In reliability-based design optimization (RBDO), dependent input random variables and varying standard deviation (STD) should be considered to correctly describe input distribution model. The input dependency and varying STD significantly affect sensitivity for the most probable target point (MPTP) search and design sensitivity of probabilistic constraint in sensitivity-based RBDO. Hence, accurate sensitivities are necessary for efficient and effective process of MPTP search and RBDO. In this paper, it is assumed that dependency of input random variable is limited to the bivariate statistical correlation, and the correlation is considered using bivariate copulas. In addition, the varying STD is considered as a function of input mean value. The transformation between physical X-space and independent standard normal U-space for correlated input variable is presented using bivariate copula and marginal probability distribution. Using the transformation and the varying STD function, the sensitivity for the MPTP search and design sensitivity of probabilistic constraint are derived analytically. Using a mathematical example, the accuracy and efficiency of the developed sensitivities are verified. The RBDO result for the mathematical example indicates that the developed methods provide accurate sensitivities in the optimization process. In addition, a 14D engineering example is tested to verify the practicality and scalability of the developed sensitivity methods.
The output variability is the variability of a performance measure induced by the input variability, or randomness of the input variable. The analytical form of the output variability is difficult to obtain, while the input variability can be represented analytically using the input joint probability density function (PDF). The absence of the analytical output variability impedes the accurate evaluation of probability of failure (PoF), the effect of output variability on a product design. For this reason, reliability analysis methods have been researched to effectively approximate PoF. The first-order reliability method (FORM) and the second-order reliability method (SORM) calculate the PoF using first- and second-order Taylor series expansion, respectively [1–7]. In addition, the dimension reduction method (DRM) uses higher-order approximation with additional function evaluations [8,9]. The most probable failure point (MPFP), where the PoF is approximated, is necessary in the reliability analysis methods, and the sensitivity (gradient) of the performance measure is required to find the MPFP. Hence, the aforementioned methods can be categorized as sensitivity-based reliability methods. Based on the sensitivity-based methods, RBDO has been developed to obtain a reliable and cost-effective design under the output variability. For the sensitivity-based RBDO, the performance measure approach (PMA)  has been developed using concept of MPTP and probabilistic constraint from the aforementioned methods in Refs. [1–8]. PMA method finds RBDO optimum design showing a more stable and robust nature in the RBDO process . To find MPTP in PMA efficiently, the advanced mean value (AMV) [10,11] method is used first. Recently, more advanced methods, such as hybrid mean value (HMV) , enriched HMV (HMV+) [13,14], hybrid chaos control (HCC) , adaptive chaos control (ACC) [16,17], self-adaptive modified chaos control (SMCC) , adjusted advanced mean value (AAMV) , and relaxed mean value (RMV)  methods, have been developed to improve convergence of MPTP search in PMA for highly nonlinear performance measure.
As the source of the output variability is the input variability, an accurate statistical model of the input variable is critically important in RBDO. However, in usual RBDO practice, the input variable is assumed statistically independent, and its STD is kept constant. The dependent input variable has an impact on the input and output variability, so the RBDO optimum result changes significantly when the dependency is considered. Noh et al. [21–24] used bivariate copula to consider the input correlation and found that the correlation significantly affect RBDO optimum design. Specifically, it was found that the correlations of fatigue material properties of SAE 950X steel can be represented by copulas [22,24], and the correlations produces 13% less cost yet reliable RBDO optimum design compared to the case when the correlations are ignored [23,24]. Similar to the previous works [21–24], the input variable dependency is limited to the bivariate correlation and represented using bivariate copulas in this paper. To handle the input correlation, the MPTP should be obtained considering the correlation. Because the MPTP search is an optimization process, sensitivity for the MPTP search has to be developed considering the input correlation. As the previous works [21–24] do not explain the details of sensitivity for the MPTP search in the presence of input correlation, the sensitivity is derived using copula in this paper.
