Spatial Twist Continuum STC
Invented on the 16th September 1993 by Dr Peter Murdoch. The STC is the solution to the global connectivity constraint in an all Hexahedral mesh. It forms the basis of the Cell3 definition and is the core technology of this revolutionary new way to mesh.
It can be referred to as the Dual of a Hexahedral mesh since it is always present. It is the existence of the STC that makes automatic mesh generation so technically difficult.
There are 3 main components to the STC; the centroids that are the geometric volumetric centre of the Cell/element, the Chords which pass through the opposite faces of the Cell/element and the Continua. The continua as the name suggests are continuous surfaces that contain layers of chords. These Continua flow and twist through the mesh changing iso-parametric directions at folds.
A tetrahedral and polyhedral mixed element mesh do not contain a continuous STC and hence are relatively easy to automate.
The best way to explain the Spatial Twist Continuum STC is with a 3D representation. Let’s start with a single Hexahedral cell.
Add a point to the cell's geometric centre. This point is called the “centroid.”
Next we add a line that starts from the centre of one face, passes through the centroid and ends at the centre of the opposite face. This line is called a “chord.”
Then we add 2 more chords for the other 2 two y, z dimensions. These directions are iso-parametric relative to the cells and are the local i, j, k directions. This Chord representation is now a “Dual” definition of a Hexahedral cell.
At this point the chords have not really added anything new until we add 2 hexahedral cells together. Since they both share a face they both share a chord.
Both adjoining cells share a chord.
Now let’s add a few more cells and join up the chords.
A pattern is starting to form, the chords all appear to be coplanar, which they are and these planar surfaces are called “continua”. After adding the continua the relationship between the centroids, chords and the continua becomes evident. Chords are the result of the intersection between two continua and centroids are the intersection point of three continua. These are the rules and definition of the STC.
It starts to get interesting when we introduce the concept of folds. When 3 hexahedral cells are arranged like this, one of the continua folds its direction so that it turns in connectivity space from the i to the j direction.
This fold extends into the k direction.
The continua of the Fold
When we fold two continua in 2 iso-parametric directions and we get this.
The Chords folding.
Rows of joined cells on the surface of a Hex mesh appear as loops that are continuous around the entire mesh and always close on themselves. This is true on any surface Quad mesh.
Layers of adjacent cells form inside a Hex mesh. These layers of cells come from the surface loops. However having surface loops does not guarantee internal layers, rarely does a random generated surface quad mesh form an array of internal layers, very rarely.