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Reliability based design optimization • Probabilistic vs. deterministic design – Optimal risk allocation between two failure modes. • Laminate design example – Stochastic, analysis, and design surrogates. – Uncertainty reduction vs. extra weight. Deterministic design for safety • Like probabilistic design it needs to lead to low probability of failure. • Instead of calculating probabilities of failure use array of conservative measures. – Safety factors. – Conservative material properties. • Tests • Accident investigations – Risk allocation driven by history (accidents). Pro and cons of probabilistic design • Probabilistic design requires more data, that is often not available or expensive to get. • Probabilistic design may require to accept finite probability of death or injury and my lead to legal liabilities. • Probabilistic design may allow more economical risk allocation. • Probabilistic design may allow trading measures for compensating against uncertainty against measures for reducing it. Optimal risk allocation • If there is a single failure mode, the chances are that history has resulted in safety factors that reflect the desired probability of failure. • When there are multiple failure modes it makes sense to have excessive protection against modes that are cheap to protect against. • Adding probabilities : If one mode has failure probability p1 and a second p2, what is the system failure probability if they are independent? Pfsystem 1 (1 p1 )(1 p2 ) p1 p2 p1 p2 p1 p2 Example • An airplane wing weighs 10,000 lb and the tail weighs 1,000 lb. With a safety factor of 1.5, each has a failure probability of 1%, for a total failure probability of 2% (actually 1-0.99^2) • For each component the relation between the probability of failure and additional weight is Pf 0.5100 W /W0 Pf 0 • Reduce the failure probability to 0.5% with minimum weight. • Adding 200 lb to wing and 20 lb to tail reduces the probabilities of each by a factor of 4 for 220 lbs. • Adding 120 lb to the wing and 80 lb to the tail will lead to 0.435% wing failure probability plus 0.004% tail failure probability. Safer and lighter. • What is the optimum? FORM vs. Monte Carlo • FORM is much cheaper, but – Does not give you good estimate of system probability of failure when failure modes are strongly coupled. – Can have large errors when variables are far from normal and limit state have multiple local MPPs. – More difficult to allocate risk. • MCS usually too expensive unless you fit a surrogate to limit state function. Deterministic Design of Composite Laminates • Design of angle-ply laminate – Maximum strain failure criterion minimize h 4t1 t 2 NAxial 2 such that 1 2 12 12u c 1 c 2 0.005 t 2 1 t 1 t 2 0.005 t1 y NHoop x Load induced by internal pressure: NHoop = 4,800 lb./in., NAxial = 2,400 lb./in. . 7 Summary of Deterministic Design • Optimal ply-angles are 27 from hoop direction • Laminate thickness is 0.1 inch • Probability of failure (510-4) is high with safety factor 1.4. . 8 Reliability-based Laminate Design minimize h 4t1 t2 such that P Pt • 4 Design Variables – 1, 2, t1, t2 • 12 Normal Random Variables – – – – – 0.005 t1 0.005 t 2 Pt = 10-4 • . Tzero (CV = 0.03) 1, 2 (CV = 0.035) E1, E2, G12, 12 (CV = 0.035) 1c, 1t (CV = 0.06) 2c, 2t, 12u (CV = 0.09) First ply failure principle 9 Structural & Multidisciplinary Optimization Group Response Surface Options Design response surface approximation (DRS) – Response or Probability v.s. design variables: G=G(d) – Used in optimization Stochastic response surface approximation (SRS) – Response v.s. random variables: G=G(x) – Used in probability calculation. – Need to construct SRS at every point encountered in optimization Analysis response surfaces – Response v.s. random variables + design variables: G=G(x, d) – Advantage: improve efficiency of SRS – Challenge: • Construct RS in high dimensional space ( > 10 variables) [email protected] 10 Structural & Multidisciplinary Optimization Group Analysis Response Surfaces (ARS) Strain • Fit strains in terms of 12 variables • Design of experiments: – Latin Hypercube Sampling (LHS) R.V. Strain = g(θ1, θ2, t1, t2, E1, E2, G12, 12, 1, 2, Tzero, Tservice) D.V. ARS • Probabilities calculated by MCS based on fitted polynomials – Reduce computational cost of MCS [email protected] 11 Structural & Multidisciplinary Optimization Group Reliability-based Design Optimization ARS • Design Response Surface (DRS) – Fit to Probability in terms of 4 D.V. – Filter out noise generated by MCS – Used in RBDO DOE & MCS DRS Probability Optimization No ti i POF = p(θ1, θ2, t1, t2) [email protected] Converge? Yes Stop No 12 Structural & Multidisciplinary Optimization Group Approximation 1 2 t1 (inch) t2 (inch) 20 to 30 20 to 30 0.0125 to 0.03 0.0125 to 0.03 Design variables Range ARS Error Statistics Quadratic ARS based on LHS 182 points 2 in 1 Rsquare Adj. 0.996 Standard error (millistrain) 0.060 Mean of Response (millistrain) 8.322 FCCCD 25 points LHS 252 points quadratic 5th order Rsquare Adj. 0.686 0.998 Standard error (probability) 5.3e-4 0.12e-4 Mean of Response (probability) 3.2e-4 0.44e-4 DRS Error Statistics [email protected] 13 Structural & Multidisciplinary Optimization Group Optimization • Deterministic, Reliability-based, and Simplified designs Ply Angles Thickness (inch) Deterministic [(27.0)2/(27.0)3]S 0.10 Probability of Failure 5e-4 Reliability [(24.9)3/(25.2)3]S 0.12 0.55e-4 Simplified [(25)6]S 0.12 0.57e-4 • The thickness is high for application [email protected] 14 Structural & Multidisciplinary Optimization Group Improving Reliability-based Design • Reliability-based design – Thickness of 0.12 inch – Probability of failure of 10-4 level Must reduce uncertainties: Quality control (QC) – Reject small numbers of poor specimen – Truncate distribution of allowables at lower side (–2 ) Reduce material scatter – Reduce Coefficient of Variation (CV) – Better manufacture process (Better curing process) Improve allowables – Increase Mean Value of allowables – New materials [email protected] 15 Structural & Multidisciplinary Optimization Group Change Distribution of 2 allowable • Reduce scatter (CV) by 10% (0.12 inch) [(25)6]S Probability of failure CV = 0.09 CV = 0.081 0.57e-4 0.11e-4 • Increase allowable (Mean value) by 10% (0.12 inch) [(25)6]S [email protected] Probability of failure E(ε2u) = 0.0154 E(ε2u) = 0.01694 0.57e-4 0.03e-4 16 Structural & Multidisciplinary Optimization Group Quality Control (QC) on 2 allowable • Reduce probability of failure (0.12 inch) [(25)6]S Normal 0.57e-4 Probability of failure Truncate at -3 (14 out of 10,000) 0.001e-4 Truncate at -2 (23 out of 1,000) < 1e-7 0.10 inch Truncate at –2.8 (26 out of 10,000) 0.9e-4 0.08 inch Truncate at -1.35 (90 out of 1,000) 1e-4 • Reduce thickness (POF=1e-4) [(25)6]S [email protected] 0.12 inch Normal 0.57e-4 17 Structural & Multidisciplinary Optimization Group Tradeoff Plot 1.0E+00 Nominal Quality control to -2 Sigma 10% increase in allowable 10% reduction in variability All Series6 Failure Probability 1.0E-01 1.0E-02 1.0E-03 1.0E-04 1.0E-05 1.0E-06 1.0E-07 1.0E-08 0.06 0.08 0.1 0.12 0.14 0.16 Thickness (inch) • To be chosen by the cost of implementing these methods [email protected] 18