In addition to the input correlation, input STD determines the output variability. Because the input STD may vary as the corresponding input mean—design variable—changes in RBDO for the applications, the constant STD in the usual practice may not represent the input variability correctly for engineering applications [25,26]. For instance, a 5 mm thick steel plate has ±0.6 mm (12%) tolerance while a 15 mm thick plate has ±0.8 mm (5.33%) tolerance . Since the nominal thickness and the tolerance are represented by input mean and STD, respectively, the input STD should be represented a (nonlinear) function of input mean. Variation of the input STD forces the output variability and its consequence—PoF—to fluctuate. Due to the fluctuation of the PoF, RBDO optimum design becomes a moving target that cannot be easily obtained without accurate design sensitivity. For sensitivity-based RBDO using PMA, the design sensitivity of the probabilistic constraint has been developed [27–29]; however, the methods have not fully considered the varying STD.
In this paper, for an effective and efficient sensitivity-based RBDO process, the sensitivity for the MPTP search and the design sensitivity of the probabilistic constraint are developed considering both the input correlation and the varying STD. Using the first development, MPTP search methods for PMA, which requires the sensitivity, can consider the correlated input variable and varying STD. After finding MPTP in PMA, the second development will provide accurate direction to the next design iteration so that the sensitivity-based RBDO procedure is carried out effectively and efficiently in the presence of input correlation and varying STD. This paper consists of five main parts. In Sec. 2, the sensitivity-based RBDO using PMA is briefly revisited. In addition, the joint PDF of input variables is defined using marginal PDFs and copulas [21–30] to consider correlated input variables. In Sec. 3, the sensitivity for the MPTP search is discussed using the marginal PDFs and copulas. In Sec. 4, the design sensitivity of the probabilistic constraints in RBDO with varying STD is derived. Section 5 demonstrates the accuracy of the proposed sensitivity methods using a mathematical example. Furthermore, RBDO is carried out using the developed sensitivities for the mathematical example to verify their effectiveness. In addition, a 14D engineering example is tested to assess the scalability and practicality of the developed sensitivity methods. Finally, Sec. 6 provides the summary and conclusions of the paper.
Review of Sensitivity-Based RBDO
where X is the N-dimensional random variable vector, d is the NDV-dimensional design variable vector, dL is the lower design bound, dU is the upper design bound, is the target PoF for the jth performance measure , and NC is the number of constraints. In the RBDO formulation, the probabilistic constraint of RBDO is the PoF.
where is the MPTP for in U-space, and is the MPTP in X-space (physical space).
The optimization process to find the MPTP in Eq. (3) requires sensitivity of its cost function with respect to . Though iterative MPTP search methods, such as AMV [10,11], HMV , HMV+ [13,14], HCC , ACC [16,17], SMCC , AAMV , and RMV  can be used instead of the standard optimization process, the sensitivity is still required for the methods. Once the MPTP is obtained, the design sensitivity of the probabilistic constraint in Eq. (4) with respect to design variable is required for the RBDO process in Eq. (1). If the sensitivities are not accurate, the MPTP search and RBDO will suffer a convergence problem and fail to find the correct RBDO optimum design. Moreover, the sensitivity of the MPTP search with respect to a correlated variable is affected by the other correlated variables. In addition, the varying STD makes the probabilistic constraint to be dependent not only on the input mean but also input STD. Therefore, new sensitivities are derived in this paper to provide accurate sensitivities for sensitivity-based RBDO with the correlated input variable and the varying STD.
Sensitivity for MPTP Search
The optimization process for the MPTP search in Eq. (3) is carried out in U-space, which is the independent standard normal space. On the other hand, the performance measure is evaluated in X-space, which is the physical space. Hence, the objective function of the optimization process involves the Rosenblatt transformation  between U-space and X-space. Therefore, the derivation of sensitivity for the MPTP search should include the transformation.
The Rosenblatt Transformation for Correlated Input Variables.
Then, and , whose realizations are and , respectively, are statistically independent and follow the standard normal distribution. Thus, and are U-space variables that correspond to and .
Derivative of Performance Measure in U-Space.
The derivative with respect to can be obtained by changing indices m and n in Eq. (11). As explained earlier, the gradient of performance measure is assumed available in the sensitivity-based RBDO. Therefore, the derivatives of , and to standard normal variables and are required to calculate the sensitivity of the MPTP search.
Therefore, the sensitivity of the constraint function in Eq. (12) is readily available.
Derivative of Variable in X-Space.
where is the standard normal PDF.
where and . It is noted that the aforementioned copula density function is defined as .
