(ch_unstructured)=

# DMPlex: Unstructured Grids

This chapter introduces the `DMPLEX` subclass of `DM`, which allows
the user to handle unstructured grids (or meshes) using the generic `DM` interface
for hierarchy and multi-physics. `DMPLEX` was created to remedy a huge
problem in all current PDE simulation codes, namely that the
discretization was so closely tied to the data layout and solver that
switching discretizations in the same code was not possible. Not only
does this preclude the kind of comparison that is necessary for
scientific investigation, but it makes library (as opposed to monolithic
application) development impossible.

## Representing Unstructured Grids

The main advantage of `DMPLEX` in representing topology is that it
treats all the different pieces of a mesh, e.g. cells, faces, edges, and
vertices, in the same way. This allows the interface to be
small and simple, while remaining flexible and general. This also allows
“dimension independent programming”, which means that the same algorithm
can be used unchanged for meshes of different shapes and dimensions.

All pieces of the mesh (vertices, edges, faces, and cells) are treated as *points* (or mesh entities), which are each identified by a
`PetscInt`. A mesh is built by relating points to other points, in
particular specifying a “covering” relation among the points. For
example, an edge is defined by being covered by two vertices, and a
triangle can be defined by being covered by three edges (or even by
three vertices). This structure is known as a [Hasse Diagram](http://en.wikipedia.org/wiki/Hasse_diagram), which is a
Directed Acyclic Graph (DAG) representing a cell complex using the
covering relation. The graph edges represent the relation, which also
encodes a partially ordered set (poset).

For example, we can encode the doublet mesh as in {numref}`fig_doubletMesh`,

:::{figure} /images/manual/dmplex_doublet_mesh.svg
:name: fig_doubletMesh

A 2D doublet mesh, two triangles sharing an edge.
:::

which can also be represented as the DAG in
{numref}`fig_doubletDAG`.

:::{figure} /images/manual/dmplex_doublet_dag.svg
:name: fig_doubletDAG

The Hasse diagram for our 2D doublet mesh, expressed as a DAG.
:::

To use the PETSc API, we consecutively number the mesh pieces. The
PETSc convention in 3 dimensions is to number first cells, then
vertices, then faces, and then edges. In 2 dimensions the convention is
to number faces, vertices, and then edges.
In terms of the labels in
{numref}`fig_doubletMesh`, these numberings are

$$
f_0 \mapsto \mathtt{0}, f_1 \mapsto \mathtt{1}, \\ v_0 \mapsto \mathtt{2}, v_1 \mapsto \mathtt{3}, v_2 \mapsto \mathtt{4}, v_3 \mapsto \mathtt{5}, \\ e_0 \mapsto \mathtt{6}, e_1 \mapsto \mathtt{7}, e_2 \mapsto \mathtt{8}, e_3 \mapsto \mathtt{9}, e_4 \mapsto \mathtt{10}
$$

First, we declare the set of points present in a mesh,

```
DMPlexSetChart(dm, 0, 11);
```

Note that a *chart* here corresponds to a semi-closed interval (e.g
$[0,11) = \{0,1,\ldots,10\}$) specifying the range of indices we’d
like to use to define points on the current rank. We then define the
covering relation, which we call the *cone*, which are also the in-edges
in the DAG. In order to preallocate correctly, we first provide sizes,

```
/* DMPlexSetConeSize(dm, point, number of points that cover the point); */
DMPlexSetConeSize(dm, 0, 3);
DMPlexSetConeSize(dm, 1, 3);
DMPlexSetConeSize(dm, 6, 2);
DMPlexSetConeSize(dm, 7, 2);
DMPlexSetConeSize(dm, 8, 2);
DMPlexSetConeSize(dm, 9, 2);
DMPlexSetConeSize(dm, 10, 2);
DMSetUp(dm);
```

and then point values (recall each point is an integer that represents a single geometric entity, a cell, face, edge, or vertex),

```
/* DMPlexSetCone(dm, point, [points that cover the point]); */
DMPlexSetCone(dm, 0, [6, 7, 8]);
DMPlexSetCone(dm, 1, [7, 9, 10]);
DMPlexSetCone(dm, 6, [2, 3]);
DMPlexSetCone(dm, 7, [3, 4]);
DMPlexSetCone(dm, 8, [4, 2]);
DMPlexSetCone(dm, 9, [4, 5]);
DMPlexSetCone(dm, 10, [5, 3]);
```

