Deciding How Conservative A Designer Should Be: Simulating Future Tests and Redesign

1. Deciding How Conservative A Designer Should Be: Simulating Future Tests and Redesign Nathaniel Price1 Taiki Matsumura1 Raphael Haftka1 Nam-Ho Kim1 University of Florida, Gainesville, FL1

2. Introduction Methods Results Conclusion Presentation Outline • Introduction • Design, testing, redesign and calibration • Methods • Simulating Future Test & Future Redesign • Analytical Techniques To Remove Sampling Noise • Results • Two Redesign Strategies • Redesign to increase performance (reduce mass) • Redesign to improve safety • Discussion & Conclusion 2 / 19

3. Introduction Methods Results Conclusion Tests Prescribe Redesign and Calibration • High levels of epistemic uncertainties at the design stage would exact large weight penalties to compensate for • Tests can be used to reduce epistemic uncertainties • These tests are used to: • Calibrate analysis models and improve their accuracy • Prescribe redesign if tests indicate that design is unsafe or too conservative • There is a tradeoff between weight and redesign costs, in that conservative design is heavier but less likely to require redesign • However, this tradeoff is currently implicit and based on experience at a company level 3 / 19

4. Introduction Methods Results Conclusion Two Redesign Strategies • Aircraft designers face two challenges • How to maximize safety while minimizing mass • How to avoid high probability of redesign • Two redesign strategies • Redesign for Performance • Start with a conservative design (higher safety factor) • Redesign if test reveals that design is too conservative (too heavy) • Redesign for Safety • Start with a less conservative design (lower safety factor) • Redesign if test reveals that design is unsafe (safety factor is too low) • Future test & redesign Time Design Test Redesign 4 / 19

5. Introduction Methods Results Conclusion P A Simplest Demonstration Example • A solid bar with circular cross section under axial loading • Uncertain Parameters 5 / 19

6. Introduction Methods Results Conclusion Design & Redesign Procedure • Deterministic design optimization (DDO) is performed to minimize mass subject to a stress constraint • A single test is performed and the test will be passed if the measured factor of safety is within the lower and upper safety factor limits, SLandSU • The test result is more accurate than the calculation and therefore we can calibrate our model using the test result Sini SU SL Sre • Using the updated calculation we may wish to redesign for a new safety factor, Sre 6 / 19

7. Introduction Methods Results Conclusion Simulating a Future Test & Possible Redesign • There are errors in calculated and measured stresses • We can combine the above equations to simulate a test result (assuming we know the calculation and measurement errors) Future test: treat ecalc & emeas as a random variable • We can use Monte Carlo Simulation (MCS) of errors to generate a distribution of possible future test results • For n pairs of error samples we obtain npossible futures • For each possible future we can use first order reliability method (FORM) to calculate true probability of failure • For our simple problem an analytical solution of FORM provides exact probability of failure 7 / 19

8. Introduction Methods Results Conclusion Analytical Method to Eliminate MCS Sampling Errors • Joint PDF of errors (independent) • Heaviside functions to model the redesign event (indicator function) • After redesign using (Sini, SL, SU, Sre) • Expectations for these functions are easily calculated 8 / 19

9. Introduction Methods Results Conclusion Optimization of Safety Factors & Redesign Rules • For an individual designer, the design problem is deterministic • Safety factors are determined by regulations and additional company safety margins • However, a design group seeks the optimum set of rules to balance performance (mass) against probability of redesign • We formulate the following multi-objective objective optimization problem to minimize mass (area) and probability of redesign • Constraint on probability of redesign is varied to capture Pareto Front of optimal designs Formulation: Redesign for Performance Redesign for Safety 9 / 19

10. Introduction Methods Results Conclusion Optimization of Discrete Cases • 4 possible future scenarios • ecalc = +30%(conservative) or -30%(unconservative) • emeas = +10% or -10% • Pre= 50% 10 / 19

11. Introduction Methods Results Conclusion Graphical optimization • Redesign for safety: Start with low Sini and redesign with high Sre • Redesign for performance: Below Sini = 1.45, PF constraint cannot be satisfied 11 / 19

12. Introduction Methods Results Conclusion Distribution of mass after redesign • Redesign for performance reduces average mass by 25.2 • Redesign for safety increase average mass by 32.6 For performance For safety 12 / 19

13. Introduction Methods Results Conclusion Distribution of PF after redesign • Redesign for performance can reduce mass with almost no penalty in reliability • Redesign for safety has larger impact on reliability For performance For safety 13 / 19

14. Introduction Methods Results Conclusion Discrete Error Simulation (4 cases) • Redesign for performance yields about 3% lower average mass • Redesign for safety significantly improves PF 14 / 19

15. Introduction Methods Results Conclusion Tradeoff Curve: ±30% Calculation Error / ±10% Measurement Error • ecalc ~ U[-0.3, 0.3], emeas ~ U[-0.1, 0.1] • Redesign for performance yields about 3% lower mass Pre = 20% 114 Performance 112 Safety 110 108 Mean Area, E(A) (mm2) 106 104 102 100 0 10 20 30 40 50 Probability of Redesign, Pre(%) 15 / 19

16. Introduction Methods Results Conclusion Mass distribution after redesign • Too conservative designs are redesigned to save mass (a large reduction in mass) • Unconservative designs are redesigned to satisfy required PF For safety For performance 16 / 19

17. Introduction Methods Results Conclusion PF distribution after redesign • Redesign for safety (right) significantly changes low PF, while the change is relatively small for performance For safety For performance 17 / 19

18. Introduction Methods Results Conclusion Conclusions • Starting with a conservative initial design pays off on average on the long run • Redesign to reduce mass is like winning the lottery • There is a small chance of very large mass reduction when redesigning for weight, but most designs are heavier • Poor impression on initial design: a large mass change in redesign • Redesign to improve safety will often result in lower mass designs and when redesign is needed the increase in mass will be small • Good impression on initial design: design may fail but requires small change • But for company’s point of view, conservative design is better • Redesign for performance yields less average weight 18 / 19

19. Introduction Methods Results Conclusion THANK YOU Questions? 19 / 19