Therefore, the first variable is not related to the second U-space variable . Equations (17) and (19) have second-order derivatives of the copula in their expressions. The second-order derivatives of widely used copula functions are obtained and summarized in Tables 1 and 2. Using the derivatives obtained in this section, the sensitivity for the MPTP search can be calculated analytically.
Design Sensitivity of Probabilistic Constraint
Once the MPTP has been obtained using the developed sensitivity in Sec. 3, RBDO could be performed using the MPTP and the probabilistic constraint given in Eq. (4). The effectiveness of the RBDO process depends on the accurate design search direction, which can be provided by analytical design sensitivity. In addition, the accurate search direction can reduce the total number of MPTP searches (design iterations and line searches)—in other words, the efficiency of RBDO can be improved by the analytical design sensitivity. Therefore, the analytical design sensitivity of probabilistic constraints is derived in this paper for effectiveness and efficiency of the RBDO process. It is noted that the RBDO cost function in Eq. (1) is a function of only the deterministic design variable ; therefore, its design sensitivity is often readily obtainable.
Design Sensitivity of Probabilistic Constraint.
The MPTP point is also a function of and , because they affect the constraint function in the MPTP search in Eq. (4). However, Eq. (20) indicates that the change of is not necessarily considered in the design sensitivity of the probabilistic constraint.
The missing terms and are derived in Sec. 4.2. The design sensitivity in Eq. (21) will provide an accurate design search direction in the RBDO process. If the search direction is accurate, a correct optimum design can be achieved in the RBDO process. As explained earlier, an accurate search direction will help the RBDO process to converge with the minimum number of MPTP searches. Therefore, the developed design sensitivity will improve the effectiveness and efficiency of the sensitivity-based RBDO with the correlated input variable and the varying input STD.
Derivative of Input Variable.
Because is not a function of and , the first term in the left side of Eq. (24) vanishes. Moreover, the copula density function is not always zero, so has to be zero to hold Eq. (24). Therefore, it is found that Eqs. (22) and (23) work as well for the second correlated variable .
The distribution parameters and are uniquely determined by and . Therefore, their partial derivatives with respect to and can be derived if their analytical expressions exist. For widely used marginal distribution types, the analytical expressions of the CDF and the distribution parameters are available. Using the analytical expression, the partial derivative of CDF with respect to and and the partial derivatives of and with respect to and are obtained in the literature . Using the partial derivatives in the literature , the analytical design sensitivity for sensitivity-based RBDO with the correlated input variable and the varying input STD in Eq. (21) can be calculated.
Using a 2D mathematical example, the accuracy of the developed sensitivities is verified in this section by comparing it to the sensitivity using the finite difference method (FDM). In addition, sensitivity-based RBDO with the correlated input variable and the varying input STD is performed for the same mathematical problem using the developed sensitivities to test the effectiveness and efficiency of the sensitivities in an RBDO process. The scalability and practicality of the developed sensitivity methods are verified by carrying out an RBDO for a 14D engineering example.
Two-Dimensional Mathematical Example.
For thorough testing, five different input models are used, as shown in Table 3. The models have the same input means and STDs. The STDs of X1 and X2 follow the quadratic function as
Hence, each STD varies as the corresponding design variable di, which is the input mean , changes in the optimization process; they are calculated as design changes in each design iteration or line search for the MPTP search. The marginal distribution types, copula types, and correlation coefficient are different, as shown in Table 3. The correlation coefficients are calculated using Kendall's tau, which represents nonlinear correlation of input variables. The equation of is different for each copula and can be found in the literature . The values of Kendall's tau for the five models are 0.5, 0.5, 0.2, 0.5, and 0.3, respectively.
Accuracy Check of Sensitivity for MPTP Search.
The accuracy of the developed sensitivity for the MPTP search is verified using the first performance measure in Eq. (27), the input models in Table 3, and the input STD in Eq. (28). The sensitivity is evaluated at in U-space, and the gradient is provided for the sensitivity calculation. For a benchmark, forward FDM sensitivity is calculated by perturbing by 0.01% of its value. The results of the developed sensitivity and the FDM sensitivity are summarized in Table 4. The ratio of the developed sensitivity to the FDM sensitivity is listed in the “agreement” column of the table. The ratio is 100.0%, which indicates that the developed sensitivity for the MPTP search agrees with that of the FDM. Therefore, it can be concluded that the developed sensitivity of the MPTP search is accurate. At the same time, the developed sensitivity does not require finding an appropriate perturbation size and evaluating the performance measure at the perturbed design. Hence, the developed method is more efficient than the FDM.
