Sampling-based RBDO using the stochastic sensitivity analysis and Dynamic Kriging method

This paper presents a sampling-based RBDO method using surrogate models. The Dynamic Kriging (D-Kriging) method is used for surrogate models, and a stochastic sensitivity analysis is introduced to compute the sensitivities of probabilistic constraints with respect to independent or correlated random variables. For the sampling-based RBDO, which requires Monte Carlo simulation (MCS) to evaluate the probabilistic constraints and stochastic sensitivities, this paper proposes new efficiency and accuracy strategies such as a hyper-spherical local window for surrogate model generation, sample reuse, local window enlargement, filtering of constraints, and an adaptive initial point for the pattern search. To further improve computational efficiency of the sampling-based RBDO method for large-scale engineering problems, parallel computing is proposed as well. Once the D-Kriging accurately approximates the responses, there is no further approximation in the estimation of the probabilistic constraints and stochastic sensitivities, and thus the sampling-based RBDO can yield very accurate optimum design. In addition, newly proposed efficiency strategies as well as parallel computing help find the optimum design very efficiently. Numerical examples verify that the proposed sampling-based RBDO can find the optimum design more accurately than some existing methods. Also, the proposed method can find the optimum design more efficiently than some existing methods for low dimensional problems, and as efficient as some existing methods for high dimensional problems when the parallel computing is utilized.     

Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

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.     

Global kriging surrogate modeling for general time-variant reliability-based design optimization problems

While design optimization under uncertainty has been widely studied in the last decades, time-variant reliability-based design optimization (t-RBDO) is still an ongoing research field. The sequential and mono-level approaches show a high numerical efficiency. However, this might be to the detriment of accuracy especially in case of nonlinear performance functions and non-unique time-variant most probable failure point (MPP). A better accuracy can be obtained with the coupled approach, but this is in general computationally prohibitive. This work proposes a new t-RBDO method that overcomes the aforementioned limitations. The main idea consists in performing the time-variant reliability analysis on global kriging models that approximate the time-dependent limit state functions. These surrogates are built in an artificial augmented reliability space and an efficient adaptive enrichment strategy is developed that allows calibrating the models simultaneously. The kriging models are consequently only refined in regions that may potentially be visited by the optimizer. It is also proposed to use the same surrogates to find the deterministic design point with no extra computational cost. Using this point to launch the t-RBDO guarantees a fast convergence of the optimization algorithm. The proposed method is demonstrated on problems involving nonlinear limit state functions and non-stationary stochastic processes.     

A Critical Assessment of Kriging Model Variants for High-Fidelity Uncertainty Quantification in Dynamics of composite Shells

This paper presents a critical comparative assessment of Kriging model variants for surrogate based uncertainty propagation considering stochastic natural frequencies of composite doubly curved shells. The five Kriging model variants studied here are: Ordinary Kriging, Universal Kriging based on pseudo-likelihood estimator, Blind Kriging, Co-Kriging and Universal Kriging based on marginal likelihood estimator. First three stochastic natural frequencies of the composite shell are analysed by using a finite element model that includes the effects of transverse shear deformation based on Mindlin’s theory in conjunction with a layer-wise random variable approach. The comparative assessment is carried out to address the accuracy and computational efficiency of five Kriging model variants. Comparative performance of different covariance functions is also studied. Subsequently the effect of noise in uncertainty propagation is addressed by using the Stochastic Kriging. Representative results are presented for both individual and combined stochasticity in layer-wise input parameters to address performance of various Kriging variants for low dimensional and relatively higher dimensional input parameter spaces. The error estimation and convergence studies are conducted with respect to original Monte Carlo Simulation to justify merit of the present investigation. The study reveals that Universal Kriging coupled with marginal likelihood estimate yields the most accurate results, followed by Co-Kriging and Blind Kriging. As far as computational efficiency of the Kriging models is concerned, it is observed that for high-dimensional problems, CPU time required for building the Co-Kriging model is significantly less as compared to other Kriging variants.     