There is also an API for providing the dual relation, using
`DMPlexSetSupportSize()` and `DMPlexSetSupport()`, but this can be
calculated automatically using the provided `DMPlexSetConeSize()` and `DMPlexSetCone()` information and then calling

```
DMPlexSymmetrize(dm);
```

The "symmetrization" is in the sense of the DAG. Each point knows its covering (cone) and each point knows what it covers (support). Note that when using automatic symmetrization, cones will be ordered but supports will not. The user can enforce an ordering on supports by rewriting them after symmetrization using `DMPlexSetSupport()`.

In order to support efficient queries, we construct fast
search structures and indices for the different types of points using

```
DMPlexStratify(dm);
```

## Grid Point Orientations

TODO: fill this out with regard to orientations.

Should probably reference Hapla and summarize what's there.

## Dealing with Periodicity

Plex allows you to represent periodic domains is two ways. Using the default scheme, periodic topology can be represented directly. This ensures that all topological queries can be satisfied, but then care must be taken in representing functions over the mesh, such as the coordinates. The second method is to use a non-periodic topology, but connect certain mesh points using the local-to-global map for that DM. This allows a more general set of mappings to be implemented, such as partial twists, but topological queries on the periodic boundary cease to function.

For the default scheme, a call to `DMLocalizeCoordinates()` (which usually happens automatically on mesh creation) creates a second, discontinuous coordinate field. These values can be accessed using `DMGetCellCoordinates()` and `DMGetCellCoordinatesLocal()`. Plex provides a convenience method, `DMPlexGetCellCoordinates()`, that extracts cell coordinates correctly, depending on the periodicity of the mesh. An example of its use is shown below:

```
const PetscScalar *array;
PetscScalar       *coords = NULL;
PetscInt           numCoords;
PetscBool          isDG;

PetscCall(DMPlexGetCellCoordinates(dm, cell, &isDG, &numCoords, &array, &coords));
for (PetscInt cc = 0; cc < numCoords/dim; ++cc) {
  if (cc > 0) PetscCall(PetscPrintf(PETSC_COMM_SELF, " -- "));
  PetscCall(PetscPrintf(PETSC_COMM_SELF, "("));
  for (PetscInt d = 0; d < dim; ++d) {
    if (d > 0) PetscCall(PetscPrintf(PETSC_COMM_SELF, ", "));
    PetscCall(PetscPrintf(PETSC_COMM_SELF, "%g", (double)PetscRealPart(coords[cc * dim + d])));
  }
  PetscCall(PetscPrintf(PETSC_COMM_SELF, ")"));
}
PetscCall(PetscPrintf(PETSC_COMM_SELF, "\n"));
PetscCall(DMPlexRestoreCellCoordinates(dm, cell, &isDG, &numCoords, &array, &coords));
```

## Connecting Grids to Data Using PetscSection:

A `PetscSection` is used to describe the connection between the grid and data associated with the grid.
Specifically, it assigns a number of dofs to each mesh entity on the DAG.
See {any}`ch_petscsection` for more details.
Using the mesh from {numref}`fig_doubletMesh`, we provide an example of creating a `PetscSection` for a single field.
We can lay out data for a continuous Galerkin $P_3$ finite element method,

```
PetscInt pStart, pEnd, cStart, cEnd, c, vStart, vEnd, v, eStart, eEnd, e;

DMPlexGetChart(dm, &pStart, &pEnd);
DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);   // cells
DMPlexGetHeightStratum(dm, 1, &eStart, &eEnd);   // edges
DMPlexGetHeightStratum(dm, 2, &vStart, &vEnd);   // vertices, equivalent to DMPlexGetDepthStratum(dm, 0, &vStart, &vEnd);
PetscSectionSetChart(s, pStart, pEnd);
for(c = cStart; c < cEnd; ++c)
    PetscSectionSetDof(s, c, 1);
for(v = vStart; v < vEnd; ++v)
    PetscSectionSetDof(s, v, 1);
for(e = eStart; e < eEnd; ++e)
    PetscSectionSetDof(s, e, 2); // two dof on each edge
PetscSectionSetUp(s);
```

`DMPlexGetHeightStratum()` returns all the points of the requested height
in the DAG. Since this problem is in two dimensions the edges are at
height 1 and the vertices at height 2 (the cells are always at height
0). One can also use `DMPlexGetDepthStratum()` to use the depth in the
DAG to select the points. `DMPlexGetDepth(dm,&depth)` returns the depth
of the DAG, hence `DMPlexGetDepthStratum(dm,depth-1-h,)` returns the
same values as `DMPlexGetHeightStratum(dm,h,)`.

