Systematic design space exploration and ... - Springer Link

Journal of Mechanical Science and Technology 26 (9) (2012) 2825~2836 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-012-0735-6

Systematic design space exploration and rearrangement of the MDO problem by using probabilistic methodology† Yong-Hee Jeon1, Sangook Jun2, Seungon Kang3 and Dong-Ho Lee4,* 1

School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, 151-742, Korea BK21 School for Creative Engineering Design of Next Generation Mechanical and Aerospace Systems, Seoul National University, Seoul, 151-742, Korea 3 School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, 151-742, Korea 4 School of Mechanical and Aerospace Engineering and the Institute of Advanced Aerospace Technology, Seoul National University, Seoul, 151-742, Korea 2

(Manuscript Received October 25, 2011; Revised March 16, 2012; Accepted May 13, 2012) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract This study proposes a probabilistic approach for systematic design space exploration and rearrangement. The stochastic quantities and qualities of design space are explored using a reliability index, and the design space is rearranged to a higher feasible region by using Chebyshev inequality. Four test cases composed of algebraic functions are used to investigate the validity of the proposed method. Results show that the converged design space can include the feasible region outside the initial design space to the rearranged design space. Moreover, the results show that the proposed method is applicable to the aircraft wing design optimization problem, which considers collaborative optimization with three subsystems, namely, aerodynamics, structure, and performance. Wing design optimizations are also performed separately for the initial and converged design spaces. The results indicate that the range obtained in the converged design space improved compared with that in the initial design space. In conclusion, the proposed design space rearrangement method was shown to have the capability to search for feasible regions that are excluded in the initial design phase. Keywords: Aerodynamic-structural coupled optimization; Aircraft wing; Chebyshev inequality; Collaborative optimization; Design space exploration; Design space rearrangement; Monte-Carlo simulation; Reliability index ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction In general, the early design phase has the most impact on product development. The rapid accumulation of knowledge and information about the design problem is the most important part that should be considered to reduce design costs [1, 2]. However, only few information on a given design problem at the initial design phase are available. Even though various methods can gather some information, they cannot fully grasp the characteristics of a design problem. Therefore, the wide scope of a design problem, including its various objectives, constraints, and design variables should be carefully considered and explored to fully understand a design problem. In most design problems, a designer defines the initial design space based on intuition or experience and then searches for a feasible design solution within the initially defined design space [3]. However, majority of a design space is composed of an infeasible region and parts of the feasible region. The feasi*

Corresponding author. Tel.: +82 2 880 7386, Fax.: +82 2 880 1910 E-mail address: [email protected] † Recommended by Associate Editor Gang-Won Jang © KSME & Springer 2012

ble region tends to decrease as the number of design variables and constraints increases, and thus, defining a physically reasonable valid design space and guaranteeing the success of the optimization and the existence of the global optimum are difficult, particularly in complicated design problems, such as multidisciplinary design optimization or multilevel design optimization. In this study, the design optimization problem was investigated using the aircraft wing design problem coupled with aerodynamics, structure, and performance. Therefore, the design optimization problem in this study is more complex than that based on single-discipline analysis. Insufficient information and knowledge regarding the three disciplines make the reasonable design space difficult to define at the initial design stage. To overcome the problem above, a number of approaches were used to efficiently explore the design space and to search for the global optimum by using stochastic criteria and approximation models [4-15]. The dividing rectangles (DIRECT) optimization [4] and the efficient global optimization (EGO) [8] are representative methods. DIRECT and EGO may be used to identify the constrained global optimum on the design space

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and explore the design space. However, the design space exploration and rearrangement results by using DIRECT and EGO are limited within the initial design space in most cases. The use of DIRECT and EGO experiences difficulties in laying the feasible region outside the initial design space into the rearranged design space, making the search for better solutions that exist outside the initial design space impossible. Thus, cautious and detailed exploration of the design space must be conducted by including the feasible region outside the initial design space to improve the feasibility. To solve the problem above, Jeon et al. [16-18] presented an automatic rearrangement method for the design space by using Monte Carlo Simulation (MCS) and Chebyshev inequality. This method can search outside the initial design space, but it cannot define the design space if the initial space does not include any feasible region. Moreover, this method can only be applied to the optimization problem, which is made up of a single-discipline analysis and a single optimizer, such as the Multidisciplinary Feasible (MDF) problem coupled with aerodynamics and structure. Consequently, it cannot be easily applied to multi-level methods, such as the collaborative optimization (CO) method, because the system and subsystem of CO have incomplete compatibility information [19, 20]. This study proposes a systematic and automatic method that rearranges the design space by using statistic and stochastic approaches. The proposed method is applied to the design optimization problem of the exact function with two variables to confirm its capability of including the feasible region outside the initial design space. In addition, despite the absence of a feasible region in the initial design space, this study presents the capability of the design space to acquire a feasible region from the rearrangement of the design space. Finally, the proposed method is applied to the design optimization problem coupled with the disciplines of aerodynamics, structure, and, performance. The results show that the proposed method can search for a better solution than the optimum solution found in the initial design space.