Robust springback optimization of a dual phase steel seven-flange die assembly

This paper investigates robust springback optimization of a DP600 dual phase steel seven-flange die assembly composed of different flange designs. The optimum values of the die radius and the punch radius are sought to minimize the mean and the standard deviation of springback using surrogate based optimization. Springback values at the training points of surrogate models are evaluated using the finite element analysis code LS-DYNA. In this work, four different surrogate modeling types are considered: polynomial response surfaces (PRS) approximations, stepwise regression (SWR), radial basis functions (RBF) and Kriging (KR). Two sets of surrogate models are constructed in this study. The first set is constructed to relate the springback to the design variables as well as the random variables. It is found for the first set of surrogate models that KR provides more accurate springback predictions than PRS, SWR and RBF. The mean and the standard deviation of springback are calculated using Monte Carlo simulations, where the first set of surrogate models is utilized. The second set of surrogate models is generated to relate the mean and the standard deviation of springback to the design variables. It is found for the second set of surrogate models that PRS provides more accurate springback predictions than SWR, RBF and KR. It is also found that introducing beads increases the mean performance and the robustness. The robust optimization is performed and significant springback reductions are obtained for all flanges ranging between 7% and 85% compared to the nominal design. It is also found that a design change that decreases the mean springback also reduces the springback variation. It is observed that the optimization results heavily dependent on the bounds of the die and punch radii. In addition, optimization with multiple surrogates is investigated. Finding multiple candidates of optimum with multiple surrogates and selecting the one with the best actual performance is found to be a better strategy than optimizing using the most accurate surrogate model.     

The stochastic finite-element method

A generic stochastic finite-element method for modeling structures is proposed as a means to analyze and design structures in a probabilistic framework. Stochastic differential and difference equation theory is applied in structures discretized with the finite-element methodology.Transient structural loads, idealized as stochastic processes, are incorporated into finite-element dynamic models with uncertain parameters. An estimate of the probability of failure based on known and established procedures in second-moment reliability analysis can be made with the aid of a transformation to gaussian space of the random variables that define structural reliability.The stochastic finite-element method will facilitate the use of probabilistic mathematical structural models for structural code development or design of important structures. It will also permit better estimation of structural reliability, which, when combined with risk analysis, could lead to improved decision-making processes.     

A method for using legacy data for metamodel-based design of large-scale systems

Despite a steady increase in computing power, the complexity of engineering analyses seems to advance at the same rate. Traditional parametric design analysis is inadequate for the analysis of large-scale engineering systems because of its computational inefficiency; therefore, a departure from the traditional parametric design approach is required. In addition, the existence of legacy data for complex, large-scale systems is commonplace. Approximation techniques may be applied to build computationally inexpensive surrogate models for large-scale systems to replace expensive-to-run computer analysis codes or to develop a model for a set of nonuniform legacy data. Response-surface models are frequently utilized to construct surrogate approximations; however, they may be inefficient for systems having with a large number of design variables. Kriging, an alternative method for creating surrogate models, is applied in this work to construct approximations of legacy data for a large-scale system. Comparisons between response surfaces and kriging are made using the legacy data from the High Speed Civil Transport (HSCT) approximation challenge. Since the analysis points already exist, a modified design-of-experiments technique is needed to select the appropriate sample points. In this paper, a method to handle this problem is presented, and the results are compared against previous work.     

Multi-objective optimization for design under uncertainty problems through surrogate modeling in augmented input space

Multi-objective design under uncertainty problems that adopt probabilistic quantities as performance objectives and consider their estimation through stochastic simulation are examined in this paper, focusing on development of a surrogate modeling framework to reduce computational burden for the numerical optimization. The surrogate model is formulated to approximate the system response with respect to both the design variables and the uncertain model parameters, so that it can simultaneously support both the uncertainty propagation and the identification of the Pareto optimal solutions. Kriging is chosen as the metamodel, and its probabilistic nature (its ability to offer a local estimate of the prediction error) is leveraged within different aspects of the framework. To reduce the number of simulations for the expensive system model, an iterative approach is established with adaptive characteristics for controlling the metamodel accuracy. At each iteration, a new metamodel is developed utilizing all available training points. A new Pareto front is then identified utilizing this surrogate model and is compared, for assessing stopping criteria, to the front that was identified in the previous iteration. This comparison utilizes explicitly the potential error associated with the metamodel predictions. If stopping criteria are not achieved, a set of refinement experiments (new training points) is identified and process proceeds to the next iteration. A hybrid design of experiments is considered for this refinement, with a dual goal of global coverage and local exploitation of regions of interest, separately identified for the design variables and the uncertain model parameters.

