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Triangular Kernel

Firstly the initial front consisting of edges constituting the boundary of the surface (including inner loops) is established. Once the initial front has been set up the mesh generation continues on the basis of the edge removal algorithm according to the following steps until the front becomes empty:

  1. The first available edge in the front is made active. The edge is oriented from to and the triangle is assumed to be generated on the left side of it when viewing against the outer normal of the surface (Fig. 1).

  2. The position of the ``ideal'' point , forming the tentative triangle, is calculated as

    (1)


    where

    (2)


    The stands for the position vector, means the surface outer normal and denotes the element size (extracted from the octree). The subscript refers to a particular point in 3D.

  3. Since point is generally not constrained to the surface (Fig. 1), its projection to the surface is calculated. The projection itself is accomplished in an iterative manner using the normal , gradients and and parametric coordinates and of any nearby point (typically an already existing node). The point is firstly projected to the tangential plane given by the normal . The increments of curvilinear coordinates and , calculated using the surface gradients at point , are then used to get better approximation of point . Consequently surface normal and gradients are calculated at this point and the whole process is repeated until the desired accuracy in the approximation of point is achieved.

  4. The new position of the ideal point is evaluated taking into account local surface curvature (Fig. 2) (this point is now referred as )

    (3)


    where

    (4)


    The is reflecting the local mesh size variation (Fig. 3) and is given as

    (5)


  5. Projection of point to the surface and the corresponding mesh size are calculated.

  6. The local neighbourhood of point in terms of a set of octants is established.

  7. The neighbourhood is searched for the most suitable candidate to form a new triangle . Two auxiliary geometrical loci are constructed around point to resolve the acceptance of tentative node . Firstly, a sphere with center in and radius of . And secondly an ellipsoid derived from sphere by extension of its half-axe parallel to the active edge by a half (Fig. 4). The selection of point is driven by the following rules. Point is

    1. identical with point if there is no point inside the sphere , no point inside the ellipsoid directly connected with the active edge and there is no point inside the sphere circumscribed to the triangle .
    2. an already existing node inside the sphere circumscribed to the triangle forming the largest angle if there is no point inside the sphere and no point inside the ellipsoid directly connected with the active edge .
    3. taken as node if it falls inside the sphere and there is no other node inside the sphere circumscribed to the triangle .
    4. an already existing node inside the sphere circumscribed to the triangle forming the largest angle . The point is an already existing node inside the sphere (Fig. 4).
    5. taken as node if the already existing edge or forms sufficiently small angle with the active edge and no point is inside the sphere circumscribed to the triangle .
    6. an already existing node inside the sphere circumscribed to the triangle forming the largest angle , where point is the end point of already existing edge or being at a sufficiently small angle to the active edge .
    7. taken as node if it falls inside the ellipsoid and there is no other node inside the sphere circumscribed to the triangle . Point is the end point of already existing edge or .
    8. an already existing node inside the sphere circumscribed to the triangle forming the largest angle , where node falls inside the ellipsoid and is directly connected with the active edge .
    In cases (b), (d), (f) and (h), the validity of point must be always verified. Similarly point has to be verified in cases (c) and (d). It may happen that although the point is satisfying all the criteria in terms of metric of the real space it is still very far from point in the parametric space. In other words, the distance between points and measured on the surface may be much larger than the distance in . Acceptance of node in cases (g) and (h) improves mesh quality in the mesh size transition zones and eliminates creation of chunks of edges when front closes up. To speed up the neighbourhood searching only the half-space given by point and vector is considered and only nodes on the front are considered. Furthermore, nodes which are too far from point are ignored.

  8. The intersection check is carried out to avoid overlapping of the tentative triangle with an already existing one in the neighbourhood. If the edge or is the already existing one, the check can be omitted. This will reduce the number of intersection investigations to about a half. The intersection check is further enhanced by considering only edges on the front. In the case that an intersection is detected, the selected point is rejected and the search of the neighbourhood is repeated without taking this point into account.

  9. The front is updated to account for newly formed triangle . This means that the already existing triangle edges are removed from the front while the newly created ones are appended to the front. To avoid searching in the front for the edges to be removed from it, these edges are marked as not available, which prevents them to be made active (in the 1st step). The node , if chosen according to the rule (a), is registered into the octree.



Next: Quadrilateral Generation Up: Mixed Mesh Generation Previous: Mixed Mesh Generation

Daniel Rypl
2005-12-03