Abstract
Multidisciplinary reliability is an important part of the reliability-based multidisciplinary design optimization (RBMDO). However, it usually has a considerable amount of calculation. The purpose of this paper is to improve the computational efficiency of multidisciplinary inverse reliability analysis. A multidisciplinary inverse reliability analysis method based on collaborative optimization with combination of linear approximations (CLA-CO) is proposed in this paper. In the proposed method, the multidisciplinary reliability assessment problem is first transformed into a problem of most probable failure point (MPP) search of inverse reliability, and then the process of searching for MPP of multidisciplinary inverse reliability is performed based on the framework of CLA-CO. This method improves the MPP searching process through two elements. One is treating the discipline analyses as the equality constraints in the subsystem optimization, and the other is using linear approximations corresponding to subsystem responses as the replacement of the consistency equality constraint in system optimization. With these two elements, the proposed method realizes the parallel analysis of each discipline, and it also has a higher computational efficiency. Additionally, there are no difficulties in applying the proposed method to problems with nonnormal distribution variables. One mathematical test problem and an electronic packaging problem are used to demonstrate the effectiveness of the proposed method.
1. Introduction
Reliability-based design optimization (RBDO) is a brand new optimum design method developed on the basis of conventional optimization method. It organically combines the reliability analysis theory and mathematical programming methods [1, 2]. The RBDO of mechanical products is a computing method which takes the reliability of the products as the target or constraint conditions and gets the optimum design in the sense of probability using appropriate optimization method. Similar to the RBDO, RBMDO is a multidisciplinary design optimization considering the influence of random uncertainty on optimization results, and it includes two important parts, multidisciplinary design optimization (MDO) and multidisciplinary reliability analysis (MRA). Due to the increasingly complicated modern products, the RBMDO has been developed rapidly in recent years [3, 4].
However, the classical RBMDO has a large amount of calculation due to the multilayer nested loops. Both efficient MRA method and reasonable decouple strategy of the MDO are effective ways to improve the computational efficiency of RBMDO. For the reasonable decouple strategy of the MDO, Sues and Cesare [5] proposed a single circular order RBMDO framework, which separated the reliability analysis from the optimization calculation. Padmanabhan and Batill [6] proposed a concurrent subspace method of RBMDO, which was on the basis of the concurrent subspace optimization (CSSO) strategy and reliability index approach (RIA). Du et al. [7] presented an efficient RBMDO method, where a relationship was established between deterministic optimization part and reliability analysis part. Their method is on the basis of sequence optimization and reliability assessment (SORA). Sun et al. [8] put forward a multidisciplinary reliability design optimization with the integration of collaborative optimization (CO) strategy and performance measurement approach (PMA).
For the multidisciplinary reliability analysis part of RBMDO, because the analyzed object is the multidisciplinary system with couplings, the amount of calculation is also very large. To improve the efficiency of the multidisciplinary reliability analysis, many scholars proposed various analysis methods. Du and Chen [9] proposed collaborative multidisciplinary reliability analysis method, which combined individual discipline feasible (IDF) method and RIA method of reliability analysis. On the basis of decoupling, Liu et al. [10] presented a sequential reliability analysis method for multidisciplinary system, and CSSO and PMA were reasonably combined in the method. Huang et al. [11] established collaborative optimization method with inverse reliability analysis using CO strategy. The multidisciplinary reliability analysis methods mentioned above are mostly aimed at multidisciplinary inverse reliability analysis. This is because most RBMDO frameworks are based on SORA method. The SORA method has a high computational efficiency and strong practicability [12, 13]. The relationship of deterministic and uncertain constraints in the SORA was set up by the most probable point (MPP) of the inverse reliability analysis.
The CO strategy is proposed to solve the multidisciplinary inverse reliability analysis problems in the literature [11]. In this method, the disciplinary analysis and optimization are performed concurrently. Therefore, the computational efficiency is higher than the directly adopting multidisciplinary inverse reliability analysis based on the strategy of multidisciplinary disciplinary feasible (MDF). However, the computational efficiency of traditional CO strategy is still limited due to the inherent problems [14–16], the system level constraint Jacobian does not exist at the solution, the Lagrange multipliers either are zero or converge to zero in the subsystem level, and the system level constraints are nonsmooth functions. These existing problems may make the multidisciplinary inverse reliability analysis of integrated collaborative optimization strategy inefficient.
