System Reliability-Based Design and Multiresolution ... · System Reliability-Based Design and...

1

Dissertation Defense

System Reliability-Based Design and

Multiresolution Topology Optimization

Tam H. Nguyen

Advisors: Glaucio H. Paulino & Junho Song

Department of Civil and Environmental Engineering

University of Illinois at Urbana-Champaign

07/16/2010

2

Introduction

Multiresolution Topology Optimization (MTOP)

Improving Multiresolution Topology Optimization (iMTOP)

System Reliability-based Design Optimization (SRBDO)

System Reliability-based Topology Optimization (SRBTO)

Summary and Conclusions

Contents

Intro. MTOP iMTOP SRBDO SRBTO Conclusions

TOP RBDO Reviews MTOP Examples improving Examples Adaptive Examples MSR SRBDO//M Examples Existing Improved Examples Summary Future

3

Topology Optimization

Classical structural design optimization: the optimal sizes or shapes

for a given layout and connectivity

Topology optimization: the best topology, shape, size under a given

domain and boundary conditions



size optimization

shape optimization

4

Topology Optimization Applications

Skidmore, Owings & Merrill, LLP (SOM)

(www.altair.com)

Airbus Wing box rib

500 kg reduction/wing



5

F1 F2F3

30×10×10 B8180×60×60 B8

Coarse mesh Fine mesh

~ 1.0 mil. unknowns

Fast solver, PC, C++

Run time: ~ 45.7 hours

Question 1: How to obtain high resolution with affordable computational cost?

Large-scale Topology Optimization

Wang, de Stuler, and Paulino, (2007), IJNME

Computationally

expensive

An example using Matlab code



Low

resolution

6

Reliability-Based Design Optimization

High probability

of failure

Low probability

of failure

( , )f Xd μ

Safe

Objective function increase

B

A Unsafe

Deterministic Optimization

Reliability-Based Design

Optimization (RBDO)



7

System Reliability-Based Design Optimization

Component RBDO

,min ( , )

. . ( , ) 0 =1,...,

,

t

i i

L U L U

f

s t P g P i n

X

Xd μ

X X X

d μ

d X

d d d μ μ μ

System RBDO

,

sys

min ( , )

. . ( )= g ( , ) 0

,

k

t

i sys

k i C

L U L U

f

s t P E P P

X

Xd μ

X X X

d μ

d X

d d d μ μ μ

?

Question 2: How to handle system probability in RBDO?



8

1. To obtain high resolution with affordable

computational cost in topology optimization.

2. To handle system probability in Reliability-

Based Design Optimization (RBDO).

3. To apply RBDO framework in topology

optimization (RBTO).

Objectives



9

Multiresolution Topology

Optimization



10

Topology Optimization Procedure

Problem formulation

Solid and Isotropic Material

with Penalization (SIMP)

0( ) ρ( ) pE Eψ ψ

T

ρ

min

min (ρ, )

. . : (ρ)

(ρ) ρ( )

0 ρ ρ( ) 1

d d

d

s

C

s t

V dV V

u f u

K u f

ψ

ψ

Optimizers

Optimality Criteria (OC)

Method of Moving Asymptotes

(MMA)

Finite Element Analysis

Objective Function & Constraints

Converged?

Result

Sensitivities Analysis

Update Material Distribution

Initial guess

Yes

No

Filtering (Projection) Technique

P

computationally

expensive



11

Large-scale (high resolution) TOP

Large number of finite elements

Computationally expensive

Existing high resolution TOP

Parallel computing (Borrvall and Petersson, 2000)

Fast solvers (Wang et al. 2007)

Approximate reanalysis (Amir et al. 2009)

Adaptive mesh refinement (de Stuler et al. 2008)

High Resolution Topology Optimization



12

Parallel computing:

Domain decomposition

TOP (1): Parallel Computing

A stool (884,736 B8/U)

96x96x96

40x120x120

A cross-shaped section (320,000 B8/U)

Borrvall and Petersson, (2000), IJNME



13

Fast iterative solvers

Use precondition Krylov subspace methods with recycling

Reduce computational time for FEA

TOP (2): Fast Solvers

Coarse mesh: 32x12x12

Fine mesh: 180x60x60

Configuration

Solution on a PC with approx. 1 million unknowns

Wang, de Stuler, and Paulino, (2007), IJNME

p



14

Reduce the number of FEA solutions

FEA at an interval of iterations

Approximate at other iterations

Efficiency factor: 1 ~ 5 times

TOP (3): Approximate Reanalysis

MBB: 60x20 Q4/UCantilever: 48x16x16 B8/U


or

Approximate the Displacement


Converged?

