Reliability-Based Design Optimization via a Cell Evolution Method

逢甲大學化工系

陳 奇 中ctchen@fcu.edu.tw

Outline

1. Introduction2. Reliability-Based Design Optimization (RBDO) 2.1 Problem formulation 2.2 Traditional solution methods for RBDO

- Double Loop - Single Loop

3. A Cell Evolution Method for RBDO 3.1 Single objective optimization 3.2 Multi-objective optimization

4. Design Examples5. Conclusions

1

2

1 2

1 2

( , )min

s.t.

( , ) 0, 1, ,

( , ) 0, 1, ,

where

, , , : decision variables

, , , : parameters

i

j

L U

Tm

Tl

f

g i n

h j n

d d d

p p p

d

d p

d p

d p

d d d

d

p

Deterministic Design Optimization

- no uncertainties involved in the design

Introduction

Uncertainties ?

Sources of uncertainties

- modeling errors- physical parameter

variations- change of environments- unknown dynamics

…Deterministic design Not reliable

uncertainties

Uncertainty is

everywhere.

Optimal Design Under Uncertainties

1

2

1 2

1 2

1 2

determinist

( , )min

s.t.

( , ) 0, 1, ,

( , ) 0, 1, ,

,

where

, , , : dic ecision variable

, , , : decision variable

, , , : parameters

i

j

L U L U

m

n

l

f

g i n

h j n

d d d

x x x

p p p

x,d

x,d p

x,

uncertain

uncertain

d p

x,d p

x x x d d d

d

x

p

Deb et al. (2009)

Deterministic solution vs. Reliable solution

*Deterministic optimum

Reliable solution

Stochastic constraint

Diwekar (2002)

Stochastic Programming frameworks- Here and Now (1/2)

Optimal solution

Stochastic Programming frameworks- Wait and See (2/2)

Diwekar (2002)

Distribution of optimal design

Objective function and constraints

(Scenario)

Reliability-Based Design Optimization (RBDO)

,

1

2

min ( , , )

s.t.

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

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

,

i i

j

L U L U

f

G R i n

g j n

xx p

d μ

x p

x x x

d μ μ

d x p

d μ μ

d d d μ μ μ

~ ,,,n NR x xxx μ σ

~ ,,,q NR p ppp μ σ

Pr( ) Probability function

iR Design reliability

where

The failure probability and reliability index

,( , , ) 0Pr ( , , ) 0 ( , )

ii G

G d d

x pd x pd x p x p x p

, ( , ) x p x p joint probability density function

Reliability level 1i iR P

Failure probability Pr ( , , ) 0i iP G d x p

i iP First-order approximation

iReliability index Standard normal cumulative dist. Func.

Traditional solution methods for RBDO - Double-loop method

Shan and Wang (2008)

(1/2)

Optimization loop

Reliability analysis loop

Reliability analysis loop

Reliability analysis loop

Reliability analysis loop (inner loop) (1/2)

A. RIA (reliability index approach)

s.t.

min

0jG

U

U

U

*for reliability: ,NOTE jU

MPP

NOTE: MPP denotes the “most probable point.”

jG > 0

Reliability analysis loop (inner loop) (2/2)

B. PMA (performance measure approach)

1

s.t.

where

"reliability index"

standard normal density function

: U-space, ~ (0,

min ( )

1)

j

j

j

jR

N

G

U

U

U MPP

*for reliability: ,TE 0NO .iG U

jG > 0

,

22

22

min ( , , )

s.t.

( , ) 0, 1,2, ,i i i

r ii i

i i

iri i

i i

L U

L U

f

g i n

G

G G

G

G G

x

x pd μ

xx

x p

pp

x p

X X X

d μ μ

d x ,p

x

p

d d d

μ μ μ

- convert inner reliability loop by using a deterministic optimization problem KKT optimality conditions

Traditional solution methods for RBDO - Single-loop method

(2/2)

approximation

Comparisons of RBDO Solution methods

Method Advantage Disadvantage

Double-loop accuracy long computation time

Single-loop computationally fast less accuracy

Motivation: accuracy and computational efficiency? New solution method ?

PMA-based RBDO problem

,

* 11

2

min ( , , )

s.t.