For $P_3$ elements there is one degree of freedom at each vertex, 2 along
each edge (resulting in a total of 4 degrees of freedom along each edge
including the vertices, thus being able to reproduce a cubic function)
and 1 degree of freedom within the cell (the bubble function which is
zero along all edges).

Now a PETSc local vector can be created manually using this layout,

```
PetscSectionGetStorageSize(s, &n);
VecSetSizes(localVec, n, PETSC_DETERMINE);
VecSetFromOptions(localVec);
```

When working with `DMPLEX` and `PetscFE` (see below) one can simply get the sections (and related vectors) with

```
DMSetLocalSection(dm, s);
DMGetLocalVector(dm, &localVec);
DMGetGlobalVector(dm, &globalVec);
```

### DMPlex-specific PetscSection Features:

The following features are built into `PetscSection`.
However, their usage and purpose is best understood through `DMPLEX`.

#### Closure:

Closure information can be attached to a `PetscSection` to allow for more efficient closure information queries.
This information can either be set directly with `DMPlexCreateClosureIndex()` or generated automatically for a `DMPLEX` via `DMPlexCreateClosureIndex()`.

#### Symmetries: Accessing data from different orientations

While mesh point orientation information specifies how one mesh point is oriented with respect to another, it does not describe how the dofs associated with that mesh point should be permuted for that orientation.
This information is supplied via a `PetscSectionSym` object that is attached to the `PetscSection`.
Generally the setup and usage of this information is handled automatically by PETSc during setup of a Plex using `PetscFE`.

#### Closure Permutation:

A permutation of the dof closure of a k-cell may be specified.
This allows data to be returned in an order that might be more efficiently processed than the default (breadth-first search) ordering.
For example, for tensor cells such as quadrilaterals, closure data can be permuted to lexicographic order (i.e. a tensor-product ordering).
This is most commonly done via `DMPlexSetClosurePermutationTensor()`.
Custom permutations can be set using `PetscSectionSetClosurePermutation()`.

## Data Layout using DMPLEX and PetscFE

A `DM` can automatically create the local section if given a description of the discretization, for example using a `PetscFE` object. We demonstrate this by creating
a `PetscFE` that can be configured from the command line. It is a single, scalar field, and is added to the `DM` using `DMSetField()`.
When a local or global vector is requested, the `DM` builds the local and global sections automatically.

```
DMPlexIsSimplex(dm, &simplex);
PetscFECreateDefault(PETSC_COMM_SELF, dim, 1, simplex, NULL, -1, &fe);
DMSetField(dm, 0, NULL, (PetscObject) fe);
DMCreateDS(dm);
```

Here the call to `DMSetField()` declares the discretization will have one field with the integer label 0 that has one degree of freedom at each point on the `DMPlex`.
To get the $P_3$ section above, we can either give the option `-petscspace_degree 3`, or call `PetscFECreateLagrange()` and set the degree directly.

## Partitioning and Ordering

In the same way as `MatPartitioning` or
`MatGetOrdering()`, give the results of a partitioning or ordering of a graph defined by a sparse matrix,
`PetscPartitionerDMPlexPartition` or `DMPlexPermute` are encoded in
an `IS`. However, the graph is not the adjacency graph of the matrix
but the mesh itself. Once the mesh is partitioned and
reordered, the data layout from a `PetscSection` can be used to
automatically derive a problem partitioning/ordering.

### Influence of Variables on One Another

The Jacobian of a problem represents the influence of some
variable on other variables in the problem. Very often, this influence
pattern is determined jointly by the computational mesh and
discretization. `DMCreateMatrix()` must compute this pattern when it
automatically creates the properly preallocated Jacobian matrix. In
`DMDA` the influence pattern, or what we will call variable
*adjacency*, depends only on the stencil since the topology is Cartesian
and the discretization is implicitly finite difference.

