Nguyen Enhancing Single-loop SRBTO AIAA

American Institute of Aeronautics and Astronautics

1

Enhancing Single-loop Approach for Component and

System Reliability-Based Topology Optimization

Tam H. Nguyen1, Junho Song

2, and Glaucio H. Paulino

3

University of Illinois at Urbana-Champaign, Urbana, IL, 61801

This paper introduces new single-loop algorithms developed for component and system

reliability-based design and topology optimization. A single-loop reliability-based

design/topology optimization (RBDO/RBTO) algorithm replaces the inner-loop iterations

that evaluate probabilistic constraints by a non-iterative approximation. The proposed

single-loop algorithms account for statistical dependence between the limit-states by using a

matrix-based system reliability (MSR) method to compute the system failure probability and

its parameter sensitivities. The SRBTO/MSR approach is applicable to general system

events including series, parallel, cut-set and link-set systems and provides the gradients of

the system failure probability to facilitate gradient-based optimization. In most RBTO

applications, probabilistic constraints are evaluated by use of the first-order reliability

method (FORM) for efficiency. In order to improve the accuracy of the reliability

calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new

single-loop RBDO scheme utilizing the second-order reliability method (SORM). Moreover,

in order to overcome challenges in applying the proposed algorithm to computationally

demanding topology optimization problems, we utilize the multiresolution topology

optimization (MTOP) method, which achieves computational efficiency in topology

optimization by assigning different levels of resolutions to three meshes representing finite

elements, design variables, and material density distribution, respectively. The paper

provides a numerical example of three-dimensional topology optimization to demonstrate

the proposed CRBTO and SRBTO algorithms and applications. Monte Carlo simulations

are also performed to verify the accuracy of the failure probabilities computed by the

proposed approach.

Nomenclature

d = vector of design variables

C = compliance

volfrac = volume fraction

i = density element i

dn = design variable n

rmin = minimum length scale

c = “event” vector

p = “probability” vector ˆ

iα = negative normalized gradient vector

i = reliability index

βt

i = target reliability index

Q4/n25 = MTOP Q4 element with 25 density elements and 25 design variables

B8/n125 = MTOP B8 element with 125 density elements and 125 design variables

1 Doctoral Student, Department of Civil and Environmental Engineering, University of Illinois, Urbana, IL, 61801,

(AIAA Student Member) 2 Assistant Professor, Department of Civil and Environmental Engineering, University of Illinois, Urbana, IL, 61801

3 Donald Biggar Willett Professor in Engineering, Department of Civil and Environmental Engineering, University

of Illinois, Urbana, IL, 61801, (AIAA Member)


2

I. Introduction

tructural design/topology optimization aims to obtain an optimal set of structural design variables or topology

that fulfills the objective(s) of a structural design while satisfying given design constraints. In spite of active

research efforts and significant technological advances in the past decades1,2

, there still exist impediments to

adopting such optimization techniques in design practice. In particular, most of the efforts have been conducted in a

deterministic manner although uncertainties in loads or material properties may result in significant likelihood of

violating design constraints. In this study, this approach is referred to as deterministic topology optimization (DTO).

Recently, active research has been performed to achieve optimal topologies with acceptable probability of satisfying

given constraints. This approach is often termed as reliability-based topology optimization (RBTO). It is noted that

most research efforts on RBTO have been focused on satisfying the probabilistic constraint given for each failure

mode. In the current study, this approach is referred to as component reliability-based topology optimization

(CRBTO). A generic formulation for CRBTO problems is given as follows.

min (ρ( ; ))

. . [ (ρ( ; ), ) 0] , 1,...,

(ρ( ; ))

ti i

d

L U

f

s t P g P i n

dψ d

ψ d X

K ψ d u f

d d d

(1)

where dk is the vector of deterministic design variables; ( ;d) is the material density at the position

2 or

3 that is generally determined by a projection function fp(.) and the design variables, i.e. ( ;d)=fp(d); f(.) is the

objective function that often describes the volume, compliance or displacement of the structure; Xm is the vector

of random variables representing the uncertainties in the problem; gi(.), i=1,..,n is the i-th “limit-state function” that

indicates violating a design constraint given in terms of volume, displacement, or compliance by its negative sign,

i.e. gi(.) 0; t

iP is the constraint on the probability of the i-th limit state; K, ud and f respectively denote the stiffness

matrix, displacement vector, and load vector in the equilibrium condition; and dL and d

U are the lower and upper

bounds on d, respectively. For simplicity, the equilibrium condition and the bounds on the design variables will be

omitted in the following RBTO formulations of the paper. The probability constraint in Eq. (1) is described in terms

of either the reliability index (RIA)3 or the performance function, i.e. the

t

iP - quantile of the limit-state function

(PMA)4, which is obtained by use of a structural reliability analysis method such as the first-order reliability method

(FORM).

