Geology Reference
In-Depth Information
T
v
∇
λ
1
(Sigmund and Petersson 1998). Checkerboard
problem refers to the formation of alternating
solid and void elements in a checkerboard-like
pattern resulting in artificially high stiffness. Mesh
dependency refers to obtaining different optimal
topologies for the same problem using different
mesh sizes. Local minima refers to the problem
of obtaining different optimal topologies using the
same mesh but different algorithmic parameters
and/or initial design.
One of the simplest yet effective approaches
to overcome checkerboard and mesh dependence
problems is filtering sensitivities (Sigmund and
Petersson 1998). In this approach the calculated
sensitivities are replaced by filtered sensitivities
which are calculated as a weighted average of
the sensitivities of the neighboring elements. A
simple linear filter takes the form
Γ =
(1.26)
T
v v
If the boxing conditions are all inactive, i.e. if
0 <
x
e
< 1, we can use Equation (1.26) in Equa-
tion (1.24) and then Equation (1.24) in Equation
(1.22) to write
T
v v
v v
∇
λ
T
=
(
)
∇ −
∆
λ
λ
λ
(1.27)
1
1
1
T
It can be easily verified that the right hand side
of Equation (1.27) is a form of Cauchy-Bunya-
kovsky-Schwarz inequality and thus
∆
λ
1
≥
.
Based on this discussion, we propose the fol-
lowing update scheme to solve Problem (1.15)
(
)
0
,
x
−
η
,
( )
k
N
∂
∂
λ
x
=
max
∑
1
−
p
x
w
(
k
+
1
)
(
)
p
min ,
1
x
+
η
,
x
+
η
D D
j
ij
x
∂
∂
λ
( )
( )
( )
( )
k
k
k
k
e
=
1
j
=
(1.29)
(1.28)
N
x
∑
x
w
i
i
ij
j
=
1
Here the subscripts denote the iteration num-
ber and
η
is a tuning parameter defining the move
limit. The vector
D
is defined in Equation (1.24).
Note that
p
used here is the previously defined
penalty power for stiffness. The value of the
Lagrange multiplier Γ can be calculated using
bisection method in an inner loop. In finding Γ,
one should note that
∂ ∂ <
in which
w
ij
= max{0,
R
-
d
ij
}.
R
is known as the
filtering radius and
d
ij
denotes the distance between
the centers of the elements
i
and
j
. The filtering
scheme (1.29) can be activated by choosing the
filtering radius
R
bigger than the size of elements
h
. This can eliminate the checkerboard problem.
In this scheme, the filtering radius
R
imposes a
local minimum length scale to the solutions. More
precisely, using this sensitivity filter, the width of
bars appearing in the resulting topologies could
not be smaller than 2
R
. This property is useful in
achieving mesh independency. By defining
R
as
a ratio of the actual length of the design domain,
the mesh dependence problem can be rectified.
Unlike the first two types of numerical insta-
bilities, the local minima problem is mostly due to
the use of gradient-based optimization algorithm
which can be trapped in local minima of usually
non-convex objective functions. On the other
v
Γ
0.
Note that the same algorithm can be used
to maximize any of the natural frequencies. To
maximize the
k
th eigenvalue, for example, one
needs to replace
λ
1
and
φ
1
by
λ
k
and
φ
k
respectively.
4.3. Numerical Instabilities
Most of the material distribution techniques,
including homogenization, SIMP and BESO
methods, are known to be prone to three major
numerical instabilities, namely checkerboard
problem, mesh dependency, and local minima
Search WWH ::
Custom Search