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regions in the C-spaces of the inpidual channels.
2.2. C-space construction of inpidual cooling channels
When an inpidual channel Ci is considered alone, it has
three degrees of freedom, say X1 and X2 for its center location
and X3 for its length. As the ideal center location and length
have already been specified in the preliminary design, it is
reasonable to assume a fixed maximum allowable variation ∆c
for X1, X2, and X3. The initial free region in the C-space
of channel Ci is thus a three-dimensional cube Bi with the
dimensions ∆c × ∆c × ∆c.
To avoid any possible interference with a mould component
Oi when channel Ci is built into the mould insert by drilling,
a drilling diameter D and drilling depth along X3 have to be
considered. To account for the diameter D, Oi is first offset
by D/2 + M to give O0i
, where M is the minimum allowable
distance between the channel wall and the face of a component.
This growing of Oi in effect reduces channel Ci to a line Li .
Consider the example illustrated in Fig. 4. Fig. 4(a) shows a
channel Ci and three mould components, O1, O2, and O3, that
may interfere with Ci . Fig. 4(b) shows the offsets O0
1, O02,and O03 of the mould components, and the reduction of Ci to
a line segment Li that is coincident with the axis of Ci . If
there is no intersection between Li and the offsets of the mould
components, then the original channel Ci will not intersect with the mould components. This growing or offset of an obstacle is
a standard technique in the C-space method [15].
A channel is formed by drilling from a face of the mould
insert, and any obstacle Oi within the drilling depth will affect
the construction of the channel. To account for the drilling
depth, the offset O0iof Oi is swept along the drilling direction
until the opposite face of the mould insert is reached to generate
O00i. This sweeping of O0i
in effect reduces line Li to a point Pi
located at the end of Li . As shown in Fig. 4(c), if the point Pi
is outside O00i
, the drilling along Li to produce Ci is feasible.
The free region FRi of channel Ci is obtained as follows.
First, the initial free region Bi is constructed with its center
at Pi as shown in Fig. 4(d). Bi then intersects with the mould
insert to obtain B0
i. B0irepresents all of the possible variations
of Ci when only the geometric shape of the mould insert is
considered. Then, FRi is obtained by subtracting from B0
itheO00iof all of the obstacles. Fig. 4(e) and (f) show the subtraction
and the resulting FRi of the example.
2.3. Basic approach to the construction of the C-space of
cooling system
To determine the free region FRF in the C-space of a
cooling system, the free regions of each cooling channel have
to be “intersected” in a proper manner so that the effect of
the obstacles to all of the channels are properly represented
by FRF . However, the standard Boolean intersection between
the free regions of two different channels cannot be performed
because their C-spaces are in general spanned by different sets
of axes. Referring to the example in Fig. 3, the C-spaces of
C1 and C2 are spanned by {X1, X2, X3} and {X1, X3, X4},
respectively. To facilitate the intersection between free regions
in different C-spaces, the projection of a region from the C-
space of one channel to that of another channel is needed. The
following notations are first introduced and will be used in
the subsequent discussions on projections and the rest of the
paper.
Notations used in describing high-dimensional spaces
Sn denotes an n-dimensional space spanned by the set of axes
¯ Xn = {X1, X2, . . . , Xn}.