2. Numerical approaches In general, each design problem has various types of design variables, which are either continuous or non-continuous. In this study, only continuous design variables are examined, and the design space is defined and surrounded by its lower and upper limits. The arrangement method of the design space is performed by first understanding the feasible region based on the constraints information and estimating the mean and deviation values. The results are applied to Chebyshev inequality to enable the design space to cover the feasible region automatically. Fig. 1 shows the overall flowchart of this study. The solid line iteration loop shows the procedure of the design space automatic rearrangement method developed by Jeon et al. [16, 17]. The hatched box contains the proposed procedure that will replace the calculation procedure for the probabilistic

Fig. 1. Flow chart of the automatic design space rearrangement method.

value [18]. The previously proposed design space rearrangement method based on the Chebyshev inequality condition is performed along with the solid line procedure. The initial step is the definition of the problem and the initial range of the design space. Then, the surrogate model of the defined problem is constructed to consider the efficiency of evaluation during the entire design process. The MCS is used to obtain the probabilistic quantities and qualities of the design space with the constructed surrogate model. The probabilistic and statistic quantities of the design space, such as the probability of success (POS), reliability index value, and mean or deviation of design variables, can be calculated using the MCS. In the case of the previously proposed method, the convergence check procedure is directly conducted using the calculated POS and the mean values of the present design space. The rearrangement iteration finishes instantly and the optimization process starts once the present design space satisfies the convergent criteria. If the present design space does not satisfy the convergent criteria, the rearrangement procedure based on Chebyshev inequality condition will be conducted to update the entire range of the design variables based on the calculated mean and deviation values of the design variables. The next iteration will then start for the updated design space. The previously proposed method is a relatively simple iterative algorithm, but probabilistic evaluation and rearrangement of the design space can be successively conducted using Chebyshev inequality. Although probabilistic distributions of the design variables are completely unknown, the Chebyshev inequality condition for uniformly distributed random variables provides a reasonable basis for design space rearrangement. However, if the feasible region does not exist within the initial design space, the mean and deviation values of the feasible region also do not exist. Therefore, the Chebyshev inequality cannot be applied in such case. Furthermore, if a very small feasible region exists within the initial design space, the iteration number unnecessarily increases until it finds a converged design space. The reliability index, which includes geometric information among the whole design points and constraint functions, was introduced to overcome the problems above. As can be seen in Fig. 1, the hatched box shows

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the replaced procedure for obtaining the mean and deviation by using the reliability index. Even though the feasible region does not exist within the design space, the mean and deviation values of the design variables can be evaluated and determined efficiently using the proposed method. A detailed description of the proposed method is provided in the following. A rapid collection of the entire design space information is important to enable the MCS to be applied to an approximated model. The function, which is composed of an algebraic expression, is not a significant issue. However, the solution for the partial differential equation (PDE)-like Euler equations can take a few days to obtain a single output. Therefore, the construction of a surrogate model is efficient when the function is not an algebraic expression, e.g., 2nd-order polynomial or neural network. The constructed approximate models, the ratio of the occupation of the feasible region in the design space, the POS, and the reliability index (k) of the each sample can be calculated using the MCS. POS can be calculated based on the number of samples that satisfy all the given constraints by spraying 1,000,000 samples into the design space. If the feasible region is very small or even inexistent, several millions of samples are still insufficient to search for the exact feasible region in the design space. However, the reliability index in the proposed method is calculated from the samples, as shown in Eq. (1). The exact feasible region cannot be searched, but its approximate location can be inferred. ki = −

gi

σ gi

,

i = 1,⋯ ,m

(1)

