6 SARAH D. OLSON AND ANITA T. LAYTON

2.1.3. Forces and Energy Functions. The force density that the elastic structure

exerts on the surrounding fluid will vary greatly based on the properties of the given

structure. The stress and deformation of the elastic structure are determined by a

given constitutive law and this is then transmitted to the fluid through a localized

force density term in the momentum equations, either Eq. (2.1a) or (2.1b). To

highlight a few basic principles, we will briefly describe a few types of force densities

that may be included at a given point in time on the structure. Since the immersed

structure is assumed to be neutrally buoyant, traditional IB methods do not account

for gravitational forces.

A structure can be assigned a prescribed motion. This has been used to rep-

resent structures with zero or minimal movement as well as structures that have

a time dependent prescribed motion [8, 39, 92]. In this formulation, we think of

a stiff spring with a restoring force that attempts to keep the given point at the

desired or ‘tethered’ configuration. The discretized form of this force density is,

(2.8) Fk = −ST (Xk − Xk

tether)

where ST is a coeﬃcient and Xk tether is the tethered or desired location for the kth

immersed boundary point. We can view a tether force density as a spring with zero

resting length connecting Xk tether and Xk. This tends to penalize deviations from

the desired configuration.

In the IB method, the immersed structure is generally assumed to be elastic.

In order to describe the stretching of this elastic structure, we idealize elastic links

connecting the points via stiff springs that are assumed to be governed by Hooke’s

law (linear spring force). Between the points Xk and Xk+1, the spring connecting

these two points is generating force at each of these two points, trying to maintain

the specified separation. This corresponds to the following discretized form of the

force density F exerted on the surrounding fluid at the point Xk,

F(Xk) = − SH (||Xk − Xk−1|| − )

Xk − Xk−1

||Xk − Xk−1||

− SH (||Xk+1 − Xk|| − )

Xk − Xk+1

||Xk+1 − Xk||

(2.9)

where SH is a spring constant or stiffness coeﬃcient, || · || denotes the Euclidean

norm, and is the resting spring length, which corresponds to the immersed bound-

ary spacing in many problems. In Eq. 2.9, the first term corresponds to the spring

force due to the spring connecting Xk−1 and Xk and the second term corresponds

to the spring force between Xk and Xk+1. If a larger value of SH is specified, this

will cause the rest length to be more strictly enforced. We can also approximate

inextensible materials by using a very large stiffness coeﬃcient SH . We can also

formulate the force in terms of the tension in a fiber or section of the immersed

structure. The fiber is assumed to only sustain tension in the direction of the fiber,

τ. Then, force balance on a given segment of the fiber can be used to write the

force density as

F = −

∂

∂s

(Tτ) , (2.10a)

τ =

∂X

∂s

∂X

∂s

, (2.10b)