestroy(&dms[r]);CHKERRQ(ierr);
    }
    ierr = PetscFree(dms);CHKERRQ(ierr);
  } else {
    for (r = 0; r < refine; ++r) {
      DM refinedMesh;
      PetscPointFunc coordFunc = ((DM_Plex*) dm->data)->coordFunc;

      ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dm);CHKERRQ(ierr);
      ierr = DMRefine(dm, PetscObjectComm((PetscObject) dm), &refinedMesh);CHKERRQ(ierr);
      /* Total hack since we do not pass in a pointer */
      ierr = DMPlexReplace_Static(dm, refinedMesh);CHKERRQ(ierr);
      ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dm);CHKERRQ(ierr);
      if (coordFunc) {
        ierr = DMPlexRemapGeometry(dm, 0.0, coordFunc);CHKERRQ(ierr);
        ((DM_Plex*) dm->data)->coordFunc = coordFunc;
      }
      ierr = DMDestroy(&refinedMesh);CHKERRQ(ierr);
    }
  }
  /* Handle DMPlex coarsening */
  ierr = PetscOptionsBoundedInt("-dm_coarsen", "Coarsen the mesh", "DMCreate", coarsen, &coarsen, NULL,0);CHKERRQ(ierr);
  ierr = PetscOptionsBoundedInt("-dm_coarsen_hierarchy", "The number of coarsenings", "DMCreate", coarsen, &coarsen, &isHierarchy,0);CHKERRQ(ierr);
  if (coarsen && isHierarchy) {
    DM *dms;

    ierr = PetscMalloc1(coarsen, &dms);CHKERRQ(ierr);
    ierr = DMCoarsenHierarchy(dm, coarsen, dms);CHKERRQ(ierr);
    /* Free DMs */
    for (r = 0; r < coarsen; ++r) {
      ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dms[r]);CHKERRQ(ierr);
      ierr = DMDestroy(&dms[r]);CHKERRQ(ierr);
    }
    ierr = PetscFree(dms);CHKERRQ(ierr);
  } else {
    for (r = 0; r < coarsen; ++r) {
      DM coarseMesh;

      ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dm);CHKERRQ(ierr);
      ierr = DMCoarsen(dm, PetscObjectComm((PetscObject) dm), &coarseMesh);CHKERRQ(ierr);
      /* Total hack since we do not pass in a pointer */
      ierr = DMPlexReplace_Static(dm, coarseMesh);CHKERRQ(ierr);
      ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dm);CHKERRQ(ierr);
      ierr = DMDestroy(&coarseMesh);CHKERRQ(ierr);
    }
  }
  /* Handle */
  ierr = DMSetFromOptions_NonRefinement_Plex(PetscOptionsObject, dm);CHKERRQ(ierr);
  ierr = PetscOptionsTail();CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode DMCreateGlobalVector_Plex(DM dm,Vec *vec)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMCreateGlobalVector_Section_Private(dm,vec);CHKERRQ(ierr);
  /* ierr = VecSetOperation(*vec, VECOP_DUPLICATE, (void(*)(void)) VecDuplicate_MPI_DM);CHKERRQ(ierr); */
  ierr = VecSetOperation(*vec, VECOP_VIEW, (void (*)(void)) VecView_Plex);CHKERRQ(ierr);
  ierr = VecSetOperation(*vec, VECOP_VIEWNATIVE, (void (*)(void)) VecView_Plex_Native);CHKERRQ(ierr);
  ierr = VecSetOperation(*vec, VECOP_LOAD, (void (*)(void)) VecLoad_Plex);CHKERRQ(ierr);
  ierr = VecSetOperation(*vec, VECOP_LOADNATIVE, (void (*)(void)) VecLoad_Plex_Native);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode DMCreateLocalVector_Plex(DM dm,Vec *vec)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMCreateLocalVector_Section_Private(dm,vec);CHKERRQ(ierr);
  ierr = VecSetOperation(*vec, VECOP_VIEW, (void (*)(void)) VecView_Plex_Local);CHKERRQ(ierr);
  ierr = VecSetOperation(*vec, VECOP_LOAD, (void (*)(void)) VecLoad_Plex_Local);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

static PetscErrorCode DMGetDimPoints_Plex(DM dm, PetscInt dim, PetscInt *pStart, PetscInt *pEnd)
{
  PetscInt       depth, d;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMPlexGetDepth(dm, &depth);CHKERRQ(ierr);
  if (depth == 1) {
    ierr = DMGetDimension(dm, &d);CHKERRQ(ierr);
    if (dim == 0)      {ierr = DMPlexGetDepthStratum(dm, dim, pStart, pEnd);CHKERRQ(ierr);}
    else if (dim == d) {ierr = DMPlexGetDepthStratum(dm, 1, pStart, pEnd);CHKERRQ(ierr);}
    else               {*pStart = 0; *pEnd = 0;}
  } else {
    ierr = DMPlexGetDepthStratum(dm, dim, pStart, pEnd);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode DMGetNeighbors_Plex(DM dm, PetscInt *nranks, const PetscMPIInt *ranks[])
{
  PetscSF           sf;
  PetscInt          niranks, njranks, n;
  const PetscMPIInt *iranks, *jranks;
  DM_Plex           *data = (DM_Plex*) dm->data;
  PetscErrorCode    ierr;

  PetscFunctionBegin;
  ierr = DMGetPointSF(dm, &sf);CHKERRQ(ierr);
  if (!data->neighbors) {
    ierr = PetscSFSetUp(sf);CHKERRQ(ierr);
    ierr = PetscSFGetRootRanks(sf, &njranks, &jranks, NULL, NULL, NULL);CHKERRQ(ierr);
    ierr = PetscSFGetLeafRanks(sf, &niranks, &iranks, NULL, NULL);CHKERRQ(ierr);
    ierr = PetscMalloc1(njranks + niranks + 1, &data->neighbors);CHKERRQ(ierr);
    ierr = PetscArraycpy(data->neighbors + 1, jranks, njranks);CHKERRQ(ierr);
    ierr = PetscArraycpy(data->neighbors + njranks + 1, iranks, niranks);CHKERRQ(ierr);
    n = njranks + niranks;
    ierr = PetscSortRemoveDupsMPIInt(&n, data->neighbors + 1);CHKERRQ(ierr);
    /* The following cast should never fail: can't have more neighbors than PETSC_MPI_INT_MAX */
    ierr = PetscMPIIntCast(n, data->neighbors);CHKERRQ(ierr);
  }
  if (nranks) *nranks = data->neighbors[0];
  if (ranks) {
    if (data->neighbors[0]) *ranks = data->neighbors + 1;
    else                    *ranks = NULL;
  }
  PetscFunctionReturn(0);
}

PETSC_INTERN PetscErrorCode DMInterpolateSolution_Plex(DM, DM, Mat, Vec, Vec);

static PetscErrorCode DMInitialize_Plex(DM dm)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  dm->ops->view                            = DMView_Plex;
  dm->ops->load                            = DMLoad_Plex;
  dm->ops->setfromoptions                  = DMSetFromOptions_Plex;
  dm->ops->clone                           = DMClone_Plex;
  dm->ops->setup                           = DMSetUp_Plex;
  dm->ops->createlocalsection              = DMCreateLocalSection_Plex;
  dm->ops->createdefaultconstraints        = DMCreateDefaultConstraints_Plex;
  dm->ops->createglobalvector              = DMCreateGlobalVector_Plex;
  dm->ops->createlocalvector               = DMCreateLocalVector_Plex;
  dm->ops->getlocaltoglobalmapping         = NULL;
  dm->ops->createfieldis                   = NULL;
  dm->ops->createcoordinatedm              = DMCreateCoordinateDM_Plex;
  dm->ops->createcoordinatefield           = DMCreateCoordinateField_Plex;
  dm->ops->getcoloring                     = NULL;
  dm->ops->creatematrix                    = DMCreateMatrix_Plex;
  dm->ops->createinterpolation             = DMCreateInterpolation_Plex;
  dm->ops->createmassmatrix                = DMCreateMassMatrix_Plex;
  dm->ops->createinjection                 = DMCreateInjection_Plex;
  dm->ops->refine                          = DMRefine_Plex;
  dm->ops->coarsen                         = DMCoarsen_Plex;
  dm->ops->refinehierarchy                 = DMRefineHierarchy_Plex;
  dm->ops->coarsenhierarchy                = DMCoarsenHierarchy_Plex;
  dm->ops->adaptlabel                      = DMAdaptLabel_Plex;
  dm->ops->adaptmetric                     = DMAdaptMetric_Plex;
  dm->ops->globaltolocalbegin              = NULL;
  dm->ops->globaltolocalend                = NULL;
  dm->ops->localtoglobalbegin              = NULL;
  dm->ops->localtoglobalend                = NULL;
  dm->ops->destroy                         = DMDestroy_Plex;
  dm->ops->createsubdm                     = DMCreateSubDM_Plex;
  dm->ops->createsuperdm                   = DMCreateSuperDM_Plex;
  dm->ops->getdimpoints                    = DMGetDimPoints_Plex;
  dm->ops->locatepoints                    = DMLocatePoints_Plex;
  dm->ops->projectfunctionlocal            = DMProjectFunctionLocal_Plex;
  dm->ops->projectfunctionlabellocal       = DMProjectFunctionLabelLocal_Plex;
  dm->ops->projectfieldlocal               = DMProjectFieldLocal_Plex;
  dm->ops->projectfieldlabellocal          = DMProjectFieldLabelLocal_Plex;
  dm->ops->projectbdfieldlabellocal        = DMProjectBdFieldLabelLocal_Plex;
  dm->ops->computel2diff                   = DMComputeL2Diff_Plex;
  dm->ops->computel2gradientdiff           = DMComputeL2GradientDiff_Plex;
  dm->ops->computel2fielddiff              = DMComputeL2FieldDiff_Plex;
  dm->ops->getneighbors                    = DMGetNeighbors_Plex;
  ierr = PetscObjectComposeFunction((PetscObject)dm,"DMPlexInsertBoundaryValues_C",DMPlexInsertBoundaryValues_Plex);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)dm,"DMPlexInsertTimeDerviativeBoundaryValues_C",DMPlexInsertTimeDerivativeBoundaryValues_Plex);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)dm,"DMSetUpGLVisViewer_C",DMSetUpGLVisViewer_Plex);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)dm,"DMCreateNeumannOverlap_C",DMCreateNeumannOverlap_Plex);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)dm,"DMPlexGetOverlap_C",DMPlexGetOverlap_Plex);CHKERRQ(ierr);
  ierr = PetscObjectComposeFunction((PetscObject)dm,"DMInterpolateSolution_C",DMInterpolateSolution_Plex);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PETSC_INTERN PetscErrorCode DMClone_Plex(DM dm, DM *newdm)
{
  DM_Plex        *mesh = (DM_Plex *) dm->data;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  mesh->refct++;
  (*newdm)->data = mesh;
  ierr = PetscObjectChangeTypeName((PetscObject) *newdm, DMPLEX);CHKERRQ(ierr);
  ierr = DMInitialize_Plex(*newdm);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*MC
  DMPLEX = "plex" - A DM object that encapsulates an unstructured mesh, or CW Complex, which can be expressed using a Hasse Diagram.
                    In the local representation, Vecs contain all unknowns in the interior and shared boundary. This is
                    specified by a PetscSection object. Ownership in the global representation is determined by
                    ownership of the underlying DMPlex points. This is specified by another PetscSection object.