     

Robust optimization of foam-filled thin-walled structure based on sequential Kriging metamodel

Deterministic optimization has been successfully applied to a range of design problems involving foam-filled thin-walled structures, and to some extent gained significant confidence for the applications of such structures in automotive, aerospace, transportation and defense industries. However, the conventional deterministic design could become less meaningful or even unacceptable when considering the perturbations of design variables and noises of system parameters. To overcome this drawback, a robust design methodology is presented in this paper to address the effects of parametric uncertainties of foam-filled thin-walled structure on design optimization, in which different sigma criteria are adopted to measure the variations. The Kriging modeling technique is used to construct the corresponding surrogate models of mean and standard deviation for different crashworthiness criteria. A sequential sampling approach is introduced to improve the fitness accuracy of these surrogate models. Finally, a gradient-based sequential quadratic program (SQP) method is employed from 20 different initial points to obtain a quasi-global robust optimum solution. The optimal solutions were verified by using the Monte Carlo simulation. The results show that the presented robust optimization method is fairly effective and efficient, the crashworthiness and robustness of the foam-filled thin-walled structure can be improved significantly.     

Stochastic optimization of nonlinear energy sinks

Nonlinear energy sinks (NES) are a promising technique to achieve vibration mitigation. Through nonlinear stiffness properties, NES are able to passively and irreversibly absorb energy. Unlike the traditional Tuned Mass Damper (TMD), NES absorb energy from a wide range of frequencies. Many studies have focused on NES behavior and dynamics, but few have addressed the optimal design of NES. Design considerations of NES are of prime importance as it has been shown that NES dynamics exhibit an acute sensitivity to uncertainties. In fact, the sensitivity is so marked that NES efficiency is near-discontinuous and can switch from a high to a low value for a small perturbation in design parameters or loading conditions. This article presents an approach for the probabilistic design of NES which accounts for random design and aleatory variables as well as response discontinuities. In order to maximize the mean efficiency, the algorithm is based on the identification of regions of the design and aleatory space corresponding to markedly different NES efficiencies. This is done through a sequence of approximated sub-problems constructed from clustering, Kriging approximations, a support vector machine, and Monte-Carlo simulations. The refinement of the surrogates is performed locally using a generalized max-min sampling scheme which accounts for the distributions of random variables. The sampling scheme also makes use of the predicted variance of the Kriging surrogates for the selection of aleatory variables values. The proposed algorithm is applied to three example problems of varying dimensionality, all including an aleatory excitation applied to the main system. The stochastic optima are compared to NES optimized deterministically.     

Performance study of gradient-enhanced Kriging

The use of surrogate models for approximating computationally expensive simulations has been on the rise for the last two decades. Kriging-based surrogate models are popular for approximating deterministic computer models. In this work, the performance of Kriging is investigated when gradient information is introduced for the approximation of computationally expensive black-box simulations. This approach, known as gradient-enhanced Kriging, is applied to various benchmark functions of varying dimensionality (2D-20D). As expected, results from the benchmark problems show that additional gradient information can significantly enhance the accuracy of Kriging. Gradient-enhanced Kriging provides a better approximation even when gradient information is only partially available. Further comparison between gradient-enhanced Kriging and an alternative formulation of gradient-enhanced Kriging, called indirect gradient-enhanced Kriging, highlights various advantages of directly employing gradient information, such as improved surrogate model accuracy, better conditioning of the correlation matrix, etc. Finally, gradient-enhanced Kriging is used to model 6- and 10-variable fluid–structure interaction problems from bio-mechanics to identify the arterial wall’s stiffness.     

