A NEW STUDY ON RELIABILITY-BASED DESIGN...
Transcript of A NEW STUDY ON RELIABILITY-BASED DESIGN...
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A NEW STUDY ON RELIABILITY-BASEDDESIGN OPTIMIZATION
FOR FATIGUE LIFE
Jian Tu, K.K. Choi, Young Ho Park, and Byengdong Youn
Center for Computer Aided Design
College of Engineering
The University of Iowa
Iowa City, IA 52242
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MOTIVATION
z Due to increasing global competitive market, engineering designsare pushed to the limit of the design constraint boundaries usingdesign optimization, leaving very little or no room for tolerances inmodeling and simulation uncertainties and/or manufacturingimperfections.
z Optimum designs obtained without consideration of uncertaintycould lead unreliable or even catastrophic designs.
z The Reliability-Based Design Optimization (RBDO) methodologywill provide not only improved designs but also a confidence rangeof the simulation-based optimum designs -- Robust Designs.
z The fatigue life is very much sensitive to uncertainties in materialproperties, empirical fatigue models, and external loads.
z The proposed method is applicable to general RBDO problems.
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MOTIVATION (cont.)
In ARC Phase I research, a RBDO method was developed using theconventional Reliability Index Approach (RIA) and successfully applied to ashape design of road arm fatigue of an M1A1 tank. However, the requiredcomputational time was extremely large -- optimization of optimization(s).
Function Description Pf = Φ(−β) Pf = Φ(−β) Changes
at “Optimum” 2 RBDO IterationsCost Volume 436.722 in3 447.691 in3 2.5%
Constraint 1 Life at node 1216 0.476% 0.532% 0.056
Constraint 2 Life at node 926 3.24% 0.992% −2.2
Constraint 3 Life at node 1544 3.21% 0.998% −2.2
Constraint 4 Life at node 1519 0.83% 0.721% −0.11
Constraint 5 Life at node 1433 0.023% 0.018% −0.005
InitialDesign
Optim alDesign
Volum e (in3) 515.1 522.1(+1.4% )
Fatigue Life(Hrs)
2189 69623(+3080% )
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RBDO PROBLEM
Gi (X) : performance function
Pfi = Φ(−βti): prescribed failure probability
βti: target safety reliability index for Gi (X)
Φ(•): monotonically increasing Cumulative Distribution
Function (CDF)
Design variables: mean d = [µ1, µ2, …, µn]T of
non-normally distributed random parameter X = [X1, X2, …, Xn]T
min cost f(d)
s.t. P(Gi(X)≤0) ≤Pfi, i=1-m
dL ≤d ≤dU
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RBDO METHODOLOGIES
z Comparison study of the above methods done by Wei Chen &Xiaoping Du, U of Illinois at Chicago, ASME, DETC99/DAC-9565.Concluded that the probabilistic feasibility formulation such as RIA isthe best method.
z Methods Not Requiring Probability and Statistical Analyses� Worst Case Analysis
� Corner Space Evaluation
� Variation Patterns Formulation
z Methods Requiring Probability and Statistical Analyses� Probabilistic Feasibility Formulation -- Distributional input parameters ⇒
Distributional output responses
� Moment matching Formulation -- Simplistic approach to reduce thecomputational cost
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GENERAL PROBABILISTIC CONSTRAINT
P(G(X)≤0) ≤ Φ(−βt)
FG(g) = Φ(−βG): βG is generalized reliability index
FG G( )g = ( ) , x( ) ... ...g P(G(X) ) f dx dx x xn iL
i iU= ≤ II ≤ ≤≤ X x 1x g
Uncertainty of G(X) is characterized by its CDF FG(g) and the randomsystem joint Probability Density Function (PDF) fX(x), with outcomeof interest g, as
So the Probabilistic Constraint becomes
FG(0) ≤ Φ(−βt)
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FG(g) = Φ(−βG) ⇒ βG(g) = −Φ−1(FG(g)) or g(βG) = FG−1
(Φ(−βG))
NONLINEAR g~ββββG RELATIONSHIP
Probabilistic Constraint can be described in two forms as:
βG(0) = −Φ−1(FG(0)) ≥ βt: Reliability Index Approach (RIA)
g(βt) = FG−1(Φ(−βt)) ≥ 0: Performance Measure Approach (PMA)
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FIRST ORDER RELIABILITY METHOD(FORM)
RIA: β β βs G n tf dx dx≡ = − II ≥−≤( ) ( ... ...0 1
1Φ X ( ) )xG(x) 0
FORM : transform random system vector X to an independent,standardized normal vector u: ui = Φ−1(FX(xi))
g g ( ) ) 0G( ) ** ( ) ( ... ...≡ = II ≥−≤β t nF f dx dxG
11X xx gPMA:
These integrations are difficult to evaluate.