Recently, an alternative formulation of CO named CLA-CO is proposed by Li et al. [17]. In CLA-CO framework, the equality consistency constraints are replaced by accumulative linear approximations with the response of subsystem level. Some application cases have validated the efficiency of CLA-CO. In this study, on the basis of this method, we propose a multidisciplinary inverse reliability analysis method based on the CLA-CO. In the proposed method, each disciplinary analysis is treated as a form of equality constraint for each subsystem optimization. As the optimization process goes on, the approximations to subsystem responses are continually added to system level as the replacement of the consistency equality constraint. The disciplines analyses are implemented in parallel with the CLA-CO, and the system level constraints are no longer the subsystem object function. The multidisciplinary inverse reliability method based on the CLA-CO, which is proposed in this study, further improves the efficiency of multidisciplinary inverse reliability analysis.
The remainder of this paper is organized as follows. In Section 2, the multidisciplinary reliability analysis based on the probability model is introduced, including RIA and PMA. In Section 3, the CLA-CO is briefly introduced. Then, the proposed multidisciplinary inverse reliability analysis is provided in Section 4. In Section 5, two test problems are used to demonstrate the effectiveness of the proposed method, and conclusions are in Section 6.
2. Multidisciplinary Reliability Analysis (MRA)
For simplicity, a three-discipline system [18] is used to present the MRA. Figure 1 shows the three-coupled-discipline system with random variables, where each box represents the discipline analysis. In this system, the vector of the shared design variables, and (, 2, and 3) are the vector of the input variables of disciplinary ; are the vector of the coupling variables, and the notation is the vector of the general representation of the coupling variables, which are the vector of the output of discipline and the input of discipline ; are the outputs of discipline . Each coupling variable can be obtained through solving the following system of equations:
Figure 1 also shows the propagation of the random uncertainty in multidisciplinary system. Due to the uncertainty of the input variable, the coupling variables and the output of system are no longer determinate. The multidisciplinary reliability analysis is to evaluate whether the output of system satisfies the design requirement under the uncertainty. In RBMDO the design requirement under the uncertainty is given by the form of constraint: that is, where represents the probability and is the limit state function. Consider , and is desired probability.
The left-hand side of (2) is the reliability , and it can be written as
The reliability is generally calculated by the following integral:where denotes the joint probability density function (PDF) and .
Due to the multidimensional random variables and nonlinear integration boundary in engineering, it is very difficult or even impossible to obtain an analytical solution to the probability integration in (4); therefore, presently the first order reliability method (FORM) [19, 20] is still a main technique for its approximate solution [21]. In the FORM, the original random variables and the limit state function need to be transformed to standard normal space ( space) [22]:where is the cumulative distribution function (CDF) of standard normal distribution and is the CDF of . is the inverse of normal distribution, and represents a probability transformation function based on (5).
The integration in (4) is now rewritten aswhere is PDF of the standard normal variables .
To easily compute the above integration, the reliability index , which is defined as the shortest distance from the origin to a point on the limit state surface in space, is used to assess reliability. Then, the reliability can be expressed as
Both RIA and PMA are well-known approaches to search .
2.1. Reliability Index Approach (RIA)
In RIA, the most probable point (MPP) is searched on the limit state surface. It can be formulated as a minimization problem aswhere is the limit state equation. It should be noted that, for multidisciplinary system, solving optimization problem (9) needs the inner loop of discipline analysis to obtain the coupling variables . The inner loop that solves the coupling is given by the following system of simultaneous equations:
2.2. Performance Measurement Approach (PMA)
The reliability analysis in PMA is formulated as the inverse of reliability analysis in RIA; namely,where is the target reliability index and the percentile performance is the function value at the solution [23]. For multidisciplinary system, solving the optimization problem (11) also needs the inner loop procedure which is given by (10). This direct integration of MPP search and inner loop is usually called MPP search using MDF method. The inverse reliability MPP search with multidisciplinary systems is shown in Figure 2.
PMA is superior to RIA at the computational robustness, efficiency, and convenience of sensitivity analysis; therefore, it is widely used in reliability-based design optimization [24, 25]. In this paper, we mainly pay attention to the multidisciplinary inverse reliability analysis. Nonetheless, the proposed method is also applicable to the multidisciplinary reliability analysis in RIA.
