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Gridap.Adaptivity

CurrentModule = Gridap.Adaptivity

The adaptivity module provides a framework to work with adapted (refined/coarsened/mixed) meshes.

It provides

  • A generic interface to represent adapted meshes and a set of tools to work with Finite Element spaces defined on them. In particular, moving CellFields between parent and child meshes.
  • Particular implementations for conformally refining/coarsening 2D/3D meshes using several well-known strategies. In particular, Red-Green refinement and longest-edge bisection.

Interface

The following types are defined in the module:

RefinementRule
AdaptivityGlue
AdaptedDiscreteModel
AdaptedTriangulation

The high-level interface is provided by the following methods:

refine
coarsen
adapt

Edge-Based refinement

The module provides a refine method for UnstructuredDiscreteModel. The method takes a string refinement_method that determines the refinement strategy to be used. The following strategies are available:

  • "red_green" :: Red-Green refinement, default.
  • "nvb" :: Longest-edge bisection (only for meshes of TRIangles)
  • "barycentric" :: Barycentric refinement (only for meshes of TRIangles)
  • "simplexify" :: Simplexify refinement. Same resulting mesh as the simplexify method, but keeps track of the parent-child relationships.

Additionally, the method takes a kwarg cells_to_refine that determines which cells will be refined. Possible input types are:

  • Nothing :: All cells get refined.
  • AbstractArray{<:Bool} of size num_cells(model) :: Only cells such that cells_to_refine[iC] == true get refined.
  • AbstractArray{<:Integer} :: Cells for which gid ∈ cells_to_refine get refined

The algorithms try to respect the cells_to_refine input as much as possible, but some additional cells might get refined in order to guarantee that the mesh remains conforming.

  function refine(model::UnstructuredDiscreteModel;refinement_method="red_green",kwargs...)
    [...]
  end

CartesianDiscreteModel refining

The module provides a refine method for CartesianDiscreteModel. The method takes a Tuple of size Dc (the dimension of the model cells) that will determine how many times cells will be refined in each direction. For example, for a 2D model, refine(model,(2,3)) will refine each QUAD cell into a 2x3 grid of cells.

  function refine(model::CartesianDiscreteModel{Dc}, cell_partition::Tuple) where Dc
    [...]
  end

Macro Finite-Elements

The module also provides support for macro finite-elements. From an abstract point of view, a macro finite-element is a finite-element defined on a refined polytope, where polynomial basis are defined on each of the subcells (creating a broken piece-wise polynomial space on the original polytope). From Gridap's point of view, a macro finite-element is a ReferenceFE defined on a RefinementRule from an array of ReferenceFEs defined on the subcells.

Although there are countless combinations, here are two possible applications:

  • Linearized High-Order Lagrangian FESpaces: These are spaces which have the same DoFs as a high-order Lagrangian space, but where the basis functions are linear on each subcell.
  • Barycentrically-refined elements for Stokes-like problems: These are spaces where the basis functions for velocity are defined on the barycentrically-refined mesh, whereas the basis functions for pressure are defined on the original cells. This allows for exact so-called Stokes sequences (see here).

The API is given by the following methods:

  MacroReferenceFE

Adaptive Mesh Refinement

One of the main uses of mesh refinement is Adaptive Mesh Refinement, where the mesh is refined only in regions of interest.

The typical AMR workflow is the so-called solve-estimate-mark-refine loop. Since estimators will generally be problem-dependent, we only aim to provide some generic tools that can be combined by the user:

  DorflerMarking
  mark
  estimate

Notes for users

Most of the tools provided by this module are showcased in the tests of the module itself, as well as the following tutorial (coming soon).

However, we want to stress a couple of key performance-critical points:

  • The refining/coarsening routines are not optimized for performance. In particular, they are not parallelized. If you require an optimized/parallel implementation, please consider leveraging specialised meshing libraries. For instance, we provide an implementation of refine/coarsen using P4est in the GridapP4est.jl library.

  • Although the toolbox allows you to evaluate CellFields defined on both fine/coarse meshes on their parent/children mesh, both directions of evaluation are not equivalent. As a user, you should always try to evaluate/integrate on the finest mesh for maximal performance. Evaluating a fine CellField on a coarse mesh relies on local tree searches, and is therefore a very expensive operation that should be avoided whenever possible.

Notes for developers

RefinementRule API

Given a RefinementRule, the library provides a set of methods to compute the mappings between parent (coarse) face ids and child (fine) face ids (and vice-versa).

The most basic information (that can directly be hardcoded in the RefinementRule for performance) are the mappings between parent face ids and child face ids. These are provided by:

get_d_to_face_to_child_faces
get_d_to_face_to_parent_face

On top of these two basic mappings, a whole plethora of additional topological mappings can be computed. These first set of routines extend the ReferenceFEs API to provide information on the face-to-node mappings and permutations:

ReferenceFEs.get_face_vertices
ReferenceFEs.get_face_coordinates
ReferenceFEs.get_vertex_permutations
ReferenceFEs.get_face_vertex_permutations

We also provide face-to-face maps:

get_cface_to_num_own_ffaces
get_cface_to_own_ffaces
get_cface_to_ffaces
get_cface_to_own_ffaces_to_lnodes
get_cface_to_ffaces_to_lnodes
get_cface_to_fface_permutations
aggregate_cface_to_own_fface_data
get_face_subface_ldof_to_cell_ldof

AdaptivityGlue API

get_n2o_reference_coordinate_map
get_old_cell_refinement_rules
get_new_cell_refinement_rules
get_d_to_fface_to_cface
n2o_reindex
o2n_reindex

New-to-old field evaluations

When a cell is refined, we need to be able to evaluate the fields defined on the children cells on the parent cell. To do so, we bundle the fields defined on the children cells into a new type of Field called FineToCoarseField. When evaluated on a Point, a FineToCoarseField will select the child cell that contains the Point and evaluate the mapped point on the corresponding child field.

FineToCoarseField
FineToCoarseDofBasis
FineToCoarseRefFE