In `DMPLEX`,
we allow the user to specify the adjacency topologically, while
maintaining good defaults. The pattern is controlled by two flags. The first flag, `useCone`,
indicates whether variables couple first to their boundary [^boundary-footnote]
and then to
neighboring entities, or the reverse. For example, in finite elements,
the variables couple to the set of neighboring cells containing the mesh
point, and we set the flag to `useCone = PETSC_FALSE`. By contrast,
in finite volumes, cell variables first couple to the cell boundary, and
then to the neighbors, so we set the flag to `useCone = PETSC_TRUE`.
The second flag, `useClosure`, indicates whether we consider the
transitive closure of the neighbor relation above, or just a single
level. For example, in finite elements, the entire boundary of any cell
couples to the interior, and we set the flag to
`useClosure = PETSC_TRUE`. By contrast, in most finite volume methods,
cells couple only across faces, and not through vertices, so we set the
flag to `useClosure = PETSC_FALSE`. However, the power of this method
is its flexibility. If we wanted a finite volume method that coupled all
cells around a vertex, we could easily prescribe that by changing to
`useClosure = PETSC_TRUE`.

## Evaluating Residuals

The evaluation of a residual or Jacobian, for most discretizations has
the following general form:

- Traverse the mesh, picking out pieces (which in general overlap),
- Extract some values from the current solution vector, associated with this
  piece,
- Calculate some values for the piece, and
- Insert these values into the residual vector

DMPlex separates these different concerns by passing sets of points from mesh traversal routines to data
extraction routines and back. In this way, the `PetscSection` which
structures the data inside a `Vec` does not need to know anything
about the mesh inside a `DMPLEX`.

The most common mesh traversal is the transitive closure of a point,
which is exactly the transitive closure of a point in the DAG using the
covering relation. In other words, the transitive closure consists of
all points that cover the given point (generally a cell) plus all points
that cover those points, etc. So in 2d the transitive closure for a cell
consists of edges and vertices while in 3d it consists of faces, edges,
and vertices. Note that this closure can be calculated orienting the
arrows in either direction. For example, in a finite element
calculation, we calculate an integral over each element, and then sum up
the contributions to the basis function coefficients. The closure of the
element can be expressed discretely as the transitive closure of the
element point in the mesh DAG, where each point also has an orientation.
Then we can retrieve the data using `PetscSection` methods,

```
PetscScalar *a;
PetscInt     numPoints, *points = NULL, p;

VecGetArrayRead(u,&a);
DMPlexGetTransitiveClosure(dm,cell,PETSC_TRUE,&numPoints,&points);
for (p = 0; p <= numPoints*2; p += 2) {
  PetscInt dof, off, d;

  PetscSectionGetDof(section, points[p], &dof);
  PetscSectionGetOffset(section, points[p], &off);
  for (d = 0; d <= dof; ++d) {
    myfunc(a[off+d]);
  }
}
DMPlexRestoreTransitiveClosure(dm, cell, PETSC_TRUE, &numPoints, &points);
VecRestoreArrayRead(u, &a);
```

This operation is so common that we have built a convenience method
around it which returns the values in a contiguous array, correctly
taking into account the orientations of various mesh points:

```
const PetscScalar *values;
PetscInt           csize;

DMPlexVecGetClosure(dm, section, u, cell, &csize, &values);
// Do integral in quadrature loop putting the result into r[]
DMPlexVecRestoreClosure(dm, section, u, cell, &csize, &values);
DMPlexVecSetClosure(dm, section, residual, cell, &r, ADD_VALUES);
```

A simple example of this kind of calculation is in
`DMPlexComputeL2Diff_Plex()` (source).
Note that there is no restriction on the type of cell or dimension of
the mesh in the code above, so it will work for polyhedral cells, hybrid
meshes, and meshes of any dimension, without change. We can also reverse
the covering relation, so that the code works for finite volume methods
where we want the data from neighboring cells for each face:

```
PetscScalar *a;
PetscInt     points[2*2], numPoints, p, dofA, offA, dofB, offB;

VecGetArray(u,  &a);
DMPlexGetTransitiveClosure(dm, cell, PETSC_FALSE, &numPoints, &points);
assert(numPoints == 2);
PetscSectionGetDof(section, points[0*2], &dofA);
PetscSectionGetDof(section, points[1*2], &dofB);
assert(dofA == dofB);
PetscSectionGetOffset(section, points[0*2], &offA);
PetscSectionGetOffset(section, points[1*2], &offB);
myfunc(a[offA], a[offB]);
VecRestoreArray(u, &a);
```

This kind of calculation is used in
TS Tutorial ex11.

