Biomedical Engineering Reference
In-Depth Information
In Eq. (
18.15
) the double dot product (
∇
w
)
T
:
σ
is used. The definition of the
double dot product of two tensors
A
and
B
is
A
:
B
=
tr(
A
·
B
)
=
A
ij
B
ji
,
(18.16)
where the Einstein summation convention has been used. With respect to a Carte-
sian basis and using index notation it is straightforward to prove Eq. (
18.15
). First
of all notice that
σ
·
w
=
σ
ij
w
j
e
i
.
(18.17)
Consequently
∂
∂
x
i
(
∇·
(
σ
·
w
)
=
σ
ij
w
j
) .
(18.18)
Application of the product rule of differentiation yields
∂
∂
x
i
(
∇·
(
σ
·
w
)
=
σ
ij
w
j
)
=
∂σ
ij
∂
x
i
w
j
+
σ
ij
∂
w
j
.
(18.19)
∂
x
i
With the identifications
(
∇·
σ
)
·
w
=
∂σ
ij
∂
x
i
w
j
,
(18.20)
and
(
∇
w
)
T
:
σ
=
∂
w
j
∂
x
i
σ
ij
,
(18.21)
the product rule according to Eq. (
18.15
) is obtained.
Use of this result in Eq. (
18.14
) yields
w
·
fd
=
0.
∇·
(
σ
·
w
)
d
−
(
∇
w
)
T
:
σ
d
+
(18.22)
The first integral may be transformed using the divergence theorem Eq. (
16.5
).
This yields
w
·
(
σ
·
n
)
d
+
w
·
fd
.
(
∇
w
)
T
:
σ
d
=
(18.23)
18.4
Galerkin discretization
Within the context of the finite element method, the domain
is split into a num-
ber of non-overlapping subdomains (elements)
e
, such that this integral equation
is rewritten as
