Geology Reference
In-Depth Information
5.3. Solution Algorithm
We now substitute Equation (1.35) in Equation
(1.40) and use Equation (1.6) to write
Assume that
Φ
is an
N
d
×
m
matrix whose columns
are the eigenvectors
φ
1
,…,
φ
m
. Also consider the
vector
d
= (
d
1
,
d
2
,…,
d
N
)
T
defined as
m
K
M
∂
∂
∂
∂
…
m
∑
=
α
ϕ
T
−
λ
ϕ
−
µ δ
0,
s
=
1,
jk
s
k
j
sk
x
x
k
=
1
i
i
(1.41)
=
∂
∂
K
∂
∂
M
d
−
λ
,
e
=
1
,
…
,
N
(1.45)
e
This linear system can be solved to calculate the
coefficients
α
jk
. A non-trivial solution only exists if
x
x
e
e
Then the subeigenvalue problem of Equation
(1.43) can be stated in the following matrix form
∂
∂
K
∂
∂
M
1 …,
m
λ
=
det
ϕ
T
−
ϕ
−
µ δ
0
,
s k
,
=
,
s
k
j
sk
x
x
i
i
(1.42)
A
−
µ
0
(1.46)
This subeigenvalue problem can be solved to
calculate the sensitivities
μ
j
,
j
= 1,…,
m
due to the
increment of the
i
i-th optimization variable.
If the vector of design variables undergo an
increment of the form
ε
e
with
e
= (∆
x
1
, ∆
x
2
,…,
∆
x
N
)
T
and ||
e
|| = 1, one can easily generalize Equa-
tion (1.42) to
where
A
=
Φ
T
Φd
T
e
is a symmetric
m
×
m
matrix
and
I
m
×
m
is the unity matrix.
Following Cox and Overton (1992) and Over-
ton (1992), the necessary optimality conditions to
solve problem (1.15) is that there exists an
m
×
m
symmetric positive semidefinite matrix
Λ
with
trace(
Λ
) = 1 such that
=
T
det
f e
sk
−
µδ
0
,
s k
,
=
1
(1.43)
,
,
m
sk
D
(
)
−
T
=
Λ Φ Φ
:
d
Γ
v
=
γ
e
e
e
e
N
N
=
∑
∑
where
Γ
v
−
x v
0
;
v
−
x v
≥
0
;
Γ
≥
0
e e
e e
e
=
1
e
=
1
x
0
0
=
⇒
γ
γ
γ
≤
T
(
(
)
(
)
(
)
)
f
T
=
ϕ
T
∂
∂
K
−
λ
∂
∂
M
ϕ ϕ
,
T
∂
∂
K
−
λ
∂
∂
M
ϕ
,
…
,
ϕ
T
∂
∂
K
−
λ
∂
M
x
ϕ
,
e
e
sk
s
x
x
k
s
x
x
k
s
x
N
k
1
1
2
2
N
0
x
1
0
< <
⇒
=
s k
,
=
1
,
…
,
m
e
e
x
1
0
(1.44)
=
⇒
≥
e
e
e
1
,
…
,
N
=
are known as the generalized gradient vectors
(Seyranian et al. 1994). Note that
f
sk
are vectors
of length
N
, thus
f e
sk
(1.47)
T
are scalars. Also note that
due to symmetry of the stiffness and mass matri-
ces
f
sk
=
f
ks
.
The solutions of subeigenvalue problem of
Equation (1.43) are the sensitivities of the multiple
eigenvalues. This equation was initially introduced
by Bratus and Seyranian (1983).
where the Frobenius matrix inner product is de-
fined as
A
:
B
= trace(
A
T
B
).
The proof of optimality conditions in (1.47)
will not be presented here. Enthusiast reader is
referred to Cox and Overton (1992), Overton
(1992), and Seyranian et al. (1994).
Note that for the case of simple eigenvalues (
m
= 1), one should have Λ = 1 and optimality condi-
tions in (1.47) reduce to (1.21). Comparing (1.47)
with (1.21), one may note that the only difference
is that the sensitivities ∂
λ
/∂
x
e
in (1.21) have been
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