GridapROMs.ParamFESpaces

module ParamGeometry

Parametric geometry — mapped grids and embedded triangulations for parameter-dependent domains.

Supports PDEs on domains that deform with the parameters, as required by, e.g., shape-optimisation or fluid–structure-interaction problems. Two main areas:

• Mapped grids — PhysicalMap and DisplacementMap represent the mapping from a reference domain to the physical domain at a given parameter sample. ParamGrid, ParamMappedGrid, ParamUnstructuredGrid, and ParamMappedDiscreteModel wrap the corresponding Gridap geometry objects so that the physical coordinates are ParamArrays (one set per sample). mapped_grid is the primary constructor.

• Embedded / level-set triangulations — ParamSubCellData and ParamSubFacetData carry cut-cell geometry (produced by GridapEmbedded's LevelSetCutters) as ParamArrays so that the cut geometry can vary with the parameters. These are consumed by the Extensions module for unfitted FEM.

The module depends on Utils (domain helpers) and ParamDataStructures (ParamArray, Realisation), and is used by ParamFESpaces and Extensions.

struct DisplacementMap <: GridMapStyle end

Trait for displacement maps d: Ω ↦ (I+d)∘Ω, with I being the identity; here, each coordinate x of Ω is mapped to x + d(x). Note: d may be parameterised (see AbstractParamFunction)

abstract type GridMapStyle end

Abstraction for maps used to modify a Grid in Gridap. Subtypes:

• PhysicalMap
• DisplacementMap

abstract type ParamGrid{Dc,Dp} <: Grid{Dc,Dp} end

Abstraction for mapped grids, whenever the map is parameterised. Subtypes:

• ParamMappedGrid
• ParamUnstructuredGrid

struct ParamMappedDiscreteModel{Dc,Dp} <: DiscreteModel{Dc,Dp}
  model::DiscreteModel{Dc,Dp}
  mapped_grid::ParamMappedGrid{Dc,Dp}
end

Represents a model with a ParamMappedGrid grid. See also MappedDiscreteModel in Gridap.

struct ParamMappedGrid{Dc,Dp,A} <: ParamGrid{Dc,Dp}
  grid::Grid{Dc,Dp}
  node_coords::A
end

Standard implementation of a ParamGrid. The field grid represents a Grid in a reference configuration, while node_coords is a structure collecting the mapped coordinates

struct ParamSubCellTriangulation{Dc,Dp,T,A} <: Triangulation{Dc,Dp}
  subcells::ParamSubCellData{Dc,Dp,T}
  bgmodel::A
  subgrid::ParamUnstructuredGrid{Dc,Dp,T,NonOriented,Nothing}
end

Parameterized version of a SubCellTriangulation in GridapEmbedded

struct ParamSubFacetTriangulation{Dc,Dp,T,A} <: Triangulation{Dc,Dp}
  subfacets::ParamSubFacetData{Dp,T}
  bgmodel::A
  subgrid::ParamUnstructuredGrid{Dc,Dp,T,NonOriented,Nothing}
end

Parameterized version of a SubFacetTriangulation in GridapEmbedded

struct ParamUnstructuredGrid{Dc,Dp,Tp,O,Tn} <: ParamGrid{Dc,Dp}
  node_coordinates::ParamBlock{Vector{Point{Dp,Tp}}}
  cell_node_ids::Table{Int32,Vector{Int32},Vector{Int32}}
  reffes::Vector{LagrangianRefFE{Dc}}
  cell_types::Vector{Int8}
  orientation_style::O
  facet_normal::Tn
  cell_map
end

Parameterized version of a Gridap UnstructuredGrid. Note: this struct could be avoided in future versions, by simply generalizing the type of the node_coordinates field in UnstructuredGrids.

struct PhysicalMap <: GridMapStyle end

Trait for standard physical maps φ: Ω ↦ φ∘Ω, i.e. each coordinate x of Ω is mapped to φ(x). Note: φ may be parameterised (see AbstractParamFunction)

mapped_grid(style::GridMapStyle,grid::Grid,f) -> Grid

Returns a Grid by applying the map f on a reference configuration grid, according to the strategy provided by style