Geology Reference
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=
∂
∂
λ
suitable gradient-based techniques. Noting that the
number of design variables (number of elements)
can be very large, one should adopt a solution
method capable of solving large-scale problems.
The method of moving asymptotes (MMA)
proposed by Svanberg (1987) is a well-known
solution method used in topology optimization
problems. Another common approach is using
optimality criteria (OC) based algorithms. In the
following, after deriving the optimality criteria
for the optimization problem (1.15), we propose
a heuristic iterative fixed-point algorithm to solve
the optimization problem based on the optimality
criteria.
The eigenvalue equation,
Kφ
j
=
λ
j
Mφ
j
can be
satisfied separately using finite element analysis.
Excluding this equation, the Lagrangian of Prob-
lem (1.15) takes the form
D
1
−
Γ
v
=
γ
e
e
e
x
e
N
N
=
∑
∑
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
≤
e
e
0
x
1
0
< <
⇒
=
e
e
x
1
0
=
⇒
≥
e
e
e
1
,
,
N
=
(1.21)
To increase the fundamental frequency, we add
a vector of increments ∆
x
= (∆
x
1
, ∆
x
2
,…, ∆
x
N
)
T
to the design variables
x
= (
x
1
,
x
2
,…,
x
N
)
T
. The
subsequent change in the fundamental frequency
and the design volume can then be evaluated as
)
T
x
=
(
∆
λ
λ
∆
(1.22)
1
1
N
N
+
∑
∑
(
)
+
∆
v
=
v
T
∆
x
(1.23)
= +
λ
Γ
v
−
x v
γ
1
−
x
γ
x
1
e e
e
e
e
e
e
=
1
e
=
1
(1.19)
where
∇ =
(
)
T
∂
∂
λ
∂
∂
λ
∂
∂
λ
,
,
,
λ
is the gradient
1
1
1
2
1
1
x
x
x
N
where
Γ
,
γ
e
and
γ
e
are Lagrange multipliers.
Using Karush-Kuhn-Tucker results (Karush 1939;
Kuhn and Tucker 1951), the necessary optimal-
ity conditions for Problem (1.15) can be expressed
as follows
vector of λ
1
and
v
=
( ,
1 2
.
Let us now define the increments of design
variables as
v v
,
,
v
N
)
T
∆
x D
= = ∇ −
λ
1
Γ
v
(1.24)
∂
∂
x
=
∂
∂
λ
1
−
Γ
v
− + =
γ
γ
0
If the volume constraint is inactive, we will
have Γ = 0 and thus
∆
x
= ∇
λ
1
which results in
∆
λ
e
e
e
x
e
e
N
N
=
∑
∑
≥
Γ
v
−
x v
0
;
v
−
x v
0
;
Γ
≥
0
e e
e e
T
=
(
)
∇ ≥
λ
λ
0
after substituting in
e
=
1
e
=
1
1
1
1
(
)
=
γ
1
−
x
0
;
1
− ≥
x
0
;
γ
≥
0
,
e
=
=
1,
N
1
,
N
Equation (1.22).
If the volume constraint is active, on the other
hand, we will have
e
e
e
e
γ
x
=
0
;
x
≥
0
;
γ
≥
0
,
e
e
e
e
e
(1.20)
= −
and use it in Equa-
tion (1.20), we can rewrite the optimality criteria
as
If we define
γ
γ
γ
∆
v
=
T
∆
x
0
=
(1.25)
v
e
e
e
Using Equation (1.24) in Equation (1.25) and
solving for Γ we obtain
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