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.