|Computer Modeling in Engineering & Sciences|
RBMDO Using Gaussian Mixture Model-Based Second-Order Mean-Value Saddlepoint Approximation
1School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu, 611731, China
2Failure Mechanics & Engineering Disaster Prevention and Mitigation, Key Laboratory of Sichuan Province, Sichuan University, Chengdu, 610065, China
3Yangzhou Yangjie Electronic Technology Co., Ltd., Yangzhou, 225008, China
4Sichuan Special Equipment Inspection and Research Institute, Chengdu, 610100, China
5Food Safety Inspection Technology Center of Administration for Market Regulation of Sichuan Province, Chengdu, 610017, China
*Corresponding Author: Tao Lin. Email: firstname.lastname@example.org
Received: 10 December 2021; Accepted: 13 January 2022
Abstract: 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.
Keywords: Uncertain factors; reliability-based multidisciplinary design optimization; saddlepoint approximate; gaussian mixture model; collaborative optimization
Multidisciplinary Design Optimization (MDO) is a methodology that deals with complex and coupled engineering system design problems [1,2]. MDO uses the cooperative mechanism of interaction to design complex systems and subsystems to improve product performance [3,4]. In actual engineering, the exchange of information between coupled disciplines will lead to the spread of uncertain factors [5–8]. However, in the original MDO method, the influence of uncertain factors is ignored, which may lead to product failure [9–11]. The Reliability-Based Multidisciplinary Design Optimization (RBMDO) method is proposed to deal with the influence of uncertain factors [12–23]. In RBMDO, reliability analysis methods mainly include three methods: the analytical method, the approximate analytical method, and the simulation method [24–27]. The analytical method can be used to handle simple reliability constraints. The simulation method requires lots of samples to obtain accurate analysis results . The approximate analytical method is widely used because of its balance between calculation accuracy and efficiency [29,30].
Approximate analytical methods in multidisciplinary reliability analysis methods include First-Order/Second-Order Mean-Value Reliability Analysis (FOMVRA/SOMVRA) methods . However, both of them require the Rosenblatt transformation, which may add to the complexity of the Limit State Function (LSF), resulting in low accuracy for uncertainty analysis [32,33]. To tackle this problem, Huang et al.  proposed a First-Order Mean-Value Saddlepoint Approximation (FOMVSA) method. Papadimitriou  proposed a Second-Order Mean-Value Saddlepoint Approximation (SOMVSA) method to increase the calculated precision of the FOMVSA method. In SOMVSA, however, only when the distribution of input random variables is Gaussian distribution can the output Moment Generating Function (MGF) be directly expressed by the input MGF.
To deal with this problem, Papadimitriou et al.  proposed a Gaussian Mixture Model-based Second-Order Mean-Value Saddlepoint Approximation (GMM-SOMVSA) method by using a Gaussian mixture model instead of randomly distributed random variables. Thus, in this study, the GMM-SOMVSA method is introduced and is integrated with the Collaborative Optimization (CO) method [37–39] to solve RBMDO problems. Furthermore, the formula and procedure of RBMDO using GMM-SOMVSA-Based CO (GMM-SOMVSA-CO) are proposed.
The other parts of this paper are organized as follows. In Section 2, MDO and RBMDO models are introduced. Current reliability assessment methods in RBMDO are also reviewed. In Section 3, the computational procedure of GMM-SOMVSA is introduced. In Section 4, the proposed method is explained, namely, the RBMDO method combining GMM-SOMVSA and CO (GMM-SOMVSA-CO). In Section 5, the calculated precision of the proposed method is proved using an engineering example. Finally, Section 6 concludes the paper.
2 The Emulation and Approximation Ways for the Reliability Assessment in RBMDO Issues
2.1 The RBMDO Model
The RBMDO model is shown in Eq. (1).
where: represents the inequality constraint. represents the design goal of the design optimization problem. represents the deterministic shared variable. represents the deterministic local variable. represents the equality constraint. represents the coupling variables between different disciplines. represents the uncertainty shared variable. represents the uncertainty local variable. represents the reliability constraint in the design optimization problem. represents LSF. represents the reliability index in the optimization objective. is mean value of random parameters.
2.2 Emulation and Approximation Ways for the Reliability Assessment
The reliability can be expressed as following:
where: is random design variables and . is the joint Probability Density Function (PDF) of .
