Local refinement

2D - Triangulate

The unsuitable mechanism is used to refine the 2D triangulation on a domain, for instance surrounding a geometric object (e.g., a point or line). Refinement is triggered when a condition for a given triangle is satisfied. For expamle when the distance between a point of interest and the center (barycentric coordinates) of the triangle falls below a threshold defined by the triangle’s area.

Example usage:

• One could define the unsuitable function with 6 points and the area of the triangle (x1, y1, x2, y2, x3, y3, area). This is the default signature in Triangulate.jl:

    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)
        return area > 0.01 * dist ? 1 : 0
    end

• As a simpler alternative, one could directly use the barycentric coordinates and the area of the triangle (center, area):

    refinement_center = [0.5, 0.5]
    function unsuitable(center, area)
        dist = norm(center - refinement_center)
        return area > 0.01 * dist ? 1 : 0
    end

To hand this function over to the builder:

    options!(builder; unsuitable = unsuitable)

3D - TetGen

The unsuitable mechanism is also used to refine the 3D triangulation on a domain, for instance surrounding a geometric object (e.g., a point or line). Refinement is triggered when a condition for a given tetrahedron is satisfied. For example when the distance between the point of interest and the barycentric coordinates of a tetrahedron falls below a threshold defined by the tetrahedron’s volume.

Example usage:

• One could define the unsuitable function with 4 points (pa, pb, pc, pd).

This is the default signature in TetGen.jl:

    refinement_center = [0.5, 0.5, 0.5]
    function unsuitable(pa, pb, pc, pd)
        vol = det(hcat(pb - pa, pc - pa, pd - pa)) / 6
        center = 0.25 * (pa + pb + pc + pd) 
        return vol > 0.05 * norm(center - refinement_center)^2.5
    end

• As a simpler alternative, one could directly use the barycentric coordinates and the volume of the tetrahedron (center, volume):

    refinement_center = [0.5, 0.5, 0.5]
    function unsuitable(center, vol)
        return vol > 0.05 * norm(center - refinement_center)^2.5
    end

To hand this function over to the builder:

    options!(builder; unsuitable = unsuitable)