Reliability-based design optimization (RBDO) is an active field of research with an ever increasing number of contributions. Numerous methods have been proposed for the solution of RBDO, a complex problem that combines optimization and reliability analysis. Classical approaches are based on approximation methods and have been classified in review papers. In this paper, we first review classical approaches based on approximation methods such as FORM, and also more recent methods that rely upon surrogate modelling and Monte Carlo simulation.We then propose a general framework for the solution of RBDO problems that includes three independent blocks, namely adaptive surrogate modelling, reliability analysis and optimization.These blocks are non-intrusive with respect to each other and can be plugged independently in the framework. After a discussion on numerical considerations that require attention for the framework to yield robust solutions to various types of problems, the latter is applied to three examples (using two analytical functions and a finite element model). Kriging and support vector machines together with their own active learning schemes are considered in the surrogate model block. In terms of reliability analysis, the proposed framework is illustrated using both crude Monte Carlo and subset simulation. Finally, the covariance-matrix adaptation -evolution scheme (CMA-ES), a global search algorithm, or sequential quadratic programming (SQP), a local gradient-based method, are used in the optimization block. The comparison of the results to benchmark studies show the effectiveness and efficiency of the proposed framework. of risk-based or life-cycle cost optimization (Beck and Gomes, 2012;Frangopol and Maute, 2003).Alternative formulations for RBDO have henceforth emerged. One, for instance, consists in maximizing the reliability, or equivalently minimizing the failure probability, under given cost constraints (See for instance Kuschel and Rackwitz (1997); Royset et al. (2001); Taflanidis and Beck (2008)).The approach we consider in this paper, which is also the prevalent in current research, consists in minimizing an initial cost under some probabilistic constraints (Hilton and Feigen, 1960). Let d ∈ R M d be an M d −dimensional vector defining some design parameters of the structure. In the presence of uncertainties, the variability of these parameters can be described by introducing random variables X ∈ X ⊂ R M d ∼ f X|d . These variables are called design parameters with uncertainty, or by slight abuse, random design variables in the sequel. In some cases, design parameters are purely deterministic. To simplify the notation, they are equally considered as X (d) at this stage, where the related random variables would simply have a

Uncertainties are inherent to real-world systems. Taking them into account is crucial in industrial design problems and this might be achieved through reliability-based design optimization (RBDO) techniques. In this paper, we propose a quantile-based approach to solve RBDO problems. We first transform the safety constraints usually formulated as admissible probabilities of failure into constraints on quantiles of the performance criteria. In this formulation, the quantile level controls the degree of conservatism of the design. Starting with the premise that industrial applications often involve high-fidelity and time-consuming computational models, the proposed approach makes use of Kriging surrogate models (a.k.a. Gaussian process modeling).Thanks to the Kriging variance (a measure of the local accuracy of the surrogate), we derive a procedure with two stages of enrichment of the design of computer experiments (DoE) used to construct the surrogate model. The first stage globally reduces the Kriging epistemic uncertainty and adds points in the vicinity of the limit-state surfaces describing the system performance to be attained. The second stage locally checks, and if necessary, improves the accuracy of the quantiles estimated along the optimization iterations. Applications to three analytical examples and to the optimal design of a car body subsystem (minimal mass under mechanical safety constraints) show the accuracy and the remarkable efficiency brought by the proposed procedure.

Abstract. An estimated 0.5–1 billion people globally have inadequate intakes of selenium (Se), due to a lack of bioavailable Se in agricultural soils. Deposition from the atmosphere, especially through precipitation, is an important source of Se to soils. However, very little is known about the atmospheric cycling of Se. It has therefore been difficult to predict how far Se travels in the atmosphere and where it deposits. To answer these questions, we have built the first global atmospheric Se model by implementing Se chemistry in an aerosol–chemistry–climate model, SOCOL-AER (modeling tools for studies of SOlar Climate Ozone Links – aerosol). In the model, we include information from the literature about the emissions, speciation, and chemical transformation of atmospheric Se. Natural processes and anthropogenic activities emit volatile Se compounds, which oxidize quickly and partition to the particulate phase. Our model tracks the transport and deposition of Se in seven gas-phase species and 41 aerosol tracers. However, there are large uncertainties associated with many of the model's input parameters. In order to identify which model uncertainties are the most important for understanding the atmospheric Se cycle, we conducted a global sensitivity analysis with 34 input parameters related to Se chemistry, Se emissions, and the interaction of Se with aerosols. In the first bottom-up estimate of its kind, we have calculated a median global atmospheric lifetime of 4.4 d (days), ranging from 2.9 to 6.4 d (2nd–98th percentile range) given the uncertainties of the input parameters. The uncertainty in the Se lifetime is mainly driven by the uncertainty in the carbonyl selenide (OCSe) oxidation rate and the lack of tropospheric aerosol species other than sulfate aerosols in SOCOL-AER. In contrast to uncertainties in Se lifetime, the uncertainty in deposition flux maps are governed by Se emission factors, with all four Se sources (volcanic, marine biosphere, terrestrial biosphere, and anthropogenic emissions) contributing equally to the uncertainty in deposition over agricultural areas. We evaluated the simulated Se wet deposition fluxes from SOCOL-AER with a compiled database of rainwater Se measurements, since wet deposition contributes around 80 % of total Se deposition. Despite difficulties in comparing a global, coarse-resolution model with local measurements from a range of time periods, past Se wet deposition measurements are within the range of the model's 2nd–98th percentiles at 79 % of background sites. This agreement validates the application of the SOCOL-AER model to identifying regions which are at risk of low atmospheric Se inputs. In order to constrain the uncertainty in Se deposition fluxes over agricultural soils, we should prioritize field campaigns measuring Se emissions, rather than laboratory measurements of Se rate constants.

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Metamodeling techniques have been widely used as substitutes of high-fidelity and timeconsuming models in various engineering applications. Examples include polynomial chaos expansions, neural networks, Kriging or support vector regression. This papers attempts to compare the latter two in different case studies so as to assess their relative efficiency on simulation-based analyses. Similarities are drawn between these two metamodels types leading to the use of anisotropy for SVR. Such a feature is not commonly used in the SVR related literature. A special care is given to a proper automatic calibration of the model hyperparameters by using an efficient global search algorithm, namely the covariance matrix adaptation-evolution scheme (CMA-ES). Variants of these two metamodels, associated with various kernel or auto-correlation functions, are first compared on analytical functions and then on finite-element-based models. From the comprehensive comparison, it is concluded that anisotropy in the two metamodels clearly improves their accuracy. In general, anisotropic L 2-SVR with the Matérn kernels is shown to be the most effective metamodel.

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