3. Collaborative Optimization with Combination of Linear Approximations (CLA-CO)
3.1. The Framework of CO
For a clear understanding of geometric and intrinsic properties, the simple one is the best choice compared with the practical MDO problem [16, 26]. The simple optimization problem is stated as follows:
Problem (12) is decomposed into a bilevel CO architecture as shown in Figure 3. In the system level, the design variables and of problem (12) are denoted by and , respectively, and the vector form of system level variables is denoted by . The object function is still the object function of initial problem (12). However, the constraints of system level optimization are no longer the constraints of problem (12), and . It is the consistency constraints to the response of subsystem. In the subsystem level, there are two independent optimization problems; the design variables and of problem (12) are denoted by and in subsystem 1 and denoted by and in subsystem 2. The vector form of variables for subsystem 1 and subsystem 2 is denoted by and , respectively. The original constraints of problem (12), and , are assigned to subsystem 1 and subsystem 2, respectively. For each subsystem, the object is as much as possible to meet the consistency, which is as the constraints in system level. In some sense, the consistency constraint is the bridge between system level and subsystem level.
3.2. The Geometric Analysis of CO
Figure 4 shows the geometry of the above two subsystem optimizations at the th iteration. For any subsystem in Figure 4, the object function and constraint share the same tangent line denoted as at the optimal point . is the linear approximation of . The constraint and object function should satisfy the Kuhn tucker condition at the optimal point:where is a real number that represents the relation between the modules of the two gradient vectors.
At the optimum point of the subsystem, the linear approximation of constraint can be approximated by Taylor’s theorem as follows:
At the optimum point of the subsystem in Figure 4, the value of the objective function is equal to zero. Therefore, the linear approximation in the subsystem can be written as follows:
For the subsystem objective functions that have uniform expressions, that is, the sum of squared terms, the gradients of the subsystem objective functions can be obtained by the following equation:
Instead of the consistency constraint, the linear approximation as the constraint is added to the system level. The linear approximation constraint is expressed as follows:
Given that is a positive parameter, can be eliminated in (17) in the calculation process; that is,
3.3. The Formulation of CLA-CO
On the basis of the analysis in Section 3.2, the formulation of CLA-CO is stated in (19a) and (19b). In the system level of the CLA-CO, the constraints are no longer nonsmooth function, and the inconsistencies are treated explicitly by the accumulation of the linear approximations. The system optimization is restored back to the initial design optimization within the accumulative linear approximations. Thus, the computational efficiency of CLA-CO is very high.
System level iswhere is the global object; is the vector of shared design variables, and are copies of the local design variables passed to the system level; and are the number of the subsystem and the iteration, respectively, and stands for the linear approximation of th iteration in subsystem .
Subsystem level is where are copies of the shared design variables passed to subsystem and is the vector of the local design variables of subsystem ; is the vector of the constraints in subsystem .
It should be noted that, when the optimal values of subsystem are equal to the allocated target values of system, the gradients of the subsystem objection function do not exist. For this situation, no linear approximation can be constructed due to no information of the constraint bound, and no linear approximation can be added to system level.
4. Multidisciplinary Inverse Reliability Analysis under the Framework of CLA-CO
In order to improve the efficiency of multidisciplinary inverse reliability analysis, a multidisciplinary inverse reliability analysis, which combines inverse reliability analysis with CLA-CO, is proposed in this section. It has also to be noted that elements of all the random variables are assumed to be independent in our proposed method.
4.1. Model of Multidisciplinary Inverse Reliability Analysis with CAL-CO
The proposed multidisciplinary inverse reliability analysis based on CLA-CO and PMA can be formulated as follows.
System level is
Subsystem level is
In the system level, the gradients of the subsystem objective functions for the linear approximation can be obtained by the following equation:
In the proposed method, the coupled multidisciplinary analysis is decomposed into several individual disciplinary analyses. Each disciplinary analysis belongs to a subsystem (or disciplinary) optimization and is treated as a form of equality constraint in the belonging subsystem optimization. The disciplinary analysis is conducted independently and in parallel. The system level optimization is to search the MPP of the inverse reliability, and it is under the constraints of the target reliability index and the cumulative linear approximation to the subsystem responses. Therefore, the computational difficulty of multidisciplinary inverse reliability analysis with CO strategy is solved. The proposed method not only decomposes coupling among disciplines, but also can improve the computational efficiency of searching the MPP of multidisciplinary inverse reliability. The model of the integration of CLA-CO and inverse reliability for MPP is shown in Figure 5.
4.2. Procedures of Multidisciplinary Inverse Reliability Analysis with CAL-CO
The procedures of the proposed method are shown in Figure 6. Details of the procedures are described in six steps.