## Saving and Loading DMPlex Data with HDF5

PETSc allows users to save/load `DMPLEX`s representing meshes,
`PetscSection`s representing data layouts on the meshes, and
`Vec`s defined on the data layouts to/from an HDF5 file in
parallel, where one can use different number of processes for saving
and for loading.

### Saving

The simplest way to save `DM` data is to use options for configuration.
This requires only the code

```
DMViewFromOptions(dm, NULL, "-dm_view");
VecViewFromOptions(vec, NULL, "-vec_view")
```

along with the command line options

```console
$ ./myprog -dm_view hdf5:myprog.h5 -vec_view hdf5:myprog.h5::append
```

Options prefixes can be used to separately control the saving and loading of various fields.
However, the user can have finer grained control by explicitly creating the PETSc objects involved.
To save data to "example.h5" file, we can first create a `PetscViewer` of type `PETSCVIEWERHDF5` in `FILE_MODE_WRITE` mode as:

```
PetscViewer  viewer;

PetscViewerHDF5Open(PETSC_COMM_WORLD, "example.h5", FILE_MODE_WRITE, &viewer);
```

As `dm` is a `DMPLEX` object representing a mesh, we first give it a *mesh name*, "plexA", and save it as:

```
PetscObjectSetName((PetscObject)dm, "plexA");
PetscViewerPushFormat(viewer, PETSC_VIEWER_HDF5_PETSC);
DMView(dm, viewer);
PetscViewerPopFormat(viewer);
```

The `DMView()` call is shorthand for the following sequence

```
DMPlexTopologyView(dm, viewer);
DMPlexCoordinatesView(dm, viewer);
DMPlexLabelsView(dm, viewer);
```

If the *mesh name* is not explicitly set, the default name is used.
In the above `PETSC_VIEWER_HDF5_PETSC` format was used to save the entire representation of the mesh.
This format also saves global point numbers attached to the mesh points.
In this example the set of all global point numbers is $X = [0, 11)$.

The data layout, `s`, needs to be wrapped in a `DM` object for it to be saved.
Here, we create the wrapping `DM`, `sdm`, with `DMClone()`, give it a *dm name*, "dmA", attach `s` to `sdm`, and save it as:

```
DMClone(dm, &sdm);
PetscObjectSetName((PetscObject)sdm, "dmA");
DMSetLocalSection(sdm, s);
DMPlexSectionView(dm, viewer, sdm);
```

If the *dm name* is not explicitly set, the default name is to be used.
In the above, instead of using `DMClone()`, one could also create a new `DMSHELL` object to attach `s` to.
The first argument of `DMPlexSectionView()` is a `DMPLEX` object that represents the mesh, and the third argument is a `DM` object that carries the data layout that we would like to save.
They are, in general, two different objects, and the former carries a *mesh name*, while the latter carries a *dm name*.
These names are used to construct a group structure in the HDF5 file.
Note that the data layout points are associated with the mesh points, so each of them can also be tagged with a global point number in $X$; `DMPlexSectionView()` saves these tags along with the data layout itself, so that, when the mesh and the data layout are loaded separately later, one can associate the points in the former with those in the latter by comparing their global point numbers.

We now create a local vector associated with `sdm`, e.g., as:

```
Vec  vec;

DMGetLocalVector(sdm, &vec);
```

After setting values of `vec`, we name it "vecA" and save it as:

```
PetscObjectSetName((PetscObject)vec, "vecA");
DMPlexLocalVectorView(dm, viewer, sdm, vec);
```

A global vector can be saved in the exact same way with trivial changes.

