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141
IBRACON Structures and Materials Journal • 2013 • vol. 6 • nº 1
V. S. ALMEIDA
|
H. L. SIMONETTI
|
L. OLIVEIRA NETO
where
j
A
is the area,
0*
j
W
is the mean energy of deformation,
all for elements j, totalizing
m
elements,
0*
W
is the mean energy
of deformation of the structure, where “*” represents the point of
minimization or a basic feasible solution. In short, each iteration
path has become a new basic solution within a feasible region to
the linearized problem. In view of eq. [3], the element that has the
average energy of deformation close to the deformation energy of
the structure can be said to have its partial derivative equal to zero,
indicating that a stationary point has been reached.
3. Smoothing evolutionary structural
optimization (SESO)
A relaxation condition, a “soft-kill” procedure or Smoothing ESO, is
applied to the ESO method, in which the elements that should be
removed by the ESO criterion - following inequality [1a] - are ar-
ranged in
n
groups and allocated in order of increasing tensions
being weighted by a function
1 )( 0
≤ ≤
j
η
. Then, a defined p% of
these
n
groups is removed, and the groups that contain the ele-
ments with the least stress (
Γ
LS
domain), and (1-p%) are returned
to the structure, the
Γ
GS
domain. This removal and return of ele-
ments to the structure is performed by a function, either linear or
hyperbolic, that weights the rate
max
vm vm
e
s
s
within the
Γ
domain;
that is, it allows the high-stress elements (closest to
max
vm vm
e
s
s
but
fulfilling the ESO constraint in the
Γ
GS
domain) to be reintegrated
into the structure at each iteration path.
The minimization of the objective function is achieved by finding
a stationary region, and this is achieved when all the terms have
the value zero gradient vector, that is, if the average energy of
deformation of the element j
0*
(
)
j
W
tends to the average energy
of strain of the structure
0*
(
)
j
W
, the term
(
)
*0
*0
1
WW
j
−
in eq. [3]
tends to zero. Thus, each term is understood to represent a vector
element of the discretized structure. Tanskanem [17] also highlights
the fact that the removal of an element can affect the convergence
of the optimization procedure, because the criteria for withdrawal in
the ESO is indicated by the attendance of inequality [1a], which can
often be extreme, since there are elements that are left in the vicinity
of this condition, which are numerically excluded, but they have strain
energy equivalent to the structure; the gradient is thus also zero, but
it should compose the gradient vector that defines the stationary point
cited by Tanskanem [17]. Thus, the removal of an element drastically
may unduly affect the way the optimum; one way to correct this devia-
tion would be the possibility of inserting the element in the structure
again. In this sense, a variant of the ESO, the BESO - Bidirectional
Evolutionary Structural Optimization stands out, Querin [18]. SESO
comes from this mathematically consistent philosophy, weighting the
Young’s modulus (E), making the strain energy of the element in-
creases, tending to the strain energy of the structure then the gradient
tends to zero and the direction of the minimum is restored.
The elements near the limit maximum stress are maintained in the
structure, defining the procedure for no “hard-kill” withdrawal, but
so smoothing. The “soft kill” procedure used in the SESO tech-
nique can be interpreted as follows:
(4)
ﮭ
i
0
D ( )
( )
0
i
j
GS
LS
D if j
j
D if j
if j
where
s
vm
and
s
vm
are, respectively, the principal Von Mises stress
of element “e” and the maximum stress effective structure in itera-
tion “i” , RR is called the rejection ratio, which is an input datum that
is updated using the evolutionary rate: ER.
In each iteration path, the elements that satisfy the inequality [1a]
are removed from the structure, Figure [1]. The RR factor is ap-
plied to control the removal process in the structure (0.0 ≤ RR ≤
1.0). The same cycle of removing elements by inequality [1a] is re-
peated until there are no more elements that satisfy the inequality
[1a]. When this occurs, a steady state is reached. The evolutionary
process is defined by adding the ER. Thus, a new cycle begins,
until there are no more elements to be eliminated with this new
RR value. The RR factor will be updated according to equation
[1b], to obtain an optimal configuration, achieved by controlling a
performance parameter, called the performance index (PI). This
procedure is known as “hard kill” and can be interpreted as follows:
(2)
ﮭ
if j
( )
0 if j
D
D j
�
where
( )
D j
is the constitutive matrix of point
Ω∈
j
,
0
D
is the initial constitutive matrix,
i
i
Γ+Γ=Ω
is the structure
domain,
{ / ( /
( )) RR }
MAX
i
e
VM
i
s s
Γ = Ω
Ω ≥
is the amount of ele-
ments that will not be removed from the structure (solid), and
{ / ( /
( ))<RR }
MAX
i
i
e
VM
i
s s
Γ = Ω − Γ = Ω
Ω
is the set of elements
that are removed from the structure (creation of void), in the i-th iteration.
In the removal heuristic via ESO, when the element is removed
from the design domain during the evolutionary process, elements
that remain in the structure represent a basic solution: the terms of
the gradient vector are null. As reported by Tanskanen [17], the
objective function is written in terms of thickness and it can be
minimized such as:
ln[
({ })] ln[ ({ })]
ext
f
W t
V t
=
+
, with
ext
W
being
the work of external forces and V is the total volume. The partial
derivative of the objective function to thickness
t
of element j is
given by:
(3)
m j
where
W
W
V
A
t
f
j
j
j
,..., 2,1
1
*0
*0
*
*
Figure 1 – Evolutionary algorithm: based
on element removal from the grid