0, 1,...,

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

,

ii G i

j

L U L U

f

G F i n

g j n

xx pd μ

x p

x x x

d μ μ

d μ μ

d d d μ μ μ

iGF

where

cumulative distribution function

*iG Calculated from PMA reliability optimization problem

MPP

jG > 0

Reliability-test cells- Determination of MPPs

1=0G

2 =0G

3 =0G

31mpp 32mpp

33mpp

11mpp12mpp

13mpp

21mpp22mpp

23mpp

1x

2x

A cell generation method

Step 1: Sobol quasi-random sequence (Sobol, 1967; Bratley and Fox, 1988)

Step 2: Spherical parameterization method (Watson, 1983; Zayer et al., 2006)

--- sampling method

Some template reliability-test cells (1/2) 2D cells in U-space

β 1, 100N β 1, 1000N

β 3, 100N β 3, 1000N

Some template reliability-test cells (2/2) 3D cells in U-space

β 1, 1000N

β 3, 1000N

β 1, 10000N

β 3, 10000N

RS Operation

DBX Operation

Stop criteria met?

k = k+1

No

Start

Stop

Initialize cell population

Replacement Operation

Yes

Std.( F(Ɵ) ) ≤ ε ?Yes

NoAlleviate premature

stagnation

DRM Operation

For each paired parents, r > λ ?

Yes

No

A cell evolution algorithm

Cell generation

1=0G

2 =0G

3 =0G

31mpp 32mpp

33mpp

11mpp12mpp

13mpp

21mpp22mpp

23mpp

1x

2x

A real-coded genetic algorithm(Chuang and Chen, 2011)

+

What is genetic algorithm (GA)?

GA is a particular class of evolutionary algorithm Initially developed by Prof. John Holland "Adaptation in natural and artificial systems“, University of Michigan press, 1975

Based on Darwin’s theory of evolution

“Natural Selection” & “Survival of the fittest”

Imitate the mechanism of biological evolution - Crossover - Mutation

- Reprodution

物競天擇 適者生存 不適者淘汰

Organisms produce a number of offspring similar to themselves but can have variations due to:

(a) Crossover (Sexual reproduction )

Evolution in biology (1/3)

Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf

Parents offspring

IMG from http://www.tulane.edu/~wiser/protozoology/notes/images/ciliate.gif

(b) Mutations (Random changes in the DNA sequence)

Evolution in biology (2/3)

Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf

Before After

IMG from http://www.tulane.edu/~wiser/protozoology/notes/images/ciliate.gif

IMG from http://offers.genetree.com/landing/images/mutation.png

Some offspring survive, and produce next generations, and some don’t:

Evolution in biology (3/3)

Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf

http://www.ugobe.com/Home.aspx

Ugobe Inc. Pelo

All variables of interest must be encoded as binary digits (genes) forming a string (chromosome).

Gene – a single encoding of part of the solution space.

Chromosome – a string of genes that represent a solution.

Traditional GA- binary-coded

IMG from http://static.howstuffworks.com/gif/cell-dna.jpg

1

1 1 0 1 0

gene

chromosome

All genes in chromosome are real numbers- suitable for most systems.

- genes are directly real values during genetic

operations. - the length of chromosomes is shorter than that in

binary-coded, so it can be easily performed.

Real-coded GA (RCGA)

1.1

1.1 0.1 15 10 0.12

gene

chromosome

IMG from http://static.howstuffworks.com/gif/cell-dna.jpg

The cell evolution method- Survival and elimination of cells according to their fitness

Illustrative examples- Example 1 (Liang et al., 2004)

1 2min f

21 2

1

2 2

1 2 1 22

21

1 2

1

23

120

5 121

30 12080

18 5

0 10, 1,2

0.3,

3, 1,

Pr ( ) 0

2,

,3

3

, 1, 2i i

i

jj

x xG

x x x xG

Gx x

i

j

G R i

R

x

x

x

x

Methods DLP/PMAa Single loopb

The Proposed

Design variables

     

3.4391 3.4391 3.4391  

3.2866 3.2864 3.2866  

Objective function

       

6.7257 6.7255 6.7257  

Constraints        

0 0 0  

0 0 0  

-0.5 -0.5097 -0.5096  

CPU time (s) 138 8.89 11.76aResults are from Du and Chen [8]. bResults are from Liang et

al. [7]. 