Result



Initial guess

Yes

No


Amir, Bendsoe, and Sigmund, (2009), IJNME



15

AMR TOP

Refine the solid and surface regions

Reduce the total number of FEs

Obtain resolution as fine uniform mesh (efficiency factor 3)

TOP (4): Adaptive Mesh Refinement

Configuration: 2:1:1 Uniform mesh: 128x64x64

524,288 B8/U

AMR: initial mesh 64x32x32

initial 65,520 B8/U

final 228,692 B8/U

de Stuler, Wang, and Paulino, (2008), IASS-IACM

P



16

Large-scale TOP

Fine mesh: Large number of finite elements

FEA cost increases

Remarks on High Resolution TOP

Existing approaches:

Powerful computing resources: many processors

Reduce cost associated with FEA:

• Fast solvers

• Approximate reanalysis

• Adaptive mesh refinement

Same discretization for analysis and design



17

Conventional element-based approach (Q4/U)

Same discretization for displacement and density

Proposed Multiresolution TOP (MTOP)

Displacement Density/design variable

Proposed MTOP approach (Q4/n25)

Different discretizations for displacement and density/design variables

Displacement Density Design variable



18

Stiffness matrix

MTOP: Integration of Stiffness Matrix

Te

e

dK B DB

Numerical integration

1

nT

e ii

i

AK B DB

SIMP model

T

0

1 1

( ) ( )n nN N

p p

e i e e i i iiii i

AK B D B I

i

iA

Vi

i

Sensitivity

1 1

(ρ )ρ ρ ρ

(ρ )ρ ρ

nNp

j j

j pe e i i ii i

n i n i n n

pd d d d

IK K

I

Q4/n25

B8/n125



19

Compute density from design variables

Minimum length-scale (Guest et al. 2004, Almeida et al. 2009)

MTOP: Projection (filtering)

( )i nf d

( )

( )

i

i

n nin S

i

mim S

d w r

w r

minmin

min

if ( )

0 otherwise

mimi

mi

r rr r

rw r

ρ ( )

( )i

i ni

n mim S

w r

d w r



20

MTOP Examples: 2D Cantilever Beam

Configuration MTOP Q4/n25 FE mesh 48x16

(C=208.23)

Q4/U FE mesh 48x16

(C=205.57)

Q4/U FE mesh 240x80

(C=210.68)

Convergence history

Objective: minimum compliance

Constraint: volfrac = 0.5

Length scale: rmin = 1.2

16

48

Nguyen, Paulino, Song, and Le, (2010), JSMO



21

MTOP Examples: 2D Michell Truss

Domain (3:2) Analytical solution

Sigmund solution (Sigmund, 2000) MTOP: 180x120 Q4/n25 elements



22

MTOP: 3D Cross-shaped Section

Borrvall & Petersson (2000)

320,000 B8/U elements

MTOP

5,000 B8/n125 elements



23

MTOP: 3D Bridge Design

Configuration

MTOP B8/n125

36,000 elements

10x120x30

Existing design

(http://www.sellwoodbridge.org)

6L

q

L

2L/3

L

L

L

non-designable layer Non-designable layer



24

Can MTOP’s efficiency be improved?

MTOP approach

Density/design variable:

same fine mesh

FE mesh: coarse

Reduce cost (ρ) d K u f



Converged?

Result



Initial guess

Yes

No


MTOP

???Improving MTOP efficiency?

Different discretizations for

density & design variable?

Q4/n25



25

MTOP approach (Q4/n25) or (Q4/n25/d25)Displacement Density Design variable

Proposed iMTOP approach (Q4/n25/d9) and (Q4/n25/d16)


Q4/n25/d16Q4/n25/d9




26

ρi

iV

jd jd

B8/n125/d8 B8/n125/d15

jd

jd

ρi

iV

TET4/n64/d8 TET4/n64/d10


ρi

iV

di

B8/n125

MTOP iMTOP

ρi

iV

di

TET4/n64




27

Compute density from design variables

Minimum length-scale (Guest et al. 2004, Almeida et al. 2009)

iMTOP: Projection (Q4/n25/d9)