After saving, we destroy the `PetscViewer` with:

```
PetscViewerDestroy(&viewer);
```

The output file "example.h5" now looks like the following:

```
$ h5dump --contents example.h5
HDF5 "example.h5" {
FILE_CONTENTS {
 group      /
 group      /topologies
 group      /topologies/plexA
 group      /topologies/plexA/dms
 group      /topologies/plexA/dms/dmA
 dataset    /topologies/plexA/dms/dmA/order
 group      /topologies/plexA/dms/dmA/section
 dataset    /topologies/plexA/dms/dmA/section/atlasDof
 dataset    /topologies/plexA/dms/dmA/section/atlasOff
 group      /topologies/plexA/dms/dmA/vecs
 group      /topologies/plexA/dms/dmA/vecs/vecA
 dataset    /topologies/plexA/dms/dmA/vecs/vecA/vecA
 group      /topologies/plexA/labels
 group      /topologies/plexA/topology
 dataset    /topologies/plexA/topology/cells
 dataset    /topologies/plexA/topology/cones
 dataset    /topologies/plexA/topology/order
 dataset    /topologies/plexA/topology/orientation
 }
}
```

### Saving in the new parallel HDF5 format

Since PETSc 3.19, we offer a new format which supports parallel loading.
To write in this format, you currently need to specify it explicitly using the option

```
-dm_plex_view_hdf5_storage_version 3.0.0
```

The storage version is stored in the file and set automatically when loading (described below).
You can check the storage version of the HDF5 file with

```
$ h5dump -a /dmplex_storage_version example.h5
```

To allow for simple and efficient implementation, and good load balancing, the 3.0.0 format changes the way the mesh topology is stored.
Different strata (sets of mesh entities with an equal dimension; or vertices, edges, faces, and cells) are now stored separately.
The new structure of `/topologies/<mesh_name>/topology` is following:

```
$ h5dump --contents example.h5
HDF5 "example.h5" {
FILE_CONTENTS {
 ...
 group      /topologies/plexA/topology
 dataset    /topologies/plexA/topology/permutation
 group      /topologies/plexA/topology/strata
 group      /topologies/plexA/topology/strata/0
 dataset    /topologies/plexA/topology/strata/0/cone_sizes
 dataset    /topologies/plexA/topology/strata/0/cones
 dataset    /topologies/plexA/topology/strata/0/orientations
 group      /topologies/plexA/topology/strata/1
 dataset    /topologies/plexA/topology/strata/1/cone_sizes
 dataset    /topologies/plexA/topology/strata/1/cones
 dataset    /topologies/plexA/topology/strata/1/orientations
 group      /topologies/plexA/topology/strata/2
 dataset    /topologies/plexA/topology/strata/2/cone_sizes
 dataset    /topologies/plexA/topology/strata/2/cones
 dataset    /topologies/plexA/topology/strata/2/orientations
 group      /topologies/plexA/topology/strata/3
 dataset    /topologies/plexA/topology/strata/3/cone_sizes
 dataset    /topologies/plexA/topology/strata/3/cones
 dataset    /topologies/plexA/topology/strata/3/orientations
 }
}
```

Group `/topologies/<mesh_name>/topology/strata` contains a subgroup for each stratum depth (dimension; 0 for vertices up to 3 for cells).
DAG points (mesh entities) have an implicit global numbering, given by the position in `orientations` (or `cone_sizes`) plus the stratum offset.
The stratum offset is given by a sum of lengths of all previous strata with respect to the order stored in `/topologies/<mesh_name>/topology/permutation`.
This global numbering is compatible with the explicit numbering in dataset `topology/order` of previous versions.

For a DAG point with index `i` at depth `s`, `cone_sizes[i]` gives a size of this point's cone (set of adjacent entities with depth `s-1`).
Let `o = sum(cone_sizes[0:i]])` (in Python syntax).
Points forming the cone are then given by `cones[o:o+cone_sizes[i]]` (in numbering relative to the depth `s-1`).
The orientation of the cone with respect to point `i` is then stored in `orientations[i]`.

