Reliability-Based Multidisciplinary Design Optimization under Correlated Uncertainties

Complex mechanical system is usually composed of several subsystems, which are often coupled with each other. Reliabilitybased multidisciplinary design optimization (RBMDO) is an efficient method to design such complex system under uncertainties. However, the present RBMDO methods ignored the correlations between uncertainties. In this paper, through combining the ellipsoidal set theory and first-order reliability method (FORM) for multidisciplinary design optimization (MDO), characteristics of correlated uncertainties are investigated. Furthermore, to improve computational efficiency, the sequential optimization and reliability assessment (SORA) strategy is utilized to obtain the optimization result. Both a mathematical example and a case study of an engineering system are provided to illustrate the feasibility and validity of the proposed method.


Introduction
The conventional MDO is a deterministic method, which considers the loads, material properties, geometry dimensions, and so on as deterministic factors.However, uncertainties often exist during entire life cycle of the mechanical systems.The MDO results might be inaccurate due to these uncertainties and their propagations between subsystems.For instance, Du and Chen [1] proposed an integrated MDO method by using the system uncertainty analysis (SUA) and concurrent subsystems uncertainty analysis (CSSUA) methods [2].Gu et al. [3,4] proposed an implicit uncertainty delivery method to estimate the uncertainties.Sankararaman et al. [5,6] utilized Bayesian statistics to solve the presence of incomplete information in actual design situations.Yao et al. [7] presented a RBMDO procedure based on combined probability and evidence theory to solve the problem under aleatory and epistemic uncertainties.Jiang et al. [8] proposed a spatial-random-process (SRP) based on multidisciplinary uncertainty analysis (MUA) method to address both aleatory and epistemic uncertainties.For more details on how to conduct mechanical design under uncertainties, including physics-based and reliability-based design, one can refer to [9][10][11][12][13][14][15][16][17][18].
Recently, reliability analysis in MDO has been paid more attention.Agarwal et al. [15] quantified uncertainties using evidence theory and then applied the sequential approximate optimization approach to drive the design optimization.Du and Chen [16] presented a modified concurrent subsystem uncertainty analysis (MCSSUA) method for uncertainty analysis in MDO. Park et al. [17,18] combined reliability-based design optimization, possibility-based design optimization, and robust design optimization methods with the MDO method to obtain the reliable results.Lin and Gea [19] developed a probabilistic gradient-based transformation method (PGTM) to solve the complex design optimization problems under design uncertainties.Ahn and Kwon [20] presented an efficient reliability-based multidisciplinary design optimization strategy using BLISS.For ensuring the operational reliability of engineering critical components, Zhu et al. [21][22][23] studied a unified failure criterion for structural integrity analysis and life assessment of critical components operating under harsh conditions.
The above-mentioned reliability-based multidisciplinary design optimization (RBMDO) methods intend to quantify and model various uncertainties with an assumption that these uncertainties are independent of each other.However, uncertainties are often correlated in real mechanical design

Correlated Uncertainty Analysis
where  is the nominal value of design variable,  is the symmetric characteristic matrix which determines the shape and principal axis direction of the ellipsoid, and  is the radius of the ellipsoid.Let   ,   be the low and upper bounds of , respectively, and the nominal value  can be obtained by The ellipsoidal model under correlated uncertainties is derived as follows: where Cov(  ,   ) is the covariance of   and   .     is the correlation coefficient of   and   ; the correlation between uncertainties   and   can be expressed as where    is the standard deviation of   and (  ) is the variance of   .In this analysis, take two correlated uncertainties as an example:  = [ 1 ,  2 ], and  1 ∈ [1,5],  2 ∈ [1,5].The correlation coefficient matrix  0 is as follows: and the covariance matrix  0 is ) .
Using (3), the corresponding ellipsoidal model  0 is constructed as follows: ) According to the minimal volume method [27], we have  = 3.The corresponding schematic diagram of these two correlated uncertainties is shown in Figure 1.