Sampling-based approach for design optimization in the presence of interval variables

This paper proposes a methodology for sampling-based design optimization in the presence of interval variables. Assuming that an accurate surrogate model is available, the proposed method first searches the worst combination of interval variables for constraints when only interval variables are present or for probabilistic constraints when both interval and random variables are present. Due to the fact that the worst combination of interval variables for probability of failure does not always coincide with that for a performance function, the proposed method directly uses the probability of failure to obtain the worst combination of interval variables when both interval and random variables are present. To calculate sensitivities of the constraints and probabilistic constraints with respect to interval variables by the sampling-based method, behavior of interval variables at the worst case is defined by the Dirac delta function. Then, Monte Carlo simulation is applied to calculate the constraints and probabilistic constraints with the worst combination of interval variables, and their sensitivities. A merit of using an MCS-based approach in the X-space is that it does not require gradients of performance functions and transformation from X-space to U-space for reliability analysis, thus there is no approximation or restriction in calculating sensitivities of constraints or probabilistic constraints. Numerical results indicate that the proposed method can search the worst case probability of failure with both efficiency and accuracy and that it can perform design optimization with mixture of random and interval variables by utilizing the worst case probability of failure search.     

Performance of an ensemble of ordinary, universal, non-stationary and limit Kriging predictors

The selection of stationary or non-stationary Kriging to create a surrogate model of a black box function requires apriori knowledge of the nature of response of the function as these techniques are better at representing some types of responses than others. While an adaptive technique has been previously proposed to adjust the level of stationarity within the surrogate model such a model can be prohibitively expensive to construct for high dimensional problems. An alternative approach is to employ a surrogate model constructed from an ensemble of stationary and non-stationary Kriging models. The following paper assesses the accuracy and optimization performance of such a modelling strategy using a number of analytical functions and engineering design problems.     

Random linear-quadratic mathematical models: Computing explicit solutions and applications

This paper deals with the study of linear random population models and with a random logistic model (where parameters are random variables). Assuming appropriate conditions, the stochastic processes solutions are obtained under closed form using mean square calculus. Expectation and variance expressions for the stochastic processes solutions are given and illustrative examples are included.     

面向复杂装备精密产品质量特性的Kriging-RBDO可靠性优化设计

不确定设计参数情形下的复杂装备柔顺机构精密产品质量特性波动与可靠性疲劳退化是精密微机电系统领域的基础性工程难题.针对这一基础性工程难题,提出一种面向复杂装备柔顺机构精密产品可靠性优化设计模型.利用拉丁超立方试验设计(Latin hypercube design,LHD)构建试验设计组合方案,通过有限元数值模拟获取各试验设计组合方案的质量特性值.据此,采用Kriging代理模型建立质量特性与不确定设计参数之间复杂非线性函数关系模型.在此基础上,引入基于可靠性优化设计(Reliability-based design optimization,RBDO)策略,构建面向复杂装备柔顺机构精密产品Kriging-RBDO可靠性优化设计模型.算例表明,所提出的方法在不确定设计参数情形下的复杂装备柔顺机构精密产品早期质量设计方面具有良好的抗疲劳退化特性.     

Unified uncertainty representation and quantification based on insufficient input data

Uncertainty quantification accuracy of system performance has an important influence on the results of reliability-based design optimization (RBDO). A new uncertain identification and quantification methodology is proposed considering the strong statistical variables, sparse variables, and interval variables simultaneously. Maximum likelihood function and Akaike information criterion (AIC) methods are used to identify the best-fitted distribution types and distribution parameters of sparse variables. The interval variables are represented with evidence theory. Finally, a unified uncertainty quantification framework considering the three types of uncertain design variables is put forward, and then the failure probability of system performance is quantified with belief and plausibility measures. The Kriging metamodel and random sampling method are used to reduce the computational complexity. Three examples are illustrated to verify the effectiveness of the proposed methodology.     

An adaptive hybrid surrogate model

The determination of complex underlying relationships between system parameters from simulated and/or recorded data requires advanced interpolating functions, also known as surrogates. The development of surrogates for such complex relationships often requires the modeling of high dimensional and non-smooth functions using limited information. To this end, the hybrid surrogate modeling paradigm, where different surrogate models are combined, offers an effective solution. In this paper, we develop a new high fidelity surrogate modeling technique that we call the Adaptive Hybrid Functions (AHF). The AHF formulates a reliable Crowding Distance-Based Trust Region (CD-TR), and adaptively combines the favorable characteristics of different surrogate models. The weight of each contributing surrogate model is determined based on the local measure of accuracy for that surrogate model in the pertinent trust region. Such an approach is intended to exploit the advantages of each component surrogate. This approach seeks to simultaneously capture the global trend of the function as well as the local deviations. In this paper, the AHF combines four component surrogate models: (i) the Quadratic Response Surface Model (QRSM), (ii) the Radial Basis Functions (RBF), (iii) the Extended Radial Basis Functions (E-RBF), and (iv) the Kriging model. The AHF is applied to standard test problems and to a complex engineering design problem. Subsequent evaluations of the Root Mean Squared Error (RMSE) and the Maximum Absolute Error (MAE) illustrate the promising potential of this hybrid surrogate modeling approach.     

