Reliability-based Topology Optimization of Continuum Structure

Zhaokun Li

， Jinying Chen

1 b

，Lijuan Shi

and Huamei Bian

1 d

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

minmin =⇔SE

(1)

Where

is the displacement vector and

is the

sum of all the external force vectors.

Using the SIMP approach, the relative density

of material in each element is a design variable. The

-vector containing the design variables is denoted

. 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:

()

min

min ( )

subj ect t o

ee e

fx xuku

fxv

∗

⎧

⎪

⎨

=⋅ = ≤

⎪

<≤≤

⎪

⎩

∑

UKU

FKU

（2）

where

-vector containing the element

volume,

is the upper bound on material volume

and

min

is an

-vector with the minimum values

of the densities,

is the volume. K is the tangent

stiffness matrix.

The constitutive tensor for element

with

intermediate densities

ijkl

can be expressed as:

()

ijkl

x CC =

(3)

The second Piola-Kirchhoff stresses are

calculated:

()

klijkl

x εCs

（4）

Where

is stress tensor,

is Green-

Lagrange strain tensor,

is the penalization factor

(typically

3=p

ijkl

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

, the random variables

, and the

normalized variables

. 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

iir

=≤≤yx

（5）

Where

[]

iir

PGP ≤≤ 0),( yx

and

are the

probability of constraint violation and the allowable

probability violation ,respectively, and

),( yxG

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

and resisting force

and active states according to a limit state function

),( yxG

can be expressed as [18]:

⎪

⎩

⎪

⎨

⎧

=−=

>−=

<−=

statelimit0),(),(),(

statesafety0),(),(),(

statefailure0),(),(),(

yxyxyx

SRG

(6)

The uncertainties of

and

which is mostly

not statistical information are modeled by a vector of

stochastic physical variables. No matter what

regularities of distribution of

and

, distribution

characteristics generally can be described using

mean values and standard deviation. Defining mean

values and standard deviation of

and

are

、

and

、

, supposed that

and

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

is then calculated by

[]

))((0),(

SRRSf

σσμμΦGPP +−=≤= yx

(9)

Reliability index

is defined as:

)(

SRSRZZ

σσμμσμβ +−==

(10)

The relation between the probability of failure

and reliability index is expressed as:

)()(1)(

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

−=

(12)

where

)(βΦ

is increasing with the enlargement

, and

is corresponding to

. Since reliability

index

is corresponding to the probability of failure

, we may solve the reliability level by

introducing reliability index

.according to the

formula (11) (12) and (5), the reliability constraint

can be transformed into:

0-0)(-)(

≤⇒≥−−

βββΦβ

(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

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

to standard normal

is given by

)( ys T=

, or )(

−

= T (14)

where

(.)T

is generally a non-linear mapping

that depends on the type of random distribution of

In the case of a normal distribution, normal random

variables

can be transformed into a standard

normal random variable

jjjj

σμys )( −=

(15)

where

is the

-th random variable, with

mean value

and standard-deviation

and

the number of selected random variables. The

reliability index

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

sss

dβ

(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:

ββtosubject

dβ

≥

)(

)()(min)(min

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:

min

min ( )

()

(,,) 0

(,,)

subj ect t o

ββ

⎧

⎪

≥

⎪

⎨

≤

⎪

<≤≤

⎪

⎩

Rx y s

Vx y s V

(18)

where

and

are the reliability index of the

system and the target reliability index, respectively,

and

is the normalized variable.

In the evaluation of the reliability index, the

derivative of

with respect to normalized variables

can be written as [18]:

∂

∑

2)(

(19)

The resulting

of problem in (19) will be used

to evaluate the random variable

jj j

.using (15) with the standard deviations given by

μσ 1.0=

Reliability analysis design proceed:

First, determine design variables and random

variables. The design variables

based on variable

density method are relative density

x in finite

element, considering the uncertainty of geometric

size and action load, random variables

are action

load

, discrete units nelx and nely in the horizontal

and vertical direction and volumetric ratio

Secondly, using the above random variable

the initial value and constructing the mean vector

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 j

∂∂ =ΔΔ = +Δ − Δ

μμμμμ

（）（

（20）

where

001.

Δ=

μμ

Thirdly, using the formula (17) to calculate the

standardized variable

under the constraint of the

reliability index, the standardized variable

is used

to modify the random variable

by using the

formula (15). The revised random variable

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

Input force(KN) 1

Poisson ratio

0.3

Young modulu E/Gpa 1

V/V

0.5

Table 2. Random variable parameters.

Type Random variables Mean value

normalized value

Modified value

Geometry

dimension

Horizon size(

μm

) 60 2 72

Vertical size (

μm

) 20 -2 16

Volume ratio 0.5 -2.003 0.4

applied loa

load (K

) 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

n domain size S/

μm

Input force(KN) 1

Poisson ratio

0.3

Youn

modulu E/Gpa 1

V/V0 0.4

Table 4. Random variable parameters.

pe Random variables Mean value

normalized value

Modified value

Geometry

dimension

Horizon size(

μm

) 32 1.56 37

Vertical size (μ

) 20 -1.5 17

Volume ratio 0.4 -1.48 0.34

applied loa

load (K

) 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

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；

（3）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.

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) and（bgzyky201723z），the supports are

greatly acknowledged.

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