IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
7016804
Composite First-Order Reliability Method for Efficient Reliability-Based Optimization of Electromagnetic Design Problems Dong-Wook Kim, Nak-Sun Choi, Chang-Uk Lee, and Dong-Hun Kim Department of Electrical Engineering, Kyungpook National University, Daegu 702-701, Korea This paper proposes a composite first-order reliability analysis method to effectively perform the reliability-based optimization of electromagnetic (EM) design problems. The proposed method utilizes both of two different ways, reliability index approach (RIA) and performance measure approach (PMA), for reliability analysis of probabilistic constraints. However, each performs a distinct function: the first merely checks the feasibility status of probabilistic constrains, and the second evaluates the failure probability of only active constraints selected from the feasibility identification. Such a unique scheme can substantially enhance computational efficiency during optimization process. The proposed method is applied to two EM design problems, and its numerical efficiency and accuracy are examined by comparison with existing methods. Index Terms— Electromagnetics, optimization, reliability theory, sensitivity analysis.
I. I NTRODUCTION
T
HE existence of uncertainty in electromagnetic (EM) devices due to manufacturing process, material property, and operational condition causes subsequent variances in product performances [1]–[3]. To systematically incorporate the uncertainty into an early design stage, a probabilistic design method called reliability-based design optimization (RBDO) has been developed in other engineering fields, such as mechanics and aerodynamics, for the last decade [4]–[6]. The RBDO formulation involves an objective function as deterministic optimization and also contains probabilistic constraints for considering the probability of the satisfaction/failure of output performances. In RBDO, the quantitative assessment of performance reliability is an essential step. Thus, the failure probability of constraint functions has to be evaluated at each design point by executing iterative simulations with the given probabilistic information of design variables. Although our community is in the early research stage of this issue, various attempts have been made in other fields. The common reliability analysis methods are first-order reliability method (FORM), moment method, and Monte Carlo simulation (MCS). Among them, FORM has been widely used to incorporate probabilistic constraints into the RBDO process [5], [6]. In FORM, the random design variables are first transformed into the independent and standard normal probability distributions. The probability of failure is then evaluated using the first-order Taylor series approximation. It turns out that an optimization problem, of which the goal is to find the most probable failure point (MPP) in a normal design space, has to be solved. According to the MPP search algorithm, there are two different approaches; reliability index approach (RIA), and performance measure approach (PMA). It was revealed that RIA often demonstrates instability, whereas PMA is robust in evaluating probabilistic constraints [6]. The reliability analysis methods require a significant computational cost in RBDO.
Manuscript received June 27, 2013; revised August 14, 2013; accepted September 20, 2013. Date of current version February 21, 2014. Corresponding author: D.-H. Kim (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2013.2283601
To improve computational efficiency of FORM-based RBDO, a composite reliability analysis method is proposed. It exploits both of RIA and PMA, but each performs a distinct function in RBDO. At an initial design point, the feasibility status of probabilistic constraints is once estimated by RIA. In accordance with a certain condition of the feasibility status, reliability analysis of PMA is carried out only for selected constraints at intermediate designs. Such composite analysis scheme can effectively eliminate unnecessary reliability computations from RBDO without sacrificing numerical accuracy and stability. Using two EM design problems, numerical efficiency and accuracy of the proposed method are examined by comparison with conventional RBDO methods adopting either RIA or PMA. II. C OMPOSITE R ELIABILITY METHOD In this section, a RBDO model based on FORM is briefly summarized, and the proposed feasibility check scheme is explained. Lastly, the program architecture, which integrates the two reliability methods into RBDO, is presented. A. Reliability Analysis of FORM In the system parameter design, the RBDO model is formulated as minimize subject to
f (d) P f (gi (x) > 0) ≤ Pt,i , i = 1, 2, . . . , np d L ≤ d ≤ dU , d ∈ R n (1)
where f is the objective function, d is the design variable vector given by d = μ(x), μ denotes the mean of a random vector x, and Pt,i is the target probability of failure with respect to the i th constraint gi in np constraints. The symbols, d L and dU , mean the lower and upper bounds given by a designer. The failure probability P f of gi is calculated by integrating the joint probability density function, f x (x), over the infeasible/fail region as Pf = (2) f x (x) dx. gi (X)>0
0018-9464 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
7016804
IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
where βt is the prescribed reliability [5], [6]. The MPP is determined by minimizing the constraint while satisfying the spherical equality constraint. Finally, the failure probability of (3) is approximated by P f = P (gi (u) > 0) ≈ (−β)
(6)
where (·) is the standard normal cumulative distribution function. It means the larger the reliability is, the smaller the failure probability is.