基于Kriging模型的航空发动机管道系统有限元模型修正

为获得精确可靠的航空发动机外部管道结构动力学模型,采用将Kriging模型与多目标遗传算法(MOGA)相结合的模型修正方法进行有限元模型修正.首先进行管道模型的模态试验和有限元建模,分别获得模态参数的试验值和有限元分析值；然后在合理的参数选取和试验设计(DOE)的基础上,拟合得到Kriging模型；最后基于Kriging模型采用多目标遗传算法进行有限元模型修正,并对比了不同修正方法的精度和修正效果.结果表明：采用Kriging模型进行有限元模型修正可以有效提升修正效果,获得更为准确的有限元模型；对于航空发动机管道系统,基于Kriging模型的模型修正方法相较于基于灵敏度分析的模型修正方法具有更高的修正效率和修正精度.     

A Critical Appraisal of Design of Experiments for Uncertainty Quantification

Surrogate models are widely used to predict response function of any system and in quantifying uncertainty associated with the response function. It is required to have response quantities at some preselected sample points to construct a surrogate model which can be processed in two way. Either the surrogate model is constructed using one shot experimental design techniques, or, the sample points can be generated in a sequential manner so that optimum sample points for a specific problem can be obtained. This paper addresses a comprehensive comparisons between these two types of sampling techniques for the construction of more accurate surrogate models. Two most popular one shot sampling strategies: Latin hypercube sampling and Sobol sequence, and four different type sequential experimental designs (SED) namely, Monte Carlo intersite projected (MCIP), Monte Carlo intersite projected threshold (MCIPT), optimizer projected (OP) and LOLA-Voronoi (LV) method are taken for the present study. Two most widely used surrogate models, namely polynomial chaos expansion and Kriging are used to check the applicability of the experimental design techniques. Three different types of numerical problems are solved using the two above-mentioned surrogate models considering all the experimental design techniques independently. Further, all the results are compared with the standard Monte Carlo simulation (MCS). Overall study depicts that SED performs well in predicting the response functions more accurately with an acceptable number of sample points even for high-dimensional problems which maintains the balance between accuracy and efficiency. More specifically, MCIPT and LV method based Kriging outperforms other combinations.     

A surrogate modeling approach for reliability analysis of a multidisciplinary system with spatio-temporal output

Reliability analysis of a multidisciplinary system is computationally intensive due to the involvement of multiple disciplinary models and coupling between the individual models. When the system inputs and outputs are varying over time and space, the reliability analysis is even more challenging. This paper proposes a surrogate model-based method for the reliability analysis of a multidisciplinary system with spatio-temporal output. The transient characteristics of the multidisciplinary system output under time-dependent variability are analyzed first. Based on the transient analysis, surrogate models are built for individual disciplinary analyses instead of a single surrogate model for the fully coupled analysis. To address the challenge introduced by the high-dimensionality of spatially varying inter-disciplinary coupling variables, a data compression method is first employed to convert the high-dimensional coupling variables into low-dimensional latent space. Kriging surrogate modeling is then used to build surrogates for the individual disciplinary models in the latent space. Based on the individual disciplinary surrogate models, reliability analysis of the coupled multidisciplinary system under time-dependent uncertainty is investigated. Further, epistemic uncertainty sources, such as data uncertainty and model uncertainty, lead to uncertainty in the reliability estimate. Therefore, an auxiliary variable approach is used to efficiently include the epistemic uncertainty sources within the reliability analysis. An aircraft panel subjected to hypersonic flow conditions is used to demonstrate the proposed method. The analysis involves four interacting disciplinary models, namely, aerodynamics, aerothermal analysis, heat transfer, and structural analysis. The results show that the proposed method is able to effectively perform reliability analysis of a multidisciplinary system with spatio-temporal output.