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Reliability Based DesignOptimization

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Outline

RBDO problem definition

Reliability Calculation

Transformation from X-space to u-space RBDO Formulations

Methods for solving inner loop R!" # $M"%

Methods of M$$ estimation

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Terminologies

X & vector of uncertain variables

' & vector of certain variables

( & vector of distribution parameters ofuncertain variable X means # s)d)%

d & consists of * and ' +hose values can be

changed

p & consists of * and ' +hose values can not

be changed

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Terminologies(contd..)

,oft constraint& depends upon ' only)

ard constraint& depends upon both X*%

and '

.*#'/ 0 .d#p/

Reliability 0 1 2 probability of failure

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RBDO problem

Optimization problem min F X#'% ob3ective

f i '% 4 5

g 3 X# ' % 4 5

RBDO formulation

min F d#p% ob3ective

f i d#p% 4 5 soft constraints

$ g 3 d#p % 4 5% 4 $t hard constraints

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Comparison b/w RBDO and Deterministic Optimization

Deterministic Optimum

Reliability Based Optimum

Feasible Region

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Basic reliability

problem

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$robability of failure

Reliabilty Calculation

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Reliability inde6

Reliability Inde

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!ormulation o" structural reliabilityproblem

x1

x2

0

Ω s

Ω f

g( x ) = 0

f X ( x ) = const.

Vector of basic random variables

represents basic uncertain 7uantities that define the state of the

structure# e)g)# loads# material property constants# member si8es)

Limit state function

Safe domain

Failure domain

Limit state surface

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#eometrical interpretation

u R

failure domain

∆ f

∆ S

safe domain

uS

0

limit state surface

Transformation to thestandard normal space

Distance from the origin .u R# uS / tothe

linear limit state surface

Cornell reliability inde6

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$aso"er%&ind reliability inde 9 :ac; of invariance# characteristic for the Cornell reliability

inde6# can be resolved by e6panding the Taylor seriesaround a point on the limit state surface) ,ince alternative

formulation of the limit state function correspond to the same

surface# the lineari8ation remains invariant of the formulation)

9 The point chosen for the lineari8ation is one +hich has theminimum distance from the origin in the space of

transformed

standard random variables ) The point is ;no+n as the

design point or most probable point since it has the highest li;elihood among all points in the failure domain)

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For the linear limit state function# the absolute value of the

reliability inde6# defined as # is e7ual to the distance

from the origin of the space standard normal space% to

the limit state surface)

#eometrical interpretation

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δ ∗ = β

u*

G1( u ) = 0G2 ( u ) = 0

G3 ( u ) = 0

∆ f

∆ S

failure domain

safe domain

0

u1

u2

$aso"er%&ind reliability inde

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RBDO "ormulations

RBDO Methods

Double Loop Decoupled Single Loop

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Double loop 'etod

Ob3ective

function

Reliability

<valuationFor 1st

constraint

Reliability

<valuationFor mth

constraint

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Decoupled metod(OR* )

Deterministic optimi8ation loop

Ob3ective function & min Fd#=6%

,ub3ect to & fd#=6% > 5

gd#p#=6-si#% > 5

!nverse reliability analysis for

<ach limit state

d;#=6;

; 0 ;?1

si 0 =6; 2 6;

mpp

6;mpp #pmpp

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ingle &oop

'etod

:o+er level loop does not e6ist) min @ F=6% A

f i =6% 5 deterministic constraints

gi 6% 5 +here

x- =6 0 -tEEG

0gradgud#6%%HIIgradgud#6%%II =6l =6 =6u

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Inner &e+el Optimization(Cec,ing Reliability Constraints)

Reliability !nde6

"pproachR!"%

min IIuII

sub3ect to giu#=6%05

if min IIuII 4tfeasible%

$erformance Measure

"pproach$M"%

min gi u#=6 %

sub3ect to IIuII 0 t

!f guE# =6 %45feasible%

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'ost -robable -oint('--)

The probability of failure is ma6imum corresponding

to the mpp)

For the $M" approach # -gradg% at mpp is parallel

to the vector from the origin to that point) M$$ lies on the -circle for $M" approach and on

the curve boundary in R!" approach)

<6act M$$ calculation is an optimi8ation problem)

M$$ esimation methods have been developed)

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'-- estimation

inactive

constraint

activeconstraint

RI

!""

"! !""

"! !""

RI !""

U Space

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'etods "or reliabilitycomputation

First Order Reliability Method FOR!%

,econd Order Reliability Method SOR!%

,imulation methods& !onte #arlo# Importance Sampling

$umerical computation of t%e integral in definition forlarge number of random variables n 4 J% is e6tremely difficult

or even impossible) !n practice# for the probability of failure

assessment the follo+ing methods are employed&

!OR' !i t O d R li bilit

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u 2

G ( u ) = 0

∆ s

∆ f

l ( u ) = 0

δ *

u*

0 u 1

region of most

contribution toprobability integral

ϕ n ( u ,0 , I ) = const

!OR' !irst Order Reliability 'etod

OR' d O d R li bilit

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u 2

Gv ( v ) = f v

( v ) – v n = 0

∆ s

∆ f

0 u 1

v n≡ v n

v ~

~

v n = sv ( v )

v *

′

~

OR' econd Order Reliability 'etod

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#radient Based 'etod "or nding '--

find 0 -gradu;%HIIgradu;%II

u;?10tE

!f Iu;?1

-u;

I>K# stop u;?1 is the mpp point

else goto start

!f gu;?1%4gu;%# then perform an arc search+hich is a uni-directional optimi8ation

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*bdo%Rac,witz%!iessleralgoritm

Rac;+it8-Fiessler iteration formula

find

sub3ect to

Lradient vector in the standard space&

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+here is a constant > 1# is the other indication

of the point in the RF formula)

for every and

Convergence criterion

ery often to improve the effectiveness of the RF algorithm

the line search procedure is employed

Merit function proposed by "bdo

*bdo%Rac,witz%!iessleralgoritm

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*lternate -roblem 'odel

solution to &

min Nf s)t

atleast 1 of the reliability constraint is e6actly

tangent to the beta circle and all others are satisfied) "ssumptions&

minimum of f occurs at the aforesaid point

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*lternate -roblem 'odel

Reliability based

optimum

&'

&(

x'

x(

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cope "or !uture Researc

Developing computationally ine6pensive

models to solve RBDO problem

The methods developed thus far are not

sufficiently accurate !ncluding robustness along +ith reliability

Developing e6act methods to calculate

probability of failure