Reliability Based Design Optimization. Outline RBDO problem definition Reliability Calculation...
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Transcript of Reliability Based Design Optimization. Outline RBDO problem definition Reliability Calculation...
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