Reliability Analysis under Correlated Uncertainties
In most engineering optimization problems, designers assume that uncertainties are independent of each other.However, correlations often exist among multisource uncertainties.The conventional reliability methods, such as first and second reliability methods, cannot be used to solve such problems directly.
where  is the objective function,  is a vector of random design variables,  is a vector of design parameters,   (, ) ≤ 0 is the th constraint, which is also called the limit state function or performance function in reliability analysis, [  ] represents the minimum reliability accepted by designers, and [(, ) ≤ 0] denotes the probability of design variable  in the feasible region.First-order reliability method (FORM) is a traditional method to calculate the reliability.FORM simplifies the integral operation and approximates limit state function by using the mean and variance of random variables [28,29].FORM requires that random variables are independent standard normal distributions.However, the uncertainties are often correlated in real engineering issues.Accordingly, this situation needs to transform correlated uncertainties into independent uncertainties during reliability analysis [30,31].
For independent uncertainties , we need to transform it into a standard normal distribution  by Then, the Taylor approximation with respect to limit state function can be derived.In order to improve calculation accuracy of Taylor approximation, the value of the probability density at the expansion point ( * ) should be as follows: the bigger the better.When leaving this point the value of probability density function will decline at a faster rate.Thus,  * is also called the most probable point (MPP).The mathematical model for solving MPP is shown as follows: where ‖‖ = √ 2 1 +  2 2 + ⋅ ⋅ ⋅ +  2  ; in geometric space MPP is the closest point on the limit state function to the origin of coordinates.Let  = ‖ * ‖, which is the reliability index.
Inverse reliability strategy is introduced to solve reliability optimization problems [32].The strategy tries to find percentage performance   in the case of a given probability : Then, this leads to a new performance function: The MPP should satisfy the following formula: The reliability problem can be transformed to find the MPP of the maximum performance function on the circle , and the following optimization model can be established: max  (  , )

Transformation of Correlated Uncertainties.
As mentioned before, current reliability methods cannot be applied to handle correlated uncertainties directly.When uncertainties are statistically correlated, Rosenblatt transformation, Nataf transformation, and orthogonal transformation are often used to handle these uncertainties in reliability analysis.Among them, Rosenblatt transformation needs to know the joint cumulative distribution function of variables in advance.Nataf transformation takes into account the change of correlation caused by transformation process.Orthogonal transformation is relatively simple and easy to be implemented, but it has a premise that the transformation process is basically not to change the relevance of the variables.More details on these three methods can be found in [30].
Since orthogonal transform has shown higher computation efficiency, in this paper, it is used to deal with the correlated uncertainties.
Orthogonal transformation can be applied to handle correlated random variables.Let vector  present the normal correlated random uncertainties:  = ( 1 ,  2 , . . .,   )  .The covariance  of  can be expressed as follows: Let the matrix  consist of regularized eigenvectors of .
The matrix  should satisfy the condition that  can be transformed to a diagonal matrix through matrix operation   .Let  = , since  −1 = ; this leads to where  is the linear independent vector and the covariance of The mean value and covariance can be expressed by the following formulas, respectively: Linear combination of normal random variables is still a normal random variable.Since the uncorrelated normal random variables are equivalent to the independent normal random variables,  is the independent normal random variable.Then, the performance function is According to (16), the reliability can be calculated by utilizing FORM.

The Effect of Correlated Uncertainties on Reliability.
The correlated uncertainties generally exist in real mechanical design problems.Not considering the correlations between uncertainties might lead to inaccurate design solutions.In this section, a simple mathematical model is taken as an example to illustrate the effect of correlated uncertainties on reliability.Let  1 and  2 be design variables, and The limit state function is given as follows: If the correlation between the two uncertainties is not considered, the safe region and failure region of limit state function can be obtained as shown in Figure 2.
Assuming that these two uncertainties are correlated and the correlation coefficient   1  2 is 0.3, the corresponding safe region and failure region are shown in Figure 3. Let  1 and  2 be normal distributions with mean 3 and variance 0.3; the correlation coefficient   1  2 is 0.3.Firstly, the orthogonal transformation method is utilized to transform correlated  to independent ; then  is transformed to the standard normal space  for reliability analysis by using FORM.Figure 4 shows that when the correlations between uncertainties are not considered, the corresponding reliability index is  1 = 2.26 and when the correlations of uncertainties are considered, the corresponding reliability index is  2 = 3.27.From this simple mathematical example, a conclusion can be drawn that correlations of uncertainties have shown an influence on reliability.

