Biomedical Engineering Reference
In-Depth Information
Conversion of the fluid and structural set of equations to discrete nodal locations
is referred to as discretisation. The fluid equations are typically discretised using
finite difference and finite volume which are based on the Taylor series expansions
of the differentials. Although the finite element method may also be used for the
fluid equations, its use for structural equations is more prevalent. A characteristic
stiffness matrix is used to connect each node for the finite elements. This was shown
for a simple one-dimensional spring-mass system and the plane stress for a two-
dimensional triangle was presented.
After discretisation, a set of algebraic equations is formed which are solved nu-
merically by either an iterative or direct solution method. For complicated geom-
etries the matrices of equations can be extremely large which take some time to
solve. Simple examples were provided to highlight the nature solving the matrices.
If the pressure exerted by blood flow onto the vessel artery wall is accounted for,
then a one-way FSI coupling has been achieved. Typically if the wall deformation is
small and has little bearing on the blood flow, then the coupling is sufficient. How-
ever in cases where large deformations are found, then two-coupling is required.
The interface boundary separating fluid from structure is used for matching mesh
nodes to transfer the force loadings between the two domains. FSI is very prone to
unstable simulations where large deformations cause problems in the mesh which is
related back to the discretised domain onto nodal points. Strategies to help conver-
gence, such as under-relxation, and time step selections were discussed.
This chapter provided an introduction into the numerical aspects of the com-
putational modelling required to satisfactorily achieve FSI simulations. Following
on from an understanding of the discretisation of the physical domain into discrete
nodes, the next chapter continues this theme by presenting methods for producing a
computational mesh that contain the discrete nodes.
5.8
Review Questions
1. Write down the relation for the total mass flow rate for the four artery outlets
shown in the figure on the right. If the mass flow fraction for
m
1
and
m
2
is 0.4,
what does this mean for the flow in
m
3
and
m
4
?
2. What factors contribute to the mass flow?
3. In a converging nozzle, the flow accelerates due to the narrowing geometry. Dis-
cuss the changes in the velocity gradients
∂
∂
v
y
during the flow (assume
a constant density). What is its corresponding pressure gradient?
4. Write a force balance equation for all the forces acting on a 2D differential con-
trol volume. What is its equivalent 3D form?
5. Discuss the difference between the local and advection acceleration in the fluid
momentum equation
u
x
and
∂
∂
∂
∂
v
t
∂
∂
v
x
∂
∂
v
y
+
u
+
v
local
advection
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