Mesh generation in three dimensions - Finite Element Mesh Generation - page 278

boundary surface. Hence, only the geometry of the boundary surface is recovered, but not the
original topology of the boundary mesh, and usually, the recovered boundary surface consists
of more triangular facets connected in a different mesh pattern. Relative to a fully constrained
boundary recovery, boundary surface recovered only in geometry without identical element con-
nections is termed as a semi-constrained or conforming boundary mesh.

Algorithm: Semi-constrained boundary recovery

1. Input boundary surface triangular mesh S = {Ti, i , i = 1, N s }.

2. Create a DT using nodes on the boundary surface S.

3. Collect missing edges ME = {Ei, i , i = 1, N E }.

4. If N E > 0, then (i) divide missing edges by bisection of longest edges; (ii) update the
boundary surface S taking into the account the division of edges; (iii) go to step 2 with
additional nodes introduced to missing edges and other edges in the bisection process.

5. Collect missing faces MF = {Fi, i , i = 1, N F }.

6. If N F > 0, then (i) divide missing faces by bisection of longest edges; (ii) update the
boundary surface S taking into the account the division of edges; (iii) go to step 2 with
additional nodes introduced to missing faces and other edges in the bisection process.

7. Semi-constrained boundary surface tetrahedral mesh completed.

Remarks: Missing edges can be determined by first finding out all the tetrahedral elements
connected to each node (Section 2.5.3). To check whether edge AB exists in the DT, simply
verify if B is in the tetrahedra connected to node A. Similarly, to check whether face ABC
is in the DT, all we have to do is to verify if nodes B and C are in one of the tetrahedra
connected to node A. For each division of line segment by the introduction of a mid-side
node, as shown in Figure 5.56, the boundary surface mesh S has to be updated as follows:
delete triangles ABC and ACD from surface S, and add triangles PAB, PBC, PCD and PDA
to surface S. It is noted that the number of triangles in S will increase by two for each edge
division by introducing an additional point P. That's why the number of points and triangles
in S may increase quite rapidly in the boundary recovery process.

In some fluid mechanics and field problems, the adjacent space has to be meshed as well. In
this case, the MG can be taken as a constrained Delaunay insertion problem by introducing
nodes to the ambient space according to the node space function taking into account the con-
straints of the boundary surface, as discussed in Section 5.6. Alternatively, the boundary sur-
face S can also be inserted directly to an existing mesh M, which is typically a DT prepared
based on the specified element-size distribution. Let there be N M points in mesh M and N B
points on boundary surface S; a Delaunay mesh can be created for this set of points N = N M +
N B sequentially. The process can be repeated with the updated set of points N* = N M + N B * in
which N B * is the number of points in the modified (subdivided) boundary surface S* for edge
and face recovery. If computer memory is not an issue, mesh M needs to be generated only
once, which can be stored and retrieved as the starting point for each Delaunay insertion of

D

C

P

A

B

Figure 5.56 Introducing point P and surface mesh updating.