( )i nf d

( )

( )

i

i

n nin S

i

mim S

d w r

w r

minmin

min

if ( )

0 otherwise

mimi

mi

r rr r

rw r

ρ ( )

( )i

i ni

n mim S

w r

d w r

Displacement

Design variable

Density

rmin

rni

in

Si

w(r)

1

rni

rmin



28

iMTOP: MBB Beam

300x100 Q4/U 60x20 Q4/U

60x20 Q4/n25/d25 60x20 Q4/n25/d16

60x20 Q4/n25/d9 60x20 Q4/n25/d4

20 40 60 80 1000

200

400

600

800

1000

Iteration

Com

pli

ance

conventional (60x20 Q4/U)

iMTOP (60x20 Q4/n25/d4)

iMTOP (60x20 Q4/n25/d9)

iMTOP (60x20 Q4/n25/d16)

MTOP (60x20 Q4/n25)

conventional (300x100 Q4/U)

1 2 3 4 5 60

1

2

3

4

5

6

7

8

9

10

11

norm

aliz

ed c

om

puta

tional

tim

e

1: conventional (60x20 Q4/U)

2: iMTOP (60x20 Q4/n25/d4)

3: iMTOP (60x20 Q4/n25/d9)

4: iMTOP (60x20 Q4/n25/d16)

5: MTOP (60x20 Q4/n25)

6: conventional (300x100 Q4/U)

convergence

Efficiency

?

60x20



Messerschmitt-Bolkow-Blohm (MBB) beam

29

iMTOP: A Cube with Concentrated Load

MTOP B8/n125/d125

(C=29.04)

iMTOP B8/n125/d64

(C=29.06)

iMTOP B8/n125/d27

(C=29.08)

iMTOP B8/n125/d8

(C=29.33)

LP

L

L=24

24x24x24

convergence

1 2 3 40

1

2

3

4

5

6

7

8

9

10

11

no

rmal

ized

co

mp

uta

tio

nal

tim

e

1: iMTOP B8/n125/d8

2: iMTOP B8/n125/d27

3: iMTOP B8/n125/d64

4: MTOP B8/n125

10 20 30 40 500

50

100

150

200

Iteration

Co

mp

lian

ce

iMTOP B8/n125/d8

iMTOP B8/n125/d27)

iMTOP B8/n125/d64)

MTOP B8/n125)

Efficiency



Nguyen, Paulino, Song, and Le, (submitted), IJNME

30

Why Adaptive MTOP?

Further improve the efficiency

Reduce the number of density elements and design variables

during optimization process?

Adaptive MTOP

Adaptive MTOP (e.g. Q4/U & Q4/n25/d4)

Q4/n25/d4 requires more computational cost than Q4/U

Q4/n25/d4 provides higher resolution

Use Q4/n25/d4 where and when needed only, otherwise Q4/U

Unchanged the Finite Element Mesh during optimization process



31

Adaptive MTOP Procedure

No

Element type array

A(i,j) = 1

(all Q4/U elements)

if k < L or U < k , k=1,25

Q4/U Q4/n25/d4

A(i,j) = 1

if L < < U

A(i,j) = 25

Q4/U Q4/n25/d4

A(i,j) = 25 A(i,j) = 1

Q4/U

Q4/n25/d4

A(i,j) = 1

A(i,j) = 25



Converged?

Result



Initial guess

Yes


Check & update element type array

Element type



32

Optimal topologies by iMTOP and adaptive MTOP

Adaptive MTOP: 2D Cantilever

2L

L

P

Configuration 160x80 Q4/U 32x16 Q4/U

initial iteration intermediate iteration final iteration

Q4/U Q4/n25/d4

Mesh (element types)

Topology

32x16 Q4/n25/d4 32x16 - adaptive

Q4/n25/d4 & Q4/U

Adaptive MTOP optimization process



33

Adaptive MTOP: 3D Cantilever Beam

Configuration 24x12x12

B8/U

24x12x12

B8/n125/d8

L

2L

L

P

24x12x12

B8/n125/d8 & B8/U

Adaptive MTOP optimization process

Initial iteration

(3,456 B8/U)Intermediate iteration

(2,288 B8/U & 1,168 B8/n125/d8)

Final iteration

(2,072 B8/U & 1,384 B8/n125/d8)

(FE mesh : 24x12x12 unchanged)



34

System Reliability-Based

Design/Topology Optimization



35

RBDO Formulation

Component RBDO

,min ( , )

. . ( , ) 0 =1,...,

,

t

i i

L U L U

f

s t P g P i n

X

Xd μ

X X X

d μ

d X

d d d μ μ μ

System RBDO

,

sys

min ( , )

. . ( )= g ( , ) 0

,

k

t

i sys

k i C

L U L U

f

s t P E P P

X

Xd μ

X X X

d μ

d X

d d d μ μ μ

?



