Biomedical Engineering Reference
In-Depth Information
and introduce the shape functions in matrix form as
⊡
⊣
⊤
⊦
,
N
1
00
N
n
en
00
...
N
1
N
n
en
0
N
1
N
n
en
N
˕
=
...
N
ˆ
=
0
0
0
...
.
(28)
00
N
1
00
N
n
en
...
∇
x
˕
∇
x
ˆ
˕
Similarly, the spatial gradients
and
of the placement
and the non-local
ˆ
damage variable
are approximated by means of
B
˕
·
˕
e
,
B
ˆ
·
ˆ
e
,
h
h
∇
x
˕
=
∇
x
ˆ
=
(29)
where we express the spatial gradients of the shape functions
N
n
en
,
x
:= ∇
x
N
n
en
in
matrix form as
⊡
⊤
N
1
,
x
00
N
n
en
,
x
...
0
0
⊣
N
1
,
y
N
n
en
,
y
⊦
⊡
⊤
0
0
...
0
0
N
1
,
x
N
n
en
,
x
...
00
N
1
,
z
N
n
en
,
z
...
0
0
⊣
⊦
.
N
1
,
y
...
N
n
en
,
y
B
˕
=
B
ˆ
=
,
N
1
,
y
N
1
,
x
N
n
en
,
y
N
n
en
,
x
0
...
0
N
1
,
z
N
n
en
,
z
...
N
1
,
z
N
1
,
x
N
n
en
,
z
N
n
en
,
x
0
...
0
N
1
,
z
N
1
,
y
...
N
n
en
,
z
N
n
en
,
y
0
0
(30)
According to the Bubnov-Galerkin method, we apply an analogous approximation
for the variations of field variables and corresponding gradients. The discretised
weak form on element level is characterised by the difference of the element-specific
internal and external virtual work related terms
ʴ
˕
W
e
int
and
ʴ
˕
W
e
ext
, so that
ʴ
˕
W
e
=
ʴ
˕
W
e
int
−
ʴ
˕
W
e
ext
=
0
∀
ʴ
˕
e
,
(31)
ʴ
ˆ
W
e
=
ʴ
ˆ
W
e
int
−
ʴ
ˆ
W
e
ext
=
0
∀
ʴ
ˆ
e
,
(32)
where the discrete representations take the following forms
f
e
int
with
f
e
int
=
B
˕
t
v
d
ʴ
˕
W
e
int
=
ʴ
˕
e
·
·
˃
v,
(33)
B
t
f
e
ext
with
f
e
ext
=
N
˕
t
·
b
d
N
˕
t
·
t
d
a
ʴ
˕
W
e
ext
=
ʴ
˕
e
·
v
+
,
(34)
B
t
∂B
t
t
f
e
int
with
f
e
int
=
B
ˆ
ʴ
ˆ
W
e
int
=
ʴ
ˆ
e
·
·
y
d
v,
(35)
B
t
t
f
e
ext
with
f
e
ext
=
N
ˆ
ʴ
ˆ
W
e
ext
=
ʴ
ˆ
e
·
y
d
v,
(36)
B
t
