The fluid surface is based on the position of the tetrahedra as defined inT. The saturation parameterδ adjusts the position of the tetrahedra faces to allow a varied volume of fluid to exist in the agglomerate. The state δ = 0 corresponds to a minimum state where the fluid segments are connected to the centre of the primary particles. For δ > 0 the external faces are moved in the direction of their outward pointing normal vectors up to a maximum value ofδ = 1. For δ = 1 the faces are displaced a distanceR to match the radius of the spheres. The position of the spheres remain fixed as δ is varied.
Figure 4.6 shows a fluid segment formed for a particulart∈T with vertices [P1P2P3P4].
To illustrate the expansion of the faces,~n1 is used to denote the outward pointing normal
vector of face f1 with vertices [P1 P2 P3]. The expanded vertices due to this face are
shown in Figure 4.6(b) as u1 =P1+δ~n1, u2 =P2+δ~n1 and u3 =P3+δ~n1. For δ 6= 0,
cylinders of radiusδ connect adjacent faces together along their edges as shown in Figure 4.6(c). In Section 4.5 a computational geometry toolbox is used to calculate properties of the agglomerates. The toolbox requires the fluid segments (and spheres) to be convex and bounded. The objects formed in Figure 4.6(c) are convex. They may be bounded by using a routine from the toolbox to intersect the fluid segments with half-spaces which pass through the ui vertices (Section 4.5 discusses half-spaces). To illustrate, Figure
4.6(c) shows the vertices u6, u9 and u10 that the half-space passes through due to the
tetrahedron vertex P4. The inequality of the half-space is determined by including the
mean point of the fluid segment following the intersection. The completed fluid segment is shown in Figure 4.6(d). The fluid segments are required to be in the same saturation state δ for a given agglomerate. The fluid segment corresponding to tetrahedron tj is
4.4 Defining the Fluid Surface 105
(a) Sphere A is the position for plac- ing one particle between faces [1 3 4] and [1 2 4]. In this case two tetra- hedra are required : tetrahedra (with vertices) [1 3 4A] and [1 2 4A].
(b) Spheres B and C are the place- ment (as determined by optimisa- tion), for placing two particles be- tween faces [1 3 4] and [1 2 4]. In this case, three tetrahedra are re- quired : [1 3 4B], [1 4B C], and [1 4C2].
(c) Diagram of the contact faces. (d) The actual position for the 6th sphere, as determined by optimisation.
Figure 4.4: Adding a particle to a five particle agglomerate. A contact face comparison between faces [1 3 4] and [1 2 4] is shown in figures (a) and (b). A single contact face exists for this case ([1 3 4]), with contact spheres 1, 3 and 4. Figure (c) illustrates the placement of the new particle.
(a) Adding a particle to an 8 particle agglomerate. The initial contact facef
is [1 5 2].
(b) The result after adding particle to the contact faces [1 5 2] and [7 1 2]
(c) The result after adding parti- cle to the contact faces [5 1 2] and [7 1 2]
Figure 4.5: Adding a new particle to a 8 particle agglomerate. In this case, there are 2 contact faces, and 2 tetrahedra are therefore added: [5 1 2 9] and [7 1 2 9]. (The 8th particle, attached to spheres 2, 3 and 4, is not visible in this view).
4.4 Defining the Fluid Surface 107
denoted byTj.
Originally a convex hull routine from the toolbox was used to form the half-spaces for the spheres and fluid segments. Since the objects were already convex, however, it is more efficient to form the objects in face-vertex format. In this format the half-spaces may be readily obtained. We show how this was completed for the fluid segments. Faces are defined on the segments due to the four expanded faces, shown in Figure 4.6(b), and, by defining vertices on the cylinders, faces for the cylinders are defined by using triangular mapping. Vertices are defined on the cylinders along circular arcs between adjacent ui
points as shown in Figure 4.7(b). We show how vertices are defined for one of the arcs. Figure 4.7(a) shows a view from the point P4 to the points P1, P2 and P3 which lie in
the same plane. The arc γ1 centred at P2 exists between the pointsu5 = P2+δ~n2 and u11 =P2+δ~n3 and has the equation|x−P2|=δ wherex ∈γ1. The angle between the
faces f2 and f3 is calculated as θ= acos
³
~ n2·~n3 |~n2||~n3|
´
and an incremental angle ∆θ = kθ is defined where k is the number of vertices along the arcs. The vector ~n∗ is introduced which is orthogonal to~n2 as shown in Figure 4.7(a). The points alongγ1 are given by
vi =P2+δ[cos (i∆θ)~n2+ sin (i∆θ)~n∗] (4.10)
wherei= 1,2, ..., k. But now
~n3 = cosθ~n2+ sinθ~n∗. (4.11)
Therefore
vi=P2+δ[cos (i∆θ)~n2+ sin (i∆θ) (cscθ~n3−cotθ~n2)]
=P2+δ[(cos (i∆θ)−sin (i∆θ) cotθ)~n2+ sin (i∆θ) cscθ~n3].
(4.12)
The arc at the opposite end of the cylinder is denoted byγ2. Points on this arc are also
denoted by vi but for i=k+ 1, ...,2k. The steps above are repeated for the remaining
cylinders and superscripts are introduced forvi to label the corresponding edges with the
above case being denoted by(2,4)vi.
The face-vertex matrix for the fluid segments is defined by
u1 u4 u7 u10 (1,2)v1 (1,2)v1 . . . (1,2)vk−1 . . . (2,4)v1 (2,4)v1 . . . u2 u5 u8 u11 (1,2)vk+2 (1,2)v2 . . . (1,2)vk . . . (2,4)vk+2 (2,4)v2 . . . u3 u6 u9 u12 (1,2)vk+1 (1,2)vk+2 . . . (1,2)v2k . . . (2,4)vk+1 (2,4)vk+2 . . . t .
The box in the above matrix represents the shaded face shown in Figure 4.7(b). After the objects are formed they are bounded by intersecting them with half-spaces as described
(a) An unexpanded binder tetra- hedra corresponding toδ= 0.
(b) Fluid tetrahedra expanded byδ= 0.7.
(c) Cylinders added to satura- tion stateδ= 0.7.
(d) Half-space cuts are used to produce a bounded convex ob- ject.
Figure 4.6: Creation of a binder fluid segment
earlier.