Step 1 (initialization). Provide original random variables , coupling relationship of multidiscipline system, and distribution parameters of the original random variables.
Step 2 ( space transformation). When the original random variables (in the original design space) all follow normal distribution, the random variables can be transformed into a set of normalized random variables (in space) by the following equation:where and are the mean value and standard deviation of , respectively.
Notably, the distributions of random variables are varied and include the uniform, Weibull, and normal distributions. The normal distribution is among the most widely used ones, and other distributions can be converted into it using a certain approach [27]. The most commonly used transformation is given by Rackwitz and Flessler [28] aswhere presents the specified point which is needed to perform transformation, and are the mean value and standard deviation of normal distribution, respectively, and and are the PDF of normal distribution and PDF of original random variables.
Therefore, for the original random variable which does not follow the normal distribution, it can be converted into normal distribution through (23), and then the obtained normal distribution variable can be converted into a normalized random variable through (22).
After the transformation, each element of follows the standard normal distribution.
Step 3 (searching for MPP). With the transformation of Step 2, the original multidisciplinary reliability assessment problem (in the original design space) is transformed into a multidisciplinary optimization problem (in space). Based on the CLA-CO method the MPP in space can be quickly found. The circulation of Step 3 is listed as follows: Input: initial value of the parameters in space Output: MPP of multidisciplinary inverse reliability in space Repeat 3.1: Distribute system level objectives to each subsystem For each subsystem (or disciplinary) (in parallel), do 3.2: Subsystem optimization End for 3.3: Construct the system level constraint. At each subsystem optimization point, construct linear approximations of the responses of subsystem, and add them to system level. 3.4: Do system level optimization with constructed constraint Until 3.4 → 3.1: System optimization has converged
Step 4 (checking convergence). The convergence criterion of the procedures is set as follows:where and are the th and the th iteration system level optimal objective values, respectively, and is a predetermined small positive parameter.
If the obtained MPP satisfies (24), go to Step 6. Otherwise, go to Step 5.
Step 5 (original design space transformation). In this step, the obtained MPP from Step 4 is transformed into the original design space by the following equation:
It has also to be noted that if the original random variables are normally distributed variables, this transformation actually does not need to be performed, because the transformation of Step 5 and transformation of Step 2 are mutual transformation for normally distributed variables.
Step 6 (multidisciplinary reliability assessment). With the obtained MPP, the percentile performance (where stands for the limit state function) can be calculated at the MPP, and the reliability of the multidisciplinary system can be assessed through the percentile performance. The reliability assessment by percentile performance is given as follows.
For a given reliability assessmentwhere is the target or required reliability while denotes a probability.
If the percentile performance , it stands for the fact that the limit state function satisfies the reliability requirement at the specified design point ; otherwise, the limit state function will not satisfy the reliability requirement.
5. Illustrative Examples
In this section, two test problems are used to demonstrate the proposed method. To verify the effectiveness of the proposed method, two aspects, efficiency and accuracy, are mainly considered. In each example, the total number of individual disciplinary analyses needed for the MDF method and CO method is compared with those for the proposed method, and the results of the reliability analysis from the proposed method, MDF method, and CO method are also compared in computational accuracy. The sequential quadratic program (SQP) algorithm is adopted as the optimizer.
5.1. Mathematical Test Problem
This numerical example is modified from [29]. It includes five random variables and two disciplines. The two-discipline system is as shown in Figure 7.
In discipline 1, the functional relationships are represented as
In discipline 2, the functional relationships are represented as
The design point is selected for the reliability analysis. In order to show the ability of handling nonnormal distribution variables, in this example, it is assumed that the design variable follows a lognormal distribution with logarithmic mean value and logarithmic standard deviation ; the design variable follows a type I extreme value distribution with distribution parameter and ; the design variables , , and are normally distributed with . The system output is treated as the limit state function. The target reliability index is 3. The proposed method is utilized to perform the multidisciplinary reliability analysis.
For simplification of presentation, only the transformations of nonnormal distribution variable are presented as follows.
The transformation of design variable is obtained through (23) and represented aswhere is the CDF of lognormal distribution and is given by
is the PDF of lognormal distribution and is given by
The transformation of design variable is obtained through (23) and represented aswhere is the CDF of type I extreme value distribution and is given by
is the PDF of type I extreme value distribution and is given by
After the above transformations all the design variables are normal distribution variables, and then these design variables can be transformed into a set of normalized random variables through (22). Therefore, the formulations of the system and subsystem for our proposed model are obtained as follows.