In Eq. (1), g i is the i-th constraint, σ gi is the deviation of g i , and m is the number of constraints. Each g i value at sampled design points can easily be evaluated using MCS with the surrogate model. A negative g i value satisfies the constraint, and σ gi shows the variation of g i caused by input disturbance. Therefore, the reliability index ( ki ) physically indicates the distance of a sample from the boundary of g i . In other words, if ki of a sample is positive, the g i constraint is satisfied, and the sample is farther from the boundary of g i because the value of ki is large. On the other hand, if ki is negative, the g i constraint is not satisfied, and the sample is farther from the region that satisfies g i because the value of ki is small. From ki , which has these characteristics, each sample should choose the minimum ki value as a representative to estimate whether the sample exists in the feasible region or not. Temporary mean µ x should be chosen based on the decided reliability index, as shown in Eq. (2). In other words, the input value, which has the largest k value among the reliability indexes ( k ) of each sample, is taken as the mean ( µ x ) of the feasible region and at the same time, the largest k is chosen among the critical k values.

µx =

arg max x∈{input of MCS sample}

( min ki )

,

i = 1,⋯ ,m

(2)

Design Space

Mean of Real Feasible Region

Feasible Region in Design Space

Temporary Mean from Reliability Index

Mean of Input

(a) The design space exactly includes feasible region

Design Space

Mean of Real Feasible Region

Feasible Region in Design Space Temporary Mean from Reliability Index

Mean of Input

(b) The design space does not exactly include the feasible region Fig. 2. Mean of the input and temporary mean from reliability index with respect to the design space.

If the design space completely includes the feasible region, the mean of the feasible region, the mean of the input, and the mean from reliability index almost correspond with one another, as shown in Fig. 2(a). However, if the design space does not cover the real feasible region, the mean of the input could differ with the mean of the real feasible region, whereas the mean defined in Eq. (2) could approach the mean of the real feasible region, as shown in Fig. 2(b). Therefore, deciding the mean in Eq. (2) is a more efficient way of searching the real feasible region compared to merely using the input mean. The mean is chosen from the feasible region in the given design space, and thus, the variance ( σ 2 ) can be calculated from the sample in the feasible region. However, there are cases when the feasible region does not exist in the design space or it is too small that the sample does not exist for calculating the variance. In such cases, the variance is chosen from the range of the prescribed design space. To prevent the design space from growing up indefinitely, the range of the given design space is specified as µ − nσ to µ + nσ , which is the confidence interval of Chebyshev inequality to enable the variance ( σ 2 ) of the previously chosen mean to be calculated. The mean and the variance of the feasible region were calculated in the previous step, and the Chebyshev inequality condition was applied to the rearrangement of the design space to facilitate the inclusion of the feasible region of the design space [16]. The Chebyshev inequality condition can be written as follows:

{

}

P x − µ ≤ ε ≥1−

σ2 ε2

(3)

where ε is a small arbitrary value. If x has normal distri-

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bution and ε is equal to 2σ , the probability can be calculated as follows:

Local optima

}

P x − η ≤ 2σ = 2G (2) − 1 = 0.9545 .

1

(4)

In other words, the probability that x is inside the interval ( −2σ , 2σ ) is 95%. However, if the distribution of the random variable is unknown, the adjusted Chebyshev inequality condition is used for any f ( x) to indicate that the probability that x is inside the interval ( −3σ , 3σ ) is at least 8/9 [21]. This process is repeated using the method above until the POS calculated during the rearrangement of the design space and the variation of the mean of the feasible region becomes less than 0.1%. This procedure can be used to define the design space of the problem with a single optimizer, such as the MDF. Two subjects are considered when such procedure is applied to a multilevel MDO problem, such as the CO. First, the search for the feasible region in the system level of CO is difficult because the constraints are equality conditions. Therefore, the common region of the design spaces, which is supplied by each subsystem, should be considered as the feasible region in the system level. The common region is the space that satisfies the compatibility condition, which is the constraint of the system. Second, all subsystems should not only satisfy its own constraints but also exist inside the design space of the system. Thus, in a subsystem, the design space of the system should be considered together with the constraints because the design space of the system also contains information of other subsystems.

x2

{

2

0

Feasible region -1

Global optimum -2

-1

0

1

2

x1

Fig. 3. The Goldstein function and the constraints. x 1, x 2

Discipline 1. y 1 = x1 + x2 g1(x 1, x 2)= 0.5(x 1+1.4) 2+x 2