### Loading

To load data from "example.h5" file, we create a `PetscViewer`
of type `PETSCVIEWERHDF5` in `FILE_MODE_READ` mode as:

```
PetscViewerHDF5Open(PETSC_COMM_WORLD, "example.h5", FILE_MODE_READ, &viewer);
```

We then create a `DMPLEX` object, give it a *mesh name*, "plexA", and load
the mesh as:

```
DMCreate(PETSC_COMM_WORLD, &dm);
DMSetType(dm, DMPLEX);
PetscObjectSetName((PetscObject)dm, "plexA");
PetscViewerPushFormat(viewer, PETSC_VIEWER_HDF5_PETSC);
DMLoad(dm, viewer);
PetscViewerPopFormat(viewer);
```

where `PETSC_VIEWER_HDF5_PETSC` format was again used. The user can have more control by replace the single load call with

```
PetscSF  sfO;

DMCreate(PETSC_COMM_WORLD, &dm);
DMSetType(dm, DMPLEX);
PetscObjectSetName((PetscObject)dm, "plexA");
PetscViewerPushFormat(viewer, PETSC_VIEWER_HDF5_PETSC);
DMPlexTopologyLoad(dm, viewer, &sfO);
DMPlexCoordinatesLoad(dm, viewer, sfO);
PetscViewerPopFormat(viewer);
```

The object returned by `DMPlexTopologyLoad()`, `sfO`, is a
`PetscSF` that pushes forward $X$ to the loaded mesh,
`dm`; this `PetscSF` is constructed with the global point
number tags that we saved along with the mesh points.

As the `DMPLEX` mesh just loaded might not have a desired distribution,
it is common to redistribute the mesh for a better distribution using
`DMPlexDistribute()`, e.g., as:

```
DM        distributedDM;
PetscInt  overlap = 1;
PetscSF   sfDist, sf;

DMPlexDistribute(dm, overlap, &sfDist, &distributedDM);
if (distributedDM) {
  DMDestroy(&dm);
  dm = distributedDM;
  PetscObjectSetName((PetscObject)dm, "plexA");
}
PetscSFCompose(sfO, sfDist, &sf);
PetscSFDestroy(&sfO);
PetscSFDestroy(&sfDist);
```

Note that the new `DMPLEX` does not automatically inherit the *mesh name*,
so we need to name it "plexA" once again. `sfDist` is a `PetscSF`
that pushes forward the loaded mesh to the redistributed mesh, so, composed
with `sfO`, it makes the `PetscSF` that pushes forward $X$
directly to the redistributed mesh, which we call `sf`.

We then create a new `DM`, `sdm`, with `DMClone()`, give it
a *dm name*, "dmA", and load the on-disk data layout into `sdm` as:

```
PetscSF  globalDataSF, localDataSF;

DMClone(dm, &sdm);
PetscObjectSetName((PetscObject)sdm, "dmA");
DMPlexSectionLoad(dm, viewer, sdm, sf, &globalDataSF, &localDataSF);
```

where we could also create a new
`DMSHELL` object instead of using `DMClone()`.
Each point in the on-disk data layout being tagged with a global
point number in $X$, `DMPlexSectionLoad()`
internally constructs a `PetscSF` that pushes forward the on-disk
data layout to $X$.
Composing this with `sf`, `DMPlexSectionLoad()` internally
constructs another `PetscSF` that pushes forward the on-disk
data layout directly to the redistributed mesh. It then
reconstructs the data layout `s` on the redistributed mesh and
attaches it to `sdm`. The objects returned by this function,
`globalDataSF` and `localDataSF`, are `PetscSF`s that can
be used to migrate the on-disk vector data into local and global
`Vec`s defined on `sdm`.

We now create a local vector associated with `sdm`, e.g., as:

```
Vec  vec;

DMGetLocalVector(sdm, &vec);
```

We then name `vec` "vecA" and load the on-disk vector into `vec` as:

```
PetscObjectSetName((PetscObject)vec, "vecA");
DMPlexLocalVectorLoad(dm, viewer, sdm, localDataSF, localVec);
```

where `localDataSF` knows how to migrate the on-disk vector
data into a local `Vec` defined on `sdm`.
The on-disk vector can be loaded into a global vector associated with
`sdm` in the exact same way with trivial changes.

After loading, we destroy the `PetscViewer` with:

```
PetscViewerDestroy(&viewer);
```

The above infrastructure works seamlessly in distributed-memory parallel
settings, in which one can even use different number of processes for
saving and for loading; a more comprehensive example is found in
DMPlex Tutorial ex12.

(dm_adaptor_table)=

## Metric-based mesh adaptation

`DMPLEX` supports mesh adaptation using the *Riemannian metric framework*.
The idea is to use a Riemannian metric space within the mesher. The
metric space dictates how mesh resolution should be distributed across
the domain. Using this information, the remesher transforms the mesh such
that it is a *unit mesh* when viewed in the metric space. That is, the image
of each of its elements under the mapping from Euclidean space into the
metric space has edges of unit length.

