Sampling-based Reliability-based Design Optimization Research Articles

A New Framework for Efficient Sequential Sampling-Based RBDO Using Space Mapping

Abstract In engineering applications of sampling-based reliability-based design optimization (RBDO), the Monte Carlo simulation (MCS) using a surrogate model of the performance function is mainly used for the probability of failure calculation and sensitivity analysis. However, if an inaccurate surrogate model is used, the calculation result using MCS will also be inaccurate, so it is essential to improve the accuracy of the surrogate model using sequential sampling. Hence, various sampling-based RBDO methods and sequential sampling methods have been proposed and used in various fields, and space mapping may also be a new framework for sequential sampling. In this paper, sampling-based RBDO with the Gaussian process regression (GPR) and space mapping is proposed. Space mapping generally attempts to utilize high-fidelity samples to update the low-fidelity model in multi-fidelity model conditions. However, in the proposed method, it is used for sequential sampling to improve the accuracy of the existing surrogate model. The major advantage of the proposed space mapping-based RBDO is that the existing surrogate model and the finally updated surrogate model can be formulated with simple matrix and vector calculations. In particular, when there is only a surrogate model that has been built due to the loss of existing sample information since the space mapping updates the model, the accuracy of the surrogate model can be improved by sequential sampling. The proposed method is compared with sequential sampling-based RBDO using GPR, and the calculation accuracy and efficiency are demonstrated through a 2D highly nonlinear example and an engineering problem.

Design method for radar absorbing structures using reliability-based design optimization of the composite material properties

Functional composites dispersed with nanofillers are used to enhance the characteristics of radar absorbing structures (RAS). In this study, the relative permittivity of the RAS substrate is measured under various environmental conditions of temperature and humidity. The probability distributions of the dielectric materials used for RAS are determined using maximum likelihood estimation based on raw data. With the determined distributions, 107 samples are generated and sampling-based reliability-based design optimization (RBDO) for reflection loss of a double slab RAS is performed based on transmission line theory. Two objectives considering absorption properties and bandwidth that represent the performance of the RAS are presented. The results of deterministic optimization (DO) and RBDO are compared considering the failure probability of the application system under operating conditions. Although the total thickness increases slightly, the RBDO provides a significant reduction in the failure probability.

Probabilistic analytical target cascading using kernel density estimation for accurate uncertainty propagation

Probabilistic analytical target cascading (PATC) has been developed to incorporate uncertainty of random variables in a hierarchical multilevel system using the framework of ATC. In the decomposed ATC structure, consistency between linked subsystems has to be guaranteed through individual subsystem optimizations employing special coordination strategies such as augmented Lagrangian coordination (ALC). However, the consistency in PATC has to be attained exploiting uncertainty quantification and propagation of interrelated linking variables that are the major concern of PATC and uncertainty-based multidisciplinary design optimization (UMDO). In previous studies, the consistency of linking variables is assured by matching statistical moments under the normality assumption. However, it can induce significant error when the linking variable to be quantified is highly nonlinear and non-normal. In addition, reliability estimated from statistical moments may be inaccurate in each optimization of the subsystem. To tackle the challenges, we propose the sampling-based PATC using multivariate kernel density estimation (KDE). The framework of reliability-based design optimization (RBDO) using sampling methods is adopted in individual optimizations of subsystems in the presence of uncertainty. The uncertainty quantification of linking variables equivalent to intermediate random responses can be achieved by multivariate KDE to account for correlation between linking variables. The constructed KDE based on finite samples of the linking variables can provide accurate statistical representations to linked subsystems and thus be utilized as probability density function (PDF) of linking variables in individual sampling-based RBDOs. Stochastic sensitivity analysis with respect to multivariate KDE is further developed to provide an accurate sensitivity of reliability during the RBDO. The proposed sampling-based PATC using KDE facilitates efficient and accurate procedures to obtain a system optimum in PATC, and the mathematical examples and roof assembly optimization using finite element analysis (FEA) are used to demonstrate the effectiveness of the proposed approach.

Reliability-based design optimization of wind turbine drivetrain with integrated multibody gear dynamics simulation considering wind load uncertainty

This study aims to develop an integrated computational framework for the reliability-based design optimization (RBDO) of wind turbine drivetrains to assure the target reliability under wind load and gear manufacturing uncertainties. Gears in wind turbine drivetrains are subjected to severe cyclic loading due to highly variable wind loads that are stochastic in nature. Thus, the failure rate of drivetrain systems is reported to be higher than the other wind turbine components, and improving drivetrain reliability is critically important in reducing downtime caused by gear failures. In the numerical procedure developed in this study, a wide spatiotemporal variability for wind loads is considered using 249 sets of wind data to evaluate probabilistic contact fatigue life in the sampling-based RBDO. To account for wind load uncertainty in evaluation of the tooth contact fatigue, multiple drivetrain dynamics simulations need to be run under various wind load scenarios in the RBDO process. For this reason, a numerical procedure based on the multivariable tabular contact search algorithm is applied to the modeling of wind turbine drivetrains to reduce the overall computational time while retaining the precise contact geometry required for considering the gear tooth profile optimization. An integrated computational framework for the wind turbine drivetrain RBDO is then developed by incorporating the wind load uncertainty, the rotor blade aerodynamics model, the drivetrain dynamics model, and the probabilistic contact fatigue failure model. It is demonstrated that the RBDO optimum for a 750 kW wind turbine drivetrain obtained using the procedure developed in this study can achieve the target 97.725% reliability (2 sigma quality level) with only a 1.4% increase in the total weight from the baseline design, which had a reliability of 8.3%. Furthermore, it is shown that the tooth profile optimization, tip relief introduced as a design variable, prevents a large increase of the face width that would result in a large increase in the weight (cost) of the drivetrain in order to satisfy the target reliability against the tooth contact fatigue failure.

