## Section: New Results

### Reliability

Participants : Yves Auffray, Gilles Celeux, Rémy Fouchereau, Patrick Pamphile, Jana Kalawoun.

In 2014, in the framework of a CIFRE convention with Snecma-SAFRAN Rémy Fouchereau has defended a thesis on the modeling of fatigue lifetime supervised by Gilles Celeux and Patrick Pamphile. In aircraft, space and nuclear industry, fatigue test is the main basic tool for analyzing fatigue lifetime of a given material, component, or structure. A sample of the material is subjected to cyclic loading S (stress, force, strain, etc.), by a testing machine which counts N, the number of cycles to failure. Fatigue test results are plotted on a SNcurve. A probabilistic model for the construction of SN-curve is proposed. In general, fatigue test results are widely scattered for High Cycle Fatigue region and "duplex" SN-curves appears for Very High Cycle region. That is why classic models from mechanic of rupture theory on one hand, probability theory on the other hand, do not fit SN-curve on the whole range of cycles. We have proposed a probabilistic model, based on a fracture mechanic approach: few parameters are required and they are easily interpreted by mechanic or material engineers.This model has been applied to both simulated and real fatigue test data sets. The SN-curves have been well fitted on the whole range of cycles. The parameters have been estimated using the EM algorithm, combining Newton-Raphson optimisation method and Monte Carlo integral estimations. The model has been then improved taking into account production process information, thanks to a clustering approach. Thus, we have provided engineers with a probabilistic tool for reliability design of mechanical parts, but also with a diagnostic tool for material elaboration.

Since two years SELECT collaborates with CEA for the estimation of the battery State of Charge (SoC). For vehicles powered by an electric motor, SoC estimation is essential to guarantee vehicle autonomy, as well as safe utilization. The aim is to create a reliable SoC model to closely fit the battery dynamic, in embedded applications (e.g. Electric Vehicle). Jana Kalawoun started a thesis supervised by Gilles Celeux, Patrick Pamphile and Maxime Montaru (CEA) on this topic. The SoC is modeled by a Switching Markov State-Space Model. The parameters are estimated by combining the EM algorithm and Particle Filter methods. The model is validated using real-life electric vehicle data. It has been proved to be highly superior to a simple state space model. The optimal number of battery modes is then identified, using different model selection criteria as BIC or the slope heuristics.

Yves Auffray and Gilles Celeux proposed a solution to a reliability problem on Dassault's F7X business jet brakes. As the origin brake version showed poor reliability performance, an increased frequency inspection of the brakes had been decided and, after a while, a new brake version adopted. The new version has not shown any failure since its adoption. Then the question was : is it possible to relax the brakes inspection frequency ?

On the basis of first brake version failure data, the parameters of a Weibull law was estimated : $\eta =3169,\beta =1.38$. Under the hypothesis that the new brake version would follow the same Weibull law, the probability that none of them broke was $1.67\phantom{\rule{4pt}{0ex}}{10}^{-6}$. This led to reject that hypothesis.

A Weibull model for the new brakes was then estimated. The shape parameter beeing leaved conservatively unchanged, the scale parameter was estimated so that the no failure event probability amounts to 0.05. This led to $\eta =9326$.

From the resulting Weibull model, dates ${D}_{0},{D}_{1},\cdots ,{D}_{k},\cdots $ of inspection for the new brakes was established so that : $\mathbb{P}(T\le {D}_{0}+{D}_{1}+\cdots +{D}_{k}|T>{D}_{0}+\cdots +{D}_{k-1})=0.01$.

Dassault has adopted this far less constraining inspection calendar.