An enhanced Monte Carlo simulation–based design and optimization method and its application in the speed reducer design

Abstract

Uncertainty-based design and optimization is becoming one of the research focuses for high safety of complex engineering systems. Evaluating failure probability is essential and necessary in uncertainty-based design and optimization. Reliability analysis method using original Monte Carlo simulation usually shows good performance in evaluation accuracy. However, in situations for computing small failure probabilities, the calculation efficiency of Monte Carlo simulation is low generally. In this study, an enhanced Monte Carlo simulation method is utilized to solve the above challenge for assessment of the probability of rare failure events in uncertainty-based design and optimization. A mathematic example and a speed reducer design problem are given to illustrate the utilization of the proposed approach.

Keywords

Uncertainty-based design and optimization complex engineering system failure probability enhanced Monte Carlo simulation rare failure events

Introduction

Unavoidable uncertainties exist in every aspect of practical engineering. Without the tolerance space for uncertainties, these variations cause the design solutions infeasible.^1–5 In complex engineering systems, even a small variation may affect system performance by the transport between linking subsystems. To solve the above problem, uncertainty-based design and optimization (UBDO) is becoming one of research focuses.^6–16 To reduce the computation cost in uncertainty analysis in UBDO, Sues et al.¹⁷ utilize surrogate models which are created at the top level instead of the finite element analysis models. Sues and Cesare¹⁸ propose a framework for UBDO that uncertainty analysis can be divided from the optimization process. Uncertainty analysis is performed initially, followed by the execution of optimization. Moreover, a current multi-stage conduction strategy is utilized to introduce existing uncertainty measure approaches into UBDO. In the study of Meng et al.,¹² the possibility and probability analysis is introduced into UBDO, proposing the UBDO with mixed variables in order to deal with the multi-sources uncertainties at the same time. Zhang et al.¹⁹ utilize the interval design information of variables to consider the lack of knowledge and propose the reliability evaluation approach to establish the reliability of complex systems.

Although many researches are finished in UBDO, there are still some problems requiring more investments. In many situations, high-quality performances, for example, long life, are needed. To obtain the accurate uncertainty estimation of safety, Monte Carlo simulation (MCS) is usually introduced in UBDO. This method is robust to the dimension and type of inputs.^20,21 It also has the ability to give accurate evaluations of the probability if sufficient simulation samples can be given. MCS, however, usually cannot be utilized in calculating small probabilities directly. Higher reliability of safety means smaller probabilistic of rare failure events. In this situation, the information of rare simulation samples about failure is necessary for evaluating reliability, which requires a large number of simulation samples until failure events happen. The needs of the large number of simulation samples will use longer evaluation time of reliability assessment.

The UBDO problem considering the above challenges is dealt here. In this article, failure probability is denoted by a product of conditional probabilities of a series of involved failure events using the subset simulation strategy.²² Then, in the conditional probability spaces, the original problem is changed through a sequence of simulations of more frequent events. A sequence of next samples in every failure event $F_{i}$ can be generated by the enhanced metropolis algorithm. The typical application of subset simulation is different from the one used in the UBDO. It should be noted that UBDO is a design and optimization process. Compared with a simple failure probability evaluation, it is more complicated. Thus, how to apply subset simulation in UBDO will also be discussed in this study.

UBDO and commonly used uncertainty analysis methods

Here, a UBDO problem can be denoted as

\begin{matrix} min_{X_{DV}} f (d, μ_{X}) \\ s . t . P [g_{i} (d, μ_{X}) \geq 0] \geq R_{i} = 1 - p_{f} \\ d^{L} \leq d \leq d^{U}, X^{L} \leq μ_{X} \leq X^{U}, i = 1, 2, \dots, n \end{matrix}

(1)

where $f (\cdot)$ denotes the system objective function, $g_{i} (\cdot) > 0$ denotes the feasible region. $R_{i}$ is the required reliability of the probabilistic constraints $P [g_{i} (\cdot) \geq 0]$ . $X_{DV}$ is a vector of input information in optimization problem. $d$ and $X$ are deterministic and random variables, respectively. $μ$ denotes the mean value of $X$ .