Joint PDF is non-linear and random variables are multi-dimensional . Therefore, under the modern scientific system, it is not easy to directly find the solution of Eq. (2). Currently, three strategies are often used to estimate : the sampling-based methods, the Most Probable Point (MPP) based methods, and the moment matching methods [41–46]. Besides the above methods, the Saddlepoint Approximation (SA) is a novel alternative method which is widely utilized in reliability engineering [47,48]. Here, different SA methods, including FOMVSA, SOMVSA, and GMM-SOMVSA, are introduced into RBMDO.
3 The Reliability Evaluation Method with GMM-SOMVSA
FOMVSA only uses the first two sample points of random variables as the research objects . The Cumulative Distribution Function (CDF) and PDF of the LSF can be obtained by SA. And use of the MGF of the input random variables to compute the MGF of the LSF is allowed. These features make the computational procedure simple and clear; thus, it can roughly solve RBMDO problems. The detailed computational procedure is as follows:
Step I: Linearize the LSF. Linearized LSF using first-order Taylor expansion at the deterministic variable's value or the random variable's mean. The expansion expression is shown in Eq. (3) :
where: is a deterministic variable and ; represents the LSF; is a random variable and ; is the mean value of the random variable ; is the value of the deterministic variable .
Step II: Calculate the Cumulative Generating Function (CGF) of the LSF. If the CGF of the random variable is , it can be expressed by Eq. (4):
where: is the MGF of the random variable . The expression of is shown in Eq. (5):
where: is the PDF of random variables.
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 are independent, and the corresponding CGF is . Then the CGF of the response is ;
Property 2: If is random variable and its CGF is . Then the CGF of ( and d are constants) is .
By these two properties, the CGF of is showed in Eq. (6):
Find the solution where the first derivative of the above formula is equal to zero:
Step III: Calculate the PDF and CDF of the LSF. Through Eq. (7), the value of saddlepoint can be obtained. Finally, SA method is used to approximate the PDF and CDF of the LSF. The PDF of the LSF can be calculated as follows:
where: is the second derivative form of CGF.
The failure probability (CDF of the LSF) can be obtained by Eqs. (9) or (10):
where: and are the PDF and CDF of the standard normal distribution, respectively, and , ; is a piecewise function related to the saddlepoint , as shown in Eq. (11):
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 . Nevertheless, this method only uses Taylor first-order expansion, which leads to poor approximation of the LSF. This problem is especially notable when the variance of the random variables is large. Thus, this method is appropriate only when the standard deviation of random variables is small and the uncertainty of the model is low.
To improve the calculation accuracy of FOMVSA, SOMVSA method is proposed . Compared with FOMVSA, this method has a more accurate approximate solution. And it can also solve the problem that FOMVSA cannot handle the large standard deviation of random variables . The detailed computational procedure is as follows:
Step I: Linearize the LSF. Linearized LSF using second-order Taylor expansion at the deterministic variable's value or the random variable's mean. Its expression is as follows:
After simplifying the above equation, as shown in Eq. (13):
Express the above equation as a matrix polynomial form, as shown in Eq. (14):
The parameter expression is as follows:
Step II: Calculate the CGF of the LSF. Through the above derivation process, the MGF of the LSF can be expressed as follows:
where: is MGF of the LSF. take the natural logarithm of to obtain the CGF of the LSF .
Step III: Calculate the PDF and CDF of the LSF. Take the derivative of the CGF of the LSF and make it equal to zero. Solve the above equation to get saddlepoint . Then combine Eqs. (9) or (10) to get the PDF and CDF of the LSF .
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 . This method uses the Expectation Maximization (EM) algorithm to find the parameters of the Gaussian mixture model, thereby transforming the joint distribution of non-Gaussian inputs into the Gaussian mixture model. Finally, the MVSOSA method is used to solve the reliability of the system. The detailed steps are as follows:
Step I: Extract a sample of size M from the copula function . Where, .
Step II: Use the inverse marginal CDFs and produce a sample as . After selecting a Gaussian input distribution number K, the joint PDF can be expressed as follows:
where: is the number of Gaussian input distributions. is the weight of the kth Gaussian distribution. is the mean vector of the kth Gaussian distribution. is covariance matrix of the kth Gaussian distribution.
