Biology Reference
In-Depth Information
where
V
p
is the volume of particle
p
. Using this discretization, we
obtain the function approximation
s
s
h
h
Ê
Á
ˆ
˜
Ê
Á
ˆ
˜
h
r
O
O
O
(6)
u
()
x
=
u
()
x
+
=
u
()
x
+
( )
+
,
where
s
depends on the number of continuous derivatives of the mol-
lification kernel
z
,
27,85
and
h
is the interparticle distance. For a
Gaussian,
s
→∞.
From the approximation error in Eq. (6), we see that it is imper-
ative that the distance
h
between any two particles is always less than
the kernel core size
, thus maintaining
h
< 1
(7)
at all times. If this “particle overlap” condition is violated, the approx-
imation error becomes arbitrarily large, and the method ceases to be
well posed.
5.2. Operator Approximation
Two strategies are distinguished to evaluate differential operators on
particles: pure particle methods and hybrid PM methods.
5.2.1. Pure particle methods
In pure particle methods, differential operators on functions that are rep-
resented on particles are approximated by integral operators. The sums
on the right-hand side of Eq. (1) thus represent the discretized versions
of these integral operators. If we are interested in diffusion processes, for
example, the relevant differential operators are
.
(
D
) (cf. Sec. 7).
Both of them can be approximated by integral operators that allow con-
sistent evaluation on scattered particle locations and conserve mass
exactly
103,104
(cf. Sec. 7.2). The concept of this approximation method
has been extended to a general, systematic framework for approximating
∇
2
and
∇
∇
