.. _ch_unstructured: DMPlex: Unstructured Grids -------------------------- This chapter introduces the ``DMPLEX`` subclass of ``DM``, which allows the user to handle unstructured grids 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*, 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 `__, 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 .. math:: 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, .. code-block:: DMPlexSetChart(dm, 0, 11); Note that a *chart* here corresponds to a semi-closed interval (e.g :math:`[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, .. code-block:: /* 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), .. code-block:: /* 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 .. code-block:: 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 .. code-block:: DMPlexStratify(dm); 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 satisified, 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: .. code-block:: 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)); .. _sec_petscsection: Connecting Data on Grids to its Location in arrays or Vec (PetscSection) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The strongest links between solvers and discretizations are - the relationship between the layout of data (unknowns) over a mesh (or similar structure) and the data layout in arrays and ``Vec`` used for computation, - data (unknowns) partitioning, and - ordering of data (unknowns). To enable modularity, we encode the operations above in simple data structures that can be understood by the linear algebra (``Vec``, ``Mat``, ``KSP``, ``PC``, ``SNES``), time integrator (``TS``), and optimization (``Tao``) engines in PETSc without explicit reference to the mesh (topology) or discretization (analysis). Data Layout by Hand ^^^^^^^^^^^^^^^^^^^ .. TODO: This text needs additional work so it can be understood without a detailed (or any) understanding of ``DMPLEX`` because the ``PetscSection`` concept is below ``DM`` in the .. We may want to even move this introductory ``PetscSection`` material to its own pride of place in the user guide and not inside the ``DMPLEX`` discussion. Specific entries (or collections of entries) in a ``Vec`` (or a simple array) can be associated with a "location" on a mesh (or other types of data structure) using the ``PetscSection`` object. A **point** is a ``PetscInt`` that serves as an abstract "index" into arrays from iteratable sets, such as points on a mesh. ``PetscSection`` has two modes of operation. Mode 1: A ``PetscSection`` associates a set of degrees of freedom (dof), (a small space :math:`\{e_k\} 0 < k < d_p`), with every point. The number of dof and their meaning may be different for different points. For example, the dof on a cell point may represent pressure while a dof on a face point may represent velocity. Though points must be contiguously numbered, they can be in any range :math:`[\mathrm{pStart}, \mathrm{pEnd})`, which is called a **chart**. A ``PetscSection`` in mode 1 may be thought of as defining a two dimensional array indexed by point in the outer dimension with a variable length inner dimension indexed by the dof at that point, :math:`v[pStart <= point < pEnd][0 <= dof `__). 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: .. code-block:: 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 .. code-block:: DMViewFromOptions(dm, NULL, "-dm_view"); VecViewFromOptions(vec, NULL, "-vec_view") along with the command line options .. code-block:: 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: .. code-block:: 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: .. code-block:: PetscObjectSetName((PetscObject)dm, "plexA"); PetscViewerPushFormat(viewer, PETSC_VIEWER_HDF5_PETSC); DMView(dm, viewer); PetscViewerPopFormat(viewer); The ``DMView()`` call is shorthand for the following sequence .. code-block:: 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 :math:`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: .. code-block:: 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 :math:`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 assiciated with ``sdm``, e.g., as: .. code-block:: Vec vec; DMGetLocalVector(sdm, &vec); After setting values of ``vec``, we name it "vecA" and save it as: .. code-block:: 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: .. code-block:: 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//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//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//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: .. code-block:: 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: .. code-block:: 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 .. code-block:: 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 :math:`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: .. code-block:: 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 :math:`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: .. code-block:: 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 :math:`X`, ``DMPlexSectionLoad()`` internally constructs a ``PetscSF`` that pushes forward the on-disk data layout to :math:`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 assiciated with ``sdm``, e.g., as: .. code-block:: Vec vec; DMGetLocalVector(sdm, &vec); We then name ``vec`` "vecA" and load the on-disk vector into ``vec`` as: .. code-block:: 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: .. code-block:: 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 `__. 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 .. code-block:: 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 .. code-block:: 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 .. code-block:: DMPlexMetricSetMinimumMagnitude(DM dm, PetscReal h_min); DMPlexMetricSetMaximumMagnitude(DM dm, PetscReal h_max); DMPlexMetricSetMaximumAnisotropy(DM dm, PetscReal a_max); or the command line arguments .. code-block:: -dm_plex_metric_h_min -dm_plex_metric_h_max -dm_plex_metric_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 .. code-block:: 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 :math:`L^p` normalization framework, with the routine .. code-block:: DMPlexMetricNormalize(DM dm, Vec metricIn, PetscBool restrictSizes, PetscBool restrictAnisotropy, Vec metricOut, Vec determinant); There are two important parameters for the normalization: the normalization order :math:`p` and the target metric complexity, which is analogous to the vertex count. They are controlled using .. code-block:: DMPlexMetricSetNormalizationOrder(DM dm, PetscReal p); DMPlexMetricSetTargetComplexity(DM dm, PetscReal target); or the command line arguments .. code-block:: console -dm_plex_metric_p

-dm_plex_metric_target_complexity Two different metric-based mesh adaptation tools are available in PETSc: - `Pragmatic `__; - `Mmg/ParMmg `__. 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 .. code-block:: -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 .. code-block:: 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 .. [#petscsection_footnote] A ``PetscSection`` can be thought of as a generalization of ``PetscLayout``, in the same way that a fiber bundle is a generalization of the normal Euclidean basis used in linear algebra. With ``PetscLayout``, we associate a unit vector (:math:`e_i`) with every point in the space, and just divide up points between processes. .. [#boundary_footnote] The boundary of a cell is its faces, the boundary of a face is its edges and the boundary of an edge is the two vertices. .. bibliography:: /petsc.bib :filter: docname in docnames