  Options Database Keys:
+ -dm_refine_pre                     - Refine mesh before distribution
+ -dm_refine_uniform_pre             - Choose uniform or generator-based refinement
+ -dm_refine_volume_limit_pre        - Cell volume limit after pre-refinement using generator
. -dm_distribute                     - Distribute mesh across processes
. -dm_distribute_overlap             - Number of cells to overlap for distribution
. -dm_refine                         - Refine mesh after distribution
. -dm_plex_hash_location             - Use grid hashing for point location
. -dm_plex_partition_balance         - Attempt to evenly divide points on partition boundary between processes
. -dm_plex_remesh_bd                 - Allow changes to the boundary on remeshing
. -dm_plex_max_projection_height     - Maxmimum mesh point height used to project locally
. -dm_plex_regular_refinement        - Use special nested projection algorithm for regular refinement
. -dm_plex_check_all                 - Perform all shecks below
. -dm_plex_check_symmetry            - Check that the adjacency information in the mesh is symmetric
. -dm_plex_check_skeleton <celltype> - Check that each cell has the correct number of vertices
. -dm_plex_check_faces <celltype>    - Check that the faces of each cell give a vertex order this is consistent with what we expect from the cell type
. -dm_plex_check_geometry            - Check that cells have positive volume
. -dm_view :mesh.tex:ascii_latex     - View the mesh in LaTeX/TikZ
. -dm_plex_view_scale <num>          - Scale the TikZ
- -dm_plex_print_fem <num>           - View FEM assembly information, such as element vectors and matrices


  Level: intermediate

.seealso: DMType, DMPlexCreate(), DMCreate(), DMSetType()
M*/

PETSC_EXTERN PetscErrorCode DMCreate_Plex(DM dm)
{
  DM_Plex       *mesh;
  PetscInt       unit;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  ierr     = PetscNewLog(dm,&mesh);CHKERRQ(ierr);
  dm->dim  = 0;
  dm->data = mesh;

  mesh->refct             = 1;
  ierr                    = PetscSectionCreate(PetscObjectComm((PetscObject)dm), &mesh->coneSection);CHKERRQ(ierr);
  mesh->maxConeSize       = 0;
  mesh->cones             = NULL;
  mesh->coneOrientations  = NULL;
  ierr                    = PetscSectionCreate(PetscObjectComm((PetscObject)dm), &mesh->supportSection);CHKERRQ(ierr);
  mesh->maxSupportSize    = 0;
  mesh->supports          = NULL;
  mesh->refinementUniform = PETSC_TRUE;
  mesh->refinementLimit   = -1.0;
  mesh->interpolated      = DMPLEX_INTERPOLATED_INVALID;
  mesh->interpolatedCollective = DMPLEX_INTERPOLATED_INVALID;

  mesh->facesTmp = NULL;

  mesh->tetgenOpts   = NULL;
  mesh->triangleOpts = NULL;
  ierr = PetscPartitionerCreate(PetscObjectComm((PetscObject)dm), &mesh->partitioner);CHKERRQ(ierr);
  mesh->remeshBd     = PETSC_FALSE;

  mesh->subpointMap = NULL;

  for (unit = 0; unit < NUM_PETSC_UNITS; ++unit) mesh->scale[unit] = 1.0;

  mesh->regularRefinement   = PETSC_FALSE;
  mesh->depthState          = -1;
  mesh->celltypeState       = -1;
  mesh->globalVertexNumbers = NULL;
  mesh->globalCellNumbers   = NULL;
  mesh->anchorSection       = NULL;
  mesh->anchorIS            = NULL;
  mesh->createanchors       = NULL;
  mesh->computeanchormatrix = NULL;
  mesh->parentSection       = NULL;
  mesh->parents             = NULL;
  mesh->childIDs            = NULL;
  mesh->childSection        = NULL;
  mesh->children            = NULL;
  mesh->referenceTree       = NULL;
  mesh->getchildsymmetry    = NULL;
  mesh->vtkCellHeight       = 0;
  mesh->useAnchors          = PETSC_FALSE;

  mesh->maxProjectionHeight = 0;

  mesh->neighbors           = NULL;

  mesh->printSetValues = PETSC_FALSE;
  mesh->printFEM       = 0;
  mesh->printTol       = 1.0e-10;

  ierr = DMInitialize_Plex(dm);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreate - Creates a DMPlex object, which encapsulates an unstructured mesh, or CW complex, which can be expressed using a Hasse Diagram.

  Collective

  Input Parameter:
. comm - The communicator for the DMPlex object

  Output Parameter:
. mesh  - The DMPlex object

  Level: beginner

@*/
PetscErrorCode DMPlexCreate(MPI_Comm comm, DM *mesh)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidPointer(mesh,2);
  ierr = DMCreate(comm, mesh);CHKERRQ(ierr);
  ierr = DMSetType(*mesh, DMPLEX);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  DMPlexBuildFromCellListParallel - Build distributed DMPLEX topology from a list of vertices for each cell (common mesh generator output)

  Input Parameters:
+ dm - The DM
. numCells - The number of cells owned by this process
. numVertices - The number of vertices owned by this process, or PETSC_DECIDE
. NVertices - The global number of vertices, or PETSC_DECIDE
. numCorners - The number of vertices for each cell
- cells - An array of numCells*numCorners numbers, the global vertex numbers for each cell

  Output Parameter:
. vertexSF - (Optional) SF describing complete vertex ownership

  Notes:
  Two triangles sharing a face
$
$        2
$      / | \
$     /  |  \
$    /   |   \
$   0  0 | 1  3
$    \   |   /
$     \  |  /
$      \ | /
$        1
would have input
$  numCells = 2, numVertices = 4
$  cells = [0 1 2  1 3 2]
$
which would result in the DMPlex
$
$        4
$      / | \
$     /  |  \
$    /   |   \
$   2  0 | 1  5
$    \   |   /
$     \  |  /
$      \ | /
$        3

  Vertices are implicitly numbered consecutively 0,...,NVertices.
  Each rank owns a chunk of numVertices consecutive vertices.
  If numVertices is PETSC_DECIDE, PETSc will distribute them as evenly as possible using PetscLayout.
  If both NVertices and numVertices are PETSC_DECIDE, NVertices is computed by PETSc as the maximum vertex index in cells + 1.
  If only NVertices is PETSC_DECIDE, it is computed as the sum of numVertices over all ranks.

  The cell distribution is arbitrary non-overlapping, independent of the vertex distribution.

  Not currently supported in Fortran.

