Reliability Based Design Optimization

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Reliability Based Design Optimization
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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

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Terminologies

• X vector of uncertain variables
• ? vector of certain variables
• T vector of distribution parameters of
  uncertain variable X( means , s.d.)
• d consists of ? and ? whose values can be
  changed
• p consists of ? and ? whose values can not be
  changed

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Terminologies(contd..)

• Soft constraint depends upon ? only.
• Hard constraint depends upon both X(?) and ?
• ?,? d,p
• Reliability 1 probability of failure

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RBDO problem

• Optimization problem
• min F (X,?) objective
• fi (?) gt 0
• gj (X, ? ) gt 0
• RBDO formulation
• min F (d,p) objective
• fi (d,p) gt 0 soft constraints
• P (gj (d,p ) gt 0) gt Pt hard constraints

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Comparison b/w RBDO and Deterministic Optimization
Feasible Region
Reliability Based Optimum
Deterministic Optimum
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Basic reliability problem
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Reliabilty Calculation
Probability of failure
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Reliability Index
Reliability index
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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
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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
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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.

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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.
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Hasofer-Lind reliability index
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RBDO formulations
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Double loop Method
Objective function
Reliability Evaluation For 1st constraint
Reliability Evaluation For mth constraint
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Decoupled method(SORA)
Deterministic optimization loop Objective
function min F(d,µx) Subject to f(d,µx) lt
0 g(d,p,µx-si,) lt 0
k k1 si µxk xkmpp xkmpp ,pmpp
dk,µxk
Inverse reliability analysis for Each limit state
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Single Loop Method

• Lower level loop does not exist.
• min F(µx)
• fi (µx) 0 deterministic constraintsgi (x) 0
• where
• x- µx -ßtas
• agrad(gu(d,x))/grad(gu(d,x))
• µxl µx µxu

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Inner Level Optimization(Checking Reliability
Constraints)

• Performance Measure Approach(PMA)
• min gi ( u,µx )
• subject to u ßt
• If g(u, µx )gt0(feasible)

• Reliability Index Approach(RIA)
• min u
• subject to gi(u,µx)0
• if min u gtßt(feasible)

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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 ß-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.

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MPP estimation
active constraint

• inactive
• constraint
  
  

RIA MPP
PMA MPP
PMA MPP
RIA MPP
U Space
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Methods for reliability computation
Numerical computation of the integral in
definition for large number of random variables
(n gt 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

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FORM First Order Reliability Method
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SORM Second Order Reliability Method
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Gradient Based Method for finding MPP

• find a -grad(uk)/grad(uk)
• uk1ßt a
• If uk1-uklte, stop
• uk1 is the mpp point
• else goto start
• If g(uk1)gtg(uk), then perform an arc search
  which is a uni-directional optimization

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Abdo-Rackwitz-Fiessler algorithm
find
subject to
Rackwitz-Fiessler iteration formula
Gradient vector in the standard space
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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 lt 1, is
the other indication of the point
in the RF formula.
Merit function proposed by Abdo
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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

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Alternate Problem Model
x1
ß1
Reliability based optimum
ß2
x2
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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