Biomedical Engineering Reference
In-Depth Information
multiplication by a suitable, spatially continuous weighting function
i
i
12
-----------------
〈
G
u
〉
(
u
)
〈
p
〉
〈
(
)
g
〉
(15.2a)
i
i
f
t
i
i
t
-----
(
u
)
〈
i
k
p
〉
0
(15.2b)
where
th
member of a complete set of scalar functions of position (which is ultimately
truncated as part of the discretization process), in particular, the standard
〈〉
·
indicates integration over the problem space. Here,
is the
i
i
C
°
locally defined Lagrange polynomial interpolants associated with finite ele-
ments. Applying divergence and gradient integral theorems to the second
derivative terms in Equation (15.2) leads to
G
12
-----------------
(
u
)
〈
i
p
〉
〈
(
f
t
)
g
i
〉
〈
G
u
〉
i
i
G
12
G
ˆ
-----------------
ˆ
(
u
)
i
ds
u
i
ds
(15.3a)
i
-----
(
u
)
〈
k
p
i
〉
(15.3b)
k
ˆ
p
i
ds
t
where
denotes integration over the boundary which encloses the brain vol-
ume and is the outward pointing normal direction to this boundary. Spatial
discretization of Equation (15.3) is completed by expanding the unknown
displacement vector,
ˆ
, and fluid pressure, p, as sums of time-varying (but
spatially constant) coefficients multiplied by known (time-invariant) func-
tions of position which produces the coupled set of ordinary differential
equations.
u
j
G
12
p
j
j
i
〈
〉
u
j
〈
G
j
i
〉
u
j
-----------------
i
j
j
j
G
12
-----------------
ˆ
(
u
)
i
ds
G
ˆ
〈
(
)
g
〉
u
i
ds
(15.4a)
f
t
i
u
j
p
j
--------
〈
〉
〈
k
〉
k
ˆ
p
i
ds
(15.4b)
j
i
j
i
t
j
j
Equation (15.4) can be integrated in time using a simple two-point weighting
t
n
1
ft
()
d
t
[
ft
n
1
(
)
(
1
)
ft
n
()
]
(15.5)
t
n
