PSI - Issue 8

G. Arcidiacono et al. / Procedia Structural Integrity 8 (2018) 163–167

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G. Arcidiacono et al. / Structural Integrity Procedia 00 (2017) 000–000

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apply the Mate´rn covariance function introduced by Rasmussen and Williams (2006). Moreover, computational prob lems and jumps may be encountered during simulations and, to this end, a nugget parameter has been introduced. The inclusion of this coe ffi cient allows to avoid instability, e.g. it may be viewed as a noise added to the process variance. In particular, we consider a nugget equal to zero, but we also investigated for the presence of the nugget. A last remark is relating to the reliability of predictions; in general, the predicted value ˆ Y ( x ) is estimated by involving the correlation matrix and therefore by considering the correlation among the simulated point and the training set of starting data ( X ). Therefore, the error in prediction is useful for two aims: i) to evaluate the applied covariance structure for the Kriging model and ii) to evaluate the reliability of prediction, e.g. the Kriging variance, which is smaller when the predicted point is located nearby X and larger as moving away from X . In order to build the model, we have defined a strategy for representing a train in the design space, for further details see Arcidiacono et al. (2017). The main chosen characteristic is the payload distribution of the entire train. In particular, we consider three payload distributions: uniform, triangular and trapezoidal. We denote by B the length of the train and by Q the area of each distribution, corresponding to the total payload on the train. We consider that: i) wagons and the total payload are chosen a-priori and ii) wagons and the total payload are the same for every train evaluated in the Kriging model. Therefore, every distribution share the same base (B) and it has the same area (Q). To describe it mathematically, we define two discrete variables, e.g. h and x , that could univocally represent the payload distribution on the entire train. Through the input variable h , it is possible to define the shape of the payload distribution, which can gradually change from the uniform shape to the triangular one. Through the input variable x , it is possible to identify the position of the maximum load along the length of the train. The optimal solution searched through the Kriging model is the best payload distribution along the train. It guar antees an emergency braking with the minimum in-train forces between wagons, both in compression and in tensile. Therefore, the estimated response variables are the in-train forces computed at 2 m and at 10 m; at each position, the minimum value (in absolute sense) is considered of the in-train forces occurring in the previous 2 m or 10 m. The true values of compression and tensile forces are calculated to evaluate the proposed model through the TrainDy software by Cantone (2011). This software developed by Cantone et al. (2009) is internationally certified for the computation of in-train forces of freight trains. The Kriging method is applied building a model by considering the sum of compression and tensile forces as output. A cross-validation has also been carried out in order to verify the applied Kriging model for each monitored force. The prediction model has been trained through 360 simulations while the remaining 40 have been applied to validate the results. The diagnostic results, calculated for each force ( R 2 index), show that the models are highly predictive, as reported in Table 1. The Kriging model related to the sum of forces (compression and tensile at 2 m) yields the best performance. Model results are evaluated also considering the residuals. In Table 2 diagnostic results are summarized according to the Max Absolute Error (MaxAE), the Max Relative Error (MRE), the Mean Absolute Error (MAE) and the Error Mean (EM).The largest MRE value occurs for simulations that are in the boundary region of the design space. The largest MAE value, for all the observed forces, is one order of magnitude less than the simulated. These results can be consider satisfactory for this case study. Regarding the nugget parameter, it can be viewed as a constant that can be determined by statistical and numerical criteria since grants numerical stability and predictive accuracy. When considering the semivariogram function we pay our attention to the maximum value for the semivariogram when the stationarity is achieved, e.g the sill and the range, where the latter is the distance fixed by the point where sill is achieved. In the case study we have calculated di ff erent semivariograms also by adding a small noise to the process variance by choosing a nugget equal to 0.01. As shown in Table 3, we achieved a better result only for the model related to sum of forces at 2m. For all the models we reached satisfactory results in terms of small errors (Table 2) especially when considering the magnitude of forces. 2.1. Kriging modeling: input and output variables 3. Simulations and results