Erdem Acar

Improving the accuracy of vehicle crashworthiness response predictions using an ensemble of metamodels

Due to the scale and computational complexity of current simulation codes for vehicle crashworthiness analysis, metamodels have become indispensable tools for exploring and understanding the design space. Traditional application of metamodelling techniques is based on constructing multiple types of metamodels based on a common data set, selecting the most accurate one and discarding the rest. However, this practice does not take full advantage of the resources devoted for constructing different metamodels. This drawback can be overcome by combining individual metamodels in the form of an ensemble. Two case studies with a high-fidelity finite element vehicle model subject to offset-frontal and side impact conditions are presented for demonstration. The prediction accuracies of the individual metamodels and the ensemble of metamodels are compared, and it is found for all the crash responses of interest that the ensemble of metamodels outperforms all individual metamodels. It is also found that as the number of metamodels included in the ensemble increases, the prediction accuracy of the ensemble of metamodels increases.

Effects of Structural Tests on Aircraft Safety

Thispaperpresents amethodology to investigatethe effects of structuraltests onaircraft safety.Inparticular, the paper focuses on the effect of the number of coupon tests and structural element tests on the final distribution of failure stress. The mean failure stress is assumed to be predicted by a failure criterion (e.g., Tsai-Wu), and the initial distribution of this mean failure stress reflects the uncertainty in the analysisprocedure that usescoupontest datato predict structural failure. In addition to the uncertainty in the mean failure stress, there is also uncertainty in its variabilityduetothe finitenumberofcoupontests.Bayesianupdatingisusedtoupdatethefailurestressdistribution basedonresultsoftheelementtests.AMonteCarlosimulationofalargenumberofuncertaintiesandthepossibletest results are used to obtain the probability of structural failure in a certification test or in actual flight. Incorporating the Bayesian updating into the Monte Carlo simulation loop is computationally prohibitive; therefore, a surrogate procedure is devised to overcome the computational challenge. A structural design following the Federal Aviation Administration regulations is considered, and the tradeoffs between the number of tests and the weight and probability of failure in the certification test and in service are explored. To make this tradeoff analysis computationally affordable, response surface approximations are used to relate the knockdown factor to the probability of failure in service and in the certification test. It is found that it is possible to do a simultaneously probabilistic design and satisfy the Federal Aviation Administration regulations for deterministic design.

Tradeoff of Uncertainty Reduction Mechanisms for Reducing Weight of Composite Laminates

Inspired by work on allocating risk between the different components of a system for a minimal cost, we explore the optimal allocation of uncertainty in a single component. The tradeoffs of uncertainty reduction measures on the weight of structures designed for reliability are explored. The uncertainties in the problem are broadly classified as error and variability. Probabilistic design is carried out to analyze the effect of reducing error and variability on the weight. As a demonstration problem, the design of composite laminates at cryogenic temperatures is chosen because the design is sensitive to uncertainties. For illustration, variability reduction takes the form of quality control, while error is reduced by including the effect of chemical shrinkage in the analysis. Tradeoff plots of uncertainty reduction measures, probability of failure and weight are generated that could allow choice of optimal uncertainty control measure combination to reach a target probability of failure with minimum cost. In addition, the paper also compares response surface approximations to direct approximation of a probability distribution for efficient estimation of reliability.

Being Conservative with a Limited Number of Test Results

combined loads. It is a common practice to repeat the element tests and then select the lowest test result as a conservative estimate of the mean failure stress. Thispractice is equivalent to reducing the average test failure stress by a knockdown factor (one that is quite variable). Instead, we propose using the average test result with an explicit knockdown factor obtained from statistical distribution of the test data. We show reductions in the variability of the estimated mean failure stress as well as the likelihood of unconservative estimate. In addition, when the initial distribution or confidence interval of the mean failure stresses is available, we can further decrease the chance of unconservative estimate using Bayesian updating. We demonstrate the gains associated with Bayesian updating when the upper and lower bounds of errors in the analytical predictions are available. Examples with uniform and lognormal distributions of failure stresses compare the lowest-result approach with the two alternatives with the explicit knockdown factor. Both approaches significantly reduce the likelihood of unconservative estimates of the mean failure stress. The average approach reduced this likelihood by about a half and the Bayesian approach by up toanorderofmagnitude(from12.5to1%).Wealsoexaminescenariosinwhichestimatesoferrorandvariabilityare substantially inaccurate. We show that, even then, the likelihood of unconservative estimates reduces significantly. Remarkably,anunderestimateofvariabilityalsoresultsinabouta2%higheraverageoftheestimatedmeanfailure stress. Thus, we are able to simultaneously use higher average failure stress (leading to lower weight) and reduce the likelihood of unconservative estimates.

Structural Safety Measures for Airplanes

Passenger aircraft structural design is based on a safety factor of 1.5, and this safety factor alone is equivalent to a probability of failure of between 10 S2 and 10 S3 . Yet airliners are much safer, with crashes caused by structural failure being extremely rare based on accident records. The probability of structural failure of transport aircraft is of the order of 10 S8 per flight segment. This paper looks at two additional contributions to safety—the use of conservative material properties and certification tests—using a simple model of structural failure. We find that the three safety measures together might be able to reduce the calculated probability of failure to about 10 S7 . Additional measures, such as conservative load specifications, might be responsible for the higher safety encountered in practice, explaining why passenger aircraft are so structurally safe. In addition, the paper sheds light on the effectiveness of certification tests for improving safety. It is found that certification tests reduce the calculated failure probabilities by reducing the modeling error. We find that these tests are most effective when safety factors are low and when most of the uncertainty is caused by systemic errors rather than variability.