Design Sensitivity Method for Sampling-Based RBDO With Varying Standard Deviation

Conventional reliability-based design optimization (RBDO) uses the mean of input random variable as its design variable; and the standard deviation (STD) of the random variable is a fixed constant. However, the constant STD may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is present as a percentage of the mean value. For this kind of design problem, the STD of the input random variable should vary as the corresponding design variable changes. In this paper, a method to calculate the design sensitivity of the probability of failure for RBDO with varying STD is developed. For sampling-based RBDO, which uses Monte Carlo simulation (MCS) for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STD in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with varying STD. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed design sensitivity method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

A new sampling-based RBDO method via score function with reweighting scheme and application to vehicle designs

Sampling-based methods are general but time consuming for solving a reliability-based design optimization (RBDO) problem. However, one of the major drawbacks of using Monte Carlo simulation (MCS) to compute the sensitivities of reliability functions is computation intensive. In order to alleviate the computation burden, score function together with the MCS method is exploited to compute the stochastic sensitivities of reliability functions. Furthermore by reweighting an initial set of samples, the objective and constraint functions become smooth functions of changes of the parameters of probability distribution, rather than the stochastic functions obtained using MCS. A new sampling-based RBDO method by using score function with reweighting scheme is proposed to perform sensitivities analysis of reliability functions to improve the computational efficiency and accuracy. Several numerical examples are used to show the advantages of the proposed method. Comparisons to the conventional methods are made and discussed. Two large-scale industrial vehicle design problems are solved to demonstrate the feasibility of the proposed method.

Reliability-based design optimization of fluid–solid interaction problems

Using a sampling-based reliability-based design optimization method, we present a shape reliability-based design optimization method for coupled fluid–solid interaction problems. For the fluid–solid interaction problem in arbitrary Lagrangian–Eulerian formulation, a coupled variational equation is derived from a steady state Navier–Stokes equation for incompressible flows, an equilibrium equation for geometrically nonlinear solids, and a traction continuity condition at interfaces. The fluid–solid interaction problem is solved using the finite element method and the Newton–Raphson scheme. For the fluid mesh movement, we formulated and solved a pseudo-structural sub-problem. The shape of the solid is modeled using the Non-Uniform Rational B-Spline (NURBS) surface, and the coordinate components of the control points are selected as random design variables. The sensitivity of the probabilistic constraint is calculated using the first-order score functions obtained from the input distributions and from the Monte Carlo simulation on the surrogate model constructed by using the Dynamic Kriging method. The sequential quadratic programming algorithm is used for the optimization. In two numerical examples, the proposed optimization method is applied to the shape design problems of solid structure which is loaded by prescribed fluid flow, and this proves that the sampling-based reliability-based design optimization can be successfully utilized for obtaining a reliable optimum design in highly nonlinear multi-physics problems.

Conservative Surrogate Model Using Weighted Kriging Variance for Sampling-Based RBDO

In sampling-based reliability-based design optimization (RBDO) of large-scale engineering applications, the Monte Carlo simulation (MCS) is often used for the probability of failure calculation and probabilistic sensitivity analysis using the prediction from the surrogate model for the performance function evaluations. When the number of samples used to construct the surrogate model is not enough, the prediction from the surrogate model becomes inaccurate and thus the Monte Carlo simulation results as well. Therefore, to count in the prediction error from the surrogate model and assure the obtained optimum design from sampling-based RBDO satisfies the probabilistic constraints, a conservative surrogate model, which is not overly conservative, needs to be developed. In this paper, a conservative surrogate model is constructed using the weighted Kriging variance where the weight is determined by the relative change in the corrected Akaike Information Criterion (AICc) of the dynamic Kriging model. The proposed conservative surrogate model performs better than the traditional Kriging prediction interval approach because it reduces fluctuation in the Kriging prediction bound and it performs better than the constant safety margin approach because it adaptively accounts large uncertainty of the surrogate model in the region where samples are sparse. Numerical examples show that using the proposed conservative surrogate model for sampling-based RBDO is necessary to have confidence that the optimum design satisfies the probabilistic constraints when the number of samples is limited, while it does not lead to overly conservative designs like the constant safety margin approach.