While most research efforts in the literature have been focused on CRBTO, in certain circumstances, the

probabilistic constraint should be given on a system failure event, i.e. a logical (or Boolean) function of multiple

failure modes. For example, the failure of a topology design can be defined as an event that at least one of the

potential failure modes occurs. This approach is termed as system reliability-based topology optimization (SRBTO).

SRBTO introduces additional complexity to reliability calculations especially when component events are

statistically dependent, or when the system event is not a series (i.e. union of events) or parallel system (i.e.

intersection of events). A generic formulation for SRBTO is as follows.

min (ρ( ; ))

. . ( ) (ρ( ; ), ) 0 , 1,...,k

t

sys i sys

k i C

f

s t P E P g P i n

dψ d

ψ d X (2)

where P(Esys) is the probability of the system failure event; Ck is the index set of the components (limit-states) in the

k-th cut-set; and t

sysP is the constraint on the system failure probability. Any type of system event may be considered

in SRBTO but, for illustration purpose, Eq. (2) shows a cut-set system formulation that can represent series, parallel,

and cut-set systems. A limited number of studies have been performed on SRBTO5 because calculation of system

probability and its parameter sensitivities introduces additional complexity to the topology optimization that already

requires high computational cost.

In the research efforts to apply the single-loop RBTO approaches to a wide range of problems, we identified the

needs to enhance the accuracy of the failure probabilities computed by the first-order reliability method, and the

applicability of the approach to large-size or three-dimensional topology optimization problems.

S


3

II. SORM-Based Component Reliability-Based Topology Optimization

Let us consider the following formulation of a single-loop approach6 for CRBTO problems in Eq. (1):

T,

min (ρ( ; ))

. . g g ρ( ; ), ( ) 0, =1,...,

(ρ( ; ), ( ))ˆ ˆwhere β ( ) and

(ρ( ; ), ( ))

ti

i

ti iP

t t t t ii i i i

i

f

s t i n

g

g

d

xx u

x u u

ψ d

ψ d x u

ψ d x uu α α J

ψ d x u

(3)

At each step of the design iterations, iu is

obtained by use of the KKT condition6 using

the target reliability index βt

i . Then, an

approximate most probable point (MPP) t

iu is

obtained by scaling the negative normalized

gradient vector at iu u , ˆ t

iα by βt

i as shown

in Fig. 1. The validity of this approximate

MPP is checked at the final step of the design

iterations. As this single-loop CRBTO uses

the first-order reliability method (FORM), the

reliability calculations can be inaccurate for

highly nonlinear limit-states7.

We thus modify this procedure to improve

the accuracy of the above single-loop

approach. Instead of finding the approximate

MPP on the surface of the sphere with the

fixed radius βt

i, the radius is updated at each

step of the design iterations by the ratio of βt

i

to the reliability index by a more accurate

reliability method such as the second-order

reliability method (SORM). The formulation of the proposed scheme is as follows.

( ) T,

( )

min (ρ( ; ))

. . (ρ( ; ), ( )) 0 1,...,

(ρ( ; ), ( ))ˆ ˆat the -th step: β ( ) and

(ρ( ; ), ( ))

1

ti

i

ti iP

t t k t t ii i i i

i

ti

t k ti i

f

s t g g i n

gk

g

k

d

xx u

x u u

ψ d

ψ d x u

ψ d x uu α α J

ψ d x u

( 1)

( 1)( )otherwiset k

it k SORMi

(4)

where ( )βt k

i is the radius used for the scaling at the k-th step of the iterations; and ( 1)( )βt k SORM

iis the reliability index

by SORM, which is obtained based on the principal curvatures around iu u at the (k─1)th step. The convergence

of the radius ( )βt k

i indicates that the reliability index by SORM approaches the target reliability index βt

i. Compared

to similar techniques in the literature8,9,10

, our study focuses on implementation of the idea of updating into the

single-loop RBTO based on FORM. In this study, the improved CRBTO by SORM in Eq. (4) is termed as “SORM-

based CRBTO.”

