Geology Reference
In-Depth Information
and its
first derivatives is required to perform the integrations in the functional (8.32). At
the
k
,vertex we write
On the rectangular finite element
R
ij
a continuous representation of
Φ
=
δ
k
,
. Local Hermite spline
cubics, developed in Section 1.6.2, are used as support functions in
r
.Atthe
k
,ver-
tex, the cubicη
k
(
r
) supports the function in radius, and the cubic ψ
k
(
r
) supports the
radial derivative. Support in
z
, incorporating the even or odd properties, is provided
by the basis functions developed in Section 1.6.3. At the
k
, vertex, η
(
z
) supports
the function in
z
and ψ
(
z
) supports its derivative. Hence, on the finite element
R
ij
,
the representation of
Φ=
α
k
,
,
Φ
=
β
k
,
,
Φ
=
γ
k
,
,
Φ
r
z
rz
Φ
(
r
,
z
)isgivenby
=
j
+
1
k
=
i
+
1
α
k
,
η
k
(
r
)η
(
z
)
Φ
i
,
j
(
r
,
z
)
=
+
β
k
,
ψ
k
(
r
)η
(
z
)
k
=
i
=
j
+
δ
k
,
ψ
k
(
r
)ψ
(
z
)
.
+
γ
k
,
η
k
(
r
)ψ
(
z
)
(8.37)
A four-vector can be defined at each of the nodes of the finite element grid. For the
node at the
k
, vertex, the four-vector has the transpose
x
k
,
=
(α
k
,
,β
k
,
,γ
k
,
,δ
k
,
).
On the finite element
R
ij
, the vector
V
entering the functional (8.32) and the
expression (8.33) for the normal component of displacement then has the support
k
=
i
+
1
=
j
+
1
V
=
D
k
,
x
k
,
,
(8.38)
k
=
i
=
j
where
⎝
⎠
η
k
(
r
)η
(
z
)
ψ
k
(
r
)η
(
z
)
η
k
(
r
)ψ
(
z
)
ψ
k
(
r
)ψ
(
z
)
r
η
k
(
r
)η
(
z
)
r
ψ
k
(
r
)η
(
z
)
r
η
k
(
r
)ψ
(
z
)
r
ψ
k
(
r
)ψ
(
z
)
D
k
,
=
.
(8.39)
η
k
(
r
)η
(
z
)
ψ
k
(
r
)η
(
z
)
η
k
(
r
)ψ
(
z
)
ψ
k
(
r
)ψ
(
z
)
R
(8.32) gives a
symmetric quadratic form and setting its variation to zero gives the contribution
1
Substitution of the expression (8.38) for
V
in the functional
F
1
D
p
,
q
A
(
r
,
z
,σ)
D
k
,
dr dz
(8.40)
0
a
/
b
to the
p
,
q
row and
k
,column of the final matrix. The 3
3matrix
A
is a polynomial
in the dimensionless angular frequency σ and is a piecewise polynomial in
r
and
z
for the piecewise polynomials used as support functions and to interpolate the
decompression factor
f
. The individual contributions (8.40) are 4
×
4matrices.For
the local polynomial support functions used, they only interact with the support
at the four nearest nodes of the finite element grid, thus each element produces a
16
×
16 array of contributions to the final system of equations. If the grid has
M
nodes in
r
and
N
nodes in
z
, the final system of equations is generated by summing
the contributions from all (
M
×
−
1)
×
(
N
−
1) elements.
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