Accuracy Check of Design Sensitivity for Probabilistic Constraint.
The developed design sensitivity of probabilistic constraint is checked to see whether its accuracy is reasonable by comparing it with FDM design sensitivity. For the test, the second performance measure in Eq. (27), the input models in Table 3, and the input STD in Eq. (28) are used. The MPTP for is obtained at the initial design (see Table 3), and the design sensitivity is calculated at the MPTP. Forward FDM with 0.01% perturbation of is used to evaluate FDM design sensitivity. It is noted that, for the FDM design sensitivity, the MPTP should be found at both the original design and the perturbed design. Therefore, there are three MPTP searches to calculate the FDM design sensitivity, while the developed design sensitivity uses only one MPTP search. Hence, the efficiency of the developed design sensitivity is demonstrated in this test.
The comparison of the developed design sensitivity and that of the FDM is provided in Table 5. The agreement of design sensitivities is in the 99.4–100.1% range, which verifies the accuracy of the developed design sensitivity. Because the developed design sensitivity can evaluate an accurate design search direction without an additional MPTP search, it can be concluded that the developed sensitivity is efficient as well as effective. As explained earlier, the accurate search direction will reduce the total number of MPTP searches, which are required at every design iteration, and line searches in the RBDO procedure. Therefore, the RBDO process using the developed design sensitivity will be more efficient. This will be shown in Sec. 5.4.
RBDO Using Developed Sensitivities.
To find how well the developed sensitivities work in the optimization process, RBDO is performed using the RBDO formulation in Eq. (26), the input models in Table 3, and the input STD in Eq. (28). The initial design in Sec. 5.3 is used again, and it is the optimum design of deterministic design optimization (DDO). Following the idea in the enriched PMA , RBDO has been initiated from the DDO optimum design to reduce the total computational time used when performing only RBDO. Because DDO is computationally cheaper than RBDO and the DDO optimum design is usually close to the RBDO optimum design, the entire RBDO process becomes computationally efficient.
The result of RBDO is shown in Table 6. The optimization processes are converged very efficiently using a small number of design iterations and MPTP searches. The input model D uses six design iterations and eight MPTP searches. This indicates that only two additional line searches are needed in the optimization process when accurate design sensitivity is provided. Moreover, RBDO of the other input models converges in five or six design iterations with no additional line search, which shows the effectiveness of the design sensitivity. The number of function evaluations is also shown in Table 6. In each function evaluation, the cost function, the performance measures, and their gradients are provided at the same time in the optimization process. Input model B uses 43 function evaluations in five MPTP searches for three constraints. Therefore, for an MPTP search in a constraint, the model uses 2.87 function evaluations to find the MPTP in Eq. (3), which indicates that the provided sensitivity for the MPTP search is accurate.
To check that the RBDO optimums are correct, the PoF is calculated at the optimum designs using Monte Carlo simulation (MCS) with 1 × 106 MCS samples. The result is also shown in Table 6. Because the FORM is used in this example, the calculated PoFs do not exactly satisfy the 2.275% target PoF. However, we can see that the values are reasonably close to the target value. In Fig. 2, 2-σ contours of the input model C at initial design d0 and RBDO optimum design dopt are shown. The probability of failure of 3.121% in the input model C shows the largest discrepancy from the target 2.275% among all cases due to the significant convexity of and the first-order approximation in the FORM. also suffers the same problem as shown in Table 6 and Fig. 2. Consequently, the 2-σ contour at dopt in X-space does not meet the limit states tangentially. However, it can be seen that the 2-σ contour at dopt in U-space contacts the limit states of active constraints and tangentially. Therefore, the RBDO optimum design for input model C is obtained correctly. In addition, it is shown that the copula dramatically changes the input contour to represent input correlation and that the 2-σ contour changes radically from the initial design to the optimum design. In this example, changes significantly as the second design variable d2 changes. The 2-σ contours of the optimum designs rightly fit in the feasible region, so even a slight design movement would violate one or more probabilistic constraints. Therefore, without accurate design sensitivity, optimum design will not be easily obtainable. Moreover, the size of 2-σ contour is large compared to the feasible domain as shown in Fig. 2. The size of contour becomes large if large input STD, small target PoF, or both present. Both of the factors hinders the convergence of MPTP search. Hence, the large contour shows difficulty of this example, and we can see that the example thoroughly tests the developed sensitivities.