Figure 1. Approximation in the single-loop algorithm.

g , ( ) 0d X U

KKT

MPP

1U

2U

iU

ˆ t

iα

ˆβt t

i iα

βt

i

t

iU

*

iU


4

III. System Reliability-Based Topology Optimization using MSR Method

When the limit-states in an SRBTO problem are assumed to be statistically independent5, the system probability can

be computed by algebraic calculations of the probabilities of the individual limit-states. For example, let us describe

the failure probability of a series system event by use of the inclusion-exclusion formula

1

1

1 2

1 1 11

( ) ( ) ( 1) ( )n n n n

n

i i i j n

i i j ii

P E P E P E E P E E E (5)

where Ei denotes the failure event of the i-th limit-state. When these events are statistically independent of each

other, each joint probability in Eq. (5) can be computed by the product of the component probabilities, e.g.

P(E1E2E3)=P(E1)P(E2)P(E3). If there exists significant statistical dependence between limit-states due to shared or

correlated random variables, one needs to use system reliability analysis methods that can account for statistical

dependence in computing the system failure probability. It is desirable to obtain parameter sensitivities of the system

failure probability to facilitate the use of gradient-based optimization algorithm. However, computation of the

parameter sensitivities of a system failure probability is challenging when component events are statistically

dependent or the system event is not a series or parallel system.

The authors have recently applied the matrix-based system reliability (MSR) method to general reliability-based

design optimization problems11

. The current study aims to use the MSR method for topology optimization. The

MSR method12

computes the probability of a general system including series, parallel, cut-set and link-set system

and its parameter sensitivities by systematic matrix calculations. Consider a system event whose i-th component,

i=1,…,n has two distinct states, e.g., failure or survival. Then, the sample space can be subdivided into N=2n

mutually exclusive and collectively exhaustive (MECE) events, denoted by ej, j=1,..,N. Then, any system event can

be represented by an “event” vector c whose j-th element is 1 if ej belongs to the system event and 0 otherwise. Let

pj=P(ej), j=1,…,N denote the probability of ej. Due to the ej’s mutual exclusiveness, the probability of the system

event Esys, i.e. P(Esys) is the sum of the probabilities of ej’s that belong to the system event. Therefore, the system

probability is computed by the inner product of the two vectors, that is

T

T

(independent components)

( )( ) ( ) (dependent components)sysP E

f dS

s

c p

c p s s s (6)

where p is the “probability” vector that contains pj’s, j=1,…,N; S denotes the random variables identified as the

sources of statistical dependence between components, termed as common source random variables (CSRVs)12

. For

a given outcome of CSRVs, the component events are conditionally independent of each other, which allows us to

use the efficient procedure to construct the probability vector that is applicable to independent components; p(s)

denotes the probability vector constructed by use of the conditional failure probabilities of the limit-states given S=s,

i.e. Pi(s)=P(Ei|S=s) instead of Pi=P(Ei); and fs(s) is the joint probability density function (PDF) of S. Song and

Kang12

developed matrix-based procedures to construct the vectors c and p efficiently; to compute conditional

probabilities and component importance measures; and to evaluate parameter sensitivities of the system failure

probability.

A single-loop SRBTO approach13

employing the MSR method is formulated as follows.

,

T

T

T

min (ρ( ; ))

. . (ρ( ; ), ( )) 0 1,...,

( ) ( ) dependent ( ; )

independent

ˆ ˆwhere β ( ) and

t

ti

ti iP

t tsyst

syst t

sys

t t ti i i i

f

s t g g i n

f d PP E

P

d P

Ss

ψ d

ψ d x u

c p s s sP

c p

u α α ,

(ρ( ; ), ( ))

(ρ( ; ), ( ))i

t i

i

g

g

xx u

x u u

ψ d x uJ

ψ d x u

(7)


5

IV. SORM-Based System Reliability-Based Topology Optimization

The single-loop SRBTO/MSR approach in Eq. (7) is improved by enhancing the accuracy of component reliability

analysis results that are used for the system reliability analyses. The formulation of the SORM-based SRBTO/MSR

is as follows.

