A dynamic reliability analysis method based on support vector machine and Monte Carlo simulation

Abstract

An dynamic reliability analysis method based on support vector machine and finite element combined with Monte Carlo is presented for the parts in complex external conditions, it is difficult to built a dynamic reliability model including the models of the stress distribution and the joint probability density for the parts operating in an uncertain environment, the support vector machine has a good generalization ability and prediction ability, the integration algorithm based on support vector machine, finite element and Monte Carlo is excellently used to dynamic reliability simulation and calculation for complex and certain system. The method is applied to the reliability analysis of catenary parts in the high-speed electrified railway, the integration algorithm mathematic model of reliability analysis for the catenary suspension system is built, and the system reliability is analyzed by the model.

Keywords

Monte Carlo support vector machine complex external surroundings dynamic reliability analysis

1. Introduction

Many components operates in complex external conditions or surroundings, they will fail under cyclic loading. The reliability analysis and designr of the components is an important way to ensure their safe operation [1].

An analysis method based on support vector machine and finite element combined with Monte Carlo is applied for the force-bearing components in complex external conditions or surroundings, it is difficult to built a dynamic reliability model of the components in complex the external conditions or surroundings and it is also difficult to decide stress and intensity distribution and joint probability density because they operate in an uncertain dynamic environment, the support vector machine has a good generalization ability and prediction ability, the integration algorithm based on support vector machine, finite element and Monte Carlo can solve the questions and can excellently use for dynamic reliability simulation and calculation for the complex and certain systems. An example are given for dynamic reliability calculation and analysis of the catenary suspension system in this paper, the integration algorithm mathematic model of dynamic reliability analysis is built, and the reliability influencing on the catenary suspension system are analyzed by the model for the outside parameters.

2. An integration algorithm mathematic model of dynamic reliability analysis

2.1 A stress-intention interference model

A mechanism part maybe fail under all kinds of complex static loads and dynamic loads because the internal stress exceed the material intensity limit finally, the failure probability can be obtained by Eq. (1) according to the stress-intensity interference theory [2, 3, 4].

$\displaystyle P_{f}=P(\delta-S\leqslant 0)=\int\!\!\!\int\limits_{D}{f_{\delta S% }}(\delta,s)d\delta ds.$ (1)

Where $\delta$ is the structural intensity, $S$ is the stress. $\delta=\delta(X_{\delta 1},X_{\delta 2},X_{\delta 3},\ldots,X_{\delta i})$ , $S=S(X_{S1},X_{S2},X_{S3},\ldots,X_{Sj})$ . $f_{\delta S}(\delta,s)$ is stress and intensity joint probability density of every components, $X_{\delta i}$ is i-th parameter of the structural intensity, $X_{sj}$ is j-th parameter of the stress.

If $X=(X_{1},X_{2},\ldots,X_{n})^{T}$ is random parameter vector, then the state function is represented by the Eq. (2).

$\displaystyle g(X)=\delta(X)-S(X).$ (2)

The components will fail if $g(X)\leqslant 0$ according to Eq. (2) and the probability and statistics theory, the failure probability and the reliability can be calculated in a certain amount of random numbers of $\delta$ and $S$ .

2.2 Dynamic reliability simulation step of the integration algorithm

It is difficult to built the dynamic reliability model of the parts in complex external conditions. In this paper, the dynamic reliability analysis method based on SVM and finite element combined with Monte Carlo is presented, the integration algorithm process is as follows [5, 6, 7, 8].

1.
Critical factors that influence the reliability are decided firstly.
2.
N groups of random data are generated according to statistical distribution for the selected parameters, the maximum stress in dangerous section for each group of parameters are calculated by the finite element [5]. After selecting the fragment structure unit, characteristic analysis of typical units is necessary, the relationship of any point displacement is derived by nodal displacements Eq. (3).

$\displaystyle\{w\}=[N]\{u\}^{(e)}$ (3)

Where $\{w\}$ is column vector of any point displacement in the unit, $\{u\}^{(e)}$ is the column vector of nodal displacement, $[N]$ is the shape function matrix. The relationship of unit strain and unit stress and unit balance equation are obtained by the Eqs (4)–(6).

$\displaystyle\{\varepsilon\}=[B]\{u\}^{(e)}$ (4) $\displaystyle\{\sigma\}=[D][B]\{u\}^{(e)}$ (5) $\displaystyle\{P\}^{(E)}=[k]^{(e)}\{u\}^{(e)}$ (6)

Figure 1.
Reliability calculation based on SVM and Monte Carlo numerical simulation.

Where $\{\varepsilon\}$ is the strain column vector of any point in the unit, $[B]$ is the unit strain matrix, $[D]$ is the elasticity matrix, $[k]^{(e)}=\int\!\!\!\int\!\!\!\int{[B]^{T}}[D][B]dxdydz$ , $\{P\}^{(e)}$ is the force column vector of unit equivalent nodes.
3.
N groups of data are used as SVM training samples, the input-output relation between external parameters and the interior stress is built by the support vector machine intelligent algorithm, a SVM objective optimized function is defined by the following Eq. (7) .