RBDO of 14D Engineering Example.
To demonstrate the scalability and the practicality of the developed method, RBDO has been carried out for a 14D engineering example. The example is a car noise, vibration, and harshness (NVH) and crash-safety problem with weight of a car body as the cost function and 11 performance measures. The performance measures consist of full frontal impact (,), 40% offset frontal impact ( ∼ ), and NVH (,) [25,32]. Surrogate models for the cost function and the performance measures have been provided by Ford Motor Company.
All 14 input random variables represent the thickness of the plates in the car body. Their marginal distribution types are normal, lognormal, Weibull, and gamma. The design variables ∼ are the means of the variables, and the coefficients of variation (CoVs) of the variables are constants (i.e., ). The initial design of the optimization is baseline design and the design variables can move in lower bound and upper bound . There are two correlated pairs of – and – , whose correlations are represented using the Clayton ( and Gaussian ( copulas, respectively. The values of Kendall's tau for and are 0.5 and 0.3, respectively. The input model information is summarized in Table 7.
The cost function, the weight of car body, is a deterministic function of design variable vector . At the same time, the performance measures are a random function of the input random variable vector . They are safe if their values are less than the baseline values. Target PoF is 10% for all constraints in this example. As explained earlier, for the efficient optimization process, DDO is performed first, and RBDO is carried out from the DDO optimum design. By doing this, the weight is minimized, and active constraints are determined in DDO with less computational cost than RBDO. Then, the RBDO increases the reliability (reduces the PoF) of the active constraints with minimum increment of the weight. By starting RBDO from the DDO optimum design, the RBDO optimum design could be obtained efficiently.
In this example, both the FORM and the DRM have been used for the sensitivity-based RBDO. The DRM provides more accurate PoF estimation than the FORM using additional function evaluations . In this example, three Gaussian quadrature points (two more function evaluations in each dimension) are used for the DRM. Because the DRM does not require sensitivity information, we do not need to develop additional sensitivity methods for it. The MPTP is obtained using the FORM and the sensitivity developed in Sec. 3. Then, the MPTP point is updated using the DRM and additional function evaluations without the sensitivity. Design sensitivity at the updated MPTP uses the same method used for the FORM [9,29]. Hence, the design sensitivity developed in Sec. 4 is evaluated at the updated MPTP for the RBDO procedure. Therefore, the DRM can be carried out using the developed sensitivity methods in this paper.
Using the FORM and the DRM, sensitivity-based RBDO with varying STD is completed. Reliability-based design optimization using the FORM uses 29 MPTP searches for each probabilistic constraint. Among the 29 searches, 19 are design iterations and 10 are additional line searches. The DRM requires 26 MPTP searches, including 20 design iterations and six additional line searches. We can see that both methods use a small number of additional line searches. This indicates that the design sensitivity developed in Sec. 4 is accurate and that the design sensitivity works well for the DRM. Moreover, compared to the sampling-based RBDO that uses 19 design iterations and 11 additional line searches , the sensitivity-based RBDO using the developed design sensitivity shows comparable performance. The number of function evaluations for the RBDO using the FORM is 626. This is for 11 performance measures and 29 MPTP searches for each performance measure. Hence, an MPTP search uses 1.96 function evaluations on average, which is very small. Therefore, it can be seen that the sensitivity for MPTP search developed in Sec. 3 is very effective. The DRM uses 4614 function evaluations, which indicates that approximately 4000 evaluations are used for the accurate evaluation of the PoF. It is noted that the 4000 evaluations do not require the gradient of performance measure since DRM method does not use the gradient information [8,9].
The optimum designs are shown in Table 8. The weight values and the PoFs, which are estimated using MCS with 200,000 samples, at the optimal designs are listed in Table 9. During the DDO process, the weight is significantly reduced compared to the weight at the baseline design , while the PoFs remain similar. To find the light weight DDO design, – and reach to their lower bounds since their effect on the weight function is more significant than their effect on the performance measure. On the other hand, moves to its upper bound to lessen the PoF in compensation for the reduced design variables because is the most effective variable to suppress increase of PoF with small increment of the weight. In addition, PoFs G3, G4 and G10 at the DDO optimum design is much smaller than 50% which means that they are inactive constraints.