,

T ( )

( )

T ( )

min (ρ( ; ))

. . (ρ( ; ), ( )) 0 1,...,

( ) ( ) dependent ( ; )

independent

where

t

ti

ti iP

t SORM tsyst SORM

syst SORM t

sys

i

f

s t g g i n

f d PP E

P

d P

Ss

ψ d

ψ d x u

c p s s sP

c p

uT

,

(ρ( ; ), ( ))ˆ ˆβ ( ) and

(ρ( ; ), ( ))i

t t t t ii i i

i

g

g

xx u

x u u

ψ d x uα α J

ψ d x u

(8)

where ( )t SORM

P is the vector of the component failure probabilities by SORM at the approximate MPPs,

, 1,...,t

i i nu ; ( ) ( )t SORMp s and ( )t SORM

p respectively denote the conditional and marginal probability vectors

constructed by use of the SORM reliability indexes ( )t SORM

iinstead of t

i; and

iu is obtained by use of the KKT

condition using βt

i. The only change from Eq. (7) is that the probability vector is constructed by use of the SORM

reliability indexes instead of the FORM reliability indexes at the approximate MPPs.

V. RBTO using Multiresolution Topology Optimization

A main challenge in adopting topology optimization technique in design practice is its large computational cost.

Topology optimization often employs the material distribution method, which rasterizes the domain via the density

of pixels/voxels. This approach often requires a large number of design variables, especially in three-dimensional

applications. Most of the research efforts to overcome this challenge focus on finite element analysis since it

constitutes the dominant computational cost in topology optimization. For example, researchers make use of

powerful computing resources such as parallel computing14

or fast iterative solvers15

. These studies employ the same

resolution for the finite elements mesh and the design mesh during optimization process. The authors recently

developed a multiresolution topology optimization (MTOP) approach16

for handling large-scale problems with

relatively low computational costs. This approach uses distinct meshes for finite element, density and design

variables. In particular, the analysis is performed on a coarser finite element mesh, optimization is performed on a

fine design variable mesh, and element densities are defined on a fine mesh. This multiresolution approach enables

high resolution optimal designs with significantly reduced computational costs. This paper introduces a framework

to integrate the RBTO approach using MTOP to handle large-scale topology problem.

Topology optimization of a continuum structure aims to find the optimal distribution of the material densities in

a specific domain, which are considered as design variables. The basic formulation of “minimum compliance” (i.e.

maximum stiffness) topology optimization problem is expressed in the following discrete form using finite element

method:

Tmin (ρ( ; )) (ρ( ; ), )

. . (ρ( ; )) ρ( ; )

d d

s

f C

s t V dV V

dψ d ψ d u f u

ψ d ψ d (9)

where T( , ) dC u f u is the compliance of the continuum; ( )V is the total volume; and Vs is the prescribed volume

constraint. A desirable solution of topology optimization specifies the density at every point in the domain as either

0 (void) or 1 (solid). However, since it is impractical to perform such an integer optimization, the problem is relaxed

such that the density can have any value between 0 and 1. For example, in the Solid Isotropic Material with

Penalization (SIMP) approach17,18

, the constitutive matrix is parameterized using solid material density as follows


6

0( ) ρ( ; )p

D ψ ψ d D (10)

where 0D is the constitutive matrix of the material in the solid phase, corresponding to the density ( ) 1x ; and p is

the penalization parameter. To prevent singularity of the stiffness matrix, a small positive lower bound, e.g.

min=103, is placed on the density. Using the penalization parameter p>1, the intermediate density approaches either

0 (void) or 1(solid). In the conventional element-based approach, the density of each element is represented by one

value e and the global stiffness matrix K in Eq. (1) is expressed as

T

1 1

(ρ ) (ρ )el el

e

N N

e e e

e e

dK K B D B (11)

where Ke( e) is the stiffness matrix of the element e; B is the strain-displacement matrix of shape function

derivatives; and D( e) is the constitutive matrix.

Different from conventional element-based approaches which use the same mesh for finite element analysis and

design (Fig. 2a), the MTOP approach utilizes three different meshes: a relatively coarse finite element mesh to

perform the analysis, a fine design variable mesh to perform the optimization, and a fine density mesh to represent

material distribution and compute the stiffness matrices. The density mesh is finer than the finite element mesh so

that each finite element consists of a number of density elements (sub-elements). Within each density element, the

material density is assumed to be uniform. For example, Fig. 2a shows a conventional element-based approach Q4/U

element while Fig. 2b shows an MTOP Q4/n25 element where “n25” indicates that the number of density elements

per a Q4 element is 25 (coincides with the number of design variables).

