Evaluation and assessment of non-normal output during robust optimization
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Abstract
A robustness criterion that employs skewness of output is presented for a metamodel-based robust optimization. The propagation of a normally distributed noise variable via nonlinear functions leads to a non-normal output distribution. To consider the non-normality of the output, a skew-normal distribution is used. Mean, standard deviation, and skewness of the output are calculated by applying an analytical approach. To show the applicability of the proposed method, a metal forming process is optimized. The optimization is defined by an objective and a constraint, which are both nonlinear. A Kriging metamodel is used as nonlinear model of that forming process. It is shown that the new robustness criterion is effective at reducing the output variability. Additionally, the results demonstrate that taking into account the skewness of the output helps to satisfy the constraints at the desired level accurately.
Keywords
Robust optimization Skewness parameter Non-normal output distribution1 Introduction
Optimization in the presence of uncertainty of input parameters has become a common exercise in various fields of study (Picheny et al. 2017; Zhou et al. 2018; Marton et al. 2015; Yazdi 2017). When considering variability in input parameters, one makes sure that the output meets requirements even under the influence of scatter in the input. Robust optimization is one of the methods that is used to minimize the variation of output originating from the scatter of input parameters (Bertsimas et al. 2011). In this approach, the input parameters are categorized as design parameters (x) and as noise parameters (z). Design parameters can be adjusted during the process but noise parameters are those that are difficult or impossible to control. The goal is to find a design point at which the output is least sensitive to the variation of noise variables and the output mean is closest to the target value. Moreover, the constraints must be handled properly because of input uncertainties (Park et al. 2006).
Both the constraints and the robustness measure are usually defined using the mean and standard deviation of the output. Considering only the mean and standard deviation induces large errors for a non-normal output distribution. There have been attempts to capture the non-normality of the output by applying various methods and include it in the optimization in the presence of uncertainty. For instance, Anderson and Mattson (2012) used first-order and second-order Taylor series expansion to approximate the propagation of skewness and kurtosis by applying engineering models. Zhang (2017) developed a method to include high-order moments of output distribution for reliability analysis in the method of moments. They reported a higher accuracy in predicting the reliability index by using higher order moments. Mekid and Vaja (2008) performed higher order uncertainty propagation using Taylor expansions and compared their approach with other methods that included MC sampling from a Gaussian input distribution. They concluded that the necessity of considering higher order moments depends on the non-linearity of the response and other input factors.
In this article, we introduce an approach for robust optimization that takes into accounts the skewness of the output. Skewness is used for evaluation of the robustness measure and constraints. An existing analytical method (Nejadseyfi et al. 2017, 2018) is extended to calculate the mean, standard deviation, and skewness of the output by propagating a Gaussian input noise through a Kriging metamodel. The analytical method is used instead of MC sampling from a Gaussian distribution and it is faster, more accurate, and improves the efficiency of optimization.
This article is organized as follows: Section 2 describes the analytical method to find the mean, standard deviation, and skewness of the output, based on the propagation of a normally distributed input through a nonlinear metamodel. Section 3 introduces the new robustness measure and the method to apply the constraints at a desired level of reliability. Sections 4.1 and 4.2 discuss the finite element (FE) simulation of a stretch-bending process as a case study. In Section 4.3, the optimization problem is formulated on the basis of the new approach. The results are presented and then discussed in Section 5.
2 Analytical method for uncertainty propagation
3 Robust optimization including skewness measure
Once the mean, standard deviation, and skewness of the output have been calculated, they can be used to evaluate the robustness measure and the constraints. In this section, a new approach is proposed to include the skewness measure during robust optimization. Even though the preferred method of calculating the output moments in this article is by following an analytical approach, any other method for calculation of noise propagation (e.g., MC) can be used to evaluate those moments and include them during the robust optimization. In this section, first a distribution that can be formulated using the first three moments is introduced. Then, it will be shown how to use this distribution to implement the skewness during robust optimization.
3.1 A probability distribution with skewness measure
3.2 Including skewness measure in robustness measure and constraints
It can be seen that for ND, the robustness measure reduces to (27). As shown in Fig. 4, the new criterion sets μ_{r}(x) + ((U(γ;n) + L(γ;n))/2)σ_{r}(x) on target and minimizes ((U(γ;n) − L(γ;n))/2)σ_{r}(x).
The fitted curves and the accuracy of the fits for L and U as a function of γ
Coefficients of fit (aγ^{3}+ bγ^{2}+ cγ + d) | R ^{2} | RMSE | ||
---|---|---|---|---|
3σ | L | a = 1.31, b= − 1.06 c = 1.29, d = − 3 | 0.998 | 0.017 |
3σ | U | a = 0.29, b = − 0.85 c = 1.55, d = 3 | 1 | 0.001 |
6σ | L | a = 4.05, b = − 3.94 c = 4.08, d = − 6 | 0.998 | 0.047 |
6σ | U | a = 2.31, b = − 5.08 c = 5.64, d = 6 | 0.999 | 0.022 |
4 A case study: stretch-bending of dual-phase steel sheet
In this section, a demonstration case is presented to investigate the significance of accounting for skewness of output during robust optimization. The stretch-bending process of DP800 steel sheet is chosen. During this process, a steel sheet is simultaneously stretched and bent to a desired curvature. After bending of the sheet material, due to elastic spring-back, the final bend angle will differ from the applied angle. The amount of spring-back depends on the mechanical properties of the sheet material. By applying simultaneous stretching, the amount of spring-back may be diminished and the accuracy of the process, in terms deviation of the final bend angle from the desired one, can be improved. On the other hand, applied stretching causes thickness reduction of the sheet, which should be limited. To find the robust optimum, normally distributed noise variables are defined for the applied material morphology and the sheet thickness. The resulting material model and the stretch-bending process are nonlinear and give rise to non-normally distributed results.