Reliability-Based Aircraft Structural Design Pays, Even with Limited Statistical Data

Probabilistic structural design tends to apply higher safety factors to inexpensive or light-weight components, because it is a more efficient way to achieve a desired level of safety. We show that even with limited knowledge about stress probability distributions we can increase the safety of an airplane by following this paradigm. This is accomplished by a small perturbation of the deterministic design that maximizes safety for the same weight. The structural optimization for safety of a representative system composed of a wing, a horizontal tail and a vertical tail is used to demonstrate the paradigm. We find that moving small amount of material from the wing to the tails leads to substantially increased structural safety. Since aircraft companies often apply additional safety factors beyond those mandated by the Federal Aviation Administration (FAA), this opens the door to obtaining probabilistic design that satisfies also the FAA code based rules for deterministic design. We also find that probabilistic design is insensitive to errors committed while assessing the stress probability distribution of the deterministic design, which is the starting point of the probabilistic design. This suggests that using the deterministic design as the basis for the probabilistic design insulates the latter from the extreme sensitivity to statistical data that have been observed in the past. Finally, we find that for independent components subject to the same failure mode, the probabilities of failure at the probabilistic optimum are approximately proportional to the weight. So a component which is ten times lighter than another should be designed to be about 10 times safer.

Effect of error metrics on optimum weight factor selection for ensemble of metamodels

Weight factor selection in ensemble of metamodels is explored.Interestingly, weight factors for minimum MAXE-CV are not good to reduce actual MAXE.MAXE-CV is mostly related with the geography of DOE, not metamodel prediction ability. Optimization of complex engineering systems is performed using computationally expensive high fidelity computer simulations (e.g., finite element analysis). During optimization these high-fidelity simulations are performed many times, so the computational cost becomes excessive. To alleviate the computational burden, metamodels are used to mimic the behavior of these computationally expensive simulations. The prediction capability of metamodeling can be improved by combining various types of models in the form of a weighted average ensemble. The contribution of each models is usually determined such that the root mean square cross validation error (RMSE-CV) is minimized in an aim to minimize the actual root mean square error (RMSE). However, for some applications, other error metrics such as the maximum absolute error (MAXE) may be the error metric of interest. It can be argued, intuitively, that when MAXE is more important than RMSE, the weight factors in ensemble should be determined by minimizing the maximum absolute cross validation error (MAXE-CV). Interestingly, it is found that the ensemble model based on MAXE-CV minimization is less accurate than the ensemble model based on RMSE-CV minimization even if the MAXE is the metric of interest. The reason is found to be that MAXE-CV is mostly related with the geography of the DOE rather than the prediction ability of metamodels.

A comparative study of design optimisation methodologies for side-impact crashworthiness, using injury-based versus energy-based criterion

The tension between occupant safety during a crash and lightweight designs continues to be an important driver of modern vehicle designs. While occupant safety may be defined and evaluated in various ways, maximising energy absorption of structural components during impact has been adopted for vehicle designs by many manufacturers. An alternative method to evaluate safety but often not directly used in the design of structural components is the use of a dummy model in the finite element (FE) simulation to estimate the forces and accelerations that would be experienced by a human in a crash environment. This paper investigates the similarities/differences between designing vehicular structural components experiencing side impacts based upon two different criteria: (1) the energy absorption of collapsed components in the absence of a dummy and (2) an injury metric–based approach with the responses of the dummy as the bases. Multi-objective optimisation methods are used with finite element analysis (FEA) in the lightweight design for side-impact crashworthiness, considering the two different criterion. The results show that the optimum designs based on the two criteria are significantly different and that the injury-based approach should be incorporated into vehicular lightweight design process when considering crashworthiness.

Shape and Sizing Optimisation of Automotive Structures With Deterministic and Probabilistic Design Constraints

This paper presents the results of a study on the combined shape and sizing optimisation of automotive structures while examining the effects of different design constraints and associated uncertainties on reliability and efficiency of the optimum designs. Nonlinear transient dynamic finite element analysis is used for full- and offset-frontal crash simulations of a full vehicle model. Surrogate models are developed for the intrusion distance and peak acceleration responses at different vehicle locations based on the material and geometric characteristics of the rail component. The obtained solutions provide insight on the effect of uncertainties in optimum design of automotive structures.

Increasing Allowable Flight Loads by Improved Structural Modeling

The tradeoffs of allowable flight loads and safety of aerospace structures via deterministic and probabilistic design methodologies are analyzed. The methodologies are illustrated by performing allowable flight load calculation of a sandwich panel used in aerospace structures. The effect of using a more accurate prediction technique for interfacial fracture toughness that combines interfacial fracture toughness with mode mixity instead of using the traditional model that disregards mode mixity is explored. It was found that by utilizing this more accurate model with the change in B-basis properties, the deterministic approach allows a 13.1% increase in the allowable flight load and a reduction of probability of failure by a factor of five. The probabilistic approach allows a 26.5% increase in allowable flight load, while maintaining the original probability of failure.