Biomedical Engineering Reference
In-Depth Information
with components:
T
x
∂ξ
∂
=
∂
∼
∼
e
∂ξ
T
∂η
=
∂
∂
x
∼
∼
e
∂η
(16.48)
T
∂
y
∂ξ
=
∂
∼
y
∼
e
∂ξ
T
∂η
=
∂
∼
∂
y
y
∼
e
.
∂η
Matrix (
16.47
) is the inverse of the matrix in Eq. (
16.46
) (this can be checked by
multiplying the two matrices, which gives the unit matrix). Accordingly
⎡
⎤
⎡
⎤
∂ξ
∂
x
∂η
∂
x
∂
y
∂η
−
∂
y
∂ξ
1
j
⎣
⎦
=
)
−
1
⎣
⎦
(
x
,
ξ
=
(16.49)
∂ξ
∂
y
∂η
∂
y
−
∂
x
∂η
∂
x
∂ξ
where
∂
x
∂ξ
∂η
−
∂
x
∂
y
∂
y
∂ξ
j
=
det(
x
,
ξ
)
=
.
(16.50)
∂η
It is an elaborate process and usually not possible to analytically compute the inte-
grals in the expressions for the element matrices or arrays, so generally they are
approximated by numerical integration. Each of the components of the matrices,
such as
K
e
etc., that need to be computed consists of integrals of a given function,
say
g
(
x
,
y
), over the domain of the element
e
. These may be transformed to an
integral over the unit square
−
1
≤
ξ
≤
1,
−
1
≤
η
≤
1, according to
1
1
f
(
x
,
y
)
d
=
f
(
x
(
ξ
,
η
),
y
(
ξ
,
η
))
j
(
ξ
,
η
)
d
ξ
d
η
,
(16.51)
e
−
−
1
1
with
j
(
) defined by Eq. (
16.50
).
The integral over the unit square may be approximated by a numerical
integration (quadrature) rule, giving
1
ξ
,
η
1
n
int
f
(
ξ
,
η
)
j
(
ξ
,
η
)
d
ξ
d
η
≈
f
(
ξ
i
,
η
i
)
j
(
ξ
i
,
η
i
)
W
(
ξ
i
,
η
i
) .
(16.52)
−
−
1
1
i
=
1
For example, in case of a two-by-two Gaussian integration rule the location of the
integration points have
ξ
,
η
-coordinates and associated weights:
−
1
√
3
,
1
√
3
,
W
1
=
1
−
ξ
1
=
η
1
=
1
√
3
,
−
1
√
3
,
W
2
=
ξ
2
=
η
2
=
1
(16.53)
