1 Introduction
1.1 Background
1.2 State-of-the-art development
1.3 Motivation, contribution and structure
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Developing four kinds of kernels in the stochastic process model to describe different disturbances, namely, the trend uncertainty, the time uncertainty, the smaller irregularities and the data sources uncertainty.
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Using stochastic variational inferences and mini-batch algorithms, the model could effectively explain every single detail of the stochastic process.
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Proposing a DSPM approach for collision dynamic simulation of railway vehicles to accurately describe the stochastic processes.
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Obtaining a higher computing accuracy and efficiency of the DSPM approach compared with that of GPR and FE methods.
2 FE simulation
2.1 Modelling
Component | Material setting | Yield strength (MPa) | Density (×10–9) | Young’s modulus E (GPa) | Poisson’s ration v |
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Bogie framework | Mat.20 | – | 46.8 | 206 | 0.30 |
Rail | Mat.20 | – | 7.85 | 306 | 0.30 |
Endwell | Mat.24 | 125 | 2.70 | 68 | 0.30 |
Car body | Mat.24 | 205 | 2.70 | 68 | 0.30 |
Underframe | Mat.24 | 245 | 2.70 | 68 | 0.30 |
Chassis | Mat.24 | 205 | 5.74 | 68 | 0.30 |
Beams | Mat.24 | 235 | 7.85 | 210 | 0.30 |
Honeycombs | Mat.3 | 292 | 2.70 | 69 | 0.30 |
2.2 Model verification
3 DSPM approach
3.1 Stochastic process analysis
3.2 Parameter Initialization with MLE
3.3 Model prediction
4 Stochastic dynamic simulation of railway vehicle collisions
4.1 Data collection and normalization
4.2 Stochastic process modelling
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A temporal trending term is to be explained by a radial basis function (RBF) kernel [32]. The RBF kernel is given bywhere \(\gamma^{2}\) represents a variance parameter, and \(w\) represents a scalar of an isotropic variant of the kernel.$$k_{{{\text{RBF}}}} \left( {{\varvec{X}},{\varvec{X}}^{{\prime }} ;{\varvec{\theta}}} \right) = \gamma^{2} {\text{exp}}\left( { - \frac{1}{2}\sum w^{2} \left( {{\varvec{X}} - {\varvec{X}}^{{\prime }} } \right)^{2} } \right),$$(12)
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A time component is expressed by the periodic ExpSineSquared kernel with a periodicity parameter \(q(q > 0)\) [33], and the kernel is given by$$k_{{{\text{ESS}}}} \left( {{\varvec{X}},{\varvec{X}}^{{\prime }} ;{\varvec{\theta}}} \right) = \gamma^{2} {\text{exp}}\left( { - 2w^{2} \sin^{2} ({\uppi }\left( {{\varvec{X}} - {\varvec{X}}} \right)^{2} /q)} \right)^{2} .$$(13)
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Smaller irregularities are to be explained by a Rational Quadratic kernel, which could be better expressed than a RBF kernel component. The Rational Quadratic can accommodate several length scales [34], and can be parameterized by a scale mixture parameter \(\alpha > 0\), the definition is as follows:At last, we applied the White kernel as the part of a sum-kernel which explains the noise as independently and identically normally distributed [35]. Overall, using these four kernels, it is possible to explain the noise present in the real-world collision process.$$k_{{{\text{RQ}}}} \left( {{\varvec{X}},{\varvec{X}}^{{\prime }} ;{\varvec{\theta}}} \right) = \gamma^{2} {\text{exp}}\left( {1 + \frac{\alpha }{2}w^{2} \left( {{\varvec{X}} - {\varvec{X}}^{{\prime }} } \right)^{2} } \right)^{ - \alpha } .$$(14)
4.3 Model training
4.4 Simulations and evaluation
5 Illustrative example analysis
5.1 Collision scenario set up
Collision scenario No. | Collision scenario | Velocity (km/h) |
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1 | Lead car to rigid wall | 10, 20, 30, 40, 50 and 60 |
2 | Lead car to lead car | 20, 40 and 50 |
5.2 Model training
5.3 Model validation
5.4 Comparative analysis
Collision scenario No | Dynamic response | Method | Training set | Testing/simulation set | ||||||||
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Time (s) | R2 | MAE | MSE | RMSE | Time (s) | R2 | MAE | MSE | RMSE | |||
Case 1: lead car to rigid wall | Displacement | FE | – | – | – | – | – | 21,336.6 | – | – | – | – |
