System Reliability Based Topology Optimization of...

Junho Chun*

University of Illinois at Urbana-Champaign, USA

Junho Song


Glaucio H. Paulino


ICOSSAR 2013

11th

International Conference on Structural Safety & Reliability

System Reliability Based Topology

Optimization of Structures under

Stochastic Excitations

Columbia University, New York, NY, June 20, 2013

2

Structural engineering under Natural hazards

and risks

San Francisco Earthquake, 1907 http://www.documentingreality.com Tacoma bridge, 1940 http://failuremag.com

Structural system Courtesy of Skidmore, Owing and Merrill, LLP

3

Motivation

http://en.wikipedia.org/wiki/John_Hancock_Center

John Hancock Center Ssiger International Plaza

Courtesy of Skidmore, Owing and Merrill, LLP

Research aims to find optimal structural system under stochastic excitations

Incorporation of random vibration theories into topology optimization Topology optimization

Stromberg et al. (2011)

Stochastic excitation

0 5 10 15 20 25 30 35 40 45 50-600

-400

-200

0

200

400

600

800HYOGOKEN NANBU EQ - KOBE-JMA3.EW 1/17/1995 DT=0.02 Amax=617.14gal

x - time / DT = 0.02

Accele

rati

on

(g

al)

4

Contents

Topology optimization

Topology optimization under stochastic excitations

System reliability-based topology optimization under

stochastic excitations

Numerical example: multi-story building structure

Conclusions & future research

5

?

Topology Optimization

Topology optimization aims to identify optimal material layouts of problems

through mathematical programming while fulfilling given design constraints

min compliance

. volume constraints t

d

Applications

Nguyen at el. (2010) Stromberg et al. (2011)

?

0 void0 1

1 solid

e

e

e

Design variable : Element density

T

0( ) e

e p

e e e

K B D B

T( ) e

q

e e e ed

M N Ne

6

Discrete Representation Method

(Der Kiureghian, 2000)

Discretization of Random process

0 2 4 6 8 10Time

f(t)

0 2 4 6 8 10Time

v

Der Kiureghian, A. (2000). The geometry of random vibrations and solutions by FORM and SORM. Probabilistic Engineering Mechanics, 15(1),: 81-90.

o v: uncorrelated standard normal random variables

o s(t): deterministic basis functions based on the

spectral characteristics of the process

1

( ) ( ) ( ) ( ) ( )n

T

i i

i

f t t v s t t t

s v

W( )t

, g g Filter

White Noise 0

( )f t

Example : Filtered white noise (earthquake)

0

1 1

T

0

1

( ) ( ) ( )

( ) ( )

2π / ( ) ( )

t

n n

i i i f i

i i

n

i f i

i

f t v s t d

v s t W h t t t

t v h t t t t

s v

Magnitude of WN impulse

Unit-impulse response

function of the filter

7

Response of Linear System to Stochastic

Excitation

o hs(t) : the unit-impulse response function of the system

Response

Linear system + Gaussian process

Duhamel’s Integral

0

( ) (τ) ( τ) τ

t

su t f h t d

T

1 10

( ) ( ) ( τ) τ ( ) ( )

t n n

i i s i i

i i

u t v s h t d v a t t

a v0

( ) ( ) ( ) , 1,...,

t

i i sa t s h t d i n

Deterministic, time-dependent

- filter + structure

Random, time-independent

a(t) in FEM settings

1 1 1 0

2 1 2 2 0

1 0 1 2 1 1 0

0 1 2 1 0

( ) ( ) 0 0 0 ( )

( ) (2 ) 0 0 ( )

( ) ( ) 0 ( )

( ) ( ) ( )

n n n n

n n n n

u t u t v a t

u t u t v v a t

u t u t t v v v a t

u t u t v v v v a t

Numerical time integration Inversely computed

. . ( ) ( , ) ( ) ( , ) ( ) ( , ) ( , )e g t t t t M ρ u ρ C ρ u ρ K ρ u ρ f ρ

8

Instantaneous Failure Probability

‘Instantaneous’ failure events of a linear system

0 0 0 0

000 0 0

0 0

0 ( ) 0

ˆ( ) β , ( )

T

T

P u t u P u t P g u

tug u u t t

t t

a v

av α v

a a

0 0 0 0

0 0 0 0 0

[ β , ]

