Sensitivity Developments for RBDO With Dependent Input Variable and Varying Input Standard Deviation

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) [2] 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 [2]. 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) [12], enriched HMV (HMV+) [13,14], hybrid chaos control (HCC) [15], adaptive chaos control (ACC) [16,17], self-adaptive modified chaos control (SMCC) [18], adjusted advanced mean value (AAMV) [19], and relaxed mean value (RMV) [20] 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 [25]. 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.

where X is the N-dimensional random variable vector, d is the NDV-dimensional design variable vector, d^L is the lower design bound, d^U is the upper design bound, $PFjtar$ is the target PoF for the jth performance measure $Gj(X)$⁠, and NC is the number of constraints. In the RBDO formulation, the probabilistic constraint of RBDO is the PoF.

where $u*$ is the MPTP for $Gj(X)$ in U-space, and $x*=x(u*,μ,σ)$ is the MPTP in X-space (physical space).

The optimization process to find the MPTP in Eq. (3) requires sensitivity of its cost function $−gj(u,μ,σ)$ with respect to $u$⁠. Though iterative MPTP search methods, such as AMV [10,11], HMV [12], HMV+ [13,14], HCC [15], ACC [16,17], SMCC [18], AAMV [19], and RMV [20] 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 $d$ 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.

where M is the number of correlated pairs. It is noted that the mean $μi$ and STD $σi$ of input variable $Xi$ are only related to the marginal PDF $fXi(xi;ai,bi)$ and the copula density function $ck(u,v;θk)$ when $Xi$ is in the kth correlated pair. The input joint PDF defined in Eq. (6) is used in the sensitivity derivations in Secs. 3 and 4.

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 $gj(u,μ,σ)≡Gj(x)$ is evaluated in X-space, which is the physical space. Hence, the objective function of the optimization process $gj(u,μ,σ)$ involves the Rosenblatt transformation [31] between U-space and X-space. Therefore, the derivation of sensitivity for the MPTP search should include the transformation.

Then, $Um$ and $Un$⁠, whose realizations are $um$ and $un$⁠, respectively, are statistically independent and follow the standard normal distribution. Thus, $Um$ and $Un$ are U-space variables that correspond to $Xm$ and $Xn$⁠.

The derivative with respect to $un$ can be obtained by changing indices m and n in Eq. (11). As explained earlier, the gradient of performance measure $Gj(x)$ is assumed available in the sensitivity-based RBDO. Therefore, the derivatives of $xi$⁠, $xm,$ and $xn$ to standard normal variables $ui$ and $um$ are required to calculate the sensitivity of the MPTP search.

Therefore, the sensitivity of the constraint function in Eq. (12) is readily available.

where $ϕ(•)$ is the standard normal PDF.

where $Ck,vv=(∂2/(∂v2))Ck$ and $Ck,vw=(∂2/(∂v∂w))Ck$⁠. It is noted that the aforementioned copula density function $ck$ is defined as $ck≡Ck,vw$⁠.

Therefore, the first variable $xm$ is not related to the second U-space variable $un$⁠. 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.

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 $cost(d)$ in Eq. (1) is a function of only the deterministic design variable $d$⁠; therefore, its design sensitivity is often readily obtainable.

The MPTP point $u*$ is also a function of $μ$ and $σ$⁠, because they affect the constraint function $g(u,μ,σ)$ in the MPTP search in Eq. (4). However, Eq. (20) indicates that the change of $u*$ is not necessarily considered in the design sensitivity of the probabilistic constraint.

The missing terms $dxi/dμi$ and $dxi/dσi$ 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.

Because $FXm(xm;am,bm)$ is not a function of $μn$ and $σn$⁠, the first term in the left side of Eq. (24) vanishes. Moreover, the copula density function $ck≡Ck,vw$ is not always zero, so $(d/(dpn))FXn(xn;an,bn)$ 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 $xn$⁠.

The distribution parameters $ai$ and $bi$ are uniquely determined by $μi$ and $σi$⁠. Therefore, their partial derivatives with respect to $μi$ and $σi$ 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 $ai$ and $bi$ and the partial derivatives of $ai$ and $bi$ with respect to $μi$ and $σi$ are obtained in the literature [25]. Using the partial derivatives in the literature [25], 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.

The contours of the cost function $cost(d)$ in Eq. (26) and the limit states of the performance measures in Eq. (27) are shown in Fig. 1.

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 $σi$ varies as the corresponding design variable d_i, which is the input mean $μi$⁠, 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 $θ1$ are different, as shown in Table 3. The correlation coefficients $θ1$ are calculated using Kendall's tau, which represents nonlinear correlation of input variables. The equation of $θ1$ is different for each copula and can be found in the literature [22]. The values of Kendall's tau for the five models are 0.5, 0.5, 0.2, 0.5, and 0.3, respectively.

The accuracy of the developed sensitivity for the MPTP search is verified using the first performance measure $G1(X)$ in Eq. (27), the input models in Table 3, and the input STD in Eq. (28). The sensitivity is evaluated at $u=[−2, 4]$ in U-space, and the gradient $dG1/dxi$ is provided for the sensitivity calculation. For a benchmark, forward FDM sensitivity is calculated by perturbing $ui$ 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.

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 $G2(X)$ in Eq. (27), the input models in Table 3, and the input STD in Eq. (28) are used. The MPTP for $G2(X)$ is obtained at the initial design $d0=[μ1,μ2]=[5.19, 0.74]$ (see Table 3), and the design sensitivity is calculated at the MPTP. Forward FDM with 0.01% perturbation of $μi$ 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.

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 $d0$ 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 [14], 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 d^opt are shown. The probability of failure of $G1(X)$ 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 $G1(X)$ and the first-order approximation in the FORM. $G2(X)$ also suffers the same problem as shown in Table 6 and Fig. 2. Consequently, the 2-σ contour at d^opt in X-space does not meet the limit states $Gi(X)=0$ tangentially. However, it can be seen that the 2-σ contour at d^opt in U-space contacts the limit states of active constraints $g1(U,μ,σ)$ and $g2(U,μ,σ)$ 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, $σ2$ 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.

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 (⁠$G1$,$G2$⁠), 40% offset frontal impact (⁠$G3$ ∼ $G9$⁠), and NVH (⁠$G10$,$G11$⁠) [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 $d1$ ∼ $d14$ are the means of the variables, and the coefficients of variation (CoVs) of the variables are constants (i.e., $σi=CoVi μi$⁠). The initial design of the optimization is baseline design $dB,$ and the design variables can move in lower bound $dL$ and upper bound $dU$⁠. There are two correlated pairs of $X6$ – $X7$ and $X25$ – $X26$⁠, whose correlations are represented using the Clayton (⁠$θ1=2)$ and Gaussian (⁠$θ2=0.454)$ copulas, respectively. The values of Kendall's tau for $θ1$ and $θ2$ are 0.5 and 0.3, respectively. The input model information is summarized in Table 7.