One of the main advantages of metric-based mesh adaptation is that it allows
for fully anisotropic remeshing. That is, it provides a means of controlling
the shape and orientation of elements in the adapted mesh, as well as their
size. This can be particularly useful for advection-dominated and
directionally-dependent problems.

See {cite}`alauzet2010` for further details on metric-based anisotropic mesh
adaptation.

The two main ingredients for metric-based mesh adaptation are an input mesh
(i.e. the `DMPLEX`) and a Riemannian metric. The implementation in PETSc assumes
that the metric is piecewise linear and continuous across elemental boundaries.
Such an object can be created using the routine

```
DMPlexMetricCreate(DM dm, PetscInt field, Vec *metric);
```

A metric must be symmetric positive-definite, so that distances may be properly
defined. This can be checked using

```
DMPlexMetricEnforceSPD(DM dm, Vec metricIn, PetscBool restrictSizes, PetscBool restrictAnisotropy, Vec metricOut, Vec determinant);
```

This routine may also be used to enforce minimum and maximum tolerated metric
magnitudes (i.e. cell sizes), as well as maximum anisotropy. These quantities
can be specified using

```
DMPlexMetricSetMinimumMagnitude(DM dm, PetscReal h_min);
DMPlexMetricSetMaximumMagnitude(DM dm, PetscReal h_max);
DMPlexMetricSetMaximumAnisotropy(DM dm, PetscReal a_max);
```

or the command line arguments

```
-dm_plex_metric_h_min <h_min>
-dm_plex_metric_h_max <h_max>
-dm_plex_metric_a_max <a_max>
```

One simple way to combine two metrics is by simply averaging them entry-by-entry.
Another is to *intersect* them, which amounts to choosing the greatest level of
refinement in each direction. These operations are available in PETSc through
the routines

```
DMPlexMetricAverage(DM dm, PetscInt numMetrics, PetscReal weights[], Vec metrics[], Vec metricAvg);
DMPlexMetricIntersection(DM dm, PetscInt numMetrics, Vec metrics[], Vec metricInt);
```

However, before combining metrics, it is important that they are scaled in the same
way. Scaling also allows the user to control the number of vertices in the adapted
mesh (in an approximate sense). This is achieved using the $L^p$ normalization
framework, with the routine

```
DMPlexMetricNormalize(DM dm, Vec metricIn, PetscBool restrictSizes, PetscBool restrictAnisotropy, Vec metricOut, Vec determinant);
```

There are two important parameters for the normalization: the normalization order
$p$ and the target metric complexity, which is analogous to the vertex count.
They are controlled using

```
DMPlexMetricSetNormalizationOrder(DM dm, PetscReal p);
DMPlexMetricSetTargetComplexity(DM dm, PetscReal target);
```

or the command line arguments

```console
-dm_plex_metric_p <p>
-dm_plex_metric_target_complexity <target>
```

Two different metric-based mesh adaptation tools are available in PETSc:

- [Pragmatic](https://meshadaptation.github.io/);
- [Mmg/ParMmg](https://www.mmgtools.org/).

Mmg is a serial package, whereas ParMmg is the MPI version.
Note that surface meshing is not currently supported and that ParMmg
works only in three dimensions. Mmg can be used for both two and three dimensional problems.
Pragmatic, Mmg and ParMmg may be specified by the command line arguments

```
-dm_adaptor pragmatic
-dm_adaptor mmg
-dm_adaptor parmmg
```

Once a metric has been constructed, it can be used to perform metric-based
mesh adaptation using the routine

```
DMAdaptMetric(DM dm, Vec metric, DMLabel bdLabel, DMLabel rgLabel, DM dmAdapt);
```

where `bdLabel` and `rgLabel` are boundary and interior tags to be
preserved under adaptation, respectively.

```{rubric} Footnotes
```

[^boundary-footnote]: In three dimensions, the boundary of a cell (sometimes called an element) is its faces, the boundary of a face is its edges and the boundary of an edge is the two vertices.

```{eval-rst}
.. bibliography:: /petsc.bib
    :filter: docname in docnames
```
