Biomedical Engineering Reference
In-Depth Information
For the right-hand side of (10) using the divergence theorem we get
g
0
dx
u
n
g
0
∂u
n
∂ν
∇
p
∇·
=
ds
|∇
u
n−
1
|
|∇
u
n−
1
|
∂p
=
q∈C
p
g
0
∂u
n
∂ν
ds.
|∇
u
n−
1
|
e
pq
So we have an integral formulation of (8):
dx
=
q∈C
p
u
n
u
n−
1
τ
g
0
∂u
n
∂ν
1
−
ds,
(11)
u
n−
1
|
u
n−
1
|
|∇
|∇
p
e
pq
expressing a “local mass balance” in the scheme. Now the exact “fluxes”
e
pq
g
0
|∇u
n
1
|
∂u
n
∂ν
ds
on the right-hand side and the “capacity function”
1
|∇u
n
1
|
on the
left-hand side (see, e.g., [59]) will be approximated numerically using piecewise
linear reconstruction of
u
n−
1
on the tetrahedral grid
−
−
T
h
. If we denote by
g
T
the
approximation of
g
0
on a tetrahedron
T
∈T
h
, then for the approximation of the
right-hand side of (11) we get
T ∈E
pq
u
q
u
p
h
pq
g
T
−
c
pq
,
(12)
u
n−
1
T
|∇
|
q∈C
p
and the left-hand side of (11) is approximated by
u
p
−
u
n−
1
p
M
p
m
(
p
)
,
(13)
τ
where
m
(
p
) is a measure in
IR
d
of co-volume
p
and
M
p
is an approximation of the
capacity function inside the finite volume
p
. For that goal we use the averaging of
the gradients in tetraherda crossing co-volume
p
, i.e.,
|
=
T ∈N
p
∩
1
m
(
T
p
)
u
n−
1
p
u
n−
1
T
M
p
=
,
|∇
|∇
|
.
(14)
u
n−
1
m
(
p
)
|∇
|
p
Then the regularization of the capacity function is given by
1
|∇
u
n−
1
M
p
=
,
(15)
|
ε
p
and if we define coefficients (where the
ε
-regularization is taken into account),
d
n−
1
p
=
M
p
m
(
p
)
,
(16)
g
T
h
pq
a
n−
1
pq
c
pq
=
,
(17)
u
n−
1
T
|∇
|
ε
T ∈E
pq