Reliability-Based MDO under Correlated Uncertainties
Since a complex designed mechanical system often involves many subsystems, in this section, reliability-based multidisciplinary design optimization is carried out by considering correlated uncertainties.

RBMDO.
In this study, we assume that a system is composed of two subsystems, which is shown in Figure 5.
According to Figure 5, the model of RBMDO is as follows: min   (  ,   ,   ,   ,   ) where   ,   are the share variables and share parameters, respectively,   is the coupled state variable from th subsystem to th subsystem,   and   are the local variables and local parameters of th subsystem, respectively,   and   represent the constraints and objective function of th subsystem, respectively, ℎ  is the equality constraint,   is the function to obtain coupled state variable   , and [] is the designed reliability.

SAND-RBMDO. Simultaneous analysis and design
(SAND) is a typical MDO method.It utilizes optimization solver to substitute the system and/or subsystem analysis, which reduces the high analysis cost.In SAND method, in order to reconcile the inconsistency caused by coupling characteristics of multidiscipline system, the coupled variables are treated as design variables.The framework of SAND-RBMDO is elaborated as shown in Figure 6.
In Figure 6,   are state variables and    are complement variables.The deterministic MDO optimization for the th iteration can be expressed as where   ,   are the nominal values, where  is the reliability index.

CC-SORA-RBMDO.
The reliability-based MDO under correlated uncertainties is investigated in this section.The correlations between uncertainties are described by ellipsoidal model in the deterministic MDO process.When the reliability analysis is implemented, the CC-FORM method is used to calculate the reliability index.In order to improve computation efficiency, SORA strategy is applied [33,34].The procedure of CC-SORA-RBMDO is shown in Figure 7.
According to Figure 7, the optimization steps of CC-SORA-RBMDO are given as follows.
Step 1.According to the correlation coefficient of any two uncertainties, obtain the covariance matrix of correlated uncertainties (see ( 4)); then construct ellipsoidal model (see (3)).The ellipsoidal model is added as constraint   (  ,   ) ≤  2 in the MDO model.The deterministic MDO (see (22)) is implemented.
Step 2. The optimization result of deterministic MDO can be obtained.The corresponding inverse reliability analysis (see (23)) is implemented by utilizing the FORM method.
Step 3. If the optimization convergence and the reliability requirements of constraints are satisfied, then end the optimization process.Otherwise implement the next step.
Step 4. According to the values of MPP, generate new mean values of correlated uncertainties and implement new deterministic MDO.
A corresponding flowchart of the proposed procedure is given in Figure 8.

Case Studies
In this section, both a mathematical example and a case study of an engineering system are utilized to illustrate feasibility and validity of the proposed method.

A Mathematical Example.
A classical MDO problem is introduced to illustrate the proposed MDO method [35].The problem has two subsystems.Coupling relationship exists between these two subsystems, as shown in Figure 9.The mathematical expression is listed as follows: ℎ i = 0; G P s , P i ≤  2 ; i = 1, 2, . . ., n s.t.Table 1: The information of design variables and design parameters.
where  is the objective function,   ,   are the sharing design variable and sharing random design parameter, respectively,  1 , ) . ( The covariance matrix   is ) ) ) The corresponding ellipsoidal model   is derived according to (3) as follows: ) ) For random design parameters, Rosenblatt transformation, orthogonal transformation, or Nataf transformation is used to convert correlated design parameters to independent design parameters.The correlation coefficient matrix   of design parameters is ) . ( The covariance matrix   is ) .
According to the following transformation we have Through converting the vector  into the linear independent vector , the covariance matrix of  is ] × .The mean value and variance of  can be expressed as   =     and   =     .The corresponding function () =   () =   (); then we can convert  to the standard normal space by Then, the FORM method is used to obtain MPP ( * ).When the deterministic MDO is implemented, the vector  can be obtained by ( 31)- (33).
The deterministic optimization problem can be solved by using SAND method.The MPP ( * ,,(