β β β β βG FORM G FORMor G( ) ( ) ) ) ( )* *g g g( g(g≈ = ≈ =u u
: Most Probable Point (MPP) for a given g -- RIA
: Most Probable Point (MPP) for a given β -- PMA
u
u
g*
*β
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RELIABILITY AND INVERSE RELIABILITYANALYSES -- MPP SEARCH
z Inverse Reliability Analysis
minimize G(u)
subject to ||u|| = β
z Reliability Analysis
minimize ||u||
subject to G(u) = g
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RIA vs. PMA
RIA -- sensitivity of reliability index at current design dk
β β β β
β β
s s dT
s t
s FORM t
( ) ( ) ( )( )
( )
( )( ),
*
*
d d d d d
d d
≈ + ∇ − ≥
+∇
∇− ≥
k k k
kT
=0
=0
kG
G
d
u
u
u
kg
kg
At active constraint, and PMA & RIA are thesame.
β βs FORM t FORMand, ,k *, kg= = 0
g g g
g G
k k k
*,k T k
* ( ) * ( ) * ( )( )
( )( )*
d d d d d
d d
≈ + ∇ − ≥
+ ∇ − ≥dT
FORM t
0
0d ukβ
PMA -- sensitivity of probabilistic constraint at current design dk
PMA and RIA are two consistent perspectives of the general probabilisticconstraint. However, they are not equivalent in solving RBDO problems.
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RBDO EXAMPLE
Transform to u: X1 = µ1 − 1 + 2Φ(u1) and X2 = µ2 − 1 + 2Φ(u2)
Ga(u) = 2Φ(u1) + 4Φ(u2) + (µ1 + 2µ2 – 13)
Gb(u) = 4Φ(u1) + 2Φ(u2) + (2µ1 + µ2 – 13): both are nonlinear
RBDO Problem:
minimize Cost(d) = µ1 + µ2
subject to P(Ga(X) = X1 + 2X2 – 10 ≤ 0) ≤ 15.87% = Φ(−1)
P(Gb(X) = 2X1 + X2 – 10 ≤ 0) ≤ 2.275% = Φ(−2)
Random variable X = [X1, X2]T, Xi~uniform [µi−1, µi+1], i=1,2
Design variable d = [µ1, µ2]T
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RBDO EXAMPLE USING RIA
Minimize Cost(d) = µ1 + µ2
Subject toβsa(d) ≥ 1
βsb(d) ≥ 2: nonlinear constraints
u u12
22+
G ( ) = 2a u Φ Φ( ) ( ) ( )u u1 2 1 24 2 13 0+ + + − =µ µk k
minimize
subject to
u u12
22+
G ( ) = 4b u Φ Φ( ) ( ) ( )u u1 2 1 22 2 13 0+ + + − =µ µk k
minimize
subject to
d k k k= [ , ]µ µ1 2TAt kth design iteration, , reliability analyses are
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RBDO EXAMPLE USING PMAMinimize Cost(d) = µ1 + µ2
Subject to ga* 0( )d ≥
gb* 0( )d ≥
ga* 0( ) .d = + − ≥µ µ1 22 11 611
gb* 0( ) .d = + − ≥2 12 5521 2µ µ
Using solutions of two inverse reliability analyses, RBDO becomesMinimize Cost(d) = µ1 + µ2
Subject to : linear constraints
At kth design iteration inverse reliability analyses ared k k k= [ , ]µ µ1 2T
u u12
22 1+ =
G ( ) = 2a u Φ Φ( ) ( ) ( )u u1 2 1 24 2 13+ + + −µ µk k
u u12
22 4+ =
G ( ) = 4b u Φ Φ( ) ( ) ( )u u1 2 1 22 2 13+ + + −µ µk k
minimizesubject to
minimizesubject to
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RESULTS OF RBDO ITERATIONSRBDO Cost µµµµ1 µµµµ2
Total Numberof RBDOIteration
Total Number ofReliability or
Inverse ReliabilityAnalyses
PMA 8.055 4.498 3.557 1 2 (Inverse RA)
RIA (SLP) 8.055 4.498 3.557 4 8 (RA)
RIA (SQP) 8.055 4.498 3.557 1 12 (RA)
z It is easier to solve inverse reliability analysis than reliability analysis.