4.1 The finite element model
4.2 A noisy material model
To model the material behavior, a micro/macro approach is implemented. In this approach, the rule of mixture is applied on a representative volume element (RVE) and the overall properties are obtained by weighted averaging of the respective properties of the constitutive phases in the material. This method is appropriate for composites or materials consisting of more than one phase, e.g., dual-phase (DP) steels. This method was developed by among others (Hill 1965; Hashin and Shtrikman 1963) and is based on the inclusion model proposed by Eshelby (1957). For an extensive treatment, see Mura (1987).
4.3 The optimization problem
Design parameters, x, in the stretch-bending process
Design parameter | Lower bound | Upper bound |
---|---|---|
Forming curvature (m^{− 1}) | 50 | 55 |
Stretch (%) | 2 | 8 |
Noise variables in the stretch-bending process
Noise variable | Mean (μ_{z}) | St. dev. (σ_{z}) | μ_{z} − 3σ_{z} | μ_{z} + 3σ_{z} |
---|---|---|---|---|
Sheet thickness (t) [mm] | 1 | 0.1/6 | 0.95 | 1.05 |
Volume fraction (VF) [%] | 35 | 10/6 | 30 | 40 |
Shape factor (α) [–] | 0.7 | 0.1 | 0.4 | 1 |
Banding parameter (β) [–] | 1.5 | 1/6 | 1 | 2 |
The output of the simulation is the final curvature and the percentage thickness reduction of the sheet. For robust optimization, we aim to obtain a target curvature and we apply a constraint on the thinning percentage. Design variables are set by defining upper and lower bounds, noise variables are taken into account through a normal probability distribution and the constraint is expressed using an inequality expression.
The target curvature is C_{r} = 50m^{− 1} and the thinning limit is C_{g} = 2.5%. After defining the objective function, input, output, and constraints, the optimization steps are followed based on Fig. 1. First, a DOE of 793 points is generated on a six-dimensional space based on the combination of a Latin hypercube design (LHS) and a full-factorial design (FFD). Responses (final curvature and thinning) are evaluated by FE analysis as explained in Section 4.1. Kriging metamodels are fit to both the final curvature and the final sheet thickness output. Mean, standard deviation, and skewness of the output are calculated using the analytical approach described in Section 2.
5 Results and discussion
To observe the influence of skewness in the robust optimization procedure, we use the proposed criterion of (30) and compare it with formulation of (27).
Robust optimum design and objective function value for conventional and new approach
Method | Conventional method | New criterion |
---|---|---|
Formed curvature,x_{1} (m^{− 1}) | 51.63 | 51.62 |
Stretch, x_{2}(%) | 3.45 | 3.58 |
Objective function value (m^{− 1}) | 0.26 | 0.23 |
Products out of (Lσ,Uσ) range | 0.017 | 0.007 |
Another important aspect about the new criterion is the treatment of the constraint, which is more accurate than for the conventional method. If we use the conventional criterion for the constraint (shown by dashed ND in Fig. 11), the upper bound of thinning is overestimated and it leads to rejection of this design. The overestimation of the right tail of the distribution using an ND violates the constraint on thinning during robust optimization although the constraint is not really violated as a result of a negative skewness. The new criterion can considerably improve the search for the robust optimum by proper predictions of the tails of the distribution.
Compared to other methods that have been developed to include high-order moments of output distribution for reliability analysis, the proposed approach has several advantages. In the proposed approach, Kriging is used which is capable of handling nonlinear relations between input and output more effectively compared to first-order or second-order approximations. The response in other studies is considered as polynomial, e.g., linear and quadratic in the first-order reliability method (FORM) and the second-order reliability method (SORM), respectively (Zhao and Ono 2001). The second advantage is that in this article, the mean, standard deviation, and skewness parameters are obtained analytically and therefore they are accurate compared to moment calculation based on MC sampling from a Gaussian input (Padmanabhan et al. 2006). Another advantage is that the derivatives of the output mean, standard deviation, and skewness can be obtained analytically and improve the search for robust optimum in a gradient-based algorithm. Nevertheless, the calculation of moments based on analytical approach has some limitations and it is not feasible for specific combinations of metamodels and input probability distributions. During analysis of the constraints and evaluation of the robustness measure, the addition of the skewness parameter is a significant improvement over considering only mean and standard deviation. However, for a highly nonlinear response that gives rise to higher order moments (Mekid and Vaja 2008), one might consider even higher than third-order moments during the search for a robust optimum. This is the topic of ongoing research by the authors.
6 Conclusions
When a normally distributed input parameter is subjected to a nonlinear process, the resulting output will be characterized by a skewed distribution. The mean, standard deviation, and skewness of the output distribution are calculated analytically. The skewness parameter is then used to find the shift of the tails of the distributions of the objective function and the constraints. The shift of the tails is embedded in the definition of the robustness measure as well as in the evaluation of constraints during robust optimization.
Application of this approach for robustness criterion and evaluation of constraints shows that the variation of output is successfully reduced and the constraints are handled more accurately compared to the conventional method of robust optimum evaluation.
Notes
Acknowledgements
This research was carried out under project number F22.1.13506 in the framework of the Partnership Program of the Materials innovation institute M2i (www.m2i.nl) and the Foundation of Fundamental Research on Matter (FOM) (www.fom.nl), which is part of the Netherlands Organization for Scientific Research (www.nwo.nl).
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