SIMULATING IMMERSED BIOLOGICAL STRUCTURES 5
(a) (b)
2 1 0 1 2
1.5
1
0.5
0
0.5
1
1.5
2 1 0 1 2
1.5
1
0.5
0
0.5
1
1.5
Figure 1. An example of an immersed boundary simulation
where forces are proportional to the curvature. In (a), the per-
turbed circle is immersed in a viscous, incompressible fluid and
visualized with the dashed lines. A subset of the Cartesian grid
points are denoted by smaller circles on the domain and the La-
grangian singular force density F(X,t) on the immersed boundary
are shown with the vectors on the structure. In (b), this is a later
time point where the Eulerian force density f(X,t) is shown on the
Cartesian grid by spreading the Lagrangian force density via
Eq. (2.3).
equations,
j even
φ(r j) =
j odd
φ(r j) =
1
2
, (2.6a)
j
(r j)φ(r j) = 0 , (2.6b)
for all real r. The first equation, (2.6a), is an even-odd and
zeroth
moment condi-
tion, which ensures that the central difference operators apply the correct weight to
the points. The second equation, (2.6b), is a first moment condition. An example
of a function φ that satisfies these conditions and others is,
(2.7) φ(r) =











0 |r| 2
1
8
(5 + 2r

12r 4r2), −2 r −1
1
8
(3 + 2r +
√−7
4r 4r2), −1 r 0
1
8
(3 2r +
√1
1 + 4r 4r2), 0 r 1
1
8
(5 2r

−7 + 12r 4r2), 1 r 2
.
In order to have expressions for mass, momentum, and torque be the same when
evaluated both in the Lagrangian and Eulerian form, this will depend on the specific
properties of the delta function used [138]. Please refer to [138] for a list of
additional conditions to determine the compactly supported delta function given in
Eq. (2.7). Recently, Liu and Mori [113] have further analyzed properties of delta
functions in reference to convergence of the IB method. They have shown that
another property, called the smoothing order, is also very important in terms of
convergence.
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