Structural & Multidisciplinary Optimization Group Deciding How Conservative A Designer Should Be: Simulating Future Tests and Redesign Nathaniel Price 1 Taiki Matsumura 1 Raphael Haftka 1 Nam-Ho Kim 1 University of Florida, Gainesville, FL 1

Structural & Multidisciplinary Optimization Group 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 IntroductionMethodsResultsConclusion 2 / 19

Structural & Multidisciplinary Optimization Group 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 IntroductionMethodsResultsConclusion 3 / 19

Structural & Multidisciplinary Optimization Group 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 DesignTestRedesign IntroductionMethodsResultsConclusion 4 / 19

Structural & Multidisciplinary Optimization Group Simplest Demonstration Example  A solid bar with circular cross section under axial loading  Uncertain Parameters DescriptionClassificationDistributionSymbolC.O.V / Bounds (%) Applied LoadAleatoryNormalP0.20 Material StrengthAleatoryNormalσ allow 0.12 Calculation ErrorEpistemicUniforme calc ±30 Measurement ErrorEpistemicUniforme meas ±10 P A IntroductionMethodsResultsConclusion 5 / 19

Structural & Multidisciplinary Optimization Group 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, S L and S U  The test result is more accurate than the calculation and therefore we can calibrate our model using the test result  Using the updated calculation we may wish to redesign for a new safety factor, S re S ini S U S L S re IntroductionMethodsResultsConclusion 6 / 19

Structural & Multidisciplinary Optimization Group 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)  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 n possible 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 Future test: treat e calc & e meas as a random variable IntroductionMethodsResultsConclusion 7 / 19

Structural & Multidisciplinary Optimization Group Analytical Method to Eliminate MCS Sampling Errors  Joint PDF of errors (independent)  Heaviside functions to model the redesign event (indicator function)  Expectations for these functions are easily calculated  After redesign using (S ini, S L, S U, S re ) IntroductionMethodsResultsConclusion 8 / 19

Structural & Multidisciplinary Optimization Group 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 Redesign for PerformanceRedesign for Safety Formulation: IntroductionMethodsResultsConclusion 9 / 19

Structural & Multidisciplinary Optimization Group Optimization of Discrete Cases  4 possible future scenarios  e calc = +30%(conservative) or -30%(unconservative)  e meas = +10% or -10%  P re = 50% Error [e calc, e meas ] [-30,-10] [ 30,-10] [-30, 10] [ 30, 10] IntroductionMethodsResultsConclusion 10 / 19

Structural & Multidisciplinary Optimization Group Graphical optimization  Redesign for safety: Start with low S ini and redesign with high S re  Redesign for performance: Below S ini = 1.45, P F constraint cannot be satisfied IntroductionMethodsResultsConclusion 11 / 19

Structural & Multidisciplinary Optimization Group Distribution of mass after redesign  Redesign for performance reduces average mass by 25.2  Redesign for safety increase average mass by 32.6 For performanceFor safety IntroductionMethodsResultsConclusion 12 / 19

Structural & Multidisciplinary Optimization Group Distribution of P F after redesign  Redesign for performance can reduce mass with almost no penalty in reliability  Redesign for safety has larger impact on reliability For performanceFor safety IntroductionMethodsResultsConclusion 13 / 19

Structural & Multidisciplinary Optimization Group Discrete Error Simulation (4 cases)  Redesign for performance yields about 3% lower average mass  Redesign for safety significantly improves P F Error [e calc, e meas ] Area Before Redesign (mm 2 ) Area After Redesign (mm 2 ) P F Before RedesignP F After Redesign Perform SafetyPerformSafetyPerformSafetyPerformSafety [-30,-10] e e-52.65e-5 [ 30,-10] e e-61.49e-56.65e-6 [-30, 10] e e-51.69e-7 [ 30, 10] e e-69.16e-86.65e-6 Mean e e-5 IntroductionMethodsResultsConclusion 14 / 19

Structural & Multidisciplinary Optimization Group Tradeoff Curve: ±30% Calculation Error / ±10% Measurement Error  e calc ~ U[-0.3, 0.3], e meas ~ U[-0.1, 0.1]  Redesign for performance yields about 3% lower mass P re = 20% Probability of Redesign, P re (%) Mean Area, E(A) (mm 2 ) Performance Safety IntroductionMethodsResultsConclusion 15 / 19

Structural & Multidisciplinary Optimization Group Mass distribution after redesign  Too conservative designs are redesigned to save mass (a large reduction in mass)  Unconservative designs are redesigned to satisfy required P F For performance For safety IntroductionMethodsResultsConclusion 16 / 19

Structural & Multidisciplinary Optimization Group PF distribution after redesign  Redesign for safety (right) significantly changes low PF, while the change is relatively small for performance For performance For safety IntroductionMethodsResultsConclusion 17 / 19

Structural & Multidisciplinary Optimization Group 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 IntroductionMethodsResultsConclusion 18 / 19

Structural & Multidisciplinary Optimization Group THANK YOU Questions? IntroductionMethodsResultsConclusion 19 / 19