Notes on connect algorithm:
Connect uses the standard point insertion method of tetrahedralization which consists of the following steps:
First an enclosing tetrahedra is constructed which contains the nodes to be tetrahedralized.
Then the nodes are processed in a random order and inserted one at a time.
To insert node n+1:
Find a tetrahedra whose circumsphere contains node n+1. Place this tetrahedron on the stack. Pop a tetrahedron off the stack. Look at all face neighbors of this containing tetrahedron to determine if their circumspheres contain node n+1. Place the containing neighbor tetrahedra onto the stack. Continue until the stack is empty. Keep track of the set of containing tetrahedra as the stack is depleted. The union of these tetrahedra is the insertion cavity into which the new node is placed; so we remove the containing tetrahedra and construct new tetrahedra by connecting the new node to the faces of the cavity.
If for some reason a node cannot be inserted, it is placed in a fail list and subsequent passes will attempt to insert the node by using looser ‘inside’ tests.
When all nodes are inserted, we inspect all edges to see if there are any edges that cross material interfaces. If so we construct a node at the intersection of the edge and interface. These new nodes (if any) are then inserted into the mesh using the algorithm described above. This process is repeated until no more ‘multi-material’ edges are encountered or until the maximum number of iterations is reached. This step is skipped if the noadd option has been specified.
Finally the initial enclosing tetrahedron vertices and the attached tetrahedra are removed.
To determine if a node is ‘inside’ a tetrahedron’s circumsphere we compare the distance from the tetrahedron’s voronoi point to a vertex of the tetrahedron (i.e. the circumsphere radius) with the distance from the voronoi point to the node in question. If the second distance is less than the first, the node is ‘inside’ the circumsphere. Obviously the comparison of these distances should involve an epsilon which is dependent upon both the machine accuracy and the problem scale. For computational efficiency we compare the distances squared.
Currently the ‘inside’ test is as follows:
Since we are comparing distances squared the appropriate test is an area test. We construct a variable called smalarea =
[ ((machine precision) *10,000) * *(boxsizex *boxsizey *boxsizez) * *(2.d0/3.d0)]
where boxsize is the (max coordinate value - min coordinate value) in each of the 3 directions,
machine precision is usually around 2 *10-16 for double precision.
So for a unit cube the test value is about 7 *10-11.
For a 1000x1000x1000 cube the test value is about .07.
For the first pass we use smalarea *100.
We loop until no more nodes can be added using this value.
For the second pass we use smalarea
For the third pass smalarea/100.
For the fourth pass we use zero
For the fifth pass we use -100 *smalarea
The inside test requires that the coordinates of the circumcenter of the new tetrahedra be calculated. As this calculation uses the squares of the coordinates of the vertices, we first translate the tetrahedron so that the coordinates one of its vertices lie at zero. This simplifies the calculation and avoids loss of precision when the values of the coordinates are very large. After calculating the coordinates of the circumcenter, the tetrahedron and its circumcenter are translated back to their original location.
As the insertion cavity is filled with new tetrahedra, the new tetrahedra are subjected to two tests. The first test checks that the volume of each new tetrahedron is positive. This test uses a volume epsilon that represents the smallest volume that can be handled computationally. We use the mesh object attribute epsilonv which is set to [ ((machine precision) *1000) * *(boxsizex *boxsizey *boxsizez)]. The second test compares the sum of the volumes of the new tetrahedra to the sum of the volumes of the removed tetrahedra. This test fails if: [ (volnewt-vololdt)/vololdt > machine precision *10 * *8].
The user can change the ‘tightness’ of the circumsphere test by adding an attribute to the mesh object called circumsphere_factor and setting this factor, e.g.:
cmo /addatt//circumsphere_factor /REAL /scalar /scalar
This factor will only be required in extreme circumstances. If a node distribution has an extremely high aspect ratio, the user might see warning messages about circumsphere problems and connect may fail to connect all nodes. In this case one might try adjusting the circumsphere_factor.
The requirement for a delaunay tessellation is that the circumcircle or circumsphere of each element enclose no other nodes in the tessellation. If nodes on the boundary of the mesh are nearly coplanar or nearly collinear, it is possible that the big tetrahedron or big triangle constructed automatically will not be large enough. For illustration purposes consider the 2D case in which there are 3 nodes on the boundary that are nearly collinear say (0,0,0) (.5,.05,0.) and (1,0,0), then the circumcircle determined by these three nodes is very large and may contain node(s) of the big triangle, and the triangle consisting of these three nodes will not be formed. In this case it is necessary for the user to supply the coordinates of the ‘big’ triangle in the connect command.