An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging

Variable-fidelity (VF) modelling methods have been widely used in complex engineering system design to mitigate the computational burden. Building a VF model generally includes two parts: design of experiments and metamodel construction. In this article, an adaptive sampling method based on improved hierarchical kriging (ASM-IHK) is proposed to refine the improved VF model. First, an improved hierarchical kriging model is developed as the metamodel, in which the low-fidelity model is varied through a polynomial response surface function to capture the characteristics of a high-fidelity model. Secondly, to reduce local approximation errors, an active learning strategy based on a sequential sampling method is introduced to make full use of the already required information on the current sampling points and to guide the sampling process of the high-fidelity model. Finally, two numerical examples and the modelling of the aerodynamic coefficient for an aircraft are provided to demonstrate the approximation capability of the proposed approach, as well as three other metamodelling methods and two sequential sampling methods. The results show that ASM-IHK provides a more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.     

Non-parametric adaptive importance sampling for the probability estimation of a launcher impact position

Importance sampling (IS) is a useful simulation technique to estimate critical probability with a better accuracy than Monte Carlo methods. It consists in generating random weighted samples from an auxiliary distribution rather than the distribution of interest. The crucial part of this algorithm is the choice of an efficient auxiliary PDF that has to be able to simulate more rare random events. The optimisation of this auxiliary distribution is often in practice very difficult. In this article, we propose to approach the IS optimal auxiliary density with non-parametric adaptive importance sampling (NAIS). We apply this technique for the probability estimation of spatial launcher impact position since it has currently become a more and more important issue in the field of aeronautics.     

Active learning method combining Kriging model and multimodal-optimization-based importance sampling for the estimation of small failure probability

A novel method which combines the active learning Kriging (ALK) model with important sampling is proposed in this paper. The main aim of the proposed method is to solve problems with very small failure probability and multiple failure regions. A surrogate limit state surface (LSS) which strikes a balance between the Kriging mean and variance is proposed. In each iteration, important samples of the surrogate LSS are generated, optimal training points are chosen, the Kriging model is updated and the surrogate LSS is refined. After several iterations, the surrogate LSS will converge to the true LSS. To obtain all the local and global most probable points (MPPs) on the surrogate LSS in each iteration, a recently proposed evolutionary algorithm from the field of multimodal optimization is introduced. In this way, none of the potential failure regions is missed and the unbiasedness of the proposed method is guaranteed. The contribution factor of each MPP is defined and a weighted multimodal instrumental sampling density is formulated. In this way, more attention is paid to the more important failure regions and training points are further saved. The performance of the proposed method is verified by six case studies.     

Efficient conditional probability theorem and importance sampling-based methods for global reliability sensitivity analysis

Global reliability sensitivity (GRS) analysis can measure the effect of random inputs on failure probability (FP). To efficiently solve GRS, two conditional probability theorem (CPT)-based methods are proposed by combining adaptive Kriging (AK) with importance sampling (IS) (CPT-AK-IS) and combining AK with Meta-IS (CPT-AK-Meta-IS) respectively. Firstly, differentiation approximation and CPT are used to convert the estimation of conditional probability density function (PDF), which is required by the existing Bayes theorem-based methods, into that of a series of probabilities. Secondly, GRS can be directly estimated by the failure samples of IS, while the existing Bayes theorem methods based on IS need to transform the failure samples of IS into those of original PDF. Both the first and second strategies can reduce the computational complexity of solving GRS. Thirdly, by selecting a suitable differentiation interval with a proposed adaptive strategy, the estimation of a series of probabilities can be accurately completed as a byproduct of one IS based simulation for solving FP without additional computational cost. Finally, by introducing AK into IS and Meta-IS, it can reduce the number of evaluating performance function and the size of candidate sample pool simultaneously. These novelties are sufficiently verified by the presented examples.     

A note on inclusion probability in ranked set sampling and some of its variations

Inclusion probabilities are design dependent and should be furnished with the design elements. Inclusion probability of an element in the population is the probability that the element will be chosen in a sample. In this paper the inclusion probabilities in the case of ranked set sampling design and some of its variations are furnished. This paper provides good and interesting examples of sampling designs for which the inclusion probabilities are not equal. M.F. Al-Saleh is currently at Qatar University (on leave), e-mail: malsaleh@qu.edu.qa.     

基于支持向量机和重要度抽样的高强度钢板冲压成形工艺稳健设计

高强度钢板成形中噪声因素的存在造成了冲压质量不稳定.提出了基于支持向量机和重要度抽样的板料成形工艺稳健设计方法,量化了噪声因素对成形质量的影响,同时结合优化算法求解即满足质量可靠性又保证质量目标最优的工艺条件.对一高强度钢板冲压实例进行了工艺优化.按优化工艺冲压成形的零件减薄率及回弹均有所改善,验证了该方法的有效性.