ˆβ , / ( ) *

P u t u u t

u t u t t

a α v

T0 0 0 0( )fE u t u t u a v

Failure Probability, P(Ef)

t

( )u t

0u

0t

Failure

Failure domain

Safe domain 0 0( ) 0u u t v i

v j

0 0( ) 0u u t

T

0 0( ) 0u t a v

MPP

( , )0 0β u t

*( , )0 0u tv

0ˆ ( )tα

Topology

Optimization

Reliability

Analysis

9

Topology Optimization of Structure under


target

min ( ( ))

. ( ) 1,...,

( ) ( , ) ( ) ( , ) ( ) ( , ) ( ))

i fi

f

s t P E P i n

with t t t f t

dρ d

M ρ u ρ C ρ u ρ K ρ u ρ Ml

Probabilistic constraints on responses of a structure

(Instantaneous failure probability)

Objective function

Stochastic topology optimization framework (Chun, Song, Paulino 2012)

Stochastic excitation (filtered Gaussian process)

5m

3aP 3bP

2aP 2bP

1aP 1bP

Design domain:

Bilinear quadrilateral element, Q4

LEVEL 04

EL: 15m

LEVEL 03

EL: 10m

LEVEL 02

EL: 5m

GROUND

EL: 0m

Frame element

0 2 4 6 8 10-50

0

50

Acce

lation

(m

/s2)

Time (seconds)

Stochastic excitation

Component failure events, Ei

A B

β 1.3 2.2

Pf 9.68% 1.39%

target( ) 1,...,i fiP E P i n

Solve optimization problem

(CRBTO)

10

SYSTEM RBTO of Building Structures under


SRBTO : System Reliability Based Topology Optimization

Motivation : further extension of the stochastic topology optimization framework

aforementioned is expected to provide more realistic means for the prediction of

a given building’s response to stochastic excitations

- Time: the first passage probability

- Location

- Failure modes: different types of design constraints such as a target tip displacement, a target

natural frequency of a structure

0

1

( ) ( max | ( ) |) ( )n

n

sys t t i

i

P E P x X t P X t x

Matrix-based System Reliability Method (MSR)(Song and Kang 2009, Kang et

al. 2012)

General system

Statistical dependency

Parameter sensitivity

T

T

( ) ( ) dependent( ; )

independent

t t

syst

syst t

sys

f d PP E

P

ssc p s s s

Pc p

Event vector Conditional probability

vector

Probability vector Joint PDF

Song, J., and W.-H. Kang (2009). System reliability and sensitivity under statistical dependence by matrix-based system reliability method. Structural Safety,

Vol. 31(2), 148-156.

Kang, W.-H., Y.-J. Lee, J. Song, and B. Gencturk (2012). Further development of matrix-based system reliability method and applications to structural

systems. Structure and Infrastructure Engineering: Maintenance, Management, Life-cycle Design and Performance. Vol. 8(5), 441-457.

11

SYSTEM RBTO of Building Structures under


,

0 0

min ( ( ))

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

( ) ( ) dependent ( ; )

independent

with ( ) ( , ) ( ) ( , ) ( ) ( , ) (

ti

ti

P

t

i iP

T t t

syst

sysT t t

sys

f

s t g g t u P i n

f d PP E

P

t t t

d

ss

ρ d

ρ

c p s s sP

c p

M ρ u ρ C ρ u ρ K ρ u ρ f , )t ρ

Probabilistic constraints solved by MSR method

System equation

𝑃𝑖𝑡 quantile of the i-th limit-state function

: Filtered density

: Mass matrix, : Damping matrix, : Stiffness matrix

( ) : Stochastic

System fa

process (e.g., for earthquake loading ( ) ( )) ( ))

ilure event: ,

g

sys sy

f t t u t f t

E P

ρ

M C K

f Ml Ml

target Target failure probab: ilitys

LEVEL k

EL : - m

LEVEL GN

EL : 0 m

LEVEL j

EL : - m

LEVEL i

EL : - m

0 1 2 3 4 5 6 7-1500

-1000

-500

0

500

1000

1500

t, sec

f(t)

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

Drift r

atio

3

2

1

Stochastic excitations

Responses

Formulation

12

Sensitivity Calculation

Formulation

( ) / , ( ) / t

i e i iP d P P s s

/ , /t ti i

t

e iP Pg d g P

1

2

1

1

2 2

1 1

( ) β ( ) 1

β ( β )

β ( ) 1

β ( β )1

β ( )1 1

( β

(

)

)