During the optimization process, system reliability can be guaranteed by probabilistic constraints. Theoretically, the reliability can be denoted using the cumulative distribution function of $g_{i} (\cdot)$ as shown in equation (2)

P [g_{i} (d, X) \geq 0] = \int_{g_{i} (d, X) \geq 0} f_{X} (X) dX \geq R_{i}

(2)

In practical engineering, different approximation or simulation approaches are widely utilized for reliability analysis in equation (2). Sampling-based methods usually are robust to utilize. Generally, accurate reliability estimation can be obtained when a large number of simulations are introduced. However, this method is not suitable for many practical challenges where computationally expensive $G$ is involved.²³ Most probably point (MPP)-based methods have good performance in reliability analysis. However, a nonlinear transformation is necessary in this method. This process can increase the difficulty of original problem design.^24–26 To solve the problems of the above problems, in this study, an enhanced MCS method will be introduced into UBDO.

The strategy of an enhanced MCS approach

The division of intermediate events

The original strategy of sequential Monte Carlo simulation (SMCS) is to change an initial probability evaluation problem into a series of conditional probability evaluation problems. Use $F$ to denote the target event and denote a sequence of events as $F_{1} ¹ F_{2} ¹ \dots ¹ F_{m} = F$ so that $F_{m} = ⋂_{j = 1}^{m} F_{j}$ , $F_{m} (= F)$ is the failure event of interest. Theoretically, the probability of $F$ can be calculated as a product of conditional probabilities ${P (F_{j + 1} | F_{j}) : j = 1, \dots, m - 1}$ and the initial failure probability $P (F_{1})$ as shown in equation (3)

P_{F} = P (F_{m}) = P (⋂_{j = 1}^{m} F_{j}) = P (F_{1}) Π_{j = 1}^{m - 1} P (F_{j + 1} | F_{j})

(3)

The enhanced metropolis algorithm and its application in SMCS

The enhanced metropolis algorithm is a powerful means which usually is utilized into performing uncertainty estimation.²⁷ In other words, random samples which have a certain distribution can be generated by this method. The significance of the metropolis method in this article is that if a simulation point follows a conditional distribution $p (\cdot | F_{j})$ , another simulation point could be given in the following state that also follows $p (\cdot | F_{j})$ . The basic strategy of metropolis method is given as follows.

For every $t = 1, \dots, N_{j}$ , let $p_{j}^{*} (ξ | X_{R, j} (t))$ , one-dimensional probability density function (PDF) for $ξ$ centered at $X_{R, j} (t)$ with $p_{j}^{*} (ξ_{j + 1} (t) | X_{R, j} (t)) = p_{j}^{*} (X_{R, j} (t) | ξ_{j + 1} (t))$ . Give a sequence of points ${X_{R, 1}, X_{R, 2}, \dots}$ from $X_{R, 1}$ thought computing $X_{R, j + 1}$ from $X_{R, j} = [X_{R, j} (1), X_{R, j} (2), \dots, X_{R, j} (N_{j})], j = 1, 2, \dots$ by the following strategy:

Step 1. Set ${\tilde{X}}_{R, j + 1}$ : for each $X_{R, j} (t), t = 1, \dots, N_{j}$ , simulate $ξ_{j + 1}$ from $p_{j}^{*} (ξ_{j + 1} (t) | X_{R, j} (t))$ . Compute the acceptance ratio

r_{j + 1} (t) = \frac{p (ξ_{j + 1} (t))}{p (X_{R, j} (t))}

(4)

Given the value of ${\tilde{X}}_{R, j + 1}$ by

{\tilde{X}}_{R, j + 1} (t) = {\begin{matrix} ξ_{j + 1} (t) with probability min (1, r_{j + 1} (t)) \\ X_{R, j} (t) with probability 1 - \min (1, r_{j + 1} (t)) \end{matrix}

(5)

Step 2. Accept or reject ${\tilde{X}}_{R, j + 1}$ : if ${\tilde{X}}_{R, j + 1} \in F_{j}$ , then $X_{R, j + 1} = {\tilde{X}}_{R, j + 1}$ ; otherwise $X_{R, j + 1} = X_{R, j}$ .