z PMA yields linear probabilistic constraints if the performancefunctions of non-normally distributed random parameter X are linear,whereas RIA converts them to nonlinear probabilistic constraints.
z If the the current design is feasiblewith a large safety margin, reliabilityanalysis may not converge for RIA.
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z (βs, 0): RIA and (βt, g*): PMA
z Required sample size L=10/Φ(−βa) for Monte Carlo methodincreases exponentially as |βa| increases
z Computational effort for RIA is less if βs< βt (infeasible) and PMA isless if βs> βt (feasible)
z In practical applications, selection between RIA and PMA for RBDOshould be based on rate of convergence and computational effort.
ADAPTIVE APPROACH FOR RBDO
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ADAPTIVE APPROACH FOR RBDO (cont.)
Adaptively choose a point (ga, βa) between (0, βs) and (g*, βt) by
ga=αg* and βa= (1−α)βs + αβt
β ββ
ββ
G a G aad
dwith RA to evaluate g and
d
dg( ) ( ) ( ( ))g
gg g F≈ + − = − −
aa
a Φ 1
g gg
g Fg
( ) ( ) ( ( ))ββ
β β ββG
GG a G a
a
G
d
dwith inverse RA toevaluate and
d
d≈ + − = −
aa
a1 Φ
g gg
(( ) ) ,
( )
ββ
β β
β α β αβ α
tG
t
a s t
d
dfor given
PMA if
= + − ≥
= − + ⇒ =
aa
a 0
1 1
Adaptive PMA:
β ββ
β α αG t a
d
dfor given RIA if( ) , *0 0= − ≥ = ⇒ =a
aag
g g gAdaptive RIA:
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ADAPTIVE APPROACH FOR RBDO (cont.)
� If βs>0, then choose βa= βs and use RIA (MPP is closest to theorigin)
ug=0*
z If the probabilistic constraint is violated (βs< βt), then
z If the probabilistic constraint is active (βs= βt), then choose βa= βt
= βs and use PMA ( )u ug o t= ==* *β β
� If βs≤0, then choose βa= 0 and select the origin as MPP
( )u 0β= =0*
z If the probabilistic constraint is inactive (βs> βt>0), then chooseβa= βt and use PMA (MPP is closest to the origin)uβ β= t
*
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CONCLUSIONS AND FUTURE DIRECTIONS
z PMA and RIA are consistent and two extreme cases of the generalprobabilistic constraint.
z Significant differences of RIA and PMA in solving RBDO is shown fora non-normally distributed system. PMA provides better convergence.
z General probabilistic constraint for a linear performance function of thenon-normally distributed random system parameters yields a linearconstraint in the proposed PMA, but it becomes a nonlinear constraintin the conventional RIA.
z Adaptive approach that considers both the convergence rate andcomputational effort will be developed.
z Design sensitivities of a probabilistic constraint of PMA and RIA aredifferent since they are approximate sensitivities. In the unified system,the exact sensitivities can be defined and computed. This exactsensitivities will be derived to provide very efficient RBDO.