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Transcript of RBDO
7/18/2019 RBDO
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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