𝑠
,  * ,,( ,  * ,,( ) of  1 and  2 can be obtained from the previous cycle of reliability analysis.Then deterministic MDO optimization can be implemented by switching random variables from the  space into the  space.The frame of deterministic MDO is as follows: The design variables DV = (  ,  1 ,  2 ,  12 ,  21 ); through substituting the optimization result into reliability analysis, the corresponding MPP of  1 can be obtained by where the design variables DV = ( ,(
The optimization results are shown in Table 2. CASE 1 is the RBMDO optimization result without considering correlations between uncertainties; CASE 2 is the RBMDO optimization result by considering correlations between design variables; CASE 3 is the RBMDO optimization result by considering correlations between design parameters; CASE 4 is the RBMDO optimization result by considering correlations between both design variables and design parameters.For this mathematical example, note from Table 2 that the optimization result of CASE 2 is similar to that of CASE 1, which means the correlations between design variables have less effect on optimization result.The optimization results of CASE 3 and CASE 4 are different from that of CASE 1, which means the correlations between design parameters have great effect on optimization result.Apparently, correlations between uncertainties have shown significant influence on the optimization results.Therefore, for accurate and informative optimization results in practical engineering, correlations must be considered during the MDO procedure.

Problem
Description.This is a design optimization problem of the four-high rolling mill stand [36].The simplified structure diagram of roller base is shown in Figure 10.
The height ℎ 1 of column cross-section, the height ℎ 2 of upper beam cross-section, the height ℎ 3 of lower beam crosssection, and the diameter  1 of supporting roll are treated as design variables: The width  1 of column cross-section, the width  2 of upper beam cross-section, and the width  3 of (37) The objective function is to minimize the stand bounce value, which is composed of six parts: (1) The bending deformation of lower and upper beams caused by the bending moment: (2) The bending deformation of the upper and lower beams caused by shear stress: (3) Tensile deformation of the columns: (5) The bending deformation of the supporting rolls caused by shear stress: (6) The squash deformation between working rolls and supporting rolls: The main constraints are the deformation, stiffness, and stress requirements.There are twelve constraints in the model: (1) The contact strength of roll: 1 = 0.89 × 10  (3) The composite tensile and bending strength of column ([  ] = 0.055 × 10 6 KN/m 2 ): (4) The bending strength of upper and lower beams: (5) The dimension restrictions: As mentioned before, the system can be divided to the three subsystems, namely, column subsystem, beam subsystem, and support roll subsystem.The frame is shown in Figure 11.
The variables limits and uncertainties distributions of four-high mill stand are shown in Tables 3 and 4, respectively.) .

Reliability-Based MDO of
(51) The optimization results of four-high rolling mill stand are shown in Table 5. CASE 1 is reliability-based MDO optimization results without considering correlations between uncertainties; CASE 2 is reliability-based MDO optimization results with correlations between uncertainties.
Note from Table 5 that, compared with optimization results obtained by CASE 1, the objective function increases 5.11% by using our proposed method.This is consistent with the observation from the mathematical example that the conventional reliability-based MDO method tends to give   inaccurate and optimistic results.Moreover, the proposed method generates results that are more informative and applicable to the engineering reality.

Conclusions
In this paper through analyzing the characteristics of correlated uncertainties, a quantitative model of correlated uncertainties is established using the ellipsoidal model.The reliability analysis method under correlated uncertainties is investigated and discussed.In addition, a new RBMDO method under correlated uncertainties is proposed.Both a mathematical example and an engineering example are introduced to validate the proposed method.

Figure 2 :Figure 3 :
Figure 2: Safe region and failure region of limit state function (without considering correlations).

Figure 10 :
Figure 10: Simplified structure diagram of four-high rolling mill stand.

Figure 11 :
Figure 11: The MDO framework of four-high mill stand.
2.1.Definition of Correlated Uncertainties.Let  present dimensional uncertainties,  ∈   , and all possible values of uncertainties can be described by a multidimensional ellipsoidal set : ,   ,   ,   are optimization results obtained from deterministic optimization process.The corresponding inverse reliability optimization formula is max   (  ,   ,    ,    ,   ,    ) s.t.     (   ,    )      = ,

Table 1 .
The mean values of   ,  1 ,  2 are 2.5, 2.5, 2.5.The correlation coefficient matrix   of design variables is 1 are the local design variable and local random design parameter of subsystem 1,  2 ,  2 are the local design variable and local random design parameter of subsystem 2, and  12 and  21 are the coupled state variables.The information of design variables   ,  1 ,  2 and design parameters  1 ,  2 ,  3 is listed in
The bending deformation of supporting rolls caused by the bending moment: