SIMULATING IMMERSED BIOLOGICAL STRUCTURES 7

where τ is the unit tangent vector. When we assume the tension T on the fiber is

a linear function of the fiber strain, we can write:

(2.11) T = S

∂X

∂s

where S is a stiffness coeﬃcient. In this particular case, the force density becomes

(2.12) F = −S

∂2X

∂s2

.

If a centered difference approximation of the derivative in Eq.(2.12) is used, one

arrives at an expression like Eq. (2.9) for the force at a given point on the immersed

structure. Thus, a spring model can be formulated as a discretization of the above

fiber model.

We can also determine force density by first postulating an energy functional

E that determines the elastic potential energy stored in the structure at a given

time point. This postulated variational energy functional is set up to ensure that

it is non-negative, translation and rotation invariant, and formulated such that the

structure will want to minimize this energy. If we consider a perturbation of the

given structure X as PX, the corresponding perturbation in the elastic energy is

PE = −

B

(F · PX(s, t))ds , (2.13a)

F = −

PE

PX

, (2.13b)

where F is the Frechet derivative of E, defined implicity and corresponding to the

amount of force that is generated by a perturbation in the elastic structure. Note

that F corresponds to the force density exerted by the structure on the surrounding

fluid and this formulation corresponds to virtual work on an elastic structure due

to a perturbation of the structure.

In this energy formulation, the extent to which the energy is minimized will

depend on the material properties of the elastic structure as well as the surrounding

fluid environment. As an example, we could assume the following elastic energy

(stretching),

(2.14) E =

B

E

∂X

∂s

ds

where E is a local stretching energy density to be specified. This energy corresponds

to the elasticity determined by the strain in the direction of the fiber, as derived

above. Using this energy, the perturbation operator can be applied to both sides

of Eq. (2.14). Assuming that tension T in Eq. (2.11) is also equivalent to the

derivative of E (||∂X/∂s||), we can use integration by parts to simplify the integral.

Thus, we can arrive at Eq. (2.12) or Eq. (2.9) using an energy argument. This

energy formulation and derivation can also be extended to more complicated elastic

structures.

For a more extensive derivation and examples, we refer the reader to [138]. Ad-

ditionally, we wish to note that the choice of parameters in these energy functions

and forces are not always determined in an ad hoc manner. For a given applica-

tion, one can choose stiffness parameters to reflect the material properties of the

structure. In order to estimate the flexural rigidity of an elastic structure, one can

follow the procedure of Lim and Peskin [108].