Reliability-based Topology Optimization of Continuum Structure
Zhaokun Li
1a
Jinying Chen
1 b
Lijuan Shi
1c
and Huamei Bian
1 d
1
Engineering of mechanical and electrical department, Beijing polytechnic college
Beijing,
100042, China
Keywords: Topology optimization; reliability analysis; continuum structure.
Abstract: The reliability-based topology optimization method of continuous structures is investigated considering
structural applied loads and the geometry description. Firstly, based on the solid isotropic material with
penalization approach, the deterministic mathematical model of topology optimization is developed, in
which minimization of the compliance is taken as objective function and the volume is taken as constraint
function; Secondly, using the relation between failure probability and reliability index, the mathematical
model of reliability topology optimization based on the reliability index constraint is established. Reliability
index probabilistic constraint problem considering uncertainties is solved using first order reliability
method, in which the reliability index constraints are transformed into random variables, and then the
modified random variables are used as deterministic variables to carry on the deterministic topology
optimization; Finally, several numerical examples are simulated to show that reliability-based topology
optimization yields structures that are more reliable than those produced by deterministic topology
optimization and also certificate the validity of the proposed method..
1 INTRODUCTION
In recent years, the research on the topology
optimization has been an attractive research area and
made great progress. Topological optimization is a
design method based on structural optimization,
which is under the action of some external forces
and constraints, seeking the optimal structure
arrangement. At present, the depth and breadth of
topological optimization research of continuous
structure has been extended from single object to
multi-objective function, from single physical field
to multi-physical field design, from material and
geometric linear problem to nonlinear problem, from
static topology to dynamic topology [1-3].
But so far, papers on the topology optimization
of continuum structure presented mostly deal with
the solution without taking into account the effects
of uncertainties. Actually, because continuum
structures may be subject to inherent uncertainties
such as external loading, material properties, and
manufacturing quality, the prototypes or
manufactured products may not satisfy the necessary
performance requirements.
Therefore, in order to reduce the mechanism
performance degradation caused by the uncertainty
in manufacturing process, these uncertainties must
be considered in topology optimization. The
reliability optimization design accounting for
uncertainties has become a research hot topic in the
field of structural design. At present, structural
reliability optimization is widely used in the field of
dimension and shape optimization [5]. However, in
the field of topology optimization of continuum
structures, there are few references to the application
of reliability optimization methods. [5-8] the
research on reliability-based topology optimization
for continuum structure is currently processed in the
initial stage and many problems need further
investigate.
In this paper, a new reliability-based topology
optimization methodology for continuum structure is
presented. Firstly, Based on the solid isotropic
material with penalization approach, the
deterministic mathematical model of topology
optimization is developed, in which minimization of
the compliance is taken as objective function and the
volume is taken as constraint function; Secondly,
using the relation between failure probability and
reliability index, the mathematical model of
reliability topology optimization based on the
reliability index constraint is established. Reliability
index probabilistic constraint problem considering
uncertainties is solved using first order reliability
method, in which the reliability index constraints are
transformed into random variables, and then the
modified random variables are used as deterministic
variables to carry on the deterministic topology
optimization; Finally, several numerical examples
are simulated to certificate the validity of the
proposed method.
2 THE DETERMINISTIC
TOPOLOGY OPTIMIZATION
OF CONTINUUM STRUCTURE
The need for the continuum structure to be stiff
enough to withstand the external load is captured as
the stiffness requirement. Maximizing the stiffness
requirement is determined by minimizing Strain
Energy
)(SE
which is equivalent to minimizing the
mean compliance
)(P
of the structures and the
formulation is defined as:
UFP
T
minmin =SE
(1)
Where
U
is the displacement vector and
F
is the
sum of all the external force vectors.