$\displaystyle\mathop{\min}\limits_{w,b,e}J(w,e)=\frac{1}{2}W^{T}W+\frac{1}{2}% \gamma\sum\limits_{K=1}^{n}{e_{k}^{2}}$ (7)

$\displaystyle s.t.\ \ {y_{k}=W^{T}}\phi(x_{k})+b+e_{k},k=1,2,\ldots,N$

Where $\phi(\cdot):R^{n}\to R^{nh}$ is the kernel space mapping function; $w\in R^{nh}$ is the weight vector; $e_{k}\in R$ is the error variable; $b$ is the deflection variable; $\gamma$ is the adjustable parameter.
4.
The material intensity distribution is decided by testing and statistical data in the documents.
5.
The parts reliability is calculated by numerical simulation, Monte Carlo. The state function g is calculated by random sample, the maximum stress exceeds limit state if $g<$ 0, the parts is considered as failure. The Monte Carlo method is that the components is failure or not by judging $g<$ 0. Total numbers N are tested in case of $N\geqslant 100/P_{f}$ , if failure total number is L, failure probability $P_{f}$ is $L/N$ and reliability $R$ is $1-P_{f}$ , the calculation flow chart is shown in Fig. 1.

3. An example analysis

The integration algorithm based on support vector machine, finite element and Monte Carlo is used for the reliability analysis of the catenary components in the high-speed electrified railway, the mathematic model of reliability analysis for catenary suspension system is built [9, 10].

The critical factors that influence the reliability of catenary suspension system are decided, they are pulling force F, ice load (ice-covering thickness d) and wind speed W, N groups of random data are generated according to statistical distribution for the selected parameters, the maximum stress in dangerous section for each group of parameters are calculated by the finite element [5]. Forty groups of basic variables are randomly generated according to the mean and variance of basic variables, then the stress S of each group of basic variables are calculated by finite element software ANSYS [11], Thirty-five groups data are used as training samples, and five groups data are used as testing data.

In finite element calculating, an entity model is turned into PARASOILD format, then it is lead into ANSYS software and solid92 tetrahedron unit is used, 121342 units 203451 nods are obtained for wrist-arm of X type and 2708 units 4976 nods are obtained for location hook by free-dividing meshes. The finite element analysis models are shown in Fig. 2. The stress variation with external loads of catenary suspension system is shown in Fig. 3 [11, 12].

Figure 2.

Finite element analysis of catenary suspension system.

Figure 3.

The stress variation with external loads.

Relative error between the testing value and the calculating value of finite element is shown in Fig. 4 after the support vector machine is trained, it is clear that the theory values and the support vector machine output values is very close.

Figure 4.

Relative error between the testing deflection and the calculating value of finite element.

Figure 5.

Relation curve between variation coefficient and reliability.

The reliability of catenary suspension system is calculated by the reliability analysis model of integration algorithm, the mean value and deflection coefficient of material intensity limit value of location hook is obtained according to according to statistics. Reliability calculating result is 0.998, relative error is only 0.02% comparison with calculating result of JC method.

It is analyzed that distribution variation coefficients of external parameters have an effect on the reliability of the catenary suspension system by the program, the relation curve is shown in Fig. 5. Distribution deflection coefficient of pulling force F has a great influence on the reliability, and the external climate has also a great influence on system reliability, so reliability optimal design is necessary to reduce catenary failures.

4. Conclusion

A dynamic reliability analysis method based on support vector machine and finite element combined with Monte Carlo is applied for the components in complex external conditions or surroundings, it is difficult to built a dynamic reliability model of the components in complex the external conditions or surroundings and it is difficult to decide the stress and intention distribution and joint probability density because they operate in an uncertain environment, the support vector machine has a good generalization ability and prediction ability, the integration algorithm based on support vector machine, finite element and Monte Carlo can solve the questions and can excellently use for reliability simulation and calculation for complex and certain system. The integration algorithm is used for the reliability analysis of catenary components in the high-speed electrified railway, the mathematic model of reliability analysis for catenary suspension system is built, and the influence of external parameters on reliability are analyzed by the model.

In dynamic reliability design models, the stress-intensity distribution model reveals clearly fault cause and the essence of reliability design, but it is difficult to decide stress and intention distribution and joint probability density function of because many components operate in a complex and uncertain environment. In this paper, the reliability analysis model of complex components is built based on support vector machine and finite element combined with Monte Carlo. The model is applied to reliability analysis of catenary suspension system, the reliability calculating result is 0.998, the relative error is only 0.02% comparison with the calculating result of JC method, and the influence of external parameters on reliability are analyzed by the model. The method provides a new way for the reliability design and research of the complex systems.

Footnotes

Acknowledgments

This work is supported by Science and Technology Planning Project of Zhejiang Province, China (2016C32103), and supported by Natural Science Foundation of Zhejiang Province in China (LY14F020013). The authors thank the members of the reliability research team for their support of this work.

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