The procedure of SMCS

Based on the enhanced metropolis method, the strategy of SMCS can be given as follows:

Step 1. Obtain the value of $P (F_{1})$ by MCS as

P (F_{1}) \approx \frac{1}{N_{1}} \sum_{k = 1}^{N_{1}} I_{F_{1}} (X_{R, 1})

(6)

where $I_{F_{1}} (\cdot)$ is an indicator function, $I_{F_{1}} (X_{R, 1}) = 1$ if $X_{R, 1} \in F_{1}$ and $I_{F_{1}} (X_{R, 1}) = 0$ otherwise. $N_{1}$ is the sample size.

Step 2. Calculate each conditional probability by SMCS based on the metropolis method. In SMCS, the beginning points of the $(j + 1) th$ subset are the offspring of the points in the $(j) th$ subset and belong to $F_{j}$ . Then, the value of $P (F_{j + 1} | F_{j})$ can be calculated as

P (F_{j + 1} | F_{j}) \approx \frac{1}{N_{j + 1}} \sum_{k = 1}^{N_{j + 1}} I_{F_{j + 1}} ({X_{R,}}_{j + 1})

(7)

where the conditional PDF of ${X_{R,}}_{j + 1}$ is $p ({X_{R,}}_{j + 1} | F_{j})$ . For convenience, the simulation samples can be chosen by their performance values $G_{j}$ .

Step 3. The value of $P_{F}$ are given based on equations (3), (6), and (7) as

P_{F} \approx Π_{j = 1}^{m} P_{j}

(8)

UBDO using SMCS in SORA (UBDO-SMCS-SORA)

The sequential optimization and reliability assessment (SORA) approach is originally developed to enhance the efficiency in reliability-based design²⁸ and introduced into UBDO.^29–33 The basic idea of SORA is to decouple a UBDO problem into reliability analysis problem and deterministic design problem, respectively. Here, the idea of SORA is introduced and the strategy of UBDO-SMCS-SORA is to be discussed in detail.

Step 1. Solve the deterministic design problem in equation (9) and obtain the value of $μ_{X}^{(0)}$ and $d$ . The initial reliability of optimal $μ_{X_{R}}^{(0)}$ is only around 0.5 since the uncertainties are not considered

\begin{matrix} min_{X_{DV}} f (d, μ_{X}) \\ s . t . g_{i} (d, μ_{X}) \geq 0 \\ d^{L} \leq d \leq d^{U} X^{L} \leq μ_{X} \leq X^{U} \end{matrix}

(9)

Step 2. Perform MCS directly based on $μ_{X}^{(0)}$ and $σ_{X}$ . Suppose $N$ samples are simulated. Then, compare all samples by their performance values $G$ . We find the smallest value $G_{s} = min {G_{i} | i = int (R \times N)}$ in the sequence $(G_{1}, G_{2}, \dots, G_{N})$ which satisfies $P [g (d, X) \geq G_{s}] = R$ . Then, obtain the MPP, $X_{MPP} = {X | g (d, X) = G_{s}}$ .

Step 3. Construct a vector $S^{(1)} = μ_{X}^{(0)} - X_{MPP}^{(0)}$ to modify the location of $μ_{X}$ in each iteration. As shown in Figure 1, MPP is transferred to the safety region to insure the feasibility of design solution.

