reliability-based design optimization using a cell evolution method ~陳奇中教授演講投影片
TRANSCRIPT
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
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