System level is
Subsystem 1 is
Subsystem 2 is
The solutions of this example from the proposed method are and . For comparisons, the results from MDF method, CO method, and the proposed method are listed in Table 1.
From Table 1, the solutions of three methods (MDF, CO, and CLA-CO) are almost identical. Both the solutions of CO method and CLA-CO method are very close to that of MDF. However, from the number of discipline analyses, the computational efficiencies of the three methods are quite different. The computational efficiency of those methods from low to high is MDF, CO, and CLA-CO. This indicates that the proposed method is more efficient than the other two methods.
5.2. Electronic Packaging Problem
The electronic packaging problem is a classical MDO numerical problem [30]. This electronic packaging problem has been modified into multidisciplinary reliability assessment problem in [9]. In this test example, it is modified into a multidisciplinary inverse reliability assessment problem to test our proposed method. The multidiscipline system contains two coupling disciplines which is shown in Figure 8:
in electronic discipline 1: input variables: , , linking variables: , outputs: ;
in thermal discipline 2: input variables: , , linking variables: , outputs: .
The state variables (linking variables) can be obtained by performing multidisciplinary analysis, which is to solve 13 equations. The 13 equations can be easily found from [9].
In this example, and are treated as the limit state function, and specified reliability is 3; it also assumes that uncertainties are associated with all the input variables, and all the input variables are described by normal distributions with the variation coefficient (the ratio between the standard deviation and the mean value) .
For simplification of presentation, the formulation of the proposed method is not provided, which can be constructed easily by reference to (20a) and (20b). The specified design point is . For comparisons, the MDF method and CO method are also used to solve this reverse reliability issue, respectively. The reliability analysis results from the adopted three methods are listed in Table 2.
For the reliability assessment of limit state function , the results of MDF, CO, and CLA-CO are , , and , respectively. These results are all smaller than zero, and therefore the limit state function does not satisfy reliability requirement at the specified design point. From Table 2, the solution of CO is much closer to that of MDF compared with CLA-CO; however, the minimal number of disciplinary analysis comes from the proposed method.
For the reliability assessment of limit state function , the results of MDF, CO, and CLA-CO are 0.1135, 0.1034, and 0.1012, respectively. These results are all larger than zero, and therefore the limit state function satisfies reliability requirement at the specified design point. For the comparisons of computational accuracy and efficiency, we have similar conclusion as the reliability assessment of limit state function .
Through the above analysis, it indicates that any one of the adopted three methods is qualified for the reliability assessment task, and although the proposed approach is slightly deficient in the accuracy, the proposed approach has a high computational efficiency.
6. Conclusions
In this paper, a multidisciplinary inverse reliability analysis method is proposed based on PMA and CLA-CO. In the proposed method, each disciplinary analysis is assigned to a subsystem-level optimization and treated as an equality constraint, and the approximations to subsystem responses are continually added to system level as the optimization process goes on. The method proposed by this paper not only can realize the parallel analysis of each discipline, but also has a higher computational efficiency. In addition, through transforming nonnormal distribution variables into the normal distribution, the proposed method can effectively address problems with nonnormal distribution variables. Results of examples illustrate that the proposed method is much more efficient than the multidisciplinary inverse reliability analysis method which is combined with the MDF method or CO method. The method has further enriched the solving strategy for multidisciplinary reliability analysis.
Although a high computational efficiency for multidisciplinary inverse reliability analysis can be obtained using the proposed method, there are still two aspects needing to be further developed. (1) The accuracy of the proposed method can be further improved. As shown in electronic packaging problem, the proposed approach is slightly deficient in the accuracy compared with MDF method or CO method. This is mainly because of the adopted linear approximations, and for improving accuracy, other high precision approximation approaches are potential solutions to this difficulty. (2) The application of the proposed method with statistically correlated variables needs to be developed. In our proposed method, all input random variables are assumed to be independent. However, in practice, there always exists the case where the input random variables are statistically correlated. For developing this application, the main difficulty is to develop a multidisciplinary analysis approach for statistically correlated variables. Additionally, to extend the present multidisciplinary inverse reliability analysis method into the RBMDO problems has been our future research plan.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is partially supported by National Natural Science Foundation of China (NSFC), Grant no. 51175019, and National Defense Basic Research Program of China, Grant no. JSZL2014210B001. The authors thank the academic editor and anonymous reviewers for their valuable comments.