In the RBDO process, as shown in Table 8, the design variables – , , and remain on their bounds. In addition, the weight increment is insignificant compared to the DDO optimum design as shown in Table 9. This indicates that the DDO optimum design is close to the RBDO optimum designs. This is why DDO process has been carried out a priori because DDO efficiently finds a design close to the RBDO optimum designs more efficient than the RBDO process. On the other hand, the PoFs are reduced to achieve the 10% target PoF as shown in Table 9. However, it can be seen that the optimum design of RBDO using FORM could not find accurate RBDO optimum design. Specifically, the 15.4% PoFs of G7 and G8 do not agree with the 10% target PoF. Because of the first-order approximation of the performance measure, the FORM is not able to find the optimum design. However, the DRM uses additional function evaluations to find a more accurate optimum design for the performance measures. In Table 9, it is shown that the PoFs at the RBDO using DRM successfully find more accurate optimum design using small increment of the weight compared to the FORM RBDO optimum design.
In a number of practical application problems, input variables could be statistically correlated, and the input STD varies according to the change of the design. Therefore, RBDO with the correlated input variable and the varying input STD should be considered to obtain reliable RBDO optimum designs. However, the MPTP search is greatly affected by the input correlation model. Moreover, input variability fluctuates due to the varying STD; and thus, the optimum design of RBDO becomes a moving target problem. Without accurate sensitivities, a correct MPTP and RBDO optimum design are not easy to obtain in RBDO with the correlated input variable and the varying input STD.
In this paper, the sensitivity for the MPTP search and the design sensitivity for RBDO have been developed for sensitivity-based RBDO with the correlated input variable and the varying input STD. The sensitivities consider the change of the input STD due to the change of design variable. In addition, correlated input variables are considered in the developed sensitivities using copula. Using a 2D mathematical example, the developed sensitivity for the MPTP search and the design sensitivity of a probabilistic constraint have been tested by comparing them with the FDM ones. The accuracy of the developed sensitivities is verified, as the sensitivities agree with the FDM sensitivities. Moreover, the developed methods are much more efficient than the FDM because they do not use additional calculations. The RBDO with the correlated input variable and the varying input STD is performed using the developed sensitivities as well. The optimization result shows fast convergence, so the effectiveness of the developed methods has been confirmed. In addition, the minimum additional line search shows that the developed methods provide an accurate design search direction, which helps the optimization process to use few design iterations. Thus, the developed methods improve the efficiency of the whole optimization process. Throughout the 14D engineering example, the developed sensitivity method provides accurate MPTP search direction and effective design sensitivity in the RBDO process. Therefore, it is shown that the developed methods work for large-scale and practical RBDO problems with correlated input variables and varying STD. Moreover, the DRM is successfully applied for the RBDO procedure using the developed sensitivity methods to provide more accurate RBDO optimum design.
The research was supported by RAMDO Solutions, LLC, Iowa City, IA.
two parameters for marginal PDF and CDF of
copula density function of the kth correlated pair
copula function of the kth correlated pair
- CDF =
cumulative distribution function
- , =
design variable, design variable vector,
- , =
lower and upper design bounds
- DDO =
deterministic design optimization
- DRM =
dimension reduction method
marginal PDF of
marginal CDF of
- FDM =
finite difference method
- FORM =
first-order reliability method
jth performance measure in U-space,
jth performance measure in X-space, indicates failure
jth probabilistic constraint for RBDO, =
- M =
number of correlated input random variable pairs
- MCS =
Monte Carlo simulation
- MPTP =
most probable target point
- N =
number of input random variables
- NC =
number of constraints
- NDV =
number of design variables
a variable which can be either or
target PoF for
- PDF =
probability density function
- PMA =
performance measure approach
- PoF =
probability of failure
- RBDO =
reliability-based design optimization
- STD =
realization of and ,
input random variable and vector in U-space,
MPTP in U-space and X-space
- U-space =
standard normal space
realization of and ,
input random variable and vector in X-space,
- X-space =
target reliability index for ,
- , =
input mean of and input mean vector of
- , =
input STD of and input STD vector of
- , =
standard normal PDF and CDF