Geology Reference
In-Depth Information
4.4. Flowchart and
Numerical Examples
hand, the extremely large size of the problems in
topology optimization is a great barrier in using
non-gradient-based optimization techniques such
as Genetic Algorithm (GA) and Neural Networks.
The
continuation
method is a simple approach
used and suggested by many researchers to over-
come this problem in gradient-based optimization
methods (Sigmund and Petersson 1998). In this
approach, one would start solving the problem in a
more relaxed form and gradually apply restrictions.
For example, one can start the solution considering
no penalty factor (
p
= 1) and gradually increase
p
upon convergence of the solution.
Apart from these three problems - which are
common in all types of topology optimization
problems - some numerical artifacts are unique
to eigenvalue problems. The 'artificial modes'
problem comes under this category. These are
localized modes appearing in regions with rela-
tively high mass to stiffness ratio. Pedersen (2000)
points that due to the interpolation schemes for
stiffness and mass (Eqs. (1.12) and (1.14) respec-
tively), the mass to stiffness ratio rises steeply for
small values of
x
which ultimately results in local-
ized modes. To overcome this problem, a modi-
fication in the stiffness interpolation scheme is
suggested by Pedersen (2000) to limit the mass
to stiffness ratio in low density areas (typically
x
<
0 .
). Following the same principle, Du and
Olhoff (2007) proposed a different approach by
modifying the mass interpolation scheme. The
latter approach is adopted here. To this end, we
replace the original mass interpolation scheme,
Equation (1.14), by
A flowchart of the proposed solution algorithm
is depicted in Figure 2. The solution starts from
an initial guess design. A uniform distribution of
material defined as
x
1
with
v
t
denoting the total volume of the design domain
is usually used as an initial design. The main loop
starts by analyzing the current design using finite
element analysis. Based on FE results, the sensi-
tivities are calculated using Equation (1.18). The
sensitivities are then filtered using Equation (1.29).
The updated variables are calculated in an inner
loop. In the inner loop, starting with a positive
value for the Lagrange multiplier Γ, the updated
variables are calculated using Equation (1.28).
The volume of this new design is checked and
the new value of Γ is adjusted using the bisection
approach. The inner loop continues until the
value of Γ converges. The updated design is then
replaces the old one and the procedure is re-
peated until a convergence criterion is satisfied.
The convergence criterion used here is defined
=
v
/
v e
,
=
,
,
N
e
t
as
k
k
−
1
∑
∑
λ
−
λ
( )
( )
i
i
i k l
= − +
1
i k l
= −
≤
ε
(1.31)
k
k
−
1
∑
∑
λ
+
λ
( )
i
( )
i
i k l
= − +
1
i k l
= −
where
λ
(
i
)
is the value of the objective function at
the
i
i-th iteration and
k
denotes the last iteration.
This condition compares the value of the objective
function in the last and second last l iterations and
assumes convergence is achieved when the rela-
tive error is smaller than a predefined tolerance
0
x
M
,
x
>
≤
0 1
.
e
e
e
M
(
x
)
=
e
e
(1.30)
q
−
1
q
10
x
M
,
x
0 1
.
e
e
e
<
ε
.
In all examples reported here
l
= 5
and
ε
= 0.001 were used.
1
with
q
= 2
p
. This ensures that the mass to stiffness
ratio cannot exceed 10
p
-1
.
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