GPR | 2581.9 | 0.999 | 0.012 | 0.0004 | 0.02 | 0.22 | 0 | 0.31 | 0.166 | 0.408 | ||
DSPM | 861 | 0.998 | 0.025 | 0.001 | 0.034 | 0.09 | 0.99 | 0.03 | 0.001 | 0.039 | ||
Velocity | FE | – | – | – | – | – | 21,336.6 | – | – | – | – | |
GPR | 1965.46 | 0.998 | 0.023 | 0.0013 | 0.03 | 0.23 | − 2.08 | 0.735 | 0.732 | 0.859 | ||
DSPM | 803.19 | 0.992 | 0.064 | 0.007 | 0.088 | 0.025 | 0.995 | 0.050 | 0.003 | 0.058 | ||
Interface force | FE | – | – | – | – | – | 21,336.6 | – | – | – | – | |
GPR | 1693.08 | 0.992 | 0.039 | 0.007 | 0.087 | 0.234 | − 1.88 | 0.741 | 0.612 | 0.782 | ||
DSPM | 792.57 | 0.961 | 0.249 | 0.138 | 0.371 | 0.036 | 0.93 | 0.162 | 0.039 | 0.199 | ||
Internal energy | FE | – | – | – | – | – | 21,336.6 | – | – | – | – | |
GPR | 2161.42 | 0.999 | 0.014 | 0.0006 | 0.025 | 0.28 | − 0.02 | 0.250 | 0.090 | 0.300 | ||
DSPM | 790.81 | 0.996 | 0.037 | 0.003 | 0.059 | 0.055 | 0.993 | 0.020 | 0.0005 | 0.024 | ||
Kinetic energy | FE | – | – | – | – | – | 21,336.6 | – | – | – | – | |
GPR | 2077.19 | 0.998 | 0.018 | 0.001 | 0.036 | 0.23 | − 0.03 | 0.473 | 0.318 | 0.564 | ||
DSPM | 786.6 | 0.987 | 0.068 | 0.012 | 0.110 | 0.042 | 0.993 | 0.038 | 0.0021 | 0.04 | ||
Case 2: lead car to lead car | Displacement of car A | FE | – | – | – | – | – | 36,300 | – | – | – | – |
GPR | 6348.61 | 0.999 | 0.015 | 0.0006 | 0.024 | 1.15 | − 0.19 | 0.58 | 0.50 | 0.707 | ||
DSPM | 795.12 | 0.999 | 0.008 | 0.0001 | 0.013 | 0.16 | 0.993 | 0.052 | 0.003 | 0.055 | ||
Displacement of car B | FE | – | – | – | – | – | 36,300 | – | – | – | – | |
GPR | 5949.48 | 0.997 | 0.02 | 0.002 | 0.02 | 1.37 | − 1.39 | 0.563 | 0.354 | 0.595 | ||
DSPM | 796.15 | 0.999 | 0.010 | 0.0002 | 0.016 | 0.13 | 0.993 | 0.046 | 0.0034 | 0.058 | ||
Velocity of car A | FE | – | – | – | – | – | 36,300 | – | – | – | – | |
GPR | 5750.14 | 0.996 | 0.032 | 0.0036 | 0.06 | 1.31 | − 0.81 | 0.617 | 0.409 | 0.639 | ||
DSPM | 793.0 | 0.944 | 0.144 | 0.05 | 0.23 | 0.168 | 0.861 | 0.130 | 0.031 | 0.17 | ||
Velocity of car B | FE | – | – | – | – | – | 36,300 | – | – | – | – | |
GPR | 6696.0 | 0.999 | 0.012 | 0.0003 | 0.017 | 1.12 | − 0.10 | 0.648 | 0.555 | 0.745 | ||
DSPM | 798.02 | 0.930 | 0.159 | 0.069 | 0.26 | 0.157 | 0.872 | 0.165 | 0.036 | 0.191 | ||
Interface force of car A & car B | FE | – | – | – | – | – | 36,300 | – | – | – | – | |
GPR | 6342.57 | 0.992 | 0.037 | 0.0077 | 0.087 | 1.41 | − 0.08 | 0.629 | 0.496 | 0.704 | ||
DSPM | 4120.0 | 0.950 | 0.52 | 0.499 | 0.70 | 0.179 | 0.843 | 0.38 | 0.25 | 0.50 | ||
Internal energy of car A | FE | – | – | – | – | – | 36,300 | – | – | – | – | |
GPR | 6048.78 | 0.999 | 0.009 | 0.0003 | 0.019 | 1.21 | − 3.89 | 0.741 | 0.563 | 0.750 | ||
DSPM | 794.04 | 0.993 | 0.047 | 0.006 | 0.082 | 0.175 | 0.851 | 0.119 | 0.018 | 0.136 | ||
Internal energy of car B | FE | – | – | – | – | – | 36,300 | – | – | – | – | |
GPR | 7571.8 | 0.999 | 0.012 | 0.0007 | 0.027 | 1.18 | − 1.78 | 0.720 | 0.548 | 0.740 | ||
DSPM | 796.69 | 0.990 | 0.052 | 0.009 | 0.09 | 0.18 | 0.873 | 0.09 | 0.009 | 0.097 | ||
Kinetic energy of car A | FE | – | – | – | – | – | 36,300 | – | – | – | – | |
GPR | 5949.48 | 0.997 | 0.02 | 0.002 | 0.04 | 1.37 | − 1.39 | 0.563 | 0.354 | 0.595 | ||
DSPM | 791.138 | 0.927 | 0.156 | 0.07 | 0.268 | 0.184 | 0.815 | 0.114 | 0.027 | 0.165 | ||
Kinetic energy of car B | FE | – | – | – | – | – | 36,300 | – | – | – | – | |
GPR | 6973.36 | 0.997 | 0.024 | 0.002 | 0.052 | 1.25 | − 2.6 | 0.53 | 0.389 | 0.62 | ||
DSPM | 798.08 | 0.885 | 0.196 | 0.114 | 0.338 | 0.211 | 0.847 | 0.173 | 0.038 | 0.194 |
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FE simulation is the most commonly used method in engineering, but its simulation time is almost 10 times longer than that of other two methods, so this approach suffers from a high computational cost.
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Generally, GPR method is used to deal with stochastic processes, because of its strong fitting ability. Otherwise, it is difficult to apply and promote in engineering due to its insufficient predictive ability and high computational complexity.
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Our proposed DSPM approach with lower error and higher R2, has an obvious advantage of efficiency and accuracy compared with FE and GPR method.