Reliability Based Design Optimization

1. Reliability Based Design Optimization

2. 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

3. 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

4. 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

5. 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

6. Comparison b/w RBDO and Deterministic Optimization Feasible Region Reliability Based Optimum Deterministic Optimum

7. Basic reliability problem

9. Reliabilty Calculation Probability of failure

10. Reliability Index Reliability index

11. 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

12. 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

13. 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.

14. 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.

15. Hasofer-Lind reliability index

16. RBDO formulations

17. Double loop Method Objective function Reliability Evaluation For 1st constraint Reliability Evaluation For mth constraint

18. 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

19. 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

20. 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)

21. 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.

22. MPP estimation active constraint • inactive constraint RIA MPP PMA MPP PMA MPP RIA MPP U Space

23. 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

24. FORM – First Order Reliability Method

25. SORM – Second Order Reliability Method

26. 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

27. Abdo-Rackwitz-Fiessler algorithm find subject to Rackwitz-Fiessler iteration formula Gradient vector in the standard space:

28. 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

29. 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

30. Alternate Problem Model x1 β1 Reliability based optimum β2 x2

31. 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