The name is absent

16

material) case, the typical convention is to represent a solid as the set of solutions to
the inequality f(x, y, z) < 0. Now, given two solids f(x, y,z) < 0 and g(x, y, z) < 0,
the union of these two solids is simply the set min(ʃ(rr, y, z),g(x, y, z~)) < 0 while the
intersection of two solids is the set max(∕(τ, y, z),g(x, у, z)) < 0.

If the functions f and g are represented by signed grids, a standard technique
for approximating their union or intersection is to take the min or max of their
associated sign grids. Our goal is to develop equivalent rules for the two-material
case that generalize to the multi-material case in a natural manner.

Our approach is as follows; consider two materials A and ->A (not A). A can be
interpreted as being the inside of a solid (i.e; negative in the implicit model) and ~<A
can be interpreted as being the outside of a solid (i.e; positive in the implicit model.).
Given a multi-material map consist of only these two materials, we can attempt to
construct rules for computing new non-negative scalars and material indices on the
grid that reproduce the operations Union and Intersection.

In particular, give a grid point with two associated pairs (sɪ, fc₁) and (s₂, k₂) (where
both the s_i are non-negative), our goal is to compute a scalar∕index pair (s, k') for
the union of the material S. This new pair can be computed using the following case
look-up given in Table 2.1.

Note that the rule for computing к is straightforward. For Union, the new material
index is A if and only if at least one of the material indices is A. For Intersection, the
new material index is A if and only if both of the material indices are A. The rule for
computing the new scalar s is only slightly more involved. The key is converted back
to the signed case and then return the result of taking the min of the converted scalars.
For example, if both material indices are A, we take the negative of both scalars s₁
and s₂, compute their min and then negate the result. These three operations are
simply the equivalent of taking the max of the original scalars. In particular, if both
s₁ and s₂ are non-negative,

max(s₁, s₂) = - min(-s₁, -s₂)

<<   23   24   25   26   27   28   29   >>

More intriguing information