1

2

f μ

1( )G x

2( )G x

3( )G x

Results Comparison

MPP determination using different sampling numbers

 

Sampling Number

MPP points

MPP1 MPP2 MPP3

50 (2.6173, 2.9168) (3.7578, 2.4438) (4.0812, 3.9152)

100 (2.6168, 2.9179) (3.7573, 2.4446) (4.0807, 3.9161)

500 (2.6179, 2.9182) (3.7581, 2.4450) (4.0819, 3.9165)

1000 (2.6179, 2.9182) (3.7581, 2.4450) (4.0819, 3.9165)

5000 (2.6179, 2.9182) (3.7581, 2.4450) (4.0819, 3.9165)

10000 (2.6179, 2.9182) (3.7581, 2.4450) (4.0819, 3.9165)

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

1

2Example 4.1

Obtained solution cells with different reliability indices (0,1,2,3)

Illustrative examples- Example 2

21 2

1

2 2

1 2 1 22

21

1 2

1

23

120

5 121

30 12080

18 5

0 10, 1,2

0.3,

3, 1,

Pr ( ) 0

2,

,3

3

, 1,2i i

i

jj

x xG

x x x xG

Gx x

i

j

G R i

R

x

x

x

x

1min f Reliability index, β

0 (0%) 7.7883 1.7928

0.5 (69.146%) 7.4476 2.1224

1 (84.134%) 7.1146 2.4269

1.5 (93.319%) 3.2346 2.6961

2 (97.725%) 3.2949 2.8974

2.5 (99.379%) 3.3634 3.0941

3 (99.875%) 3.4391 3.2866

21

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

1

2

Example 4.2

Solution cells with different reliability indices (0, 0.5, 1, 1.5, 3)

The dramatic change of the reliable solution with respect to reliability indices

Reliability index

0 (0%)

0.5 (69.146%)

1 (84.134%)

1.5 (93.319%)

2 (97.725%)

2.5 (99.379%)

3 (99.875%)

4 (99.996%)

5 (99.999%)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 53

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

β

μ2μ

1

1 2

r 1

2

min , , , , , , , , ,

s.t.

P ( ( , , ) 0) , 1, 2, ,

( , , ) 0 , 1,2, ,

k

i i

j

L U

L U

G R i n

g j n

x p x p x p

x p

x x x

f d μ μ f d μ μ f d μ μ

d x p

d

d d d

μ μ μ

Multi-objective reliability-based design optimization

~ ,,,q NR p ppp μ σ

~ ,,,n NR x xxx μ σ

Concept of multi-objective optimization

Cost (US$)

Com

fort

10 k 100 k

40%

90%

f 2

f1

Feasible objective space

Pareto-optimal front

Second level

Concept of Pareto-optimal solutions: non-dominated

A

B

CD

B dominate A

C dominate A

B, C non-dominated

D, E non-dominated

E dominate A, B, C

D dominate A, BE

(Goldberg, 1989)

Parents

Offspring

1

1

N

N

2

2

Non-dominatedsorting

Front 1

Front 2 N

Rejected

Crowding distance sorting for each front

Front 1

Front 2

Front 3

New Population

RCGA

Front 3 Front 3

Front 1

Front 2

How does multi-objective cell evolution algorithm work?

CAT

An illustrative example- Multi-objective RBDO (Deb et al., 2009)

1 1

22

1

r

1 2 1

2 2 1

1 2

min

1min

s.t.

P ( ( , , ) 0) , 1,2

9 6

9 1

0.1 1 , 0 5

0.03 , 1.28,2.0,3.0

i i

f x

xf

x

G R i

G x x

G x x

d x p

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

8

9

10

f1

f 2

= 0 = 1.28 = 2 = 3

Pareto front for the RBDO problem

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

X1

X2

= 0 = 3 = 1.28 = 2

Solutions for the RBDO problem

Reliability-based design optimizationApplications in Chemical Engineering

1. Steam pipe design2. Design of a bio-process3. Heat sink design

2 22 1

r 1 2

1 2

1 2

1 2 4 42 2 2 2

2 1

2

2

1/6

8/279/16

2 2

( )min

4s.t.

