Quantitative property of MF-discrepancy and efficient point-selection strategy for the nonlinear stochastic response analysis of structures with random parameters最大边缘扩展 F 偏差的量化特性与随机参数结构非线性随机响应分析的高效选点策略
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Chen JB, Huang X, Li J, 2024. Quantitative property of MF-discrepancy and efficient point-selection strategy for the nonlinear stochastic response analysis of structures with random parameters. Probabilistic Engineering Mechanics, 78: 103708.DOI: 10.1016/j.probengmech.2024.103708
含随机参数的高维强非线性系统响应分析一直是随机计算力学中的一项重大挑战。为应对这一挑战，已发展了一些高效点集方法，其中以低偏差为代表的全局点集方法对生成高效代表性点集至关重要。已引入了包括扩展 F 偏差和广义 F 偏差在内的几类偏差以评估代表性点集的均匀性和有效性。在此背景下，本文提出了最大边缘扩展 F 偏差，它是广义 F 偏差的扩展形式，然后详细研究了最大边缘扩展 F 偏差的性质。推导了最大边缘扩展 F 偏差的概率分布，包括对随机点集的严格证明和基于某些一般点集的假设模型。建立了广义 Koksma-Hlawka 不等式来控制最不利误差估计。给出了最大边缘扩展 F 偏差的下界限，并提出了两个直观的定量指标来衡量最大边缘扩展 F 偏差的优劣。基于下界限，提出了一类增强选点策略，并建立了统一的理论框架来对最大边缘扩展 F 偏差进行最小化。在该框架中，对最大边缘扩展 F 偏差进行局部最小化可导出两步选点方法，并提出了一种基于最大边缘扩展 F 偏差全局最小化的选点新策略，验证了该策略的有效性和鲁棒性，特别是对于高维情形。研究了几个数值算例，包括 2 层剪切框架、10 层剪切框架和有限元法建模的 10 层钢筋混凝土框架结构，验证了所提选点策略的有效性和鲁棒性。关键词: 最大边缘扩展 F 偏差, 广义 F 偏差, 扩展 F 偏差, 下界限, 选点策略, 非线性系统, 随机动力学The response analysis of high-dimensional and strongly nonlinear systems with random parameters remains a significant challenge in stochastic computational mechanics. To address this challenge, some methods based on the high-efficacy point sets have been developed, in which efficient global-point-set methods represented by low-discrepancy are of paramount importance in generating representative point sets. Several discrepancies including the extended F-discrepancy (EF-discrepancy) and the generalized F-discrepancy (GF-discrepancy) have been introduced to assess the uniformity and the efficacy of a representative point set. In such context, a maximal marginal EF-discrepancy (MF-discrepancy), which is an extended form of the GF-discrepancy, is proposed in this paper and then the properties of the MF-discrepancy are studied in detail. The probability distribution of the MF-discrepancy is derived, including a rigorous proof for random point sets and a model based on an assumption for some generic point sets. A generalized Koksma-Hlawka inequality is established accordingly to govern the worst error estimate. The lowest bound of the MF-discrepancy is given, and two intuitive quantitative indices are proposed to measure the goodness of the MF-discrepancy. Based on the lowest bound, an enhanced point-selection strategy with a unified theoretical framework for minimizing the MF-discrepancy is proposed. In this framework, locally minimizing the MF-discrepancy yields the two-step point-selection method, and a new point-selection strategy is proposed based on the global minimization of the MF-discrepancy, which is verified to be efficient and robust, especially in high-dimensional cases. Several numerical examples, including a 2-story shear frame, a 10-story shear frame, and a 10-story reinforced concrete frame structure modeled by the finite element method, are studied, verifying the efficiency and the robustness of the proposed point-selection strategy.Keywords: MF-Discrepancy; GF-Discrepancy; EF-Discrepancy; Lowest round; Point-selection strategy; Nonlinear system; Stochastic dynamics.图 1: 最大边缘扩展 F 偏差与扩展 F 偏差之间的关系Fig. 1. Relationship between MF-discrepancies and EF-discrepancies
图 2: 最大边缘扩展 F 偏差对数的概率密度函数曲线Fig. 2. PDF curves for the logarithm of the MF-discrepancy
图 3: 最大边缘扩展 F 偏差分布的蒙特卡罗模拟Fig. 3. Monte Carlo simulation of the distribution of MF-discrepancies
Fig. 4. MF-discrepancy versus the point number
图 5: 随机等权重点集与随机赋得概率点集的最大边缘扩展 F 偏差对比Fig. 5. A comparison of MF-discrepancies between RNEP and RNAP
图 6: 最大边缘扩展 F 偏差局部最小化与全局最小化相应的边缘概率分布函数Fig. 6. Marginal CDFs corresponding to the local minimization and the global minimization of the MF-discrepancy
图 7: 最大边缘扩展 F 偏差局部最小化点集与随机赋得概率点集的最大边缘扩展 F 偏差对比Fig. 7. A comparison of the MF-discrepancy between L-MMFP and RNAP
Fig. 8. 2-Story shear structure
Fig. 9. A typical hysteretic curve sample
Fig. 10. Mean and the standard errors of the mean and the standard deviation for the displacements of the first floor
Fig. 11. Relative errors of the mean and the standard deviation for the displacements of the first floor
Fig. 12. 10-story shear structure
Fig. 13. A typical hysteretic curve sample
Fig. 14. Mean, the standard errors and the relative errors for the displacements of the first floor (s =10)
Fig. 15. Mean, the standard errors and the relative errors for the displacements of the first floor (s =20)
Fig. 16. Mean, the standard errors and the relative errors for the displacements of the first floor (s = 60)
Fig. 17. 10-story frame structure
Fig. 18. Typical samples of stress-strain curves
Fig. 19. Mean, the standard deviation and relative errors for the displacement of the top floor (s = 8)
Fig. 20. Mean, the standard deviation and relative errors for the displacement of the top floor (s = 60)
Fig. 21. Sampling based on the distribution of MF-discrepancies
作者信息 | Authors
陈建兵 Jian-Bing Chen, 通讯作者 (Corresp.)同济大学 (Tongji University) 土木工程学院Email: chenjb@tongji.edu.cn
同济大学 (Tongji University) 土木工程学院
中国科学院院士
同济大学 (Tongji University) 土木工程学院
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
