SIMULATING IMMERSED BIOLOGICAL STRUCTURES 13

velocity is solved for at the structure and can be evaluated at any other points

where this information is needed. The time consuming part of this method is the

calculation of these sums, which is

O(m2)

for m immersed boundary points. This

time consuming sum could also be made more eﬃcient by making it run in parallel

or using a fast multipole method [60, 160]. This method could have decreased

computation time in comparison to the IB method with the use of a Cartesian grid

for the fluid and curvilinear mesh for the Lagrangian structure; this would depend

on the number of points on the structure as well as the number of points on the

Cartesian grid in the IB method. The numerical algorithm as presented is explicit

and the resulting ODEs are stiff when the structure is represented via springs

with large stiffness coeﬃcients; this can cause a severe time step restriction. One

could use implicit methods, but this will lead to the solution of nonlinear systems

with a dense Jacobian, which can also be computationally expensive. Another

strategy developed by Bouzarth and Minion [13] involves the use of spectral deferred

corrections, giving an explicit multirate time integration.

In the above algorithm, it was assumed that there was a function to describe

the calculation of the point forces gk. In other applications, one may know the

velocity of the moving structure, u(Xk) and may not know the value of forces gk

at each of the m immersed boundary points. For example, we may know the exact

location and speed of movement for an object from experiments and we wish to

solve for the forces along that structure. Using Eq. (2.37), we can generate a system

of linear equations that satisfy

(2.38) u(Xi) =

m

k=1

Aik(X1,..., Xn)gk

where Aik is the matrix corresponding to the coeﬃcients of gk in Eq. (2.35a).

Since the contributions of each of the point forces are felt at each of the immersed

boundary points, this ends up being a dense matrix. We can solve this system for

the forces gk. If one wishes to know the surrounding fluid velocity, we can then

solve for this on a grid using Eq. (2.35a). This type of framework can be used to

determine resulting flow from cilia or sperm with prescribed motion (velocity) as

well as to understand flow around a cylinder that has a prescribed velocity [29].

Similar to the IB method, this is not a sharp interface model. The force is

smeared to the surrounding fluid via the regularization function, which puts most

of the force within a sphere of radius approximately . Due to this fact, the error

in the method will be largest in the region around the smeared interface. The error

for the MRS in 3-d was analyzed for a particular blob function by Cortez et al. [31]

and determined to be O(

2)

far from the boundary and O( ) close to the boundary.

Gonzalez and Li [58] have also analyzed this method and determined that for proper

choice of and Δs, convergence of solutions can be achieved. When representing a

structure and regularizing point forces, there is a choice in determining the optimal

and it is unclear at this time exactly how a given corresponds to a ‘virtual

radius’ of a structure. Additionally, for a given application, the optimal choice of

ψ is not known. Recently, Barrero-Gil [4] have modified the MRS to reduce the

dependence of on the numerical results.

The MRS, as derived above, is for an infinite fluid domain. Recently, Leiderman

et al. [98,99] have extended this method to a periodic domain through the use of

periodic Green’s functions. In addition to the Stokeslet for a point force, there exist