  Level: advanced

.seealso: DMPlexBuildFromCellList(), DMPlexCreateFromCellListParallelPetsc(), DMPlexBuildCoordinatesFromCellListParallel()
@*/
PetscErrorCode DMPlexBuildFromCellListParallel(DM dm, PetscInt numCells, PetscInt numVertices, PetscInt NVertices, PetscInt numCorners, const PetscInt cells[], PetscSF *vertexSF)
{
  PetscSF         sfPoint, sfVert;
  PetscLayout     vLayout;
  PetscSFNode    *vertexLocal, *vertexOwner, *remoteVertex;
  PetscInt        numVerticesAdj, *verticesAdj, numVerticesGhost = 0, *localVertex, *cones, c, p, v, g, dim;
  PetscMPIInt     rank, size;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  PetscValidLogicalCollectiveInt(dm,NVertices,4);
  ierr = PetscLogEventBegin(DMPLEX_BuildFromCellList,dm,0,0,0);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(PetscObjectComm((PetscObject) dm), &rank);CHKERRMPI(ierr);
  ierr = MPI_Comm_size(PetscObjectComm((PetscObject) dm), &size);CHKERRMPI(ierr);
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  /* Get/check global number of vertices */
  {
    PetscInt NVerticesInCells, i;
    const PetscInt len = numCells * numCorners;

    /* NVerticesInCells = max(cells) + 1 */
    NVerticesInCells = PETSC_MIN_INT;
    for (i=0; i<len; i++) if (cells[i] > NVerticesInCells) NVerticesInCells = cells[i];
    ++NVerticesInCells;
    ierr = MPI_Allreduce(MPI_IN_PLACE, &NVerticesInCells, 1, MPIU_INT, MPI_MAX, PetscObjectComm((PetscObject) dm));CHKERRMPI(ierr);

    if (numVertices == PETSC_DECIDE && NVertices == PETSC_DECIDE) NVertices = NVerticesInCells;
    else if (NVertices != PETSC_DECIDE && NVertices < NVerticesInCells) SETERRQ2(PetscObjectComm((PetscObject) dm), PETSC_ERR_ARG_WRONG, "Specified global number of vertices %D must be greater than or equal to the number of vertices in cells %D",NVertices,NVerticesInCells);
  }
  /* Partition vertices */
  ierr = PetscLayoutCreate(PetscObjectComm((PetscObject) dm), &vLayout);CHKERRQ(ierr);
  ierr = PetscLayoutSetLocalSize(vLayout, numVertices);CHKERRQ(ierr);
  ierr = PetscLayoutSetSize(vLayout, NVertices);CHKERRQ(ierr);
  ierr = PetscLayoutSetBlockSize(vLayout, 1);CHKERRQ(ierr);
  ierr = PetscLayoutSetUp(vLayout);CHKERRQ(ierr);
  /* Count locally unique vertices */
  {
    PetscHSetI vhash;
    PetscInt off = 0;

    ierr = PetscHSetICreate(&vhash);CHKERRQ(ierr);
    for (c = 0; c < numCells; ++c) {
      for (p = 0; p < numCorners; ++p) {
        ierr = PetscHSetIAdd(vhash, cells[c*numCorners+p]);CHKERRQ(ierr);
      }
    }
    ierr = PetscHSetIGetSize(vhash, &numVerticesAdj);CHKERRQ(ierr);
    ierr = PetscMalloc1(numVerticesAdj, &verticesAdj);CHKERRQ(ierr);
    ierr = PetscHSetIGetElems(vhash, &off, verticesAdj);CHKERRQ(ierr);
    ierr = PetscHSetIDestroy(&vhash);CHKERRQ(ierr);
    if (off != numVerticesAdj) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Invalid number of local vertices %D should be %D", off, numVerticesAdj);
  }
  ierr = PetscSortInt(numVerticesAdj, verticesAdj);CHKERRQ(ierr);
  /* Create cones */
  ierr = DMPlexSetChart(dm, 0, numCells+numVerticesAdj);CHKERRQ(ierr);
  for (c = 0; c < numCells; ++c) {ierr = DMPlexSetConeSize(dm, c, numCorners);CHKERRQ(ierr);}
  ierr = DMSetUp(dm);CHKERRQ(ierr);
  ierr = DMPlexGetCones(dm,&cones);CHKERRQ(ierr);
  for (c = 0; c < numCells; ++c) {
    for (p = 0; p < numCorners; ++p) {
      const PetscInt gv = cells[c*numCorners+p];
      PetscInt       lv;

      /* Positions within verticesAdj form 0-based local vertex numbering;
         we need to shift it by numCells to get correct DAG points (cells go first) */
      ierr = PetscFindInt(gv, numVerticesAdj, verticesAdj, &lv);CHKERRQ(ierr);
      if (lv < 0) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Could not find global vertex %D in local connectivity", gv);
      cones[c*numCorners+p] = lv+numCells;
    }
  }
  /* Create SF for vertices */
  ierr = PetscSFCreate(PetscObjectComm((PetscObject)dm), &sfVert);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) sfVert, "Vertex Ownership SF");CHKERRQ(ierr);
  ierr = PetscSFSetFromOptions(sfVert);CHKERRQ(ierr);
  ierr = PetscSFSetGraphLayout(sfVert, vLayout, numVerticesAdj, NULL, PETSC_OWN_POINTER, verticesAdj);CHKERRQ(ierr);
  ierr = PetscFree(verticesAdj);CHKERRQ(ierr);
  /* Build pointSF */
  ierr = PetscMalloc2(numVerticesAdj, &vertexLocal, numVertices, &vertexOwner);CHKERRQ(ierr);
  for (v = 0; v < numVerticesAdj; ++v) {vertexLocal[v].index = v+numCells; vertexLocal[v].rank = rank;}
  for (v = 0; v < numVertices;    ++v) {vertexOwner[v].index = -1;         vertexOwner[v].rank = -1;}
  ierr = PetscSFReduceBegin(sfVert, MPIU_2INT, vertexLocal, vertexOwner, MPI_MAXLOC);CHKERRQ(ierr);
  ierr = PetscSFReduceEnd(sfVert, MPIU_2INT, vertexLocal, vertexOwner, MPI_MAXLOC);CHKERRQ(ierr);
  for (v = 0; v < numVertices;    ++v) if (vertexOwner[v].rank < 0) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Global vertex %D was unclaimed", v + vLayout->rstart);
  ierr = PetscSFBcastBegin(sfVert, MPIU_2INT, vertexOwner, vertexLocal,MPI_REPLACE);CHKERRQ(ierr);
  ierr = PetscSFBcastEnd(sfVert, MPIU_2INT, vertexOwner, vertexLocal,MPI_REPLACE);CHKERRQ(ierr);
  for (v = 0; v < numVerticesAdj; ++v) if (vertexLocal[v].rank != rank) ++numVerticesGhost;
  ierr = PetscMalloc1(numVerticesGhost, &localVertex);CHKERRQ(ierr);
  ierr = PetscMalloc1(numVerticesGhost, &remoteVertex);CHKERRQ(ierr);
  for (v = 0, g = 0; v < numVerticesAdj; ++v) {
    if (vertexLocal[v].rank != rank) {
      localVertex[g]        = v+numCells;
      remoteVertex[g].index = vertexLocal[v].index;
      remoteVertex[g].rank  = vertexLocal[v].rank;
      ++g;
    }
  }
  ierr = PetscFree2(vertexLocal, vertexOwner);CHKERRQ(ierr);
  if (g != numVerticesGhost) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Invalid number of vertex ghosts %D should be %D", g, numVerticesGhost);
  ierr = DMGetPointSF(dm, &sfPoint);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) sfPoint, "point SF");CHKERRQ(ierr);
  ierr = PetscSFSetGraph(sfPoint, numCells+numVerticesAdj, numVerticesGhost, localVertex, PETSC_OWN_POINTER, remoteVertex, PETSC_OWN_POINTER);CHKERRQ(ierr);
  ierr = PetscLayoutDestroy(&vLayout);CHKERRQ(ierr);
  /* Fill in the rest of the topology structure */
  ierr = DMPlexSymmetrize(dm);CHKERRQ(ierr);
  ierr = DMPlexStratify(dm);CHKERRQ(ierr);
  ierr = PetscLogEventEnd(DMPLEX_BuildFromCellList,dm,0,0,0);CHKERRQ(ierr);
  if (vertexSF) *vertexSF = sfVert;
  else {ierr = PetscSFDestroy(&sfVert);CHKERRQ(ierr);}
  PetscFunctionReturn(0);
}

/*@C
  DMPlexBuildCoordinatesFromCellListParallel - Build DM coordinates from a list of coordinates for each owned vertex (common mesh generator output)

  Input Parameters:
+ dm - The DM
. spaceDim - The spatial dimension used for coordinates
. sfVert - SF describing complete vertex ownership
- vertexCoords - An array of numVertices*spaceDim numbers, the coordinates of each vertex

  Level: advanced

  Notes:
  Not currently supported in Fortran.

.seealso: DMPlexBuildCoordinatesFromCellList(), DMPlexCreateFromCellListParallelPetsc(), DMPlexBuildFromCellListParallel()
@*/
PetscErrorCode DMPlexBuildCoordinatesFromCellListParallel(DM dm, PetscInt spaceDim, PetscSF sfVert, const PetscReal vertexCoords[])
{
  PetscSection   coordSection;
  Vec            coordinates;
  PetscScalar   *coords;
  PetscInt       numVertices, numVerticesAdj, coordSize, v, vStart, vEnd;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscLogEventBegin(DMPLEX_BuildCoordinatesFromCellList,dm,0,0,0);CHKERRQ(ierr);
  ierr = DMPlexGetDepthStratum(dm, 0, &vStart, &vEnd);CHKERRQ(ierr);
  if (vStart < 0 || vEnd < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "DM is not set up properly. DMPlexBuildFromCellList() should be called first.");
  ierr = DMSetCoordinateDim(dm, spaceDim);CHKERRQ(ierr);
  ierr = PetscSFGetGraph(sfVert, &numVertices, &numVerticesAdj, NULL, NULL);CHKERRQ(ierr);
  if (vEnd - vStart != numVerticesAdj) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Supplied sfVert has wrong number of leaves = %D != %D = vEnd - vStart",numVerticesAdj,vEnd - vStart);
  ierr = DMGetCoordinateSection(dm, &coordSection);CHKERRQ(ierr);
  ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
  ierr = PetscSectionSetFieldComponents(coordSection, 0, spaceDim);CHKERRQ(ierr);
  ierr = PetscSectionSetChart(coordSection, vStart, vEnd);CHKERRQ(ierr);
  for (v = vStart; v < vEnd; ++v) {
    ierr = PetscSectionSetDof(coordSection, v, spaceDim);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldDof(coordSection, v, 0, spaceDim);CHKERRQ(ierr);
  }
  ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
  ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
  ierr = VecCreate(PetscObjectComm((PetscObject)dm), &coordinates);CHKERRQ(ierr);
  ierr = VecSetBlockSize(coordinates, spaceDim);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
  ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetType(coordinates,VECSTANDARD);CHKERRQ(ierr);
  ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
  {
    MPI_Datatype coordtype;