The MTOP approach introduces a scheme to obtain the element stiffness matrix from corresponding density

elements. The stiffness matrix is computed by sum of the stiffness integration in each density element. The

integration of the stiffness in each density element is performed as standard finite element method. As a result, the

formulation for the stiffness matrix integration is expressed as follows.

0

0

1 1

T T T 0

0

1 11 1

T 0

0

ξ η (ρ ) (ρ )n n

ie

i

N Np i p

e i i i

i i

i

i

d Jd d Jd

Jd

K B DB B DB B D B I

I B D B

(12)

From these formulations, the sensitivities of the objective function and the constraint in Eq. (9) can be derived.

Figure 2. Element-based and MTOP elements: (a) Q4/U, and (b) MTOP Q4/n25

Displacement

Density

Design variable

(a) (b)


7

VI. Numerical Example

This numerical example is to demonstrate the improved accuracy of the proposed SORM-based RBTO methods.

The objective of optimization is to minimize the volume in a cube domain shown in Fig. 3a while satisfying

deterministic or probabilistic constraints on the compliances for multiple load cases. One corner is fixed in all three

directions while the other corners are restricted in the vertical direction only. The isotropic material is assumed to

have Young’s modulus of E0=1,000 and Poisson’s ratio of =0.3. A cube of edge length L=24 is divided into

12×12×12 B8/n125/d125 MTOP elements. The minimum length scale rmin=L/10, and penalization parameter p=3 are

employed. The structure is subjected to three random loads applied at five locations as shown in Fig. 3a. F1 denotes

the magnitude of the force at the center while F2 and F3 represent the loads at the midpoints between the center and

the four corner points of the top face. F1, F2 and F3 are assumed to be normal random variables with the mean values

100, 0 and 0, and with the standard deviations 10, 30 and 40, respectively.

Limit-states are defined on the compliances caused by two load combinations1 1 2( , )F FF and

2 1 3( , )F FF as

follows.

T( , ) ( , ) , 1,2t t

i i i i i i ig C C C iρ F ρ F u F (13)

where ( 120)t

iC is the constraint on the compliance; ( , )i iC ρ F is the compliance corresponding to the load case iF ;

and iF is the global force vector assembled based on the load case iF . The following three topology optimization

methods are investigated: (1) Deterministic Topology Optimization (DTO) using the mean values of the loads with

Figure 3. Topology optimization of the cube: (a) configuration, (b) DTO (volfrac=6.3%), (c) SORM-based

CRBTO ( (F1)=10, volfrac=24.4%), (d) SORM-based SRBTO ( (F1)=10, volfrac=22.3%), and (e) SORM-

based SRBTO ( (F1) = 20, volfrac = 23.9%)

(b)

(c) (d) (e)

F

1

L

F1F3F3F2

F2

L

L

(a)


8

deterministic constraints ( , ) 0i ig ρ F ; (2) CRBTO with probability constraints 1 2 0.02275t tP P , i.e. reliability

indexes 1 2 2.0t t; and (3) SRBTO with the system limit-state 1 1 2 2{( ( , ) 0) ( ( , ) 0)}sysE g gρ F ρ F with

0.04493tsysP , which is given so as to match the

system failure probability of the optimal

topology of the CRBTO.

Fig. 3 shows the optimal topologies by DTO

(Fig. 3b), SORM-based CRBTO (Fig. 3c), and

SORM-based SRBTO (Fig. 3d). The volume

fraction of DTO is lower than CRBTO and

SRBTO because the risk of high compliance

caused by the load uncertainties is ignored. It is

noteworthy that, with the same system failure

probabilities, the volume fraction of CRBTO is

10% higher than SRBTO. This is because

CRBTO approach (assigning fixed constraints

on individual components) is generally more

constrained than SRBTOs (assigning a

constraint on system event, not on the individual

components) at the same level of system failure

probability11

. Fig. 3e shows the result of

SRBTO with the standard deviation of F1

increased to 20 in order to see the impact of the load variability on the optimal topology. In summary, it is seen from

Fig. 3 that the optimal topology is affected significantly by the load variability and the failure event definitions on

the optimal topology of a structure.

