Geology Reference
In-Depth Information
9.4. Restricting the Topology
ever, the direction of the load is uncertain. Thus
one needs to consider two displacement cycles: a
↑↓↑ cycle as well as a ↓↑↓ cycle. But knowing that
the results of these two displacement cycles are
mirrored images of each other, it is not necessary
to analyze the model under both of these loading
conditions. Instead, one can add the sensitivities
of the mirrored elements together to account for
both displacement cycles. The sensitivity numbers
are thus corrected as
Topology optimization techniques like BESO
can naturally introduce new holes or fill the
current holes in the design domain. However
this behavior might produce complicated shapes
which might be costly to fabricate. To prevent the
BESO algorithm from introducing new holes, we
restrict the designable domain to the elements at
the boundaries of the shape at each iteration. The
designable domain at each iteration is defined as
α
= +
α
α
(1.121)
e
e
e
{
}
D
= ∃
e
i j
∈
B
i j
∈ ∧ ≠
e
i
j
,
:
,
(1.123)
where
α
e
is the corrected sensitivity number of
element e and
e
is the element which is located
at the same location as
e
in the mirrored model.
Like the SIMP method, the BESO method is
also prone to the formation of checkerboard pat-
terns and mesh dependency. A filtering technique,
similar to Equation (1.29) can be employed to
overcome these problems in the BESO algorithm
where
is the set of boundary nodes defined as
{
}
B
= ∃
i
e
∈
S
e
∈
V
i
∈
e
e
,
:
m
v
m
v
(1.124)
with
and
denoting the sets of solid and void
elements respectively.
N
∑
α
w
9.5. Numerical Examples
j
ij
=
α
e
=
1
(1.122)
e
N
∑
w
In the following numerical examples we fix the
volume to simplify the approach. Another benefit
of using fixed volume is that the results of different
iterations are comparable to each other.
The modulus of elasticity of the material is
considered as 206.1 GPa and the tensile yield
stress is assumed to be 334 MPa. The material
model considered is depicted in Figure 13.
ij
j
=
1
Another significance of filtering in the BESO
method is extrapolating the sensitivity numbers
to void elements. If one uses a hard kill approach,
void elements are removed from the structure and
their sensitivities cannot be evaluated directly.
In other words, all void elements will have a
sensitivity number of zero. The filtering scheme
in Equation (1.122) extrapolates the sensitivities
to the void elements in the neighborhood of the
solid elements. This extrapolation leads the BESO
algorithm to add the elements in the vicinity of the
elements with high sensitivity numbers.
Example 4: Simple Damper
We use the proposed BESO algorithm to optimize
the shape of a simple damper. The target volume
is 82% of the design domain. The initial and the
final solutions are depicted in Figures 14a and 14b
respectively. The evolution history of the objective
function is plotted in Figure 14c. It can be verified
that the energy absorption of the optimal solution
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