1 1

t

i i i

t t t t

i i i i

mt

i ik kk

t tmi i

ikk

mt

i ik kk

tm mi

ik

i

t

i

ikk k

P P P

P

r s

r

r s

r r

P

s ss

T

0 0

T

0

β 1

β β ( β )

β ( , ) 1

β ( β )

1( , )

( β )

ti

t ti i

tP Pi

t t t t

i i i i

t

i

t

P

t

i i

t

t

i

i

g g

P

u t

P

t

g

a ρ

a ρ

1

1

00T

1 10

00

1 1

( ) ( , )( , )

( , )

( , )( , , )

e

e

t t

e

i i

nP j

j j e

t n nji i

i

j i j e

n njt i

i i

e

P

j

e

i e

g

d

P a ta t

dt

a tc t P

g

d

d

ρρ

a ρ

ρρ

( )0i

e

P

d

s

Analytical derivation

Design variables: d, 𝑃𝑖𝑡

,

0 0

min ( ( ))

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

( ) ( ) dependent ( ; )

independent

with ( ) ( , ) ( ) ( , ) ( ) ( , ) (

ti

ti

P

t

i iP

T t t

syst

sysT t t

sys

f

s t g g t u P i n

f d PP E

P

t t t

d

ss

ρ d

ρ

c p s s sP

c p


Calculation of sensitivity of the terms in the constraints with respect to design

variables is an essential procedure for efficient gradient-based optimization

13

Optimization Procedure

Result: Optimal Topology

Yes

No (iterate)

Convergence Criteria

achieved?

Random Vibration Analysis using dynamic FEA,

and compute Pf using MSR method

Sensitivity Analysis of objective and constraint

functions

Update Design Variables using Mathematical

Programming

Generating Stochastic

Excitation Model

Initial Design

ρ = ρInitial 0 1 2 3 4 5 6 7 8

-2000

-1000

0

1000

2000

t, sec

p

0 1 2 3 4 5 6 7 8-4

-2

0

2

t, sec

dis

pla

cem

en

ts

14

System Failure event:

It occurs when at least one of component

events happens

LEVEL 07

EL : 30m

GROUND

EL : 0m

LEVEL 02

EL : 5m

LEVEL 05

EL : 20m

LEVEL 06

EL : 25m

5m

LEVEL 03

EL : 10m

LEVEL 04

EL : 15m

Design domain:


Frame element

Numerical example : Six-story building

7

2

isys f

i

E E

Failure event: Inter-story drift ratio

It occurs when inter-story drift ratio

exceeds a prescribed threshold value

Filtered Gaussian process using

Kanai-Tajimi filter 2

2

2

2

(2 1)( ) exp( ) sin( 1 )

1

exp( )2 cos( 1 )

f f

f f f f f

f

f f f f f f

h t t t

t t

,

0 0

min ( ( ))

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

( ) ( ) dependent ( ; )

independent

with ( ) ( , ) ( ) ( , ) ( ) ( , ) (

ti

ti

P

t

i iP

T t t

syst

sysT t t

sys

V

s t g g t u P i n

f d PP E

P

t t t

d

ss

ρ d

ρ

c p s s sP

c p


15


LEVEL 07

EL : 30m

GROUND

EL : 0m

LEVEL 02

EL : 5m

LEVEL 05

EL : 20m

LEVEL 06

EL : 25m

LEVEL 03

EL : 10m

LEVEL 04

EL : 15m

CRBTO :

βti=1.7 (Pt=4.46%)

volfrac=28.5%

(Ptsys=6.67%)

SRBTO :

βtsys=1.502 (Pt

sys=6.67%)

volfrac=28.45%;

CRBTO:

- All target reliability index βti=1.7

SRBTO: - Psys: After CTBTO the probability that

at least one of the constraints is violated

(i.e. series system) is estimated by the

MSR method

16


0 20 40 60 80 100 1202

4

6

8

10

Vol

um

e (

m3)

0 20 40 60 80 100 1200

2

4

6

Rel

iabi

lity

ind

ex

1

2

3

4

5

6

0 20 40 60 80 100 1200.06

0.08

0.1

0.12

Fai

lure

Pro

ba

bilit

y

Iteration

Pt

sysPf

sys

0 1 2 3 4 5 6 7 8 9 10-100

0

100

f(t)

(m

/s2)