    /* Need a temp buffer for coords if we have complex/single */
    ierr = MPI_Type_contiguous(spaceDim, MPIU_SCALAR, &coordtype);CHKERRMPI(ierr);
    ierr = MPI_Type_commit(&coordtype);CHKERRMPI(ierr);
#if defined(PETSC_USE_COMPLEX)
    {
    PetscScalar *svertexCoords;
    PetscInt    i;
    ierr = PetscMalloc1(numVertices*spaceDim,&svertexCoords);CHKERRQ(ierr);
    for (i=0; i<numVertices*spaceDim; i++) svertexCoords[i] = vertexCoords[i];
    ierr = PetscSFBcastBegin(sfVert, coordtype, svertexCoords, coords,MPI_REPLACE);CHKERRQ(ierr);
    ierr = PetscSFBcastEnd(sfVert, coordtype, svertexCoords, coords,MPI_REPLACE);CHKERRQ(ierr);
    ierr = PetscFree(svertexCoords);CHKERRQ(ierr);
    }
#else
    ierr = PetscSFBcastBegin(sfVert, coordtype, vertexCoords, coords,MPI_REPLACE);CHKERRQ(ierr);
    ierr = PetscSFBcastEnd(sfVert, coordtype, vertexCoords, coords,MPI_REPLACE);CHKERRQ(ierr);
#endif
    ierr = MPI_Type_free(&coordtype);CHKERRMPI(ierr);
  }
  ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(dm, coordinates);CHKERRQ(ierr);
  ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  ierr = PetscLogEventEnd(DMPLEX_BuildCoordinatesFromCellList,dm,0,0,0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateFromCellListParallelPetsc - Create distributed DMPLEX from a list of vertices for each cell (common mesh generator output)

  Input Parameters:
+ comm - The communicator
. dim - The topological dimension of the mesh
. numCells - The number of cells owned by this process
. numVertices - The number of vertices owned by this process, or PETSC_DECIDE
. NVertices - The global number of vertices, or PETSC_DECIDE
. numCorners - The number of vertices for each cell
. interpolate - Flag indicating that intermediate mesh entities (faces, edges) should be created automatically
. cells - An array of numCells*numCorners numbers, the global vertex numbers for each cell
. spaceDim - The spatial dimension used for coordinates
- vertexCoords - An array of numVertices*spaceDim numbers, the coordinates of each vertex

  Output Parameter:
+ dm - The DM
- vertexSF - (Optional) SF describing complete vertex ownership

  Notes:
  This function is just a convenient sequence of DMCreate(), DMSetType(), DMSetDimension(),
  DMPlexBuildFromCellListParallel(), DMPlexInterpolate(), DMPlexBuildCoordinatesFromCellListParallel()

  See DMPlexBuildFromCellListParallel() for an example and details about the topology-related parameters.
  See DMPlexBuildCoordinatesFromCellListParallel() for details about the geometry-related parameters.

  Level: intermediate

.seealso: DMPlexCreateFromCellListPetsc(), DMPlexBuildFromCellListParallel(), DMPlexBuildCoordinatesFromCellListParallel(), DMPlexCreateFromDAG(), DMPlexCreate()
@*/
PetscErrorCode DMPlexCreateFromCellListParallelPetsc(MPI_Comm comm, PetscInt dim, PetscInt numCells, PetscInt numVertices, PetscInt NVertices, PetscInt numCorners, PetscBool interpolate, const PetscInt cells[], PetscInt spaceDim, const PetscReal vertexCoords[], PetscSF *vertexSF, DM *dm)
{
  PetscSF        sfVert;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMCreate(comm, dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  PetscValidLogicalCollectiveInt(*dm, dim, 2);
  PetscValidLogicalCollectiveInt(*dm, spaceDim, 8);
  ierr = DMSetDimension(*dm, dim);CHKERRQ(ierr);
  ierr = DMPlexBuildFromCellListParallel(*dm, numCells, numVertices, NVertices, numCorners, cells, &sfVert);CHKERRQ(ierr);
  if (interpolate) {
    DM idm;

    ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
    ierr = DMDestroy(dm);CHKERRQ(ierr);
    *dm  = idm;
  }
  ierr = DMPlexBuildCoordinatesFromCellListParallel(*dm, spaceDim, sfVert, vertexCoords);CHKERRQ(ierr);
  if (vertexSF) *vertexSF = sfVert;
  else {ierr = PetscSFDestroy(&sfVert);CHKERRQ(ierr);}
  PetscFunctionReturn(0);
}


/*@
  DMPlexCreateFromCellListParallel - Deprecated, use DMPlexCreateFromCellListParallelPetsc()

  Level: deprecated

.seealso: DMPlexCreateFromCellListParallelPetsc()
@*/
PetscErrorCode DMPlexCreateFromCellListParallel(MPI_Comm comm, PetscInt dim, PetscInt numCells, PetscInt numVertices, PetscInt numCorners, PetscBool interpolate, const int cells[], PetscInt spaceDim, const PetscReal vertexCoords[], PetscSF *vertexSF, DM *dm)
{
  PetscErrorCode ierr;
  PetscInt       i;
  PetscInt       *pintCells;

  PetscFunctionBegin;
  if (sizeof(int) > sizeof(PetscInt)) SETERRQ2(comm, PETSC_ERR_ARG_SIZ, "Size of int %zd greater than size of PetscInt %zd. Reconfigure PETSc --with-64-bit-indices=1", sizeof(int), sizeof(PetscInt));
  if (sizeof(int) == sizeof(PetscInt)) {
    pintCells = (PetscInt *) cells;
  } else {
    ierr = PetscMalloc1(numCells*numCorners, &pintCells);CHKERRQ(ierr);
    for (i = 0; i < numCells*numCorners; i++) {
      pintCells[i] = (PetscInt) cells[i];
    }
  }
  ierr = DMPlexCreateFromCellListParallelPetsc(comm, dim, numCells, numVertices, PETSC_DECIDE, numCorners, interpolate, pintCells, spaceDim, vertexCoords, vertexSF, dm);CHKERRQ(ierr);
  if (sizeof(int) != sizeof(PetscInt)) {
    ierr = PetscFree(pintCells);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

/*@C
  DMPlexBuildFromCellList - Build DMPLEX topology from a list of vertices for each cell (common mesh generator output)

  Input Parameters:
+ dm - The DM
. numCells - The number of cells owned by this process
. numVertices - The number of vertices owned by this process, or PETSC_DECIDE
. numCorners - The number of vertices for each cell
- cells - An array of numCells*numCorners numbers, the global vertex numbers for each cell

  Level: advanced

  Notes:
  Two triangles sharing a face
$
$        2
$      / | \
$     /  |  \
$    /   |   \
$   0  0 | 1  3
$    \   |   /
$     \  |  /
$      \ | /
$        1
would have input
$  numCells = 2, numVertices = 4
$  cells = [0 1 2  1 3 2]
$
which would result in the DMPlex
$
$        4
$      / | \
$     /  |  \
$    /   |   \
$   2  0 | 1  5
$    \   |   /
$     \  |  /
$      \ | /
$        3

  If numVertices is PETSC_DECIDE, it is computed by PETSc as the maximum vertex index in cells + 1.

  Not currently supported in Fortran.

.seealso: DMPlexBuildFromCellListParallel(), DMPlexBuildCoordinatesFromCellList(), DMPlexCreateFromCellListPetsc()
@*/
PetscErrorCode DMPlexBuildFromCellList(DM dm, PetscInt numCells, PetscInt numVertices, PetscInt numCorners, const PetscInt cells[])
{
  PetscInt      *cones, c, p, dim;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscLogEventBegin(DMPLEX_BuildFromCellList,dm,0,0,0);CHKERRQ(ierr);
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  /* Get/check global number of vertices */
  {
    PetscInt NVerticesInCells, i;
    const PetscInt len = numCells * numCorners;

    /* NVerticesInCells = max(cells) + 1 */
    NVerticesInCells = PETSC_MIN_INT;
    for (i=0; i<len; i++) if (cells[i] > NVerticesInCells) NVerticesInCells = cells[i];
    ++NVerticesInCells;

    if (numVertices == PETSC_DECIDE) numVertices = NVerticesInCells;
    else if (numVertices < NVerticesInCells) SETERRQ2(PetscObjectComm((PetscObject) dm), PETSC_ERR_ARG_WRONG, "Specified number of vertices %D must be greater than or equal to the number of vertices in cells %D",numVertices,NVerticesInCells);
  }
  ierr = DMPlexSetChart(dm, 0, numCells+numVertices);CHKERRQ(ierr);
  for (c = 0; c < numCells; ++c) {
    ierr = DMPlexSetConeSize(dm, c, numCorners);CHKERRQ(ierr);
  }
  ierr = DMSetUp(dm);CHKERRQ(ierr);
  ierr = DMPlexGetCones(dm,&cones);CHKERRQ(ierr);
  for (c = 0; c < numCells; ++c) {
    for (p = 0; p < numCorners; ++p) {
      cones[c*numCorners+p] = cells[c*numCorners+p]+numCells;
    }
  }
  ierr = DMPlexSymmetrize(dm);CHKERRQ(ierr);
  ierr = DMPlexStratify(dm);CHKERRQ(ierr);
  ierr = PetscLogEventEnd(DMPLEX_BuildFromCellList,dm,0,0,0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  DMPlexBuildCoordinatesFromCellList - Build DM coordinates from a list of coordinates for each owned vertex (common mesh generator output)

  Input Parameters:
+ dm - The DM
. spaceDim - The spatial dimension used for coordinates
- vertexCoords - An array of numVertices*spaceDim numbers, the coordinates of each vertex

  Level: advanced

  Notes:
  Not currently supported in Fortran.

