2D Triangulations

These examples can be loaded into Julia (Revise.jl recommended)

These examples return either a SimplexGridBuilder struct which can be turned into a grid by calling simplexgrid(builder), or just an ExtendableGrid with default simplex grid data.

The control flags for Triangle are created based on default options provided by this module which try to ensure "good" grids for FEM and FVM computations. These are documented in default_options Occasional options! statements in the examples overwrite these defaults.

This test code is released under the license conditions of Triangulate.jl

using SimplexGridFactory
using ExtendableGrids
using LinearAlgebra
using Triangulate

Domain triangulation

Here we just describe a domain as a polygon and mesh it.

function triangulation_of_domain()

    builder=SimplexGridBuilder(Generator=Triangulate)

    p1=point!(builder,0,0)
    p2=point!(builder,1,0)
    p3=point!(builder,1,2)
    p4=point!(builder,0,1)
    p5=point!(builder,-1,2)

    facet!(builder,p1,p2)
    facet!(builder,p2,p3)
    facet!(builder,p3,p4)
    facet!(builder,p4,p5)
    facet!(builder,p5,p1)

    builder
end

Triangulation with size control and boundary markers

The previous example was a little bit bland. We miss:

• size control for the triangles
• differently marking of boundary parts

function nicer_triangulation_of_domain()

    builder=SimplexGridBuilder(Generator=Triangulate)

    p1=point!(builder,0,0)
    p2=point!(builder,1,0)
    p3=point!(builder,1,2)
    p4=point!(builder,0,1)
    p5=point!(builder,-1,2)

    facetregion!(builder,1)
    facet!(builder,p1,p2)
    facet!(builder,p2,p3)
    facetregion!(builder,2)
    facet!(builder,p3,p4)
    facet!(builder,p4,p5)
    facetregion!(builder,3)
    facet!(builder,p5,p1)

    options!(builder,maxvolume=0.01)

    builder
end

Triangulation with subregions

Here we create different subregions and apply the maxvolume constraint to the subregions

function triangulation_of_domain_with_subregions()

    builder=SimplexGridBuilder(Generator=Triangulate)

    p1=point!(builder,0,0)
    p2=point!(builder,1,0)
    p3=point!(builder,1,2)
    p4=point!(builder,0,1)
    p5=point!(builder,-1,2)

    facetregion!(builder,1)
    facet!(builder,p1,p2)
    facet!(builder,p2,p3)
    facetregion!(builder,2)
    facet!(builder,p3,p4)
    facet!(builder,p4,p5)
    facetregion!(builder,3)
    facet!(builder,p5,p1)

    facetregion!(builder,4)
    facet!(builder,p1,p4)

    cellregion!(builder,2)
    maxvolume!(builder, 0.1)
    regionpoint!(builder, -0.1,0.5)

    cellregion!(builder,3)
    maxvolume!(builder, 0.01)
    regionpoint!(builder, 0.2,0.2)

    builder
end

Direct specification of input arrays

Of course we can specify the input for Triangle directly. The aim of SimplexBuilder is to avoid the tedious and error prone counting connected with this approach.

function direct_square(Generator=Triangulate)
    simplexgrid(Generator;
                points=[0 0 ; 0 1 ; 1 1 ; 1 0]',
                bfaces=[1 2 ; 2 3 ; 3 4 ; 4 1 ]',
                bfaceregions=[1, 2, 3, 4],
                regionpoints=[0.5 0.5;]',
                regionnumbers=[1],
                regionvolumes=[0.01])
end

Local refinement

We can interface to Triangle's unsuitable mechanism

function square_localref()

    builder=SimplexGridBuilder(Generator=Triangulate)
    cellregion!(builder,1)
    maxvolume!(builder,0.01)
    regionpoint!(builder,0.5,0.5)

    p1=point!(builder,0,0)
    p2=point!(builder,1,0)
    p3=point!(builder,1,1)
    p4=point!(builder,0,1)

    facetregion!(builder,1)
    facet!(builder,p1,p2)
    facetregion!(builder,2)
    facet!(builder,p2,p3)
    facetregion!(builder,3)
    facet!(builder,p3,p4)
    facetregion!(builder,4)
    facet!(builder,p4,p1)

    refinement_center=[0.5,0.5]
    function unsuitable(x1,y1,x2,y2,x3,y3, area)
        bary=[(x1+x2+x3)/3,(y2+y2+y3)/3]
        dist=norm(bary-refinement_center)
        if area > 0.01*dist
            return 1
        else
            return 0
        end
    end
    options!(builder, unsuitable=unsuitable)
    builder
end

Domain with holes

We can generate domains with holes. This at once shall demonstrate how the chosen API approach eases bookeeping of features added to the geometry description

function swiss_cheese_2d()

    function circlehole!(builder, center, radius; n=20)
        points=[point!(builder, center[1]+radius*sin(t),center[2]+radius*cos(t)) for t in range(0,2π,length=n)]
        for i=1:n-1
            facet!(builder,points[i],points[i+1])
        end
        facet!(builder,points[end],points[1])
        holepoint!(builder,center)
    end


    builder=SimplexGridBuilder(Generator=Triangulate)
    cellregion!(builder,1)
    maxvolume!(builder,0.1)
    regionpoint!(builder,0.1,0.1)


    p1=point!(builder,0,0)
    p2=point!(builder,10,0)
    p3=point!(builder,10,10)
    p4=point!(builder,0,10)

    facetregion!(builder,1)
    facet!(builder,p1,p2)
    facet!(builder,p2,p3)
    facet!(builder,p3,p4)
    facet!(builder,p4,p1)

    holes=[8.0 4.0;
           1.0 2.0;
           8.0 9.0;
           3.0 4.0;
           4.0 6.0;
           7.0 9.0;
           4.0 7.0;
           7.0 5.0;
           2.0 1.0;
           4.0 1.0;
           4.0 8.0;
           2.0 8.0;
           3.0 6.0;
           4.0 9.0;
           9.0 1.0;
           9.0 1.0;
           6.0 9.0;
           8.0 9.0;
           3.0 5.0;
           1.0 4.0]'

    radii=[0.15, 0.15, 0.1, 0.35, 0.2, 0.3, 0.1, 0.4, 0.1, 0.4, 0.4, 0.15, 0.2, 0.2, 0.2, 0.35, 0.15, 0.25, 0.15, 0.25]

    for i=1:length(radii)
        facetregion!(builder,i+1)
        circlehole!(builder,holes[:,i], radii[i])
    end

    builder
end

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