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296 Ch a p t e r N i n e

C1

(a) Cluster of small circles with equal radius is (b) Remove small circles contained in
used to represent the particle the first big circle C1

C3 C2
C2

C1 C1

(c) Continue removing small circles in the second (d) Generate circles C1, C2, C3…
big circle C2
FIGURE 9.10 Process of replacing small circles with larger ones (Burn algorithm 1).

Step 4. For the remaining small circles, repeat Step 1 while determining the boundary
of the reamaining cluster circles, then repeat the second step and the third step, finding
Circle C 2 and removing small circles whose center lies in or on the boundary of C 2 .
Step 5. Repeat the fourth step and find C n until the radius of C n is not larger than the
radius of small circles.
Step 6. Generate new circles using the center and radius of C 1 , C 2 , …C n , respectively.
Most of the small circles are then replaced by those larger circles.

Run a similar process for each particle, so the entire aggregate skeleton can be re-
built in this way.
For a 3D case, assume the number of small balls is N s , the number of balls after ap-
plying the Burn Algorithm is N l , then the reduced number of balls N r can be calculated
as: N r = N s − N l , which is an appropriate measurement of the efficiency of the algo-
rithm.

9.3.4 Burn Algorithm 2 to Reduce the Number of Balls
Another algorithm can be effective in reducing the number of required circles or balls.
It is named burn algorithm 2. Figure 9.11 presents the illustration in 2D. The method is
applicable to 3D. The procedure consists of the following steps.
Step 1. Scan all the circles used to represent a certain particle (cross-section). Find
four-circle groups and the centers of the groups. Use the center coordinates and double
radius of the small circles to generate one bigger circle to replace the four small circles.
Step 2. Repeat Step 1 until no four-circle group can be found and replaced.
For the 3D case (Figure 9.12), the process is similar but every eight-ball group (four
balls in the top and four balls in the bottom) needs to be found and replaced. If the