Actual engineering systems will be inevitably affected by uncertain factors. Thus, the Reliability-Based Multidisciplinary Design Optimization (RBMDO) has become a hotspot for recent research and application in complex engineering system design. The Second-Order/First-Order Mean-Value Saddlepoint Approximate (SOMVSA/FOMVSA) are two popular reliability analysis strategies that are widely used in RBMDO. However, the SOMVSA method can only be used efficiently when the distribution of input variables is Gaussian distribution, which significantly limits its application. In this study, the Gaussian Mixture Model-based Second-Order Mean-Value Saddlepoint Approximation (GMM-SOMVSA) is introduced to tackle above problem. It is integrated with the Collaborative Optimization (CO) method to solve RBMDO problems. Furthermore, the formula and procedure of RBMDO using GMM-SOMVSA-Based CO(GMM-SOMVSA-CO) are proposed. Finally, an engineering example is given to show the application of the GMM-SOMVSA-CO method.

Multidisciplinary Design Optimization (MDO) is a methodology that deals with complex and coupled engineering system design problems [

Approximate analytical methods in multidisciplinary reliability analysis methods include First-Order/Second-Order Mean-Value Reliability Analysis (FOMVRA/SOMVRA) methods [

To deal with this problem, Papadimitriou et al. [

The other parts of this paper are organized as follows. In

The RBMDO model is shown in

The reliability

Joint PDF is non-linear and random variables are multi-dimensional [

FOMVSA only uses the first two sample points of random variables as the research objects [

Two important functional properties of CGF are given below to further illustrate the role of CGF in the reliability analysis method based on SA [

Property 1: If the random variables in the variable space

Property 2: If

By these two properties, the CGF of

Find the solution where the first derivative of the above formula is equal to zero:

The failure probability (CDF of the LSF) can be obtained by

In this method, the cost of calculating the limit state gradient is independent of the number of random variables. Only the first-order sensitivity derivative of the random variables needs to be input. Therefore, the most significant superiority of the FOMVSA method is that it reduces the computational cost [

To improve the calculation accuracy of FOMVSA, SOMVSA method is proposed [

After simplifying the above equation, as shown in

Express the above equation as a matrix polynomial form, as shown in

The parameter expression is as follows:

It should be noted that only when the input distribution is Gaussian input distribution can the SOMVSA method directly use the MGF of the input random variables to represent the output MGF like FOMVSA. Therefore, SOMVSA is only valid when the distribution of input variables is Gaussian distribution.

Regarding the problem that MVSOSA cannot handle non-Gaussian input variables, the GMM-SOMVSA method is proposed [

Expected steps:

Maximization steps:

The corresponding parameters are updated once every iteration. The iterative convergence conditions of the algorithm are as follows:

The MGF of LSF can be expressed as follows:

Take the natural logarithm of

The GMM-SOMVSA is integrated with the CO method to solve the RBMDO problem, namely GMM-SOMVSA-CO method.

The solving procedure of GMM-SOMVSA-CO is as follows:

The reliability analysis at the discipline-level can be divided into 6 steps: (1) Extract a sample of size M from the copula function

The GMM-SOMVSA-CO solution procedure is shown in

This example is the design optimization of the working device of a mining excavator. The actual picture and structure diagram of the mining excavator are shown in

This optimization design has 11 design variables:

Design variable | Distribution type | Standard deviation | Mean | Lower bound | Upper bound |
---|---|---|---|---|---|

Gumbel | 0.001 |
0.624 | 1.282 | ||

Normal | 0.001 |
2.136 | 3.734 | ||

Gumbel | 0.001 |
2.763 | 3.107 | ||

Normal | 0.001 |
−0.641 | 0.614 | ||

Normal | 0.001 |
9.884 | 10.8816 | ||

Lognormal | 0.001 |
0 | - | ||

Lognormal | 0.001 |
0 | - | ||

Normal | 0.001 |
12.472 | 12.815 | ||

Normal | 0.001 |
3.7416 | 7.689 | ||

Lognormal | 0.001 |
1.08 | 1.22 | ||

Lognormal | 0.001 |
−208 | 0 |

Divide this optimization problem into two disciplines, as shown in

The mathematical model of this optimization problem is shown as follows:

The system-level and discipline-level optimization mathematical model can be expressed by

The optimization model of the system layer is:

The optimization model of discipline 1 is:

The optimization model of discipline 2 is:

The target reliability is 0.98. The calculated results of GMM-SOMVSA-CO are contrasted with MCS-CO and FOMVSA-CO in

Design variable | FOMVSA-CO | GMM-SOMVSA-CO | MCS-CO | Design variable | FOMVSA-CO | GMM-SOMVSA-CO | MCS-CO |
---|---|---|---|---|---|---|---|

0.6891 | 0.6675 | 0.6747 | 18.99 | 20.08 | 20.04 | ||

3.9179 | 3.7881 | 3.8117 | 13.8193 | 12.7164 | 13.0010 | ||

3.3471 | 3.3587 | 3.3607 | 5.8691 | 6.0060 | 6.0114 | ||

−0.1791 | −0.1491 | −0.1604 | 1.6303 | 1.5844 | 1.5801 | ||

12.4315 | 11.4556 | 10.4997 | 38 | 35 | 36 | ||

9.51 | 10.05 | 10.51 |

FOMVSA-CO | GMM-SOMVSA-CO | MCS-CO | FOMVSA-CO | GMM-SOMVSA-CO | MCS-CO | ||
---|---|---|---|---|---|---|---|

0.9804 | 0.9834 | 0.9854 | 0.9844 | 0.9868 | 0.9939 | ||

0.9846 | 0.9879 | 0.9904 | 0.9815 | 0.9829 | 0.9866 | ||

0.9820 | 0.9868 | 0.9870 | 0.9891 | 0.9937 | 0.9951 | ||

0.9916 | 0.9923 | 0.9941 | 19513 | 24671 | 26413 | ||

0.9842 | 0.9902 | 0.9937 | 20134 | 26110 | 25346 | ||

0.9853 | 0.9871 | 0.9896 | 76 | 99 | 98 |

The reliability indexes of the three methods all meet the requirements of the example. However, compared with the FOMVSA-CO method, the GMM-SOMVSA-CO method and the MCS-CO method are more reliable. In addition, the solution result of the GMM-SOMVSA-CO method is closer to the solution result of the MCS-CO method, which indicates that the GMM-SOMVSA-CO method has higher accuracy.

In this paper, GMM-SOMVSA-CO is proposed to solve RBMDO issues. The proposed method introduces an EM algorithm to find the parameters of the Gaussian mixture distribution. Then the non-Gaussian input joint distribution is transformed into a Gaussian mixture distribution. Finally, the response MGF is computed to reckon the reliability of the system. This paper also gives the detailed computational procedure and mathematical model of GMM-SOMVSA-CO. An engineering example is used to show the application of the GMM-SOMVSA-CO. The GMM-SOMVSA-CO method mainly has three advantages. First, GMM-SOMVSA-CO has high solution accuracy. Second, it can solve the problem that SOMVSA cannot handle (the non-normal Gaussian distribution). Third, the optimization process of the GMM-SOMVSA-CO method is consistent with the existing engineering design division. The optimization problem of each discipline level represents a certain discipline field in the actual design problem. It is not necessary to consider the influence of other disciplines in the analysis and optimization of a single discipline.