P , 0

, , 0

0.04 0.065 , 0.075 0.12

2: 2 2

ln /

2

0.3870.6

1 0.559 /

( )(2 )

j

eq

eq

D

DD

D

r rf

G r r R

h r r K

r m r m

K T Th h r T T r C T T

r r

Kh Nu

r

RaNu

gB T T rRa

3

8

2

2, 5.67 10

v

B CT T

Steam pipe design (Ho and Chan, 2011)

1r

2r

Steam

T

2T

1T

Surrounding temperature

Min. cost

0 0.5 1 1.5 2 2.5 38.9

9

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8x 10

-3

Reliability

Optim

al fu

nctio

n va

lue

Reliable solutions

r

1

2 0

3

4 0

s.t.

P ( ( ) 0) , 1 ~ 4

:5 15

:20 50

: 50 300

: 0.0

max

5 1.0

max /

f

f B f

i i

B

L

G R i

G t

G S

G K a

P

P t

G X

S

d,x,p

Design of a bio-process (Holland, 1975)

微生物濃度

葡萄糖酸

葡萄糖酸內酯

葡萄糖基質

氧氣溶解

cos

cos Cells

Cells Glu e Oxygen More cells

Glu e Oxygen Gluconolactone

Gluconolactone Water Gluconic Acid

Reliable solutions

11

1

hs m fins

bm

fins

c fin bp

cc c

R R R

tR

kA

RN

R R R

Rh A

Thermal analysis

0.05531.09/Re1

1 1.1

11.009(

45.78{0.233 }

( 1) R

)1

e

DT

L

T D

K

K

f

SS

S

1

tanh( )

1

4

finfin fin fin

fin

bpbp bp

fin

Rh A

mH

mH

Rh A

hm

kD

Nussult Number correlation

app T Tm U N HD S

0.785 0.212

1 0.5

1/2 1/31

[0.2 exp( 0.55 )]

Re

1)

Pr

(

finfin D

f

T T L

T

C

h DNu C

k

S S SS

Friction factor correlation

Mass balance

Design of cylindrical heat sinks

- in-line (Khan et al., 2004)

11

1

hs m fins

bm

fins

c fin bp

cc c

R R R

tR

kA

RN

R R R

Rh A

1.2913.1/ 0.68/1

0.08071 0.3124

(378.6 / ) / Re

1.175( ) 0.5 ReRe

T TT D

LD

T D

f K

K

S SS

SS

1

tanh( )

1

4

finfin fin fin

fin

bpbp bp

fin

Rh A

mH

mH

Rh A

hm

kD

Nussult Number correlation

app T Tm U N HD S 0.5

1/2 1/

91 0.053

1

31

0.5

0.61

( 1) (1 2 exp( 1.09 )

r

)

Re Pfinf

T

T

n

L

T

i Df

C

h DNu C

k

S SS S

Friction factor correlation Mass balance

Design of cylindrical heat sinks

- staggered (Khan et al., 2004)Thermal analysis

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 10-3

1

1.5

2

2.5x 10

-3 For in-line H=0.01m Uapp

=2 m/s N=7x7

D (m)

Sge

n (W

/K)

Tamb=300 K

Tamb=320 K

Tamb=340 K

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 61.5

2

2.5

3

3.5

4

4.5

5x 10

-3 For in-line H=0.01m D=0.001m N=7x7

Uapp

(m/s)

Sge

n (W

/K)

Tamb=300 K

Tamb=320 K

Tamb=340 K

Heat sink performance variations under change of environmental

temperature (in-line arrangement)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

x 10-3

1

1.5

2

2.5

3

3.5x 10

-3 For staggered H=0.01m Uapp

=2 m/s N=7x7

D (m)

Sge

n (W

/K)

Tamb=300 K

Tamb=320 K

Tamb=340 K

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 62

2.5

3

3.5

4

4.5

5x 10

-3 For staggered H=0.01m D=0.001 m N=7x7

U app

(m/s)

Sge

n (W

/K)

Tamb=300K

Tamb=320K

Tamb=340K

Heat sink performance variations under change of environmental

temperature (staggered arrangement)

in-line staggered

160 180 200 220 240 260 280 300 320 3402

3

4

5

6

7

8x 10

-3

Nu熱傳係數Dfin

Sgen

(W

/K)

For in-line H=0.006m N=5x5

Uapp

=2

Uapp

=4

Uapp

=6

200 250 300 350 400 4502

2.5

3

3.5

4

4.5

5

5.5

6

6.5x 10

-3

Nu熱傳係數 Dfin

Sgen

(W

/K)

