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Refinement and Substitution of Subdomains

HighVoronoi.jl allows you to refine a mesh in two different ways:

  • refinement by locally adding addional points
  • substitution: in a given domain the existing mesh is replaced by another geometry.

Refinement using refine!

The intention of refine! is to refine a given mesh in a region where the users wants a more detailed view at a later stage of either the same code or built upon calculated and stored data.

    VG = VoronoiGeometry(VoronoiNodes(rand(5,1000),cuboid(5)))

    ## Do some fancy stuff with `VG`
    ## ...
    ## Now we want to refine the geometry with additional VoronoiNodes xs

    refine!(VG,xs,update=true)
On `update`...

The parameter update::Bool tells whether or not the volumes and intgrals of new or modified cells shall be recalculated / updated. Its default value is update=true. One could also use update=false and call integrate!(VG) instead. This will have (much) higher computational costs.

Refinement using substitute!

The intention of substitute! is fast mesh generation in high dimensions, particularly in combo with periodic_grid. In this way the algorithm will have to calculate much less verteces explicitly. As an additional benefit, using periodic_grid we can achieve a grid with rather few neighbors, resulting in a much sparser matrix than with fully random nodes.

HighVoronoi.substitute! — Method
substitute!(VG::VoronoiGeometry,VG2::VoronoiGeometry,indeces)

Takes the nodes indeces from VG2 and erases all nodes from VG within the VornoiCells of indeces. Then plugs the nodes indeces into VG and generates the full mesh for this new setting.

Domain and Boundary condition matching

The domains of the original and the substitute Geometry MUST match. HighVoronoi will not controll this but you may have strange results or even a clash. Furthermore, both domains should have the same periodic boundary conditions.

The following code will create a cubic grid with poor resolution in $\mathbb R^8$ and then refine it by a cubic grid with high resolution in the cube $(0.7,1)^8$. Furthermore, the grid will have periodic boundaries in dimensions $1$ and $5$.

    VG = VoronoiGeometry(VoronoiNodes(rand(8,1)),periodic_grid = ( dimensions=ones(Float64,dim), 
            scale=0.2*ones(Float64,dim), repeat=5*ones(Int64,dim), 
            periodic=[1,5], fast=true ))

    substitute_VG = VoronoiGeometry(VoronoiNodes(rand(8,1)),periodic_grid = ( dimensions=ones(Float64,dim), 
            scale=0.05*ones(Float64,dim), repeat=20*ones(Int64,dim), 
            periodic=[1,5], fast=true ))

    ## now pu all that stuff togetehr

    substitute_indeces = indeces_in_subset(substitute_VG,cuboid(8,periodic=[],dimensions=0.3*ones(Float64,8),offset=0.7*ones(Float64,8)))
    substitute!(VG, substitute_VG, substitute_indeces)