.seealso: DMPlexBuildCoordinatesFromCellListParallel(), DMPlexCreateFromCellListPetsc(), DMPlexBuildFromCellList()
@*/
PetscErrorCode DMPlexBuildCoordinatesFromCellList(DM dm, PetscInt spaceDim, const PetscReal vertexCoords[])
{
  PetscSection   coordSection;
  Vec            coordinates;
  DM             cdm;
  PetscScalar   *coords;
  PetscInt       v, vStart, vEnd, d;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscLogEventBegin(DMPLEX_BuildCoordinatesFromCellList,dm,0,0,0);CHKERRQ(ierr);
  ierr = DMPlexGetDepthStratum(dm, 0, &vStart, &vEnd);CHKERRQ(ierr);
  if (vStart < 0 || vEnd < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "DM is not set up properly. DMPlexBuildFromCellList() should be called first.");
  ierr = DMSetCoordinateDim(dm, spaceDim);CHKERRQ(ierr);
  ierr = DMGetCoordinateSection(dm, &coordSection);CHKERRQ(ierr);
  ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
  ierr = PetscSectionSetFieldComponents(coordSection, 0, spaceDim);CHKERRQ(ierr);
  ierr = PetscSectionSetChart(coordSection, vStart, vEnd);CHKERRQ(ierr);
  for (v = vStart; v < vEnd; ++v) {
    ierr = PetscSectionSetDof(coordSection, v, spaceDim);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldDof(coordSection, v, 0, spaceDim);CHKERRQ(ierr);
  }
  ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);

  ierr = DMGetCoordinateDM(dm, &cdm);CHKERRQ(ierr);
  ierr = DMCreateLocalVector(cdm, &coordinates);CHKERRQ(ierr);
  ierr = VecSetBlockSize(coordinates, spaceDim);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
  ierr = VecGetArrayWrite(coordinates, &coords);CHKERRQ(ierr);
  for (v = 0; v < vEnd-vStart; ++v) {
    for (d = 0; d < spaceDim; ++d) {
      coords[v*spaceDim+d] = vertexCoords[v*spaceDim+d];
    }
  }
  ierr = VecRestoreArrayWrite(coordinates, &coords);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(dm, coordinates);CHKERRQ(ierr);
  ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  ierr = PetscLogEventEnd(DMPLEX_BuildCoordinatesFromCellList,dm,0,0,0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateFromCellListPetsc - Create DMPLEX from a list of vertices for each cell (common mesh generator output)

  Input Parameters:
+ comm - The communicator
. dim - The topological dimension of the mesh
. numCells - The number of cells
. numVertices - The number of vertices owned by this process, or PETSC_DECIDE
. numCorners - The number of vertices for each cell
. interpolate - Flag indicating that intermediate mesh entities (faces, edges) should be created automatically
. cells - An array of numCells*numCorners numbers, the vertices for each cell
. spaceDim - The spatial dimension used for coordinates
- vertexCoords - An array of numVertices*spaceDim numbers, the coordinates of each vertex

  Output Parameter:
. dm - The DM

  Notes:
  This function is just a convenient sequence of DMCreate(), DMSetType(), DMSetDimension(), DMPlexBuildFromCellList(),
  DMPlexInterpolate(), DMPlexBuildCoordinatesFromCellList()

  See DMPlexBuildFromCellList() for an example and details about the topology-related parameters.
  See DMPlexBuildCoordinatesFromCellList() for details about the geometry-related parameters.

  Level: intermediate

.seealso: DMPlexCreateFromCellListParallelPetsc(), DMPlexBuildFromCellList(), DMPlexBuildCoordinatesFromCellList(), DMPlexCreateFromDAG(), DMPlexCreate()
@*/
PetscErrorCode DMPlexCreateFromCellListPetsc(MPI_Comm comm, PetscInt dim, PetscInt numCells, PetscInt numVertices, PetscInt numCorners, PetscBool interpolate, const PetscInt cells[], PetscInt spaceDim, const PetscReal vertexCoords[], DM *dm)
{
  PetscErrorCode ierr;

  PetscFunctionBegin;
  if (!dim) SETERRQ(comm, PETSC_ERR_ARG_OUTOFRANGE, "This is not appropriate for 0-dimensional meshes. Consider either creating the DM using DMPlexCreateFromDAG(), by hand, or using DMSwarm.");
  ierr = DMCreate(comm, dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMSetDimension(*dm, dim);CHKERRQ(ierr);
  ierr = DMPlexBuildFromCellList(*dm, numCells, numVertices, numCorners, cells);CHKERRQ(ierr);
  if (interpolate) {
    DM idm;

    ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
    ierr = DMDestroy(dm);CHKERRQ(ierr);
    *dm  = idm;
  }
  ierr = DMPlexBuildCoordinatesFromCellList(*dm, spaceDim, vertexCoords);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateFromCellList - Deprecated, use DMPlexCreateFromCellListPetsc()

  Level: deprecated

.seealso: DMPlexCreateFromCellListPetsc()
@*/
PetscErrorCode DMPlexCreateFromCellList(MPI_Comm comm, PetscInt dim, PetscInt numCells, PetscInt numVertices, PetscInt numCorners, PetscBool interpolate, const int cells[], PetscInt spaceDim, const double vertexCoords[], DM *dm)
{
  PetscErrorCode ierr;
  PetscInt       i;
  PetscInt       *pintCells;
  PetscReal      *prealVC;

  PetscFunctionBegin;
  if (sizeof(int) > sizeof(PetscInt)) SETERRQ2(comm, PETSC_ERR_ARG_SIZ, "Size of int %zd greater than size of PetscInt %zd. Reconfigure PETSc --with-64-bit-indices=1", sizeof(int), sizeof(PetscInt));
  if (sizeof(int) == sizeof(PetscInt)) {
    pintCells = (PetscInt *) cells;
  } else {
    ierr = PetscMalloc1(numCells*numCorners, &pintCells);CHKERRQ(ierr);
    for (i = 0; i < numCells*numCorners; i++) {
      pintCells[i] = (PetscInt) cells[i];
    }
  }
  if (sizeof(double) > sizeof(PetscReal)) SETERRQ2(comm, PETSC_ERR_ARG_SIZ, "Size of double %zd greater than size of PetscReal %zd. Reconfigure PETSc --with-precision=<higher precision>.", sizeof(double), sizeof(PetscReal));
  if (sizeof(double) == sizeof(PetscReal)) {
    prealVC = (PetscReal *) vertexCoords;
  } else {
    ierr = PetscMalloc1(numVertices*spaceDim, &prealVC);CHKERRQ(ierr);
    for (i = 0; i < numVertices*spaceDim; i++) {
      prealVC[i] = (PetscReal) vertexCoords[i];
    }
  }
  ierr = DMPlexCreateFromCellListPetsc(comm, dim, numCells, numVertices, numCorners, interpolate, pintCells, spaceDim, prealVC, dm);CHKERRQ(ierr);
  if (sizeof(int) != sizeof(PetscInt)) {
    ierr = PetscFree(pintCells);CHKERRQ(ierr);
  }
  if (sizeof(double) != sizeof(PetscReal)) {
    ierr = PetscFree(prealVC);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateFromDAG - This takes as input the adjacency-list representation of the Directed Acyclic Graph (Hasse Diagram) encoding a mesh, and produces a DM

  Input Parameters:
+ dm - The empty DM object, usually from DMCreate() and DMSetDimension()
. depth - The depth of the DAG
. numPoints - Array of size depth + 1 containing the number of points at each depth
. coneSize - The cone size of each point
. cones - The concatenation of the cone points for each point, the cone list must be oriented correctly for each point
. coneOrientations - The orientation of each cone point
- vertexCoords - An array of numPoints[0]*spacedim numbers representing the coordinates of each vertex, with spacedim the value set via DMSetCoordinateDim()

  Output Parameter:
. dm - The DM

  Note: Two triangles sharing a face would have input
$  depth = 1, numPoints = [4 2], coneSize = [3 3 0 0 0 0]
$  cones = [2 3 4  3 5 4], coneOrientations = [0 0 0  0 0 0]
$ vertexCoords = [-1.0 0.0  0.0 -1.0  0.0 1.0  1.0 0.0]
$
which would result in the DMPlex
$
$        4
$      / | \
$     /  |  \
$    /   |   \
$   2  0 | 1  5
$    \   |   /
$     \  |  /
$      \ | /
$        3
$
$ Notice that all points are numbered consecutively, unlike DMPlexCreateFromCellListPetsc()

