Biomedical Engineering Reference
In-Depth Information
5.4
Computational Methods
The conservation equations for fluid flow (i.e. mass, momentum) and structural
domain (i.e. momentum) are partial differential equations (PDEs). Computational
methods are used to transform these equations into discrete algebraic equations
which are then applied to mesh points within the fluid and structural domain. Dis-
cretisation of the fluid equations are most commonly use the finite difference or
finite volume approach, while for the structural equations, it is the finite-element
approach.
5.4.1
Finite Difference Method
In the finite difference method, the partial derivatives are approximated by alge-
braic equations at grid nodal points through the Taylor series expansion. The Taylor
series is defined as:
2
1
∂
fa
()
1
∂
f a
()
2
fx
( )
≈ +
fa
( )
(
xa
−+
)
(
xa
−
)
2
1!
∂
x
2!
∂
x
(5.35)
3
1
∂
f
()
(
a
xa
…
3
+ −+
∂
)
3!
3
x
and is rewritten in sigma notation as
()
n
∞
f
()
a
∑
n
fx
()
≈
(
x a
−
)
(5.36)
n
!
n
=
0
This is applied to a one-dimensional uniformly distributed Cartesian grid shown in
Fig.
5.21
, where location
a
is replaced by the index
i
. In two-dimensions the indices
are (
i
,
j
) and in three-dimensions (
i
,
j
,
k
).
Following the Taylor series, we substitute
f
(
a
) as
ϕ
i
and
f
(
x
) as
ϕ
i
+
1
where
ϕ
is
a generic flow field variable. This produces the variable at point (
i
+ 1) expanded
about the point (
i
). Rearranging the equation, the approximation for the partial de-
rivative becomes
∂
φ
φφ
−
∂ ∆∂ ∆
2
φ
x
3
φ
x
2
= + ∆
(
)
(
)
i
+
1
i
O
x
where
O
∆=
x
+
+
∂
x
∆
x
2
2
3
6
∂
x
∂
x
i
i
i
Truncation error
(5.37)
The term O(∆
x
) signifies the truncation error related to the finite difference ap-
proximation. It measures the accuracy of the approximation and determines the rate
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