Fig. 1. Feasibility identification of two constraint functions. (a) MPP search for g1. (b) Feasibility status of g1 (βo > βt ,). (c) Feasibility status of g2 (βo < βt ).
The first-order Taylor series is used to handle the multiple integrations in (2). Assuming that the random variable x complies with a normal distribution statistically, it is transformed to the standard normal random variable u, which has zero mean and unit standard deviation (SD). That is, the constraint gi (x) in X-space is mapped onto gi (T (x)) ≡ gi (u) in U space. The constraint is approximated by a linear function in U -space (3) gi (u) ≈ gi (u∗ ) + ∇gi (u∗ )T gi (u) − gi (u∗ ) where ∇gi (u∗ ) is the gradient vector of gi , and u∗ is the MPP on the limit state surface (gi (u)=0) (see Fig. 1). The reliability index β is interpreted as the spatial distance between the origin and MPP in U -space (||u|| = β). In RIA, MPP is obtained after solving an optimization problem as follows: minimize subject to
u gi (u) = 0.
(4)
This results in seeking a minimum distance point on the limit state surface from the origin [5], [7]. In contrast, PMA is formulated as the inverse of reliability analysis in RIA. The first-order probabilistic performance measure is given by minimize subject to
gi (u) u = βt
(5)
B. Feasibility Identification of RIA The RBDO interactively executes two optimization models: design optimization (1) in X-space and reliability analysis of either (4) or (5) in U -space. Such double-loop optimization structure requires an intensive computational cost. However, it is worth noting that unnecessary reliability evaluations may occur under certain constraints, of which the failure probabilities are relatively small with respect to a target value at either partial or whole iterative design points. Such constraints are very robust to the variations of design variables, so they do not need reliability analysis until their failure probability values become close to the target. To remove unnecessary reliability computations from RBDO effectively, RIA is exploited for the feasibility identification of probabilistic constraints. The proposed feasibility check scheme is shown in Fig. 1, where βt is the target reliability for two constraints, g1 and g2 . The origin in U0 -space corresponds to the initial design point u0 . To explain how the feasibility check works, three nominal design points marked with square symbols are included in U0 -space. Depending on the relative distance between u0 and MPP, two conditions for identifying the feasibility status are presented. The feasibility of g1 , which is far from u0 as shown in Fig. 1(a), is first examined. The MPP (u∗0 ) is explored by the search technique, called Hasofer Lind and Rackwitz Fiessler [7]. As an example of the MPP search process, the movement of search points is marked with triangle symbols. Fig. 1(b) shows the reliability βo evaluated at u0 is much larger than the target βt . In the case (βo > βt ), the feasible/safe region is divided into two regions by a circular arc with the diameter of βt : active and inactive constraint regions. The figure shows u0 belongs to the inactive constraint region. An index αk is introduced to estimate the reliability status of g1 at nominal design points without executing RIA. The value αk is defined by the inner product of two distance vectors, u∗0 and uk , where k is the number of design iterations. For instance, the first improved design point d1 in X-space is transformed to the normal design point u1 in U0 -space, and accordingly, α1 is calculated. Using the same process, the value αk at each design point is easily obtained without the time-consuming reliability analysis. From the spatial relationship between three indices, βo , βt , and αk , the feasibility status of g1 at each iterative design point can be presumed by: 1) inactive probabilistic constraint if βo − αk > βt ; 2) active probabilistic constraint if 0 < βo − αk ≤ βt. Meanwhile, in Fig. 1(c), the point u0 is close to the limit state surface of g2 and initially locates in the active constraint region. In this case of βo < βt , the index αk is not suitable for estimating the reliability status of g2 , so it is replaced with the distance between uk and u∗0 . After all, the reliability analysis of g1 or g2 is required only when satisfying the individual active probabilistic constraint conditions. It should
KIM et al.: COMPOSITE FIRST-ORDER RELIABILITY METHOD FOR EFFICIENT RELIABILITY-BASED OPTIMIZATION
7016804
TABLE I D ESIGN P OINTS AND P ERFORMANCES AT F OUR D IFFERENT D ESIGNS
Fig. 2.