Step 4. Solve the deterministic design problem with the modified constraints as shown in equation (10) and obtain $μ_{X}^{(k)}$ and $d_{X}^{(k)}$

\begin{matrix} min_{X_{DV}} f (d, μ_{X}) \\ s . t . g_{i} (d, μ_{X} - s_{X}^{(k)}) \geq 0 \\ d^{L} \leq d \leq d^{U} X^{L} \leq μ_{X} \leq X^{U} \end{matrix}

(10)

Step 5. Given $μ_{X}^{(k)}$ , define $m$ intermediate failure events which include the interesting failure event $F_{m}$ as

\begin{array}{l} F_{m} = {X : g (d, X) \leq 0}, \\ F_{j} = {X : g (d, X) \leq G_{j}, j = 1, 2, \dots, m - 1} \end{array}

(11)

The probabilistic constrain $P [g (d, X) \leq 0]$ is evaluated by SMCS as shown in equation (8) where conditional failure probabilities $P (F_{j + 1} | F_{j})$ are evaluated by SMCS as shown in equation (7).

Step 6. Once the value of $P_{F}^{(k)}$ is obtained, compare this value with the target failure probability $p_{f}$ . The optimization process converges if $P_{F}^{(k)} < p_{f}$ and the value of the objective is stable. if $P_{F}^{(k)} \geq p_{f}$ , define the desired final conditional failure probability as

\begin{array}{l} P_{desire}^{(k)} (F_{m} | F_{m - 1}) = \frac{p_{f}}{P^{(k)} (F_{1}) \prod_{j =1}^{m - 1} P^{(k)} (F_{j + 1} | F_{j})} \\ = P (G \leq G_{s} | G \leq G_{m - 1}) \end{array}

(12)

which $P^{(k)} (F_{m - 1}) \times P_{desire}^{(k)} (F_{m} | F_{m - 1}) = p_{f}$ . According to their performance values, list all SMCS samples in an ascending order as $G_{1} < G_{2} \dots < G_{N_{m}}$ . We find the smallest value $G_{s} = {G_{i} | i = int (P_{desire}^{(k)} (F_{m} | F_{m - 1}) \times N_{m})}$ in the sequence $(G_{1}, G_{2}, \dots, G_{N_{m}})$ . Define $X_{MPP} = {X | g (d, X) = G_{s}}$ and the shifting vector $S^{(k + 1)} = μ_{X}^{(k)} - X_{MPP}^{(k)}$ which points from the MPP to the current solution $μ_{X}^{(k)}$ . The process is continued until $P_{F}^{(k)} < p_{f}$ and the value of objective is stable. Finally, the design solution of UBDO-SMCS-SORA can be obtained. The flowchart of UBDO-SMCS-SORA is also shown in Figure 2.

Figure 1.

The strategy of SORA.

Figure 2.

The flowchart of UBDO-SMCS-SORA.

The speed reducer example

Here, a UBDO problem is introduced to show the effectiveness of the given approach. A structure design problem of a speed reducer is considered here, which is shown in Figure 3.^34,35 The design solutions obtained by MCS-based design and optimization (DO-MCS) are utilized as reference value.

Figure 3.

Design variables of the speed reducer design.

Here, aleatory uncertainties are associated with design variables and can be described by normal distribution.