Using the SIMP approach, the relative density
e
x
of material in each element is a design variable. The
N
-vector containing the design variables is denoted
x
. The overall topology optimization solving the
problem of distributing a limited amount of material
in the design domain such that the objective function
is minimized and the volume and input displacement
is constrained can be expressed as:
()
0
1
1
01
0
min
min ( )
subj ect t o
V
e
N
pT
ee e
x
e
N
ee
e
e
fx xuku
fxv
xx
=
=
==
=
=⋅ =
<≤
T
UKU
FKU
VV
2
where
V
is
N
-vector containing the element
volume,
*
V
is the upper bound on material volume
and
min
x
is an
N
-vector with the minimum values
of the densities,
V
0
is the volume. K is the tangent
stiffness matrix.
The constitutive tensor for element
e
with
intermediate densities
e
ijkl
C
can be expressed as:
()
0
e
e
ijkl
P
ijkl
x CC =
(3)
The second Piola-Kirchhoff stresses are
calculated:
()
klijkl
P
ij
x εCs
0
e
=
4
Where
ij
s
is stress tensor,
kl
ε
is Green-
Lagrange strain tensor,
p
is the penalization factor
(typically
3=p
),
0
ijkl
C
is the constitutive tensor for
solid isotropic material.
3 RELIABILITY-BASED
TOPOLOGY OPTIMIZATION
3.1 Reliability Analysis
In reliability-based topology optimization, three
kinds of variables will be distinguished [18]: the
design variables
x
, the random variables
y
, and the
normalized variables
s
. In contrast to the
deterministic optimization, probabilistic mechanism
design optimization can be characterized by the
probabilistic constraints. The random variables as
well as the design variables are involved in defining
the problem of probabilistic optimization [19]:
[]
),...2,1(0),(:
)(:min
miPGPtosubject
f
iir
=yx
x
5
Where
[]
iir
PGP 0),( yx
and
i
P
are the
probability of constraint violation and the allowable
probability violation ,respectively, and
),( yxG
is
defined as a limit state function . Safety is the state
in which the structure is able to fulfill all the
functioning requirements. The safety of components
depends on external loading
S
and resisting force
R
,
and active states according to a limit state function
),( yxG
can be expressed as [18]:
==
>=
<=
statelimit0),(),(),(
statesafety0),(),(),(
statefailure0),(),(),(
yxyxyx
yxyxyx
yxyxyx
SRG
SRG
SRG
(6)
The uncertainties of
S
and
R
which is mostly
not statistical information are modeled by a vector of
stochastic physical variables. No matter what
regularities of distribution of
S
and
R
, distribution
characteristics generally can be described using
mean values and standard deviation. Defining mean
values and standard deviation of
S
and
R
are
R
μ
S
μ
and
2
R
σ
2
S
σ
, supposed that
S
and
R
are
uncorrelated independent random variables then
mean value and standard deviation of a limit state
function :
SRZ
μμμ =
7
222
σσσ
SRZ
+=
(8)
The failure probability
f
P
is then calculated by
[]
))((0),(
22
SRRSf
σσμμΦGPP +== yx
(9)
Reliability index
β
is defined as:
22
)(
SRSRZZ
σσμμσμβ +==
(10)
The relation between the probability of failure
and reliability index is expressed as:
)()(1)(
1
ff
PΦββΦβΦP
===
(11)
Where
)(βΦ
is the standard normal cumulative
distribution function.
That indicates from the formula (11) that the
failure rate and reliability index are corresponding.
The permissible value of the probability of failure
may thus be expressed as:
)( βΦP
f
=
(12)
where
)(βΦ
is increasing with the enlargement
of
β
, and
β
is corresponding to
f
P
. Since reliability
index
β
is corresponding to the probability of failure
f
P
, we may solve the reliability level by
introducing reliability index
β
.according to the
formula (11) (12) and (5), the reliability constraint
can be transformed into:
0-0)(-)(
tt
βββΦβ
Φ
(13)
Therefore, we can solve the reliability index and
then solve the reliability. First order reliability
method is used.
3.2 First Order Reliability Method
The first order reliability method is developed in
which the probabilistic constraints are stated in
terms of the reliability index as a measure of the
probabilistic safety. The reliability index
β
was
introduced by Hasofer and Lind (1974), who
proposed working in the space of standard
independent Gaussian variables instead of the space
of physical variables. The transformation from the
random variable
y
to standard normal
s
is given by
)( ys T=
, or )(
1
sy
= T (14)
where
(.)T
is generally a non-linear mapping
that depends on the type of random distribution of
y
.
