Advanced Delaunay Meshing - Example 7

The term Delaunay meshing does not mean simply creating a Delaunay mesh on points, but it is about Quality Meshing i.e., creating a Delaunay mesh with triangles that have a very specific quality. This is generally achieved by adding points, that is, by refining an existing triangulation. As always you can find the below source code in the Fade download in the file “examples_2D/ex7_qualityMeshing.cpp“.

The initial zone

The function createSimpleZone(Fade_2D& dt) creates an initial shape. If you have studied the previous examples, you already know how to create a Zone2 from a polygon. Here is just a quick picture of it.
This Zone2 object will be remeshed in different ways now.

Delaunay Meshing with Default Parameters

The below code snippet sets up an initial zone. Then it creates a MeshGenParams object which contains parameters that control the meshing process. These parameters have default values which are useful for the average case. We use the defaults and call the refineAdvanced() method.

void strategy_default()
{
Fade_2D dt;
Zone2* pZone=createSimpleZone(dt);
MeshGenParams params(pZone);
dt.refineAdvanced(&amp;params);
highlightTriangles(dt,pZone,"default.ps");
}

The Parameter growFactor

One of the parameters in the MeshGenParams class is growFactor. It is used to limit the area growth of adjacent triangles. More precisely, for two adjacent triangles T0 and T1, where T1 is the larger one, area(T0) / area(T1)<growFactor. By default the value is DBL_MAX, that is, growth is not limited. In the code snippet below, we set the parameter to 3.0 to achieve smooth growth from small to large triangles.

void strategy_growFactor()
{
Fade_2D dt;
Zone2* pZone=createSimpleZone(dt);

MeshGenParams params(pZone);
params.growFactor=3.0;
dt.refineAdvanced(&params);
highlightTriangles(dt,pZone,"growFactor.ps");
}

The Parameter maxEdgeLength

Among the properties that can be controlled via MeshGenParams, the maximum length of the triangle edges is important. Use the maxEdgeLength parameter to control the length of the edges in the generated mesh. By default maxEdgeLength is set to DBL_MAX, i.e. no limit.

Mesh generation with limited edge length

void strategy_maxLength()
{
Fade_2D dt;
Zone2* pZone=createSimpleZone(dt);
MeshGenParams params(pZone);
params.maxEdgeLength=10.0;
dt.refineAdvanced(&amp;params);
highlightTriangles(dt,pZone,"maxLength.ps");
}

The Parameters gridLength and gridVector

Grid Meshing is a hybrid approach: Large areas are meshed with grid points while edge areas are meshed with classical Delaunay techniques. To enable grid meshing, set the gridLength parameter to a positive value. This value will be adapted internally to fit the shape as good as possible. By default, the grid is axis-oriented, but you can change the alignment using the gridVector parameter.

Grid Mesh aligned to the Vector2(1.0,0.3)

Fade_2D dt;
Zone2* pZone=createSimpleZone(dt);
MeshGenParams params(pZone);
params.gridLength=3.0;
params.gridVector=Vector2(1.0,0.0,0.0);
// params.gridVector=Vector2(1.0,0.3,0.0);

dt.refineAdvanced(&params);
highlightTriangles(dt,pZone,"gridMeshing.ps");

Callback methods – create a custom parameters class

It is sometimes desirable that your program can control the mesh density locally. To do this, you derive a class from MeshGenParams and override the getMaxTriangleArea() or getMaxEdgeLength() method with your own code. The example code below derives a class CustomParameters and overrides getMaxTriangleArea() to force small triangles in the range x<20, y<40.

class CustomParameters:public MeshGenParams
{
public:
	CustomParameters(Zone2* pZone):MeshGenParams(pZone)
	{
	}
	double getMaxTriangleArea(Triangle2* pT)
	{
		Point2 barycenter(pT->getBarycenter());
		if(barycenter.x()<20 && barycenter.y()<40)
		{
			// Dense meshing in the lower left corner
			return 1.0;
		}
		else
		{
			// No density restriction otherwise
			return 10.0;
		}
	}
};

// Uses CustomParameters with a custom getMaxTriangleArea method
void strategy_customParameters()
{
	Fade_2D dt;
	Zone2* pZone=createSimpleZone(dt);

	CustomParameters params(pZone);
	dt.refineAdvanced(&params);
	pZone->show("customParameters.ps",true,true);
}

Height guide triangulations in 2.5D

We are dealing with a 2D example here, but refining a triangulation would of course also work in 2.5D. By default, the z-value of a new point is then calculated from the triangle that this point is currently refining. But alternatively, you might be in the special situation of having a second triangulation with higher resolution. In this case you would use this second triangulation as a height guide.

void strategy_heightGuide()
{
	Fade_2D dt;
	Zone2* pZoneSimple=createSimpleZone(dt);

	// Another triangulation is used as height guide
	Fade_2D dt_guide;
	createHeightGuide(dt_guide);

	MeshGenParams params(pZoneSimple);
	params.maxTriangleArea=10.0;
	params.pHeightGuideTriangulation=&dt_guide;
	dt.refineAdvanced(&params);
	dt.writeWebScene("mesh_with_height");
}