36

Matrix-based System Reliability (MSR) Method

Convenient: matrix-based procedures for c and p; easy SRA calculation (inner product)

General: uniform application to series, parallel, and any general systems

Flexible: inequality-type information; incomplete information (“LP bounds” method)

Efficient: no need to re-compute “p”; replace “c” for SRA of a new event

Common Source Effect: can account for statistical dependence between components

Decision Support: parameter sensitivities, component importance measure; inferences

Song and Kang, (2009); Structural Safety

e1e3

e8

e4e6

e5

e2

E3

E2

E1

e7

mutually exclusive and collectively

exhaustive events (MECE)



37

Proposed approach: SRBDO using MSR

Adopt a single-loop RBDO (Liang et al. 2007)

Use MSR method to compute Psys and its gradients



Single-loop RBDO Double-loop RBDO

Approx. MPP

1st constraint

Karush–Kuhn–Tucker (KKT) optimality conditions

Objective

function

Approx. MPP

nth constraintReliability eval.

1st constraint

Objective

function

Reliability eval.

nth constraint

MSR method

Single-loop PMA

1, , ,...,

T

T

min ( , )

. . ( , ( )) 0 =1,...,

( ) ( ) dependent

indepdendent

t tnP P

t

i i

t

sys

syst

sys

f

s t g i n

f d PP

P

X

Xd μ

S

s

d μ

d X U

c p s s s

c p

38

Proposed approach: SRBDO using MSR (contd.)

Sensitivity w.r.t. design variables d, x}

sys T

1 2

T

1 2

( )= ( )

θ θ

( ) ( ) ( )ˆ= ( ) ( ) ... ( ) ( )θ θ θ

( ) [ ( ) ( ) ( )]

n

n

Pf d

P P P

S

s

p sc s s

p s P s P sp s p s p s P s

P s s s s

Sensitivity w.r.t. component failure probability Pit

( ) ( ) β ( ) 1

β β φ( β )

i i i i

i i i i i

P P P

P P

s s s

Use probabilities and sensitivities by component reliability analysis (FORM)



39

SRBDO of Truss System

1 6

1 2 3 4 5 6 { ,..., }

15

sys

1

1 2 3 4 5 6

min ( ) 2( )

. . = g ( , ) 0 0.001

( , ) 0.707 1, 2

0.500 3,...,6

, , , , ,

k

A A

t

i sys

k i C

i i i A

i i A

f A A A A A A

s t P P P

g A F F i

A F F i

A A A A A A

dd

d X

d X

0

Minimize total weight of the

system

Definition of system failure: at

least two members fail (cut-set

systems): effects of load re-

distributions NOT considered

L

FA

L

4

1 2

53

6

Random Variables

(Gaussian distribution)Mean Std Dev

Member strength Fi , i=1,..6 (Mpa) 745 62Applied load FA (kN) 4450 45



40

SRBDO of Truss System (contd.)

Members

Area: Ai (×103 mm2) Reliability Index: i

McDonald &

MahadevanSRBDO/MSR

McDonald &

MahadevanSRBDO/MSR

1 18.43 17.89 2.89 2.67

2 18.27 17.89 2.83 2.67

3 13.51 13.20 3.16 2.99

4 13.44 13.20 3.12 2.99

5 13.33 13.20 3.06 2.99

6 13.09 13.20 2.92 2.99

f(x) 105.24 103.36

Better optimal design (i.e. less total weight) and symmetric results

Monte Carlo simulations (c.o.v. = 0.03, 106 times) on the system

failure probability: Psys = 0.00107 (cf. MSR gives 0.001)

4

1 2

53

6

>



41


T

T

( ) 'CIM = ( | )=

( )

i sys

i i sys

sys

P E EP E E

P E

c p

c p

0.00

0.25

0.50

0.75

1.00

1 2 3 4 5 6

CIM

Components

4

1 2

53

6

Conditional probability Importance Measure (CIM)