In the case of a normal distribution, normal random
variables
j
y
can be transformed into a standard
normal random variable
j
s
by
jjjj
σμys )( =
(15)
where
j
y
is the
j
-th random variable, with
mean value
j
μ
and standard-deviation
j
σ
and
j
is
the number of selected random variables. The
reliability index
β
is defined as the minimum
distance from the origin in the standard normal
space to the limit state surface and the calculation of
the reliability index can be realized by the following
form [21]:
0),(:tosubject
)()(min
=
==
sx
sss
H
dβ
T
(16)
where
0),( =sxH
is the limit state function in
the standard space. According to the structural
reliability index, the normal random variables are
solved as follows:
t
T
ββtosubject
dβ
==
)(
)()(min)(min
s
ssss
(17)
The formula (17) directly reflects the meaning of
the structural reliability index and transforms the
reliability constraint into a correction of random
variables.
3.3 Formulation
When probabilistic constraints accounting for the
randomness of the applied loads and the description
of the geometry are estimated in terms of the
reliability index, the reliability based topology
optimization may be expressed as:
0
min
min ( )
()
(,,) 0
(,,)
01
e
x
t
e
f
subj ect t o
f
xx
ββ
=
<≤
x
s
Rx y s
Vx y s V
(18)
where
β
and
t
β
are the reliability index of the
system and the target reliability index, respectively,
and
s
is the normalized variable.
In the evaluation of the reliability index, the
derivative of
β
with respect to normalized variables
s
can be written as [18]:
β
β
j
jj
j
s
ss
s
==
2)(
2
1
2
1
2
(19)
The resulting
s
of problem in (19) will be used
to evaluate the random variable
y
:
j
jj j
=+
μ
y
σ
s
.using (15) with the standard deviations given by
jj
μσ 1.0=
.
Reliability analysis design proceed:
First, determine design variables and random
variables. The design variables
x
based on variable
density method are relative density
e
x in finite
element, considering the uncertainty of geometric
size and action load, random variables
y
are action
load
F
, discrete units nelx and nely in the horizontal
and vertical direction and volumetric ratio
f
.
Secondly, using the above random variable
y
as
the initial value and constructing the mean vector
μ
,
the influence of the mean on the objective function
is positive and negative. The least squares method is
used to analyze the sensitivity of the objective
function, considering the uncertainty of the load and
geometric dimensions:
fff f)
jj
jj j
j
∂∂ =ΔΔ = +Δ Δ
μμμμμ
μ
()
20
where
001.
j
j
Δ=
μμ
.
Thirdly, using the formula (17) to calculate the
standardized variable
s
under the constraint of the
reliability index, the standardized variable
s
is used
to modify the random variable
y
by using the
formula (15). The revised random variable
y
is the
known quantity;
Finally, the deterministic topology optimization
module is called.
Thus the reliability analysis and topology
optimization are composed of two independent
modules,
Namely, the correction of random variables and
the deterministic topology optimization. The result
depends on the given reliability index. In this
method, the influence of reliability constraint on the
optimization of the mechanism is transformed into
the range of random variable due to reliability
constraint, avoiding the cumbersome reliability
analysis in the process of topology optimization. The
optimization problem is solved using the MMA
method proposed by Svanberg [10]. Mesh-
independency scheme is used to circumvent the
problem of checkerboard patterns and mesh-
dependencies proposed by Diaz and Sigmund [9-10].
4 NUMERICAL EXAMPLES
4.1 Example 1
The first example considers a beam structure.
Figure 1 shows the half symmetric design domain.
The dimension of the design domain, the material
properties, and the input parameters for the
optimization program are shown in Table 1.