The convergence histories of the optimizations are shown in Fig. 4. The proposed single-loop SORM-based

CRBTO and SRBTO show similar rates of convergence rate, which are also comparable to that of DTO. The system

failure probability of the optimal topology found by SORM-based SRBTO/MSR, Psys= 0.04493 is verified by a

fairly close estimate of Monte Carlo simulations (MCS), Psys= 0.04515 (106 times, c.o.v = 0.005).

In order to demonstrate the improved accuracy of the SORM-based single-loop CRBTO method, the results are

compared with those by the FORM-based CRBTO with the component probability targets 1 2 0.02275.t tP P

Figure 5a shows the differences in the volume fractions of the optimal designs. Monte Carlo simulations (MCS: 106

times, c.o.v=0.005) are performed to find the component failure probabilities of the optimal topologies by the

FORM-based and SORM-based CRBTOs. The results in Fig. 5b and Fig. 5c show that the component probabilities

of SORM-based CRBTOs are fairly close to the target probabilities while the FORM-based CRBTOs show

significant errors especially when the random loads have large variability.

The accuracy of the SORM-based single-loop SRBTO method is also investigated. The FORM-based and

SORM-based SRBTO are performed with the system probability target of 0.04493 while standard deviation of load

F1 is varied from 10 to 60. Fig. 6a compares the volume fractions by the FORM and SORM-based SRBTOs. It is

seen that the FORM-based SRBTO provides unconservative designs due to the inaccuracy in reliability calculations.

The results of Monte Carlo simulations (MCS: 106 times; c.o.v=0.005) in Fig. 6b show that the proposed SORM-

based SRBTO provides improved accuracy in predicting the system failure probability.

The results in Fig. 5a and Fig. 6a show the volume fractions of the optimal designs increase significantly as the

load variability increases. It is because the variability of random load increases the uncertainty of the compliance

and thus the probability of violating given constraints.

Figure 4. Convergence history

10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

iterationvolu

me

frac

tion

DTO

CRBTO

SRBTO


9

Figure 5. CRBTOs with standard deviation of load F1: (a) volume fraction of optimal designs, (b) failure

probabilities on the first limit-state, and (c) failure probabilities on the second limit-states

Figure 6. SRBTOs with standard deviation of load F1: (a) volume fractions of optimal designs, and (b) system

failure probabilities

(a)

10 20 30 40 50 600.2

0.3

0.4

0.5

0.6

0.7

standard deviation (F1)

opti

mal

volu

me

frac

tion

FORM-based CRBTO

SORM-based CRBTO

(b)

10 20 30 40 50 600.022

0.024

0.026

0.028

0.030


com

ponen

t pro

bab

ilit

y

MCS on FORM-based

MCS on SORM-based

constraint on P1(C

1)

(c)

10 20 30 40 50 600.022

0.024

0.026

0.028

0.030


com

ponen

t pro

bab

ilit

y

MCS on FORM-based

MCS on SORM-based

constraint on P2(C

2)

(a)

10 20 30 40 50 600.2

0.3

0.4

0.5


volu

me

frac

tion

FORM-based SRBTO

SORM-based SRBTO

(b)

10 20 30 40 50 600.044

0.046

0.048

0.050

0.052


syst

em p

robab

ilit

y

MCS on FORM-based

MCS on SORM-based

constraint on Psys


10

VII. Conclusion

This paper presents three research developments for enhancing the theories and applications of component and

system reliability-based topology optimization (CRBTO and SRBTO): (1) introducing a single-loop SRBTO

approach that employs the matrix-based system reliability (MSR) method to handle the statistical dependence

between multiple limit-states; (2) developing SORM-based single-loop approaches for CRBTO and SRBTO to

improve the accuracy in evaluating probabilistic constraints; and (3) incorporating multiresolution topology

optimization (MTOP) approach to CRBTO and SRBTO in order to obtain high resolution design with a relatively

low computation cost. A computational demanding numerical example demonstrates the importance of uncertainties

on structural topologies and the improvement of the accuracy by the proposed approaches.

Acknowledgments

This research was funded in part by a grant from the Vietnam Education Foundation (VEF) and the National

Science Foundation (NSF). The supports are gratefully acknowledged. The opinions, findings, and conclusions

stated herein are those of the authors and do not necessarily reflect those of sponsors.