Relative contribution of components to the system failure

probability (can be computed efficiently by MSR method)



42


6

1 2 2 4

1 2

5 3

2 30

27 28

29 1 2 2

35

32 33

34

11

1 2 9

7

8

16

1 2 2 14

12

10 15 20

26

1 2 2

22 23

25

5

13

17 18

19

21

31

36 24

Effects of load re-distributions (sequential failures)

Effects of correlation between random variables and between

components



43

Discrete structures

Mogami et al. (2006)

Truss examples

Existing SRBTO Approaches

11

1 2

1 1 11

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

n

sys i i i j n

i i j ii

P E P E P E P E E P E E E

1 2 3 1 2 3( ) ( ) ( ) ( )P E E E P E P E P E

Continuum structures

Silvia et al. (2010)

Limit-states: statistically independent

DTO SRBTO

Ground structure Optimal structure

Objective: SRBTO for continuum structures with dependent limit-states?



Mogami et al. (2006), JSMO

Silvia et al. (2010), JSMO

44

Proposed Approach: SORM-based RBTO

SORM-based CRBTO g , ( ) 0d X U

KKT

1U

2U

iU

ˆ tiα

ˆβt ti iα

βtit

iU

*

iU

( ) t k ti i

T ( )

( )

T ( )

( ) ( )( ; )

t t

syst

sys t t

SORM

SORM

SORM

sys

f d PP E

P

Ssc p s s s

Pc p

( ) ( 1)

( 1)( )

SORM

t

t k t ki

i it k

i

T

T

( ) ( )( ; )

t t

syst

sys t t

sys

f d PP E

P

Ssc p s s s

Pc p

At the k-th step

SORM-based SRBTO

Enhance the accuracy in RBTO

First-Order Reliability Method (FORM) inaccurate for nonlinear limit-states

Propose to use Second-Order Reliability Method (SORM) to improve the accuracy

At the k-th step



45

SRBTO of a Stool

F1

L

F1F3

F3F2

F2

L=12

L

Objective: minimize volume V( )

Limit-states:

Random loads: : F ~ (F1,F2,F3) ~

N(100,10), N(0,30), N(0,40)

Load cases:

( , ) 120 ( , ), 1,2i i i ig C iρ F ρ F

1 1 2( , ),F FF 2 1 3( , )F FF

Constraints

Deterministic TO (DTO):

Component RBTO (CRBTO):

System RBTO (SRBTO):

( , ) 0, 1,2ig iρ f

( ( , ) 0) , 1,2ti i iP g P iρ F

( ( , ) 0) ti i sysP g Pρ F



46

Optimal Topologies

DTO CRBTO

volfrac = 24.4%

( F=10)

SRBTO

volfrac =22.3%

( F1=10)

SRBTO

volfrac =23.9%

( F1=20)

volfrac = 6.3%



47

Improve Accuracy by Second-Order Reliability Method

10 20 30 40 50 600.2

0.3

0.4

0.5

0.6

0.7

standard deviation (F1)

op

tim

al v

olu

me

frac

tio

n

FORM-based CRBTO

SORM-based CRBTO

10 20 30 40 50 600.022

0.024

0.026

0.028

0.030


com

po

nen

t p

rob

abil

ity

MCS on FORM-based

MCS on SORM-based

constraint on P1(C

1)

10 20 30 40 50 600.022

0.024

0.026

0.028

0.030


com

po

nen

t p

rob

abil

ity

MCS on FORM-based

MCS on SORM-based

constraint on P2(C

2)

10 20 30 40 50 600.2

0.3

0.4

0.5


vo

lum

e fr

acti

on

FORM-based SRBTO

SORM-based SRBTO

10 20 30 40 50 600.044

0.046

0.048

0.050

0.052


syst

em p

rob

abil

ity

MCS on FORM-based

MCS on SORM-based

constraint on Psys

Component RBTO

System RBTO



Nguyen, Song, and Paulino, (submitted), JSMO

48

SRBTO of a Building Core

Objective: minimize volume V( )

Limit-states:

Random loads: F ~ (P1,P2,P3,q1 ,q2,q3)

Load cases:

