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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

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outline
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
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
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
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
Comparison b/w RBDO and Deterministic Optimization

Feasible Region

Reliability Based Optimum

Deterministic Optimum

reliabilty calculation
Reliabilty Calculation

Probability of failure

reliability index
Reliability Index

Reliability index

formulation of structural reliability problem
Formulation of structural reliability problem

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
Geometrical interpretation

uS

Transformation to the standard normal space

failure domain

D f

limit state surface

safe domain

D S

uR

0

Cornell reliability index

Distance from the origin [uR, uS] to the

linear limit state surface

hasofer lind 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.
geometrical interpretation14
Geometrical interpretation

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.

double loop method
Double loop Method

Objective

function

Reliability

Evaluation

For 1st

constraint

Reliability

Evaluation

For mth

constraint

decoupled method sora
Decoupled method(SORA)

Deterministic optimization loop

Objective function : min F(d,µx)

Subject to : f(d,µx) < 0

g(d,p,µx-si,) < 0

k = k+1

si = µxk – xkmpp

xkmpp ,pmpp

dk,µxk

Inverse reliability analysis for

Each limit state

single loop method
Single LoopMethod
  • 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)

Inner Level Optimization(Checking Reliability Constraints)
most probable point mpp
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
MPP estimation

active constraint

  • inactive

constraint

RIA MPP

PMA MPP

PMA MPP

RIA MPP

U Space

methods for reliability computation
Methods for reliability computation

Numerical computationof the integralin 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:

  • First Order Reliability Method (FORM)
  • Second Order Reliability Method (SORM)
  • Simulation methods: Monte Carlo, Importance Sampling
gradient based method for finding mpp
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
Abdo-Rackwitz-Fiessler algorithm

find

subject to

Rackwitz-Fiessler iteration formula

Gradient vector in the standard space:

abdo rackwitz fiessler algorithm28
Abdo-Rackwitz-Fiessler algorithm

Convergence criterion

for every and

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

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

Merit function proposed by Abdo

alternate problem model
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 model30
Alternate Problem Model

x1

β1

Reliability based

optimum

β2

x2

scope for future research
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