Flowchart of RBDO based on the composite reliability analysis.
Fig. 3.
2-D axisymmetric configuration of a loudspeaker.
be noted that the proposed feasibility check scheme may yield inaccurate identification specifically when dealing with highly nonlinear constraint functions. C. Program Architecture The program architecture integrating the two different FORMs, RIA and PMA, into RBDO is presented in Fig. 2. It consists of three parts as follows. 1) Feasibility identification of RIA: explore MMP at an initial design and estimate the feasibility status at each design. 2) Reliability analysis of PMA: evaluate the failure probability at each design point where MPP is explored by the advanced mean value (AVM) technique [6]. 3) Design optimization: optimize an objective function subject to probabilistic constraints with the sequential quadratic programing algorithm. The main optimization program was implemented by means of MATLAB functions, of which one provides the design sensitivity based on the finite difference method. The EM simulations of performance functions marked with gray boxes were executed with a commercial code, called MagNet [8].
Fig. 4. Performance indicator of RBDO for three different reliability analysis methods.
for loudspeaker mass M. The wanted confidence level βt is 1.645 corresponding to the failure probability value of 5% (i.e., reliability of 95%). It is assumed that the random variables comply with the normal distributions, of which SD values are presented in Table I. minimize f (d) = (Bt − B(d))2 , d = μ(x) subject to P f (g(x) > 0) − (−βt ) ≤ 0 g(x) = 1 − (M(x) − Mt )/(0.05 × Mt ) (7)
III. C ASE S TUDIES Two RBDO models are tested: one is a loudspeaker with one probabilistic constraint, and the other is a superconducting magnetic energy storage (SMES) system with four probabilistic constraints. RBDO was, respectively, performed with three different reliability analysis methods: two existing methods, RIA and PMA, and the proposed composite method.
where B is the average air-gap flux at a design point d, and the target mass Mt is 7.5 kg. Starting with the same initial point, three RBDO optima were obtained. Table I shows the optimized designs produce a good agreement with the target values of the air-gap flux and mass. To accurately assess the failure probability of the constraint, MCS was executed at the four design points. The failure probability at the initial design is 44.6% (i.e., βo < βt ), and the values of RBDO optima are close to the target 5%. From this fact, it is deduced that RBDO was performed with the initial design belonging to the active probabilistic constraint region as seen in Fig. 1(c). The numbers of finite element analysis (FEA) calls and design iterations are compared in Fig. 4. It is obvious that the proposed method shows the smallest numbers. It implies the feasibility check scheme effectively eliminates unnecessary reliability computations during the RBDO process.
A. Loudspeaker Model Fig. 3 depicts the configuration of a loudspeaker consisting of a steel yoke and a permanent magnet [9]. Due to a heavy computation cost in RBDO, only four random design variables are considered. The goal of (7) is to obtain average flux density of Bt = 1.8 T subject to a probabilistic constraint
B. SMES Model The TEAM benchmark problem 22 of a SMES in Fig. 5 is considered [11]. The original problem with three design variables consists of a multiobjective function without any constraints. The problem itself is not suitable for a RBDO test example because it has no constraint, on which a probabilistic
7016804
Fig. 5.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 2, FEBRUARY 2014
Configuration of the SMES device. TABLE II
D ESIGN VARIABLES AND P ERFORMANCES AT F OUR D IFFERENT D ESIGNS
TABLE III FEA C ALLS AND D ESIGN I TERATIONS A CCORDING TO D IFFERENT
Fig. 6.
