Biomedical Engineering Reference
In-Depth Information
Fig. 5.40
Solution strategies
for FSI simulations.
a
Mono-
lithic methods
b
Partitioned
methods
computational cost of the monolithic procedure is three to four times the partitioned
procedure; however, the monolithic approach reached accuracies much greater than
the partitioned approach when weakly coupled with respect to calculated structur-
al displacements (Wong et al. 2013a). Michler (2004) hypothesized that for more
complex problems the monolithic approach would be superior in a comparison of
computational cost to accuracy. One of the benefits to the partitioned approach is its
relative simplicity in implementation and solution. The interface conditions for the
partitioned case can be applied directly, while in the monolithic case they must be
added to the system of equations as extra Jacobian terms. Also, each set of govern-
ing equations for the partitioned case can be solved independently, requiring fewer
unknowns per solve than for the monolithic case. Due to its relative ease of imple-
mentation and lower computational cost, partitioned schemes have dominated the
majority of FSI research.
Therefore, this chapter will mainly focus on partitioned methods. To match both
kinematic and dynamic conditions simultaneously in both fluid and solid equation
solvers, a successive iteration method is applied. In this partitioned approach, seg-
regated solvers for each fluid and structural field are employed and the interacting
quantities at the interface from each field are exchanged sequentially. A flow chart
demonstrating this process is given in Fig.
5.41
.
Within each component of iterations, the solutions must reach converged results
before moving to the next set of iterations. The steps are detailed as:
1.
Fluid Solver:
The fluid variables, (,,,)
uvwP
are solved based on initial or
current geometrical configuration, i.e. based on the current displacement at the
interface,
d
n
−1
at time
t
n
and coupling iteration
k
−1 . The fluid pressure and
shear forces at the interface are resolved in all three components,
x
,
y
, and
z
.
2.
Structural Solver:
The forces and boundary constraints produced from the fluid
equations are applied to the structure at the interface. The structural deformation
is determined, giving the current interface displacements
d
n
.
3.
Mesh Adapt:
The interface displacement is interpolated across the fluid interface
mesh nodes which are used to alter the mesh deformation on the fluid domain.
4. The fluid equations are solved again for the unknown fluid variables based on
the new geometrical configuration caused by
d
n
.
5.
FSI Coupling
: The process is repeated until the difference in nodal deflection
and forces exchanged in steps 3 and 1, from current and previous coupling itera-
tions, are within a specified tolerance—suggesting that at a given timestep, the
kinematic and dynamic continuity are satisfied at the interface.
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