Reliability Based Design Optimization

Outline

RBDO problem definition Reliability Calculation Transformation from X-space to u-space RBDO Formulations Methods for solving inner loop (RIA , PMA) Methods of MPP estimation

Terminologies

X : vector of uncertain variables η : vector of certain variables Θ : vector of distribution parameters of

uncertain variable X( means , s.d.) d : consists of θ and η whose values can be

changed p : consists of θ and η whose values can not

be changed

Terminologies(contd..)

Soft constraint: depends upon η only. Hard constraint: depends upon both X(θ)

and η [θ,η] = [d,p] Reliability = 1 – probability of failure

RBDO problem

Optimization problem min F (X,η) objective fi (η) > 0

gj (X, η ) > 0

RBDO formulation min F (d,p) objective fi (d,p) > 0 soft constraints

P (gj (d,p ) > 0) > Pt hard constraints

Comparison b/w RBDO and Deterministic Optimization

Deterministic Optimum

Reliability Based Optimum

Feasible Region

Basic reliability problem

Probability of failure

Reliabilty Calculation

Reliability index

Reliability Index

Formulation of structural reliability problem

x1

x2

0

s

f

g ( x ) = 0

f X ( x ) = const.

Vector of basic random variables

represents basic uncertain quantities that define the state of the structure, e.g., loads, material property constants, member sizes.

Limit state function

Safe domain

Failure domain

Limit state surface

Geometrical interpretation

uR

failure domain

f

S

safe domain

uS

0

limit state surface

Transformation to the standard normal space

Distance from the origin [uR, uS] to the

linear limit state surface

Cornell reliability index

Hasofer-Lind reliability index• Lack of invariance, characteristic for the Cornell reliability index, can be resolved by expanding the Taylor series around a point on the limit state surface. Since alternative formulation of the limit state function correspond to the same surface, the linearization remains invariant of the formulation.

• The point chosen for the linearization is one which has the minimum distance from the origin in the space of transformed standard random variables . The point is known as the design point or most probable point since it has the highest likelihood among all points in the failure domain.

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

reliability index, defined as , is equal to the distance

from the origin of the space (standard normal space) to

the limit state surface.

Geometrical interpretation

u*

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

G3 ( u ) = 0

f

S

failure domain

safe domain

0

u1

u2

Hasofer-Lind reliability index

RBDO formulations

RBDO Methods

Double Loop Decoupled Single Loop

Double loop Method

Objective function

ReliabilityEvaluation

For 1st

constraint

ReliabilityEvaluation

For mth

constraint

Decoupled method(SORA)

Deterministic optimization loopObjective function : min F(d,µx)

Subject to : f(d,µx) < 0g(d,p,µx-si,) < 0

Inverse reliability analysis for Each limit state

dk,µxk

k = k+1si = µx

k – xkmpp

xkmpp ,pmpp

Single Loop Method

Lower level loop does not exist. min { F(µx) }

fi (µx) ≤ 0 deterministic constraintsgi (x) ≥ 0

where

x- µx = -βt*α*σ

α=grad(gu(d,x))/||grad(gu(d,x))||

µxl ≤ µx ≤ µxu

Inner Level Optimization(Checking Reliability Constraints)

Reliability Index Approach(RIA)

min ||u|| subject to gi(u,µx)=0

if min ||u|| >βt(feasible)

Performance Measure

Approach(PMA)

min gi ( u,µx )

subject to ||u|| = βt

If g(u*, µx )>0(feasible)

Most Probable Point(MPP)

The probability of failure is maximum corresponding to the mpp.

For the PMA approach , -grad(g) at mpp is parallel to the vector from the origin to that point.

MPP lies on the β-circle for PMA approach and on the curve boundary in RIA approach.

Exact MPP calculation is an optimization problem. MPP esimation methods have been developed.

MPP estimation

inactive

constraint

active constraint

RIA MPP

PMA MPP

PMA MPP

RIA MPPU Space

Methods for reliability computation

First Order Reliability Method (FORM) Second Order Reliability Method (SORM) Simulation methods: Monte Carlo, Importance

Sampling

Numerical computation of the integral in definition for large number of random variables (n > 5) is extremely difficult or even impossible. In practice, for the probability of failure assessment the following methods are employed:

u 2

G ( u ) = 0

s

f

l ( u ) = 0

*

u*

0 u 1

region of mostcontribution toprobability integral

n ( u,0,I ) = const

FORM – First Order Reliability Method

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*

~

SORM – Second Order Reliability Method

Gradient Based Method for finding MPP

find α = -grad(uk)/||grad(uk)||

uk+1=βt* α

If |uk+1-uk|<ε, stop

uk+1 is the mpp point else goto start If g(uk+1)>g(uk), then perform an arc search

which is a uni-directional optimization

Abdo-Rackwitz-Fiessler algorithm

Rackwitz-Fiessler iteration formula

find

subject to

Gradient vector in the standard space:

where is a constant < 1, is the other indication of the point in the RF formula.

for every and

Convergence criterion

Very often to improve the effectiveness of the RF algorithmthe line search procedure is employed

Merit function proposed by Abdo

Abdo-Rackwitz-Fiessler algorithm

Alternate Problem Model

solution to : min ‘f’ s.t atleast 1 of the reliability constraint is exactly

tangent to the beta circle and all others are satisfied. Assumptions: minimum of f occurs at the aforesaid point

Alternate Problem Model

Reliability based

optimum

β1

β2

x1

x2

Scope for Future Research

Developing computationally inexpensive models to solve RBDO problem

The methods developed thus far are not sufficiently accurate

Including robustness along with reliability Developing exact methods to calculate

probability of failure