Comparison of magnet dimensions between two designs.
methods produce failure probabilities closer to the target 5% than RIA-based one. In Table III, FEA calls and design iterations required for RBDO are compared with each other. The proposed method contributes greatly to reducing FEA calls specifically for g3 , and g4 . The results show the proposed method significantly enhances computational efficiency of reliability analysis during the RBDO process. In Fig. 6, the dimensions of the optimized coil winding are compared with the initial ones. IV. C ONCLUSION To improve computational efficiency of reliability analysis for RBDO, a composite method consisting of RIA and PMA is proposed. The results show that the proposed method drastically reduces the numbers of FEA calls and design iterations by eliminating unnecessary reliability evaluations during the RBDO process while maintaining numerical accuracy.
R ELIABILITY A NALYSIS M ETHODS
ACKNOWLEDGMENT This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0029721). R EFERENCES condition is imposed. Therefore, the original problem was modified as seen in (8). minimize f (d) =
21
|Bstray,i (d)|2 , d = μ(x)
i=1
subject to P(gi (x) > 0) − (−βt,i ) ≤ 0 i = 1, 2, 3, 4 g1 (x) = 1 − ((E(x) − E o )/(0.05 × E o ))2 g2 (x) = (R2 − R1 ) − 12 (D2 + D1 ) g3,4 (x) = − |Jk | − 6.4 Bmax,k + 54.0 k = 1, 2 (8) where Bstray,i is the stray field calculated at the i th measurement point along line a and line b, E is the stored magnetic energy with a target value E o of 180 MJ, and the wanted confidence level βt,i at the i th constraint is 1.645. The properties of three random variables and four different designs are presented in Table II. It is observed, at the optimized designs, the magnetic energy is the same value of 181 MJ, but a 5 μT difference appears in the stray fields. Form the failure probability values at the initial design, it is inferred, RBDO started with one active probabilistic constraint, g1 , and three inactive probabilistic constraints, g2 , g3 , and g4 . Two optima based on PMA and proposed reliability
[1] F. G. Guimaraes, D. A. Lowther, and J. A. Ramirez, “Multiobjective approaches for robust electro-magnetic design,” IEEE Trans. Magn., vol. 42, no. 4, pp. 1207–1210, Apr. 2006. [2] N. Kim, D.-H. Kim, D.-W. Kim, H.-G. Kim, D. A. Lowther, and J. K. Sykulski, “Robust optimization utilizing the second-order design sensitivity information,” IEEE Trans. Magn., vol. 46, no. 8, pp. 3117–3120, Aug. 2010. [3] X. Liu, S. Wang, J. Qiu, J. Guo Zhu, Y. Guo, and Z. Wei Lin, “Robust optimization in HTS cable based on design for six sigma,” IEEE Trans. Magn., vol. 44, no. 6, pp. 978–981, Jun. 2008. [4] A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design. New York, NY, USA: Wiley, 2000. [5] B. D. Youn and K. K. Choi, “Probabilistic approaches for reliabilitybased optimization,” AIAA J., vol. 42, no. 1, pp. 124–131, 2004. [6] B. D. Youn, K. K. Choi, and L. Du, “Enriched performance measure approach for reliability-based design optimization,” AIAA J., vol. 43, no. 4, pp. 874–884, 2005. [7] Y. H. Sung, D.-W. Kim, and D.-H. Kim, “Applying reliability assessment methods to SMES designs,” IEEE Trans. Magn., vol. 47, no. 11, pp. 4623–4628, Nov. 2011. [8] MagNet User’s Manual, Infolytica Corp., Quebec, Canada, 2008. [9] L. Coelho, T. C. Bora, and L. Lebensztajn, “A chaotic approach of differential evolution optimization applied to loudspeaker design problem,” IEEE Trans. Magn., vol. 48, no. 2, pp. 751–754, Feb. 2012. [10] I. Lee and K. Choi, “Sampling-based RBDO using the stochastic sensitivity analysis and dynamic Kriging method,” Struct. Multidiscipl. Optim., vol. 44, no. 3, pp. 299–317, Sep. 2011. [11] S. Xiao, M. Rotaru, and J. K. Sykulski, “Adaptive weighted expected improvement with rewards approach in Kriging assisted electromagnetic design,” IEEE Trans. Magn., vol. 49, no. 5, pp. 2057–2060, May 2013.
