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
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