0( , ) ( , )i i i i ig C Cρ F ρ F

( , )i i iP qF5L

LL

P

L

L

P

q/2

q

P2,q2 Load

case 2

P2,q2 P2,q2

P2,q2

4 symmetry axes

P1,q1 P1,q1

P1,q1P1,q1

Load

case 1

P3,q3

Load

case 3

P3,q3 P3,q3

P3,q3

L/12L/12 10L/12

Load

Cases

P q (at top)

mean c.o.v mean c.o.v

Case 1 70.71 0.30 2.82 0.15 250

Case 2 50.00 0.15 2.00 0.30 125

Case 3 50.00 0.20 2.00 0.15 125

t

iC



49

Optimal Topologies of the Building Core

DTO

volfrac = 21.93%

SRBTO

volfrac =28.15%

(Psys=0.05)

0 0.2 0.4 0.6 0.8 10.200

0.225

0.250

0.275

0.300

0.325

target system failure probability

opti

mal

volu

me

frac

tion

SRBTO (same

=0.50, diff

=0.25)

DTO

SRBTO

volfrac =22.25%

(Psys=0.85)

volfrac v.s Psys

P1 P2 P3 Psys

same = 0.5

diff = 0.25

SRBTO 0.02731 0.02088 0.00539 0.05000

MCS 0.02747 0.02101 0.00542 0.05023

same = 0.5

diff = 0.25

SRBTO 0.26940 0.25973 0.20818 0.50000

MCS 0.26977 0.26006 0.20800 0.50008

same = 0.9

diff = 0.45

SRBTO 0.02812 0.02227 0.00625 0.05000

MCS 0.02816 0.02242 0.00638 0.05017

Probabilities: SRBTO/MSR v.s MCS



Nguyen, Song, and Paulino, (submitted), JSMO

50

pattern repetition

pattern symmetry

Building Core with Pattern Repetition

m=3 m=12m=6 m=10

DTO

1 3 6 10 120.20

0.25

0.30

0.35

0.40

0.45

number of pattern repetitions, m

opti

mal

volu

me

frac

tion

DTO

SRBTO (same

=0.50, diff

=0.25)

SRBTO (same

=0.90, diff

=0.45)

SRBTO

LL

http://www.som.com



m

51

MTOP & iMTOP:

Use three distinct displacement, density, and design variable fields

Improve efficiency, apply to large-scale problems

Adaptive MTOP:

Use MTOP and iMTOP elements where and when needed

Reduce the number of density elements and design variables

SRBDO/MSR:

Apply to general system including link-set, cut-set

Address dependence between limit-states, provide sensitivity

SRBTO

Propose accurate single-loop SORM-based CRBTO & SRBTO approaches

Include pattern repetition constraints

Summary & Conclusions



52

Optimal locations of design variables in MTOP

MTOP approach in nonlinear and stress-based problems

MTOP using Krylov subspace methods and recycling

SRBDO with multi-scale MSR approach

SRBDO with mixed continuous-discrete random variables

Future Research Topics



53

Nguyen, T. H., Paulino, G. H., Song, J., Le, C. H., (2010). "A computational paradigm for

multiresolution topology optimization (MTOP)." Structural and Multidisciplinary Optimization

41(4): 525-539.

Nguyen, T. H., Song, J., Paulino, G. H., (2010). "Single-loop system reliability-based design

optimization using matrix-based system reliability method: theory and applications." Journal

of Mechanical Design 132(1): 011005.

Sutradhar, A., Paulino, G. H., Miller, M. J., Nguyen, T. H., (2010). “Topological optimization

for designing patient-specific large craniofacial segmental bone replacements.” Proceedings

of the National Academy of Sciences 107(30) 13222-13227.

Nguyen, T. H., Paulino, G. H., Song, J., Le, C. H., "Improving multiresolution topology

optimization via multiple discretizations." International Journal for Numerical Methods in

Engineering (submitted).

Nguyen, T. H., Song, J., Paulino, G. H., "Single-loop system reliability-based topology

optimization considering statistical dependence between limit states." Structural and

Multidisciplinary Optimization (submitted).

Contributions



54

Advisors:

Glaucio H. Paulino & Junho Song

Committee:

Glaucio H. Paulino, Junho Song, Jerome F. Hajjar, C. Armando

Duarte, Ravi C. Penmetsa, William F. Baker, Alessandro

Beghini, Alok Sutradhar

Financial Support:

Vietnam Education Foundation

National Science Foundation

Computational Mechanics Group, Structural System Reliability Group

and colleagues at UIUC

Special thanks to my family

Acknowledgements

55

Thank you for your attention !

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