References 1Rozvany, G. I. N., "Aims, Scope, Methods, History and Unified Terminology of Computer-Aided Topology Optimization in

Structural Mechanics," Structural and Multidisciplinary Optimization, Vol. 21, No. 2, 2001, pp. 90-108. 2Rozvany, G. I. N., "A Critical Review of Established Methods of Structural Topology Optimization," Structural and

Multidisciplinary Optimization, Vol. 37, No. 3, 2009, pp. 217-237. 3Enevoldsen, I., Sørensen, J. D., "Reliability-Based Optimization in Structural Engineering," Structural Safety, Vol. 15, No.

3, 1994, pp. 169-196. 4Tu, J., Choi, K. K., Park, Y. H., "A New Study on Reliability-Based Design Optimization," Journal of Mechanical Design,

Vol. 121, No. 4, 1999, pp. 557-564. 5Silva, M., Tortorelli, D. A., Norato, J. A., Ha, C., Bae, H. R., "Component and System Reliability-Based Topology

Optimization Using a Single-Loop Method," Structural and Multidisciplinary Optimization, Vol. 41, No. 1, 2010, pp. 87-106. 6Liang, J., Mourelatos, Z. P., Tu, J., "A Single-Loop Method for Reliability-Based Design Optimisation," International

Journal of Product Development, Vol. 5, No. 1, 2008, pp. 76-92. 7Nguyen, T. H., "System Reliability-Based Design and Multiresolution Topology Optimization," Ph.D thesis, Department of

Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 2010. 8Royset, J. O., Der Kiureghian, A., Polak, E., "Optimal Design with Probabilistic Objective and Constraints," Journal of

Engineering Mechanics, Vol. 132, No. 1, 2006, pp. 107-118. 9Rahman, S., Wei, D., "Design Sensitivity and Reliability-Based Structural Optimization by Univariate Decomposition,"

Structural and Multidisciplinary Optimization, Vol. 35, No. 3, 2008, pp. 245-261. 10Lee, I., Choi, K. K., Gorsich, D., "System Reliability-Based Design Optimization Using the MPP-Based Dimension

Reduction Method," Structural and Multidisciplinary Optimization, Vol. 41, No. 6, 2010a, pp. 823-839. 11Nguyen, T. H., Song, J., Paulino, G. H., "Single-Loop System Reliability-Based Design Optimization Using Matrix-Based

System Reliability Method: Theory and Applications," Journal of Mechanical Design, Vol. 132, No. 1, 2010, pp. 011005-1-11. 12Song, J., Kang, W. H., "System Reliability and Sensitivity under Statistical Dependence by Matrix-Based System

Reliability Method," Structural Safety, Vol. 31, No. 2, 2009, pp. 148-156. 13Liang, J., Mourelatos, Z. P., Nikolaidis, E., "A Single-Loop Approach for System Reliability-Based Design Optimization,"

Journal of Mechanical Design, Vol. 129, No. 12, 2007, pp. 1215-1224. 14Borrvall, T., Petersson, J., "Large-Scale Topology Optimization in 3D Using Parallel Computing," Computer Methods in

Applied Mechanics and Engineering, Vol. 190, No. 46-47, 2001, pp. 6201-6229. 15Wang, S., de Sturler, E., Paulino, G. H., "Large-Scale Topology Optimization Using Preconditioned Krylov Subspace

Methods with Recycling," International Journal for Numerical Methods in Engineering, Vol. 69, No. 12, 2007, pp. 2441-2468. 16Nguyen, T. H., Paulino, G. H., Song, J., Le, C. H., "A Computational Paradigm for Multiresolution Topology Optimization

(MTOP)," Structural and Multidisciplinary Optimization, Vol. 41, No. 4, 2010, pp. 525-539. 17Bendsøe, M. P., "Optimal Shape Design as a Material Distribution Problem," Structural and Multidisciplinary

Optimization, Vol. 1, No. 4, 1989, pp. 193-202. 18Rozvany, G. I. N., Zhou, M., Birker, T., "Generalized Shape Optimization without Homogenization," Structural and

Multidisciplinary Optimization, Vol. 4, No. 3, 1992, pp. 250-252.

Nguyen Enhancing Single-loop SRBTO AIAA

Documents

Transcript of Nguyen Enhancing Single-loop SRBTO AIAA