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 f i (η) > 0 g j (X, η ) > 0 RBDO formulation min F (d,p) objective f i (d,p) > 0 soft constraints P (g j (d,p ) > 0) > P t 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 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 uR failure domain   f   S safe domain uS uS 0 limit state surface Transformation to the standard normal space Distance from the origin [ u R, u S ] 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

Hasofer-Lind reliability index

RBDO formulations RBDO Methods Double Loop Decoupled Single Loop

Double loop Method Objective function Reliability Evaluation For 1 st constraint Reliability Evaluation For m th constraint

Decoupled method (SORA ) Deterministic optimization loop Objective function : min F(d,µ x ) Subject to : f(d,µ x ) < 0 g(d,p,µ x -s i, ) < 0 Inverse reliability analysis for Each limit state dk,µxkdk,µxk k = k+1 s i = µ x k – x k mpp x k mpp,p mpp

Single Loop Method Lower level loop does not exist. min { F(µ x ) } f i (µ x ) ≤ 0 deterministic constraints g i (x) ≥ 0 where x - µ x = -β t *α*σ α=grad(g u (d,x))/||grad(g u (d,x))|| µ xl ≤ µ x ≤ µ xu

Inner Level Optimization (Checking Reliability Constraints) Reliability Index Approach(RIA) min ||u|| subject to g i (u,µ x )=0 if min ||u|| >β t (feasible) Performance Measure Approach(PMA ) min g i ( u,µ x ) subject to ||u|| = β t If g(u*, µ x )>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 RIA MPP U 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:

FORM – First Order Reliability Method

SORM – Second Order Reliability Method

Gradient Based Method for finding MPP find α = -grad(u k )/||grad(u k )|| u k+1 =β t * α If |u k+1 -u k |<ε, stop u k+1 is the mpp point else goto start If g(u k+1 )>g(u k ), 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 algorithm the 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 β1β1 β2β2 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