A New Study on Reliability-based Design Optimization for Fatigue Life

Center for Computer Aided Design

Automotive Research Centerarc

A NEW STUDY ON RELIABILITY-BASEDDESIGN OPTIMIZATION

FOR FATIGUE LIFE

Jian Tu, K.K. Choi, Young Ho Park, and Byengdong Youn


College of Engineering

The University of Iowa

Iowa City, IA 52242



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.



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




InitialDesign

Optim alDesign

Volum e (in3) 515.1 522.1(+1.4% )

Fatigue Life(Hrs)

2189 69623(+3080% )



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



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



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)



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)



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



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



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.



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



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



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



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.



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



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:



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

*



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.

A New Study on Reliability-based Design Optimization for Fatigue Life

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