  Level: advanced

.seealso: DMPlexCreateFromCellListPetsc(), DMPlexCreate()
@*/
PetscErrorCode DMPlexCreateFromDAG(DM dm, PetscInt depth, const PetscInt numPoints[], const PetscInt coneSize[], const PetscInt cones[], const PetscInt coneOrientations[], const PetscScalar vertexCoords[])
{
  Vec            coordinates;
  PetscSection   coordSection;
  PetscScalar    *coords;
  PetscInt       coordSize, firstVertex = -1, pStart = 0, pEnd = 0, p, v, dim, dimEmbed, d, off;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMGetCoordinateDim(dm, &dimEmbed);CHKERRQ(ierr);
  if (dimEmbed < dim) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Embedding dimension %D cannot be less than intrinsic dimension %d",dimEmbed,dim);
  for (d = 0; d <= depth; ++d) pEnd += numPoints[d];
  ierr = DMPlexSetChart(dm, pStart, pEnd);CHKERRQ(ierr);
  for (p = pStart; p < pEnd; ++p) {
    ierr = DMPlexSetConeSize(dm, p, coneSize[p-pStart]);CHKERRQ(ierr);
    if (firstVertex < 0 && !coneSize[p - pStart]) {
      firstVertex = p - pStart;
    }
  }
  if (firstVertex < 0 && numPoints[0]) SETERRQ1(PETSC_COMM_SELF,PETSC_ERR_ARG_WRONG,"Expected %D vertices but could not find any", numPoints[0]);
  ierr = DMSetUp(dm);CHKERRQ(ierr); /* Allocate space for cones */
  for (p = pStart, off = 0; p < pEnd; off += coneSize[p-pStart], ++p) {
    ierr = DMPlexSetCone(dm, p, &cones[off]);CHKERRQ(ierr);
    ierr = DMPlexSetConeOrientation(dm, p, &coneOrientations[off]);CHKERRQ(ierr);
  }
  ierr = DMPlexSymmetrize(dm);CHKERRQ(ierr);
  ierr = DMPlexStratify(dm);CHKERRQ(ierr);
  /* Build coordinates */
  ierr = DMGetCoordinateSection(dm, &coordSection);CHKERRQ(ierr);
  ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
  ierr = PetscSectionSetFieldComponents(coordSection, 0, dimEmbed);CHKERRQ(ierr);
  ierr = PetscSectionSetChart(coordSection, firstVertex, firstVertex+numPoints[0]);CHKERRQ(ierr);
  for (v = firstVertex; v < firstVertex+numPoints[0]; ++v) {
    ierr = PetscSectionSetDof(coordSection, v, dimEmbed);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldDof(coordSection, v, 0, dimEmbed);CHKERRQ(ierr);
  }
  ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
  ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
  ierr = VecCreate(PETSC_COMM_SELF, &coordinates);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
  ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetBlockSize(coordinates, dimEmbed);CHKERRQ(ierr);
  ierr = VecSetType(coordinates,VECSTANDARD);CHKERRQ(ierr);
  ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
  for (v = 0; v < numPoints[0]; ++v) {
    PetscInt off;

    ierr = PetscSectionGetOffset(coordSection, v+firstVertex, &off);CHKERRQ(ierr);
    for (d = 0; d < dimEmbed; ++d) {
      coords[off+d] = vertexCoords[v*dimEmbed+d];
    }
  }
  ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(dm, coordinates);CHKERRQ(ierr);
  ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@C
  DMPlexCreateCellVertexFromFile - Create a DMPlex mesh from a simple cell-vertex file.

+ comm        - The MPI communicator
. filename    - Name of the .dat file
- interpolate - Create faces and edges in the mesh

  Output Parameter:
. dm  - The DM object representing the mesh

  Note: The format is the simplest possible:
$ Ne
$ v0 v1 ... vk
$ Nv
$ x y z marker

  Level: beginner

.seealso: DMPlexCreateFromFile(), DMPlexCreateMedFromFile(), DMPlexCreateGmsh(), DMPlexCreate()
@*/
PetscErrorCode DMPlexCreateCellVertexFromFile(MPI_Comm comm, const char filename[], PetscBool interpolate, DM *dm)
{
  DMLabel         marker;
  PetscViewer     viewer;
  Vec             coordinates;
  PetscSection    coordSection;
  PetscScalar    *coords;
  char            line[PETSC_MAX_PATH_LEN];
  PetscInt        dim = 3, cdim = 3, coordSize, v, c, d;
  PetscMPIInt     rank;
  int             snum, Nv, Nc;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = MPI_Comm_rank(comm, &rank);CHKERRMPI(ierr);
  ierr = PetscViewerCreate(comm, &viewer);CHKERRQ(ierr);
  ierr = PetscViewerSetType(viewer, PETSCVIEWERASCII);CHKERRQ(ierr);
  ierr = PetscViewerFileSetMode(viewer, FILE_MODE_READ);CHKERRQ(ierr);
  ierr = PetscViewerFileSetName(viewer, filename);CHKERRQ(ierr);
  if (!rank) {
    ierr = PetscViewerRead(viewer, line, 2, NULL, PETSC_STRING);CHKERRQ(ierr);
    snum = sscanf(line, "%d %d", &Nc, &Nv);
    if (snum != 2) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Unable to parse cell-vertex file: %s", line);
  } else {
    Nc = Nv = 0;
  }
  ierr = DMCreate(comm, dm);CHKERRQ(ierr);
  ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
  ierr = DMPlexSetChart(*dm, 0, Nc+Nv);CHKERRQ(ierr);
  ierr = DMSetDimension(*dm, dim);CHKERRQ(ierr);
  ierr = DMSetCoordinateDim(*dm, cdim);CHKERRQ(ierr);
  /* Read topology */
  if (!rank) {
    PetscInt cone[8], corners = 8;
    int      vbuf[8], v;

    for (c = 0; c < Nc; ++c) {ierr = DMPlexSetConeSize(*dm, c, corners);CHKERRQ(ierr);}
    ierr = DMSetUp(*dm);CHKERRQ(ierr);
    for (c = 0; c < Nc; ++c) {
      ierr = PetscViewerRead(viewer, line, corners, NULL, PETSC_STRING);CHKERRQ(ierr);
      snum = sscanf(line, "%d %d %d %d %d %d %d %d", &vbuf[0], &vbuf[1], &vbuf[2], &vbuf[3], &vbuf[4], &vbuf[5], &vbuf[6], &vbuf[7]);
      if (snum != corners) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Unable to parse cell-vertex file: %s", line);
      for (v = 0; v < corners; ++v) cone[v] = vbuf[v] + Nc;
      /* Hexahedra are inverted */
      {
        PetscInt tmp = cone[1];
        cone[1] = cone[3];
        cone[3] = tmp;
      }
      ierr = DMPlexSetCone(*dm, c, cone);CHKERRQ(ierr);
    }
  }
  ierr = DMPlexSymmetrize(*dm);CHKERRQ(ierr);
  ierr = DMPlexStratify(*dm);CHKERRQ(ierr);
  /* Read coordinates */
  ierr = DMGetCoordinateSection(*dm, &coordSection);CHKERRQ(ierr);
  ierr = PetscSectionSetNumFields(coordSection, 1);CHKERRQ(ierr);
  ierr = PetscSectionSetFieldComponents(coordSection, 0, cdim);CHKERRQ(ierr);
  ierr = PetscSectionSetChart(coordSection, Nc, Nc + Nv);CHKERRQ(ierr);
  for (v = Nc; v < Nc+Nv; ++v) {
    ierr = PetscSectionSetDof(coordSection, v, cdim);CHKERRQ(ierr);
    ierr = PetscSectionSetFieldDof(coordSection, v, 0, cdim);CHKERRQ(ierr);
  }
  ierr = PetscSectionSetUp(coordSection);CHKERRQ(ierr);
  ierr = PetscSectionGetStorageSize(coordSection, &coordSize);CHKERRQ(ierr);
  ierr = VecCreate(PETSC_COMM_SELF, &coordinates);CHKERRQ(ierr);
  ierr = PetscObjectSetName((PetscObject) coordinates, "coordinates");CHKERRQ(ierr);
  ierr = VecSetSizes(coordinates, coordSize, PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetBlockSize(coordinates, cdim);CHKERRQ(ierr);
  ierr = VecSetType(coordinates, VECSTANDARD);CHKERRQ(ierr);
  ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
  if (!rank) {
    double x[3];
    int    val;

    ierr = DMCreateLabel(*dm, "marker");CHKERRQ(ierr);
    ierr = DMGetLabel(*dm, "marker", &marker);CHKERRQ(ierr);
    for (v = 0; v < Nv; ++v) {
      ierr = PetscViewerRead(viewer, line, 4, NULL, PETSC_STRING);CHKERRQ(ierr);
      snum = sscanf(line, "%lg %lg %lg %d", &x[0], &x[1], &x[2], &val);
      if (snum != 4) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Unable to parse cell-vertex file: %s", line);
      for (d = 0; d < cdim; ++d) coords[v*cdim+d] = x[d];
      if (val) {ierr = DMLabelSetValue(marker, v+Nc, val);CHKERRQ(ierr);}
    }
  }
  ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
  ierr = DMSetCoordinatesLocal(*dm, coordinates);CHKERRQ(ierr);
  ierr = VecDestroy(&coordinates);CHKERRQ(ierr);
  ierr = PetscViewerDestroy(&viewer);CHKERRQ(ierr);
  if (interpolate) {
    DM      idm;
    DMLabel bdlabel;

    ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
    ierr = DMDestroy(dm);CHKERRQ(ierr);
    *dm  = idm;

    ierr = DMGetLabel(*dm, "marker", &bdlabel);CHKERRQ(ierr);
    ierr = DMPlexMarkBoundaryFaces(*dm, PETSC_DETERMINE, bdlabel);CHKERRQ(ierr);
    ierr = DMPlexLabelComplete(*dm, bdlabel);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

/*@C
  DMPlexCreateFromFile - This takes a filename and produces a DM

  Input Parameters:
+ comm - The communicator
. filename - A file name
- interpolate - Flag to create intermediate mesh pieces (edges, faces)

  Output Parameter:
. dm - The DM

  Options Database Keys:
. -dm_plex_create_from_hdf5_xdmf - use the PETSC_VIEWER_HDF5_XDMF format for reading HDF5

  Use -dm_plex_create_ prefix to pass options to the internal PetscViewer, e.g.
$ -dm_plex_create_viewer_hdf5_collective

  Level: beginner

.seealso: DMPlexCreateFromDAG(), DMPlexCreateFromCellListPetsc(), DMPlexCreate()
@*/
PetscErrorCode DMPlexCreateFromFile(MPI_Comm comm, const char filename[], PetscBool interpolate, DM *dm)
{
  const char    *extGmsh    = ".msh";
  const char    *extGmsh2   = ".msh2";
  const char    *extGmsh4   = ".msh4";
  const char    *extCGNS    = ".cgns";
  const char    *extExodus  = ".exo";
  const char    *extGenesis = ".gen";
  const char    *extFluent  = ".cas";
  const char    *extHDF5    = ".h5";
  const char    *extMed     = ".med";
  const char    *extPLY     = ".ply";
  const char    *extEGADS   = ".egadslite";
  const char    *extCV      = ".dat";
  size_t         len;
  PetscBool      isGmsh, isGmsh2, isGmsh4, isCGNS, isExodus, isGenesis, isFluent, isHDF5, isMed, isPLY, isEGADS, isCV;
  PetscMPIInt    rank;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidCharPointer(filename, 2);
  PetscValidPointer(dm, 4);
  ierr = DMInitializePackage();CHKERRQ(ierr);
  ierr = PetscLogEventBegin(DMPLEX_CreateFromFile,0,0,0,0);CHKERRQ(ierr);
  ierr = MPI_Comm_rank(comm, &rank);CHKERRMPI(ierr);
  ierr = PetscStrlen(filename, &len);CHKERRQ(ierr);
  if (!len) SETERRQ(comm, PETSC_ERR_ARG_WRONG, "Filename must be a valid path");
  ierr = PetscStrncmp(&filename[PetscMax(0,len-4)], extGmsh,    4, &isGmsh);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-5)], extGmsh2,   5, &isGmsh2);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-5)], extGmsh4,   5, &isGmsh4);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-5)], extCGNS,    5, &isCGNS);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-4)], extExodus,  4, &isExodus);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-4)], extGenesis, 4, &isGenesis);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-4)], extFluent,  4, &isFluent);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-3)], extHDF5,    3, &isHDF5);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-4)], extMed,     4, &isMed);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-4)], extPLY,     4, &isPLY);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-10)], extEGADS,   9, &isEGADS);CHKERRQ(ierr);
  ierr = PetscStrncmp(&filename[PetscMax(0,len-4)], extCV,      4, &isCV);CHKERRQ(ierr);
  if (isGmsh || isGmsh2 || isGmsh4) {
    ierr = DMPlexCreateGmshFromFile(comm, filename, interpolate, dm);CHKERRQ(ierr);
  } else if (isCGNS) {
    ierr = DMPlexCreateCGNSFromFile(comm, filename, interpolate, dm);CHKERRQ(ierr);
  } else if (isExodus || isGenesis) {
    ierr = DMPlexCreateExodusFromFile(comm, filename, interpolate, dm);CHKERRQ(ierr);
  } else if (isFluent) {
    ierr = DMPlexCreateFluentFromFile(comm, filename, interpolate, dm);CHKERRQ(ierr);
  } else if (isHDF5) {
    PetscBool      load_hdf5_xdmf = PETSC_FALSE;
    PetscViewer viewer;

    /* PETSC_VIEWER_HDF5_XDMF is used if the filename ends with .xdmf.h5, or if -dm_plex_create_from_hdf5_xdmf option is present */
    ierr = PetscStrncmp(&filename[PetscMax(0,len-8)], ".xdmf",  5, &load_hdf5_xdmf);CHKERRQ(ierr);
    ierr = PetscOptionsGetBool(NULL, NULL, "-dm_plex_create_from_hdf5_xdmf", &load_hdf5_xdmf, NULL);CHKERRQ(ierr);
    ierr = PetscViewerCreate(comm, &viewer);CHKERRQ(ierr);
    ierr = PetscViewerSetType(viewer, PETSCVIEWERHDF5);CHKERRQ(ierr);
    ierr = PetscViewerSetOptionsPrefix(viewer, "dm_plex_create_");CHKERRQ(ierr);
    ierr = PetscViewerSetFromOptions(viewer);CHKERRQ(ierr);
    ierr = PetscViewerFileSetMode(viewer, FILE_MODE_READ);CHKERRQ(ierr);
    ierr = PetscViewerFileSetName(viewer, filename);CHKERRQ(ierr);
    ierr = DMCreate(comm, dm);CHKERRQ(ierr);
    ierr = DMSetType(*dm, DMPLEX);CHKERRQ(ierr);
    if (load_hdf5_xdmf) {ierr = PetscViewerPushFormat(viewer, PETSC_VIEWER_HDF5_XDMF);CHKERRQ(ierr);}
    ierr = DMLoad(*dm, viewer);CHKERRQ(ierr);
    if (load_hdf5_xdmf) {ierr = PetscViewerPopFormat(viewer);CHKERRQ(ierr);}
    ierr = PetscViewerDestroy(&viewer);CHKERRQ(ierr);

    if (interpolate) {
      DM idm;

      ierr = DMPlexInterpolate(*dm, &idm);CHKERRQ(ierr);
      ierr = DMDestroy(dm);CHKERRQ(ierr);
      *dm  = idm;
    }
  } else if (isMed) {
    ierr = DMPlexCreateMedFromFile(comm, filename, interpolate, dm);CHKERRQ(ierr);
  } else if (isPLY) {
    ierr = DMPlexCreatePLYFromFile(comm, filename, interpolate, dm);CHKERRQ(ierr);
  } else if (isEGADS) {
    ierr = DMPlexCreateEGADSFromFile(comm, filename, dm);CHKERRQ(ierr);
    if (!interpolate) {
      DM udm;

      ierr = DMPlexUninterpolate(*dm, &udm);CHKERRQ(ierr);
      ierr = DMDestroy(dm);CHKERRQ(ierr);
      *dm  = udm;
    }
  } else if (isCV) {
    ierr = DMPlexCreateCellVertexFromFile(comm, filename, interpolate, dm);CHKERRQ(ierr);
  } else SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Cannot load file %s: unrecognized extension", filename);
  ierr = PetscLogEventEnd(DMPLEX_CreateFromFile,0,0,0,0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateReferenceCellByType - Create a DMPLEX with the appropriate FEM reference cell

  Collective

  Input Parameters:
+ comm - The communicator
- ct   - The cell type of the reference cell

  Output Parameter:
. refdm - The reference cell

  Options Database Keys:
. -dm_plex_ref_type <ct> - Specify the celltyoe for the reference cell

  Level: intermediate

.seealso: DMPlexCreateReferenceCell(), DMPlexCreateBoxMesh()
@*/
PetscErrorCode DMPlexCreateReferenceCellByType(MPI_Comm comm, DMPolytopeType ct, DM *refdm)
{
  DM             rdm;
  Vec            coords;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscOptionsGetEnum(NULL, NULL, "-dm_plex_ref_type", DMPolytopeTypes, (PetscEnum *) &ct, NULL);CHKERRQ(ierr);
  ierr = DMCreate(comm, &rdm);CHKERRQ(ierr);
  ierr = DMSetType(rdm, DMPLEX);CHKERRQ(ierr);
  switch (ct) {
    case DM_POLYTOPE_POINT:
    {
      PetscInt    numPoints[1]        = {1};
      PetscInt    coneSize[1]         = {0};
      PetscInt    cones[1]            = {0};
      PetscInt    coneOrientations[1] = {0};
      PetscScalar vertexCoords[1]     = {0.0};

      ierr = DMSetDimension(rdm, 0);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 0, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_SEGMENT:
    {
      PetscInt    numPoints[2]        = {2, 1};
      PetscInt    coneSize[3]         = {2, 0, 0};
      PetscInt    cones[2]            = {1, 2};
      PetscInt    coneOrientations[2] = {0, 0};
      PetscScalar vertexCoords[2]     = {-1.0,  1.0};

      ierr = DMSetDimension(rdm, 1);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_TRIANGLE:
    {
      PetscInt    numPoints[2]        = {3, 1};
      PetscInt    coneSize[4]         = {3, 0, 0, 0};
      PetscInt    cones[3]            = {1, 2, 3};
      PetscInt    coneOrientations[3] = {0, 0, 0};
      PetscScalar vertexCoords[6]     = {-1.0, -1.0,  1.0, -1.0,  -1.0, 1.0};