The UBDO problem of speed reducer design is as follows

\begin{matrix} min f = 0.7854 x_{1} x_{2}^{2} (3.333 x_{3}^{2} + 14.933 x_{3} - 43.0934) \\ - 1.508 x_{1} (μ_{x_{6}}^{2} + μ_{x_{7}}^{2}) \\ + 7.477 (μ_{x_{6}}^{3} + μ_{x_{7}}^{3}) + 0.7854 (μ_{x_{4}} μ_{x_{6}}^{2} + μ_{x_{5}} μ_{x_{7}}^{2}) \\ s . t . g_{1} = 27 / (x_{1} x_{2}^{2} x_{3}) - 1 \leq 0, g_{2} = 397.5 / (x_{1} x_{2}^{2} x_{3}^{2}) - 1 \leq 0 \\ P_{1} [g_{3} = 1.93 μ_{x_{4}}^{3} / (x_{2} x_{3} μ_{x_{6}}^{4}) - 1 \leq 0] \geq 0.9985 \\ P_{2} [g_{4} = 1.93 μ_{x_{5}}^{3} / (x_{2} x_{3} μ_{x_{7}}^{4}) - 1 \leq 0] \geq 0.9985 \\ P_{3} [g_{5} = A_{1} / B_{1} - 1100 \leq 0] \geq 0.9985 \\ P_{4} [g_{6} = A_{2} / B_{2} - 850 \leq 0] \geq 0.9985 \\ P_{5} [g_{24} = (1.5 μ_{x_{6}} + 1.9) / μ_{x_{4}} - 1 \leq 0] \geq 0.9985 \\ P_{6} [g_{25} = (1.1 μ_{x_{7}} + 1.9) / μ_{x_{5}} - 1 \leq 0] \geq 0.9985 \\ g_{7} = x_{2} x_{3} - 40 \leq 0, g_{8} = x_{1} / x_{2} - 12 \leq 0, \\ g_{9} = - x_{1} / x_{2} + 5 \leq 0 \end{matrix}

where $A_{1} = [(745 μ_{x_{4}} / x_{2} x_{3})^{2} + 16.9 \times 10^{6}]^{0.5}$ , $A_{2} = [(745 μ_{x_{5}} / x_{2} x_{3})^{2} + 157.5 \times 10^{6}]^{0.5}$ , $B_{1} = 0.1 μ_{x_{6}}^{3}$ , $B_{2} = 0.1 μ_{x_{7}}^{3}$ .

Here, the target failure probabilistic $p_{f}$ of each probabilistic constraint is 0.0015. The original event is divided into three partial events. The conditional probability is set as $P_{i} (F_{j}) = 0.1$ , $i = 1 ~ 6, j = 1 ~ 3$ . About $6 \times 3 \times 1 0^{3}$ samples are generated in UBDO-SMCS compared with $6 \times 1 0^{5}$ samples using in DO-MCS. After $k = 3$ cycles, the design results can be obtained, which are shown in Table 1. Compared with the results from DO-MCS, UBDO-SMCS enjoys much higher efficiency with the similar accuracy.

Table 1.

Design solutions of the example.

	$x_{1}$	$x_{2}$	$x_{3}$	$μ_{x_{4}}$	$μ_{x_{5}}$	$μ_{x_{6}}$	$μ_{x_{7}}$	$f$
DO-MCS	3.6	0.6	18	7.3	8.22	3.45	5.45	3085
UBDO-SMCS	3.6	0.6	18	7.4	8.25	3.46	5.47	3100
Deterministic design	3.5	0.7	17	7.3	7.71	3.35	5.29	2994

DO-MCS: Monte Carlo simulation–based design and optimization; UBDO-SMCS: uncertainty-based design and optimization–sequential Monte Carlo simulation.

Conclusion

In this study, an enhanced MCS–based UBDO method is given to solve the efficiency problem for the assessment of the probability of rare failure events. To reduce the computation burden and enhance the reliability analysis efficiency, subset simulation is utilized and integrated within SORA. Subset simulation resolves the efficiency problem by changing original problem into the calculation of some larger conditional probabilities. The modified metropolis algorithm is introduced to obtain simulation samples in SMCS during optimization process. A structure design problem is given to illustrate the effectiveness of UBDO-SMCS-SORA. Compared with the DO-MCS, the proposed approach can provide solutions as accurate as the DO-MCS but using much less simulation samples.

Footnotes

Academic Editor: Dong Wang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is partially supported by the National Natural Science Foundation of China (grant no. 51605047); the China Postdoctoral Science Foundation (grant no. 2016M602687); the Natural Science Foundation of Guangdong Province, China (Grant No. 611228787036); and the Open Research Fund of Key Laboratory of szjj2017-100 and szjj2013-03, Xihua University.

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