For staggered H=0.006m N=5x5

Uapp=2

Uapp=4

Uapp=6

Heat sink performance variations under un-uniform heat transfer

between fins

2min ( )

gen hsamb amb

Q m PS R

T T

RBDO problem formulationSingle objective

s.t. P 0 , 1~ 9r i iG X R i

6 ( ) 12

1 ( ) 3

1 ( / ) 6

5 20

app

H mm

D mm

U m s

N

0.1

Cell population size 100 、 max. gen.100 、Sampling no. 10000

Uncertain parameter Uncertain environmental temp.

Entropy generation rate

β 0 0.5 1 1.5 2 2.5 3 3.5

N 18 18 13 11 10 8 7 6

H(m) 0.0080 0.0072 0.0091 0.0097 0.0096 0.0119 0.0120 0.0120

D(m) 0.0010 0.0010 0.0013 0.0015 0.0016 0.0020 0.0022 0.0026

Uapp(m/s)

1 1 1.1791 1.5281 1.8884 2.0699 2.4012 2.7829

Sgen(W/K)

X 100

0.0535 0.0555 0.0578 0.0696 0.0727 0.0830 0.0929 0.1060

Reliable solutions(in-line)

β 0 0.5 1 1.5 2 2.5 3 3.5

N 17 17 17 17 13 11 9 9

H(m) 0.0080 0.0076 0.0073 0.0070 0.0091 0.0105 0.0120 0.0120

D(m) 0.0010 0.0010 0.0010 0.0010 0.0013 0.0016 0.0019 0.0019

Uapp(m/s)

1 1 1 1 1 1 1 1.0824

Sgen(W/K)

X 100

0.0472 0.0479 0.0480 0.0495 0.0532 0.0567 0.0629 0.0646

Reliable solutions(staggered)

Optimal entropy generation rate with respect to reliability indices

0 0.5 1 1.5 2 2.5 3 3.55

6

7

8

9

10

11

12x 10

-4

Sg

en

(W

/K)

0 0.5 1 1.5 2 2.5 3 3.54.5

5

5.5

6

6.5x 10

-4

Sg

en

(W

/K)

in-line staggered

Heat dispersion comparisons(in-line; air velocity 0.7m/s)

Reliable design with β=3Deterministic design

(322.2 < T< 329.9) (314.5 < T< 318.1)

Heat dispersion comparisons(staggered; air velocity 0.7m/s)

Deterministic design Reliable design with β=3

(321.0 < T< 323.6) (312.3 < T< 315.9)

6 ( ) 12

1 ( ) 3

1 ( / ) 6

5 20

app

H mm

D mm

U m s

N

2( )min

$

gen hs

amb amb

Q m PS R

T T

Cost Volume

s.t. P 0 , 1~ 9r j jG X R j

0.1

Uncertain parameter Uncertain environmental temp.

RBDO problem formulationMulti-objective

Cell population size 100 、 max. gen.100 、Sampling no. 10000

Entropy generation rate

Cost

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Sgen (W/K)

Co

st

(NT

D)

Deterministic = 1.28 = 3

Obtained Pareto front of the reliable design(in-line)

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.0450.8

1

1.2

1.4

1.6

1.8

2

2.2

Sgen (W/K)

Co

st

(NT

D)

Deterministic = 1.28 = 3

Obtained Pareto front of the reliable design(staggered)

Solutions

Deterministic design Reliable design (β=3)

min Sgen min. cost min Sgen min. cost

Sgen(W/K) 0.0040 0.0363 0.0101 0.0396

Cost (NTD) 1.31 0.93 1.05 0.90

Solutions

Deterministic design Reliable design (β=3)

min Sgen min. cost min Sgen min. cost

Sgen(W/K) 0.0018 0.0078 0.0035 0.0423

Cost (NTD) 2.07 1.09 1.49 0.90

in-line

staggered

Results comparison

Single- and multi-objective cell evolution methods have been developed for reliability-based design optimization.

Simulation results reveal that the proposed method is able to achieve accurate solution for RBDO without sacrificing computational efficiency.

Application examples indicate the proposed cell evolution method is a promising approach to chemical process design under uncertainties.

Conclusions

Q & A

謝謝聆聽敬請指教