Mesh Generation/de

Top: Original BRep circular face (blue) delimited by outer circle (red) Unten: Detail oben links	Oberflächenabweichung: 50mm Winkelabweichung: 90°	Oberflächenabweichung: 50 mm Winkelabweichung: 30°	Oberflächenabweichung: 50 mm Winkelabweichung: 5°

Die Reduzierung der Winkelabweichung hat einen erheblichen Einfluss auf das Erscheinungsbild des Netzes, da sie scharfe Winkel bei der Darstellung der Gesichtskontur begrenzt. Sie ermöglicht zwar eine höhere Genauigkeit des Netzes im Verhältnis zur ursprünglichen Form, liefert jedoch keinen absoluten Wert für die maximal zulässige Abweichung zwischen beiden.

Oberflächenabweichung

Surface deviation: 50mm Winkelabweichung: 90°	Oberflächenabweichung: 10 mm Winkelabweichung: 90°	Oberflächenabweichung: 1 mm Winkelabweichung: 90°	Oberflächenabweichung: 0,1 mm Winkelabweichung: 90°

Der Einfluss der Oberflächenabweichung ist in unserem Beispiel aufgrund seiner absoluten Natur deutlich sichtbar und leicht verständlich. Er ist besonders nahe am Rand sichtbar: Hohe Werte (50 mm – 10 mm) haben keine Auswirkungen, während viele Dreiecke hinzugefügt werden, um die Toleranz von 0,1 mm sicherzustellen.

Kombinierter Einfluss von Winkel- und Oberflächenabweichung

Decreasing either angular deviation or surface deviation has a similar impact on the mesh. This redundancy can lead to undesired behavior when bad parameter combinations are used: the following example shows the generation of unnecessary, oddly shaped triangles when both angular and surface deviation are set to low values.

Surface deviation: 10mm Winkelabweichung: 30°	Oberflächenabweichung: 3 mm Winkelabweichung: 7°	Oberflächenabweichung: 1 mm Winkelabweichung: 5°	Oberflächenabweichung: 0,1 mm Winkelabweichung: 1°

Erste Synthese

Die obigen Beispiele zeigen, dass es mithilfe von Winkel- und Oberflächenabweichung möglich ist, Netze innerhalb der geforderten Toleranz der ursprünglichen Form zu erhalten und die gewünschte Glätte bei der Darstellung von Kurven zu erreichen. Sie zeigen auch, wie wichtig eine angemessene Einstellung ist und dass die einfache Einstellung niedriger Werte möglicherweise nicht zu den besten Netzen führt.

Der Teufel steckt im Detail

While our circular shape provided a good understanding of the influence of the parameters on mesh generation, it is unfortunately too simplistic compared to real-life meshing challenges. A common case is the presence of fine details on bigger shapes. Let’s add a detail to our circular shape: a smaller circular edge at the top.

Winkelabweichung / Surface deviation	90°	30°	5°
5mm
Nb of triangles	10	32	149
1mm
Nb of triangles	18	34	188
0.1mm
Nb of triangles	53	59	207

Winkelabweichung / Surface deviation	90°	30°	5°
5mm
Nb of triangles	10	32	149
1mm
Nb of triangles	18	34	188
0.1mm
Nb of triangles	53	59	207

Beispiel für einen versteckten Parameter

Geometrical nature of the detail	The shape	The mesh	A close view
Arc of circle
Spline curve

Zweite Synthese

Diese Beispiele, einschließlich Details innerhalb größerer Formen, die im realen Leben sehr häufig vorkommen, zeigen, dass die Anpassung der Diskretisierungsgenauigkeit an die Größe der zu vernetzten Objekte zu einem besseren Kompromiss zwischen Netzgenauigkeit und Komplexität führen kann. Manchmal kann dies sogar zwingend erforderlich sein, beispielsweise bei automatisierten Vernetzungsvorgängen, bei denen der Benutzer nicht eingreifen kann, um die Oberflächenabweichung basierend auf der Formgröße festzulegen.

Relative Oberflächenabweichung

Oberflächenabweichung (relativ) & Angular deviation	The mesh	A close view	A closer view
0.1 - 30°
0.001 - 30°

The evolution of the surface deviation parameter (which is now a relative parameter, so the ‘mm’ unit when setting it in FreeCAD may be confusing) applies the same way to the discretization of both the long and short edges. The side effect of using the relative surface deviation is that it does not guarantee an absolute maximum deviation between the mesh and the original shape.

Our first thoughts focused on shapes and contours because they are the first entities to be discretized and have a significant influence on face meshing. Let’s dive a little bit more in surface tesselation using the standard algorithm. It applies Delaunay triangulation, using the Watson algorithm. Delaunay (from a Russian mathematician) triangulation is a way to generate points on a surface in a way that ensures triangles formed using the points will not intersect.

In addition, it optimizes triangle generation so that the angles are not too small, making them look well-shaped. That being said, as often, practical implementation to address real-world problems is complex and presents many challenges. OCCT meshing algorithms have been designed to mesh shapes generated with the BRep modeler, most often mechanical and engineering designs, which were the primary use cases the geometric modeler was initially developed for.

Thus, dedicated surface tesselation routines are used for the most common surface types in mechanical design, e.g., planar, cylindrical, conical, spherical, … to ensure performance and optimal tesselation. If the parameter settings affect all surface types, let’s examine their influence on B-spline surface tesselation to avoid biases introduced by predetermined tesselation strategies.

If you are curious about the impact of growing the number of triangles, you can have a feel by playing with the Minimum angular deflection parameter of your favorite software (Edit → Preferences → Part/Part Design → Shape view), which defines the fidelity of meshes used for visualization. A too-low value will lead to higher loading times and jerky manipulation of complex shapes.

Another way to perceive the impact of parameters is to transform our meshes into real objects by 3D printing. Resin printing (SLA) with a 0.05mm layer height will reveal some details.

These close-up pictures reveal that focusing only on surface deviation, even at low values, may lead to visible artifacts on the part's surface (as the zoom level is high, meshing impacts are not to be confused with printing layers, which are also visible as iso-lines on the pictures). It is difficult to differentiate the 9730-triangle mesh from the 36366-triangle one in terms of visual quality. Then, if it is the criteria, the lighter mesh (487ko vs 1.8Mo) should be preferred, as the other one provides assurance that the 0.1mm tolerance is respected everywhere in the part.

As a synthesis, meshing algorithms are complex by nature, and angular deviation and surface deviation are working together in an intricate way that is not trivial to understand. One could be tempted to lower these parameters so the mesh fits seamlessly to the original BRep shape. This would lead to meshes made of many triangles, which will take time to be generated, to be processed in further applications, and will produce heavy files.

From what we saw, a typical workflow to set the parameters is to first set surface deviation as a function of required fidelity between the mesh and the original shape, if any. Then, the angular deviation parameter can be used to adjust the smoothness of discretized surfaces to the required level, for visual or touch aspect, for instance. Another approach is to use a low enough angular deviation in order to achieve the desired smoothness, with no guarantee to fulfill an absolute geometric tolerance, and often at the cost of complex meshes. In those cases where no tolerance on fidelity is needed, another option is to use the relative surface deviation, which will adapt the size of the mesh elements to the original shape and the size of the details.

As explained in the Preferences section of the function wiki, you can define your preferred surface deviation, angular deviation and relative surface deviation values as default, so you can mesh most of your shapes without even thinking about it!