      ierr = DMSetDimension(rdm, 2);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_QUADRILATERAL:
    {
      PetscInt    numPoints[2]        = {4, 1};
      PetscInt    coneSize[5]         = {4, 0, 0, 0, 0};
      PetscInt    cones[4]            = {1, 2, 3, 4};
      PetscInt    coneOrientations[4] = {0, 0, 0, 0};
      PetscScalar vertexCoords[8]     = {-1.0, -1.0,  1.0, -1.0,  1.0, 1.0,  -1.0, 1.0};

      ierr = DMSetDimension(rdm, 2);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_SEG_PRISM_TENSOR:
    {
      PetscInt    numPoints[2]        = {4, 1};
      PetscInt    coneSize[5]         = {4, 0, 0, 0, 0};
      PetscInt    cones[4]            = {1, 2, 3, 4};
      PetscInt    coneOrientations[4] = {0, 0, 0, 0};
      PetscScalar vertexCoords[8]     = {-1.0, -1.0,  1.0, -1.0,  -1.0, 1.0,  1.0, 1.0};

      ierr = DMSetDimension(rdm, 2);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_TETRAHEDRON:
    {
      PetscInt    numPoints[2]        = {4, 1};
      PetscInt    coneSize[5]         = {4, 0, 0, 0, 0};
      PetscInt    cones[4]            = {1, 3, 2, 4};
      PetscInt    coneOrientations[4] = {0, 0, 0, 0};
      PetscScalar vertexCoords[12]    = {-1.0, -1.0, -1.0,  1.0, -1.0, -1.0,  -1.0, 1.0, -1.0,  -1.0, -1.0, 1.0};

      ierr = DMSetDimension(rdm, 3);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_HEXAHEDRON:
    {
      PetscInt    numPoints[2]        = {8, 1};
      PetscInt    coneSize[9]         = {8, 0, 0, 0, 0, 0, 0, 0, 0};
      PetscInt    cones[8]            = {1, 4, 3, 2, 5, 6, 7, 8};
      PetscInt    coneOrientations[8] = {0, 0, 0, 0, 0, 0, 0, 0};
      PetscScalar vertexCoords[24]    = {-1.0, -1.0, -1.0,  1.0, -1.0, -1.0,  1.0, 1.0, -1.0,  -1.0, 1.0, -1.0,
                                         -1.0, -1.0,  1.0,  1.0, -1.0,  1.0,  1.0, 1.0,  1.0,  -1.0, 1.0,  1.0};

      ierr = DMSetDimension(rdm, 3);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_TRI_PRISM:
    {
      PetscInt    numPoints[2]        = {6, 1};
      PetscInt    coneSize[7]         = {6, 0, 0, 0, 0, 0, 0};
      PetscInt    cones[6]            = {1, 3, 2, 4, 5, 6};
      PetscInt    coneOrientations[6] = {0, 0, 0, 0, 0, 0};
      PetscScalar vertexCoords[18]    = {-1.0, -1.0, -1.0,  1.0, -1.0, -1.0,  -1.0, 1.0, -1.0,
                                         -1.0, -1.0,  1.0,  1.0, -1.0,  1.0,  -1.0, 1.0,  1.0};

      ierr = DMSetDimension(rdm, 3);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_TRI_PRISM_TENSOR:
    {
      PetscInt    numPoints[2]        = {6, 1};
      PetscInt    coneSize[7]         = {6, 0, 0, 0, 0, 0, 0};
      PetscInt    cones[6]            = {1, 2, 3, 4, 5, 6};
      PetscInt    coneOrientations[6] = {0, 0, 0, 0, 0, 0};
      PetscScalar vertexCoords[18]    = {-1.0, -1.0, -1.0,  1.0, -1.0, -1.0,  -1.0, 1.0, -1.0,
                                         -1.0, -1.0,  1.0,  1.0, -1.0,  1.0,  -1.0, 1.0,  1.0};

      ierr = DMSetDimension(rdm, 3);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_QUAD_PRISM_TENSOR:
    {
      PetscInt    numPoints[2]        = {8, 1};
      PetscInt    coneSize[9]         = {8, 0, 0, 0, 0, 0, 0, 0, 0};
      PetscInt    cones[8]            = {1, 2, 3, 4, 5, 6, 7, 8};
      PetscInt    coneOrientations[8] = {0, 0, 0, 0, 0, 0, 0, 0};
      PetscScalar vertexCoords[24]    = {-1.0, -1.0, -1.0,  1.0, -1.0, -1.0,  1.0, 1.0, -1.0,  -1.0, 1.0, -1.0,
                                         -1.0, -1.0,  1.0,  1.0, -1.0,  1.0,  1.0, 1.0,  1.0,  -1.0, 1.0,  1.0};

      ierr = DMSetDimension(rdm, 3);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    case DM_POLYTOPE_PYRAMID:
    {
      PetscInt    numPoints[2]        = {5, 1};
      PetscInt    coneSize[6]         = {5, 0, 0, 0, 0, 0};
      PetscInt    cones[5]            = {1, 4, 3, 2, 5};
      PetscInt    coneOrientations[8] = {0, 0, 0, 0, 0, 0, 0, 0};
      PetscScalar vertexCoords[24]    = {-1.0, -1.0, -1.0,  1.0, -1.0, -1.0,  1.0, 1.0, -1.0,  -1.0, 1.0, -1.0,
                                          0.0,  0.0,  1.0};

      ierr = DMSetDimension(rdm, 3);CHKERRQ(ierr);
      ierr = DMPlexCreateFromDAG(rdm, 1, numPoints, coneSize, cones, coneOrientations, vertexCoords);CHKERRQ(ierr);
    }
    break;
    default: SETERRQ1(comm, PETSC_ERR_ARG_WRONG, "Cannot create reference cell for cell type %s", DMPolytopeTypes[ct]);
  }
  {
    PetscInt Nv, v;

    /* Must create the celltype label here so that we do not automatically try to compute the types */
    ierr = DMCreateLabel(rdm, "celltype");CHKERRQ(ierr);
    ierr = DMPlexSetCellType(rdm, 0, ct);CHKERRQ(ierr);
    ierr = DMPlexGetChart(rdm, NULL, &Nv);CHKERRQ(ierr);
    for (v = 1; v < Nv; ++v) {ierr = DMPlexSetCellType(rdm, v, DM_POLYTOPE_POINT);CHKERRQ(ierr);}
  }
  ierr = DMPlexInterpolate(rdm, refdm);CHKERRQ(ierr);
  if (rdm->coordinateDM) {
    DM           ncdm;
    PetscSection cs;
    PetscInt     pEnd = -1;

    ierr = DMGetLocalSection(rdm->coordinateDM, &cs);CHKERRQ(ierr);
    if (cs) {ierr = PetscSectionGetChart(cs, NULL, &pEnd);CHKERRQ(ierr);}
    if (pEnd >= 0) {
      ierr = DMClone(*refdm, &ncdm);CHKERRQ(ierr);
      ierr = DMCopyDisc(rdm->coordinateDM, ncdm);CHKERRQ(ierr);
      ierr = DMSetLocalSection(ncdm, cs);CHKERRQ(ierr);
      ierr = DMSetCoordinateDM(*refdm, ncdm);CHKERRQ(ierr);
      ierr = DMDestroy(&ncdm);CHKERRQ(ierr);
    }
  }
  ierr = DMGetCoordinatesLocal(rdm, &coords);CHKERRQ(ierr);
  if (coords) {
    ierr = DMSetCoordinatesLocal(*refdm, coords);CHKERRQ(ierr);
  } else {
    ierr = DMGetCoordinates(rdm, &coords);CHKERRQ(ierr);
    if (coords) {ierr = DMSetCoordinates(*refdm, coords);CHKERRQ(ierr);}
  }
  ierr = DMDestroy(&rdm);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/*@
  DMPlexCreateReferenceCell - Create a DMPLEX with the appropriate FEM reference cell

  Collective

  Input Parameters:
+ comm    - The communicator
. dim     - The spatial dimension
- simplex - Flag for simplex, otherwise use a tensor-product cell

  Output Parameter:
. refdm - The reference cell

  Level: intermediate

.seealso: DMPlexCreateReferenceCellByType(), DMPlexCreateBoxMesh()
@*/
PetscErrorCode DMPlexCreateReferenceCell(MPI_Comm comm, PetscInt dim, PetscBool simplex, DM *refdm)
{
  PetscErrorCode ierr;

  PetscFunctionBeginUser;
  switch (dim) {
  case 0: ierr = DMPlexCreateReferenceCellByType(comm, DM_POLYTOPE_POINT, refdm);CHKERRQ(ierr);break;
  case 1: ierr = DMPlexCreateReferenceCellByType(comm, DM_POLYTOPE_SEGMENT, refdm);CHKERRQ(ierr);break;
  case 2:
    if (simplex) {ierr = DMPlexCreateReferenceCellByType(comm, DM_POLYTOPE_TRIANGLE, refdm);CHKERRQ(ierr);}
    else         {ierr = DMPlexCreateReferenceCellByType(comm, DM_POLYTOPE_QUADRILATERAL, refdm);CHKERRQ(ierr);}
    break;
  case 3:
    if (simplex) {ierr = DMPlexCreateReferenceCellByType(comm, DM_POLYTOPE_TETRAHEDRON, refdm);CHKERRQ(ierr);}
    else         {ierr = DMPlexCreateReferenceCellByType(comm, DM_POLYTOPE_HEXAHEDRON, refdm);CHKERRQ(ierr);}
    break;
  default:
    SETERRQ1(comm, PETSC_ERR_ARG_WRONG, "Cannot create reference cell for dimension %D", dim);
  }
  PetscFunctionReturn(0);
}