Determine the time for filling a vacuum injection mold with resin.
MODEL
The discrete model for resin flow is the following:
(1) On each discrete time step, enough resin is injected into the source (+) to fill a single cell.
(2) At t=0, this completely fills the source cell with resin, and there is no overflow for the next time step.
(3) For t=1 and later, things are a little more complicated:
(3.1) The boundary cells are all the void cells which are adjacent (either vertically or horizontally) to completely filled cells, but which are not completely filled themselves. The sink (-) cell is always effectively empty. If the sink cell is adjacent to a full cell, then it will be a boundary cell.
(3.2) Each cell on the boundary is filled by an equal fraction of the injected resin, plus any overflow from the previous injection: if there are N cells on the boundary, and A amount of overflow from the previous time step, then each is filled with (1+A)/N fraction of resin (in addition to whatever resin currently resided there)
(3.3) Any cell, except the resin sink (-), which is filled to more than capacity by this step is instead filled exactly to capacity. Because of roundoff errors, consider "full" to be a fill ratio of greater than 0.999999. The overflow resin is added to what will be injected on the next time step.
(4) After any time step, the sink cell (-) is empty, even though some resin may have flowed into it.
Here is pseudo-code for advancing the resin configuration for one generic time step. The overflow variable is considered to persist between steps, since it is the previously computed value (which should be initially zero) that is used to determine the amount of fill for this step.
subroutine move_forward_in_time
determine number of boundary cells
fill=(1+overflow)/number_of_boundary_cells
overflow=0
loop through all (i,j) which label boundary cells:
fullness(i,j)=fullness(i,j) + fill
if (fullness(i,j) > 0.999999) then
overflow = overflow + (fullness(i,j)-1)
fullness(i,j)=1
end if
end loop
fullness(sink_i,sink_j)=0
end subroutine
Fullness(i,j) is the double precision floating point number representing the fill ratio (0 for empty, 1 for full) of the cell at row i and column j.
NUMERICAL ACCURACY NOTES
Do all floating-point computations using (at least) double precision arithmetic.