Ground acceleration

0 2 4 6 8 10-0.04-0.02

00.020.04

6/L

6

Initial system

0 2 4 6 8 10-0.04-0.02

00.020.04

5/L

5

0 2 4 6 8 10-0.04-0.02

00.020.04

4/L

40 2 4 6 8 10

-0.04-0.02

00.020.04

3/L

3

0 2 4 6 8 10-0.04-0.02

00.020.04

2/L

2

0 2 4 6 8 10-0.04-0.02

00.020.04

1/L

1

Time (seconds)

0 2 4 6 8 10-0.04-0.02

00.020.04

6/L

6

Optimized system

0 2 4 6 8 10-0.04-0.02

00.020.04

5/L

5

0 2 4 6 8 10-0.04-0.02

00.020.04

4/L

4

0 2 4 6 8 10-0.04-0.02

00.020.04

3/L

3

0 2 4 6 8 10-0.04-0.02

00.020.04

2/L

2

0 2 4 6 8 10-0.04-0.02

00.020.04

1/L

1

Time (seconds)

Convergence history (SRBTO) Dynamic responses (SRBTO)

17

Conclusions & Future Research

Concluding remarks

Development of a new framework incorporating random vibration theories

into topology optimization using a discrete representation method for

stochastic processes

Development of SRBTO under stochastic excitation considering statistical

dependency of component events using MSR method.

Application of the proposed method to find optimal bracing system of the

structure satisfying probabilistic constraints defined in system level.

Future Direction

Incorporation of first passage probability in stochastic topology

optimization

Development of an efficient approach for computing sensitivity of system

reliability with a large number of component events

Karol Fellowship

National Science Foundation

Skidmore, Owings, Merrill. LLP

20

Reliability-Based Design Optimization

‘Deterministic’ design optimization (DO)

Reliability-based design optimization

(RBDO)

min ( )

. ( ) 0, 1,...,

obj

i c

lower upper

f

s t g i n

dd

d

d d d

target

min ( , )

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

,

obj,

sys i sys c

lower upper lower upper

f

s t P E P g P i n

X

Xd

X

X

d

d

d d d

Objective function

v i

v j

Limit-state functions

Safe domain

High probability of

failure

Low probability

of failure

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

0

0.05

0.1

0.15

0.2

0.25

21

System Reliability-Based Design

Optimization (SRBDO)

Component Reliability Based Design Optimization (CRBDO)

target

min ( , )

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

,

obj,

i f c


f

s t P g P i n

X

Xd

X

X

d

d

d d d

target

min ( , )

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

,

obj,

sys i sys c


f

s t P E P g P i n

X

Xd

X

X

d

d

d d d

Limit-state

functions

Song, J., and W.-H. Kang (2009). System reliability and sensitivity under statistical dependence by matrix-based system reliability method. Structural Safety,

Vol. 31(2), 148-156.

Kang, W.-H., Y.-J. Lee, J. Song, and B. Gencturk (2012). Further development of matrix-based system reliability method and applications to structural

systems. Structure and Infrastructure Engineering: Maintenance, Management, Life-cycle Design and Performance. Vol. 8(5), 441-457.

22

LEVEL 07

EL : 30m

GROUND

EL : 0m

LEVEL 02

EL : 5m

LEVEL 05

EL : 20m

LEVEL 06

EL : 25m

5m

LEVEL 03

EL : 10m

LEVEL 04

EL : 15m

Design domain:


Frame element


T T

0 , 0 ,

0

T T

0 , 0 ,

0

T T

0 ( 1), 0 ( 1),

( ( , ) ( , ) )0 for 2

2

( ( , ) ( , ) )

2

( ( , ) ( , ) ) 0 for 3

2

i

i Left i Right

f i

i

i Left i Right

i

i

i Left i Right

i

t tE u i

L

t tu

L

t ti

L

a ρ a ρ v

a ρ a ρ v

a ρ a ρ v, ..., 7

7

2

isys f

i

E E

Failure event: Inter-story drift ratio

System Failure event

Filtered Gaussian process using KT filter 2

2

2

2

(2 1)( ) exp( ) sin( 1 )

1

exp( )2 cos( 1 )

f f

f f f f f

f

f f f f f f

h t t t

t t

,

0 0

min ( ( ))

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

( ) ( ) dependent ( ; )

independent

with ( ) ( , ) ( ) ( , ) ( ) ( , ) (

ti

ti

P

t

i iP

T t t

syst

sysT t t

sys

V

s t g g t u P i n

f d PP E

P

t t t

d

ss

ρ d

ρ

c p s s sP

c p


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