The resulting optimal topology principally
depends on the reliability index value. In the case of
the target reliability index
4=β
modified
parameters are shown in Table 2.From intermediate
results during reliability analysis in Table 2, it is
noted that volume ratio is modified from 0.5to 0.4
which reduces manufacture cost and the applied load
is changed into 1.2KN more than the initial value
1KNwhich indicates that reliability based topology
mechanism may bear greater external force than the
deterministic topology mechanism. Afterwards,
topology optimization is implemented to
demonstrate the global system performances as
below.
Fig.1 beam structure. (Left) design domain and boundary conditions. (Right) Equivalent model.
Table 1. Input parameters for topology optimization.
Variable name Setting value
Design domain size S/μm 60
×
20
Input force(KN) 1
Poisson ratio
0.3
Young modulu E/Gpa 1
V/V
0
0.5
Table 2. Random variable parameters.
Type Random variables Mean value
μ
normalized value
s
Modified value
y
Geometry
dimension
Horizon size(
μm
) 60 2 72
Vertical size (
μm
) 20 -2 16
Volume ratio 0.5 -2.003 0.4
applied loa
d
load (K
N
) 1 2.003 1.2
The layouts are obtained from the topology
optimization neglecting all uncertainties and
considering uncertainties respectively shown in
Table5.From topology results, it can be seen that the
main difference between considering uncertainties
and neglecting uncertainties in this example is to
more properly redistribute the arms, which
Considering uncertainties has additional arms, which
obviously improve the reliabilities of the
mechanisms.
4.2 Example 2
Fig.2 design domain and boundary conditions of
cantilevered beam.
The second example is designing a cantilevered
beam. The design domain is sketched in Figure2.
The dimension of the design domain, the material
properties, and the other initial values are all list in
Table 3.
Table 3. Input parameters for topology optimization.
Variable name Setting value
Desi
g
n domain size S/
μm
32
×
20
Input force(KN) 1
Poisson ratio
ν
0.3
Youn
g
modulu E/Gpa 1
V/V0 0.4
Table 4. Random variable parameters.
T
y
pe Random variables Mean value
μ
normalized value
s
Modified value
y
Geometry
dimension
Horizon size(
μm
) 32 1.56 37
Vertical size (μ
m
) 20 -1.5 17
Volume ratio 0.4 -1.48 0.34
applied loa
d
load (K
N
) 1 1.48 1.15
Table5. Topological diagrams.
The resulting optimal topology principally
depends on the reliability index value. In the case of
the target reliability index
3
β
=
modified
parameters are shown in Table3.From intermediate
results during reliability analysis in Table 4, it is
noted that volume ratio is modified from 0.4 to 0.34
which reduces manufacture cost and the applied load
is changed into 1KN more than the initial value
1.15KNwhich indicates that reliability based
topology mechanism may bear greater external force
than the deterministic topology mechanism.
Afterwards, Topology optimization is implemented
to demonstrate the global system performances as
below. From topology results in Table 5 , it can be
seen that the main difference between considering
uncertainties and neglecting uncertainties in this
example is to more properly redistribute the arms,
which Considering uncertainties has additional arms,
which obviously improve the reliabilities of the
mechanisms.
5 CONCLUSIONS
1 The reliability-based topology optimization
method is investigated considering structural applied
loads and the geometry description. Using the
relation between failure probability and reliability
index, the reliability mathematical model based on
the reliability index constraint is established
2 Reliability index probabilistic constraint
problem is solved using first order reliability method,
in which the reliability index constraints are
transformed into random variables, and then the
modified random variables are used as deterministic
variables to carry on the deterministic topology
optimization
3Several numerical examples are simulated
to show that reliability-based topology optimization
yields structures that are more reliable than those
produced by deterministic topology optimization.
That is, the use of reliability-based topology can
improve the performance. Meanwhile, numerical
examples also certificate the validity of the proposed
method.
ACKNOWLEDGEMENTS
This research was supported by General scientific
research project of Beijing Municipal Education
Committee KM201810853002 ),Key research
project of Beijing Polytechnic College (bgzyky
201724z) andbgzyky201723z),the supports are
greatly acknowledged.
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