- Actually, to know by computations the behaviour of objects under physical forces,
a MESH has to be constructed.
- A better accuracy of the solution is obtained if the mesh is adapted .
- To adapt a mesh, it is necessary, at any time
. to know the location of points: inside or outside a volume, on a face, on an edge or a special point;
. to move points along edges, faces, volumes.
- Today, the majority of objects are designed from Computed Aided Design (CAD) softwares
For each of them, an internal representation of objects has been chosen.
- An efficient mesh adaptation requires a fast access to data structure and functions on the objects.
- An optimal choice is obtained if the mesher and the CAD software have the same language.
- Generally, the CAD softwares do not permit this access by a compiled language.
- But, it exists at least one exception OpenCascade : a C++ opensource C.A.D. software downloadable from the Web (cf [1])
Prerequisites:
If you have a C++ or Fortran mesher
- to subdivide a curve into segments;
- to triangulate a plane surface;
- to decompose a volume into tetrahedra;
and if you are not afraid by
- 14223 C++ include files;
- 515 directories of C++ source files;
- a tar.gz file of 160 573 067 bytes; (version 5.0 may 2003)
then, you can construct a MESHER based on OpenCascade
with C++ direct calls to C.A.D. functions
The BOUNDARY REPRESENTATION (BRep) of an object of
OpenCascade:
- A COMPSOLID is a set of SOLIDS connected by their FACES;
- A SOLID (cylinder, cone, sphere, torus, ...) is a volume limited by SHELLS;
- A SHELL is a set of FACES connected by their EDGES;
- A FACE is a MAP defined from a domain included in a rectangle of
UV parameters upon the 3d-space
(cylinder: [0,2 Pi] x [0,H] -> R3, cone, sphere, torus, Bezier surface, NURBS surface, ... );
- The boundary of a FACE is composed of WIRES;
- A WIRE is a set of connected EDGES;
- An EDGE is a MAP defined from an interval of the U
parameter upon the 3d-space
(Bezier's curve: [0,1] -> R3, NURBS curve, ... );
- A VERTEX is a limit of an EDGE.
This Boundary Representation induces a structure of GRAPH between the different SHAPES
(CompSolid, Solid, Shell, Face, Wire, Edge, Vertex).
How the mesh is made in NEF?
OpenCascade permits to explore the different SHAPES of the COMPSOLID
for each EDGE of the COMPSOLID
The U-interval of the EDGE is subdivided into segments with respect
to the edge length and deflection in 3d-space.
By the map, the segments of the U-interval give the segments in 3d-space.
==> Creation of U-mesh and 3d-mesh
for each EDGE
for each FACE of the COMPSOLID
for each WIRE of the FACE
for each EDGE of the WIRE
The U-mesh of the EDGE is assembled after scaling
in the UV-domain to constitute the WIRE
The UV-surface, limited by the segmented Wires is triangulated
with respect to the edge length and deflection in 3d-space and the triangulation is mapped upon 3d-space.
for each SOLID of the COMPSOLID
for each SHELL of the SOLID
for each FACE of the SHELL
the 3d-mesh of the FACE is assembled
to form the boundary of the SOLID
Here, the triangulation, boundary of the COMPSOLID, is done.
The volumes, limited by this triangulation, are tetrahedrized.
Now, the mesh can be saved on a file.
NEF's algorithms for the meshing:
For an EDGE, the SEGMENTATION of the U-interval
is done by dichotomy on U to obtain the ideal length of segments in 3d-space,
and is modified if the deflection between the segment
and the curve is too great.
For a FACE, the TRIANGULATION of the UV-domain
is done by subdivision of a QUAD-TREE of REGULAR TRIANGLES,
homogeneization, swapping of diagonals from Delaunay's criterion, barycentering of neighbouring vertices,
moving, adding or removing vertices ... (cf [2])
For a COMPSOLID, the TETRAHEDRIZATION of the VOLUMES
limited by the triangulation of its boundary is done by a Delaunay's technic from the vertices of a
5-14 TREE of REGULAR OCTAHEDRA and TETRAHEDRA,
homogeneization, swapping of edges-faces (mt->2mt-4), barycentering of neighbouring vertices,
moving, adding or removing vertices ... (cf [3])
The USER may enter as a CONSTANT or a FUNCTION of (X,Y,Z)
- the MAXIMUM LENGTH of segments of EDGES or of edges of triangles of FACES
- the MAXIMUM DEFLECTION between the segment and the EDGE or
between the plane defined by the 3 vertices of a triangle
and the FACE
Example: If aretemax=0 the following user's function is active:
DefFunc areteideale( x, y, z );
DefVar d; d=(x**2+y**2+(z-12)**2)/36;
if d<1 then
areteideale=0.3*(1-d)+d; {interpolation from 0.3 to 1 in a sphere of radius 6}
else
areteideale=1.0;
endif;
EndFunc;
In order to facilitate the USER's work, a USER's COORDINATE SYSTEM
and LAYERS
are available.
Demonstration: Construction of a mesh of a cardan shaft
2 points and a radius => the lower cylinder
2 points and a radius => the upper cylinder
2 points min max => the box
the upper cylinder minus the box
the fusion with the lower cylinder
2 points and a radius => the lower cylinder of Z-axis
the object minus the cylinder of Z-axis
2 points and a radius => the upper cylinder of Y-axis
the cardan minus the cylinder of Y-axis
2 points and 2 radius => the cone
the intersection of the cardan and the cone gives the upper chamfer
1 point and 2 radius => the torus
the object minus the torus
the cardan after tetrahedrization with max edge length=1
the cardan with max edge length=0.5
the cardan with max edge length=1 and max deflection=0.01
the cardan with max edge length=0 and the function
DefFunc areteideale(x,y,z);
if abs(z-12)<2 then
areteideale = 0.3;
else
areteideale = 1;
endif;
EndFunc;
the cardan with max edge length=0 and the function
DefFunc areteideale(x,y,z);
if abs(x)<2 ou abs(z-12)<2 then
areteideale = 0.3;
else
areteideale = 1;
endif;
EndFunc;
the cardan with max edge length=0 and the function
DefFunc areteideale(x,y,z);
DefVar d; d=(x**2+y**2+(z-12)**2)/36;
if d<1 then
areteideale = 0.3;
else
areteideale = 1;
endif;
EndFunc;
Bibliography:
[1] http ://www.opencascade.org The Web server of the OpenCascade
software distribution
[2] A. Perronnet, Triangulation par arbre-4 de triangles
équilatéraux et maximisation de la qualité, Laboratoire
Jacques-Louis Lions Paris, Rapport R92015
[3] A. Perronnet, Tetrahedrization by the 5-14 OTT tree technic
and the Delaunay's criterion, VIII Conference on Finite Elements
in Fluids, Barcelona, sept. 1993