Step III: Apply the EM algorithm to solve the parameters , , of the Gaussian mixture distribution. The detailed computational procedure is as follows:
where: i represents the ith cycle; represents the probability that the nth sample data belongs to the kth sub-model (); are sample points extracted from arbitrary input distribution ().
The corresponding parameters are updated once every iteration. The iterative convergence conditions of the algorithm are as follows:
where: is parameter vector, namely, ; is arbitrarily small positive number.
Step IV: Linearize the LSF. From Eq. (14), the expanded LSF can be obtained.
Step V: Calculate the CGF of the expanded LSF.
The MGF of LSF can be expressed as follows:
where: is the MGF of LSF.
Take the natural logarithm of to obtain the CGF of the LSF .
Step VI: Calculate the PDF and CDF of the LSF. First, take the derivative of the CGF of the LSF and make it equal to zero. Second, solve the above equation to get saddlepoint . Finally, the PDF and CDF of the LSF is solved by Eqs. (9) or (10).
4 The Procedure of GMM-SOMVSA-CO
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:
Step I: Set the initial value of the design variables . The letter i represents the ith discipline. k represents the kth cycle. The superscripts dis and sys represent the discipline level and system level, respectively.
Step II: System-level optimization. The optimization model is as follows:
where: represents the compatibility constraint of the ith discipline; is an arbitrarily small positive number.
Step III: Discipline-level optimization. The solution obtained by Eq. (24) is the design parameter of the ith discipline. The optimization model 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 . (2) Use the inverse marginal CDFs and produce a sample as . Choose a Gaussian input distribution number K. (3) Apply the EM algorithm to solve the parameters of the Gaussian mixture distribution. (4) Use Taylor series to expand LSF to second order at the random variable's mean. (5) Calculate the CGF of the expanded LSF. (6) Calculate the PDF and CDF of the LSF.
Step IV: Determine whether the algorithm has converged. The solution of Eq. (25) is substituted back to the system layer to calculate the compatibility constraints . If and the Objective Function (OF) of the system layer is stable, output the solution of Eq. (25); otherwise, continue to Step II and .
The GMM-SOMVSA-CO solution procedure is shown in Fig. 1.
5 Engineering Example
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 Figs. 2 and 3, respectively. The working device of the excavator consists of a lifting mechanism and a pushing mechanism. The mechanism composition and movement principle of the working device are shown in Fig. 4. The purpose of optimization is to maximize the lifting force of the working device and minimize the size of the mechanism of the working device.
This optimization design has 11 design variables: r, , , , , , , , , , . r is the radius of the hoisting sheave. is the distance from the support point of the swinging arm to the center of rotation. is the height from the support point of the swinging arm to the ground. is the distance from the center of the hoisting sheave to the center line of the swinging arm. is the length of the bucket rod. is the height of the loaded item. is the starting position of loading items. is the length of the swinging arm. is the length of the swinging arm below the pushing shaft. is the distance from the saddle seat to the centerline of the swinging arm. is the corner of the bucket rod. Superscript M means mean. The specific uncertainty information of input variables is shown in Table 1.
Divide this optimization problem into two disciplines, as shown in Fig. 5. Therefore, lifting discipline is discipline 1; pushing discipline is discipline 2.
The mathematical model of this optimization problem is shown as follows:
where: q is the standard excavating bucket capacity; is the excavating bucket full factor. is the loose coefficient of the rock; is the efficiency of the working device. Ge et al.  gave the calculation methods of the specific intermediate design variables in the above optimization model.
The system-level and discipline-level optimization mathematical model can be expressed by Eqs. (27)–(29).
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 Tables 2 and 3. represents the number of iterations of the system layer. and represent the number of iterations of the disciplines 1 and 2, respectively.
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.
Funding Statement: The support from the National Natural Science Foundation of China (Grant No. 52175130), the Sichuan Science and Technology Program (Grant No. 2021YFS0336), the China Postdoctoral Science Foundation (Grant No. 2021M700693), the 2021 Open Project of Failure Mechanics and Engineering Disaster Prevention, Key Lab of Sichuan Province (Grant No. FMEDP202104), the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2019J035), the Sichuan Science and Technology Innovation Seedling Project Funding Project (Grant No. 2021112) and the Sichuan Special Equipment Inspection and Research Institute (YNJD-02-2020) are gratefully acknowledged.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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