DU Xiu-yun, XUE Qi-wen, LIU Xu-dong. Reliability Analysis Based on Bregman Distances[J]. Applied Mathematics and Mechanics, 2016, 37(6): 609-616. doi: 10.3879/j.issn.1000-0887.2016.06.006
 Citation: DU Xiu-yun, XUE Qi-wen, LIU Xu-dong. Reliability Analysis Based on Bregman Distances[J]. Applied Mathematics and Mechanics, 2016, 37(6): 609-616.

# Reliability Analysis Based on Bregman Distances

##### doi: 10.3879/j.issn.1000-0887.2016.06.006
Funds:  The National Natural Science Foundation of China(10802015)
• Rev Recd Date: 2015-12-23
• Publish Date: 2016-06-15
• A reliability analysis model based on the homotopy algorithm was established to address probabilistic reliability problems of uncertain structures through introduction of Bregman distances. By means of the limit state equations, solution of the reliability index was transformed to a nonlinear constrained optimization problem. According to the homotopy theory and the Bregman distances, the system of homotopy equations was constructed and solved with the path-tracking algorithm. The reliability calculations for different types of functions and different degrees of nonlinear problems were discussed via numerical examples, and the results were compared with those out of previous methods. The results show that the proposed analysis model solves the probabilistic reliability problems of uncertain structures with high efficiency and good accuracy.
•  [1] Hohenbichler M, Rackwitz R. Non-normal dependent vectors in structural safety[J]. Journal of the Engineering Mechanics Division,1981,107(6): 1227-1238. [2] Zhang Y, Kiureghian A D. Two improved algorithms forreliability analysis[C]// Reliability and Optimization of Structural Systems. USA: Springer, 1995: 297-304. [3] 贡金鑫. 工程结构可靠度计算方法[M]. 大连: 大连理工大学出版社, 2003.(GONG Jin-xin. Reliability Calculation Methods for Engineering Structures [M]. Dalian: Dalian University of Technology Press, 2003.(in Chinese)) [4] YANG Di-xiong. Chaos control for numerical instability of first order reliability method[J]. Communications in Nonlinear Science and Numerical Simulation,2010,15(10): 3131-3141. [5] 亢战, 罗阳军. 计算结构可靠度指标的修正迭代算法[J]. 工程力学, 2008,25(11): 20-26.(KANG Zhan, LUO Yang-jun. A modified iteration algorithm for structural reliability index evaluation[J]. Engineering Mechanics,2008,25(11): 20-26.(in Chinese)) [6] LIN Zheng-hua, YU Bo, FENG Guo-chen. A combined homotopy interior point method for convex nonlinear programming[J]. Applied Mathematics and Computation,1997,84(2/3): 193-211. [7] 宋辉, 李芬, 徐献芝. 电池系统建模中Butler-Volmer方程的同伦分析求解[J]. 应用数学和力学, 2013,34(4): 373-382.(SONG Hui, LI Fen, XU Xian-zhi. Analytical solution of Butler-Volmer equation in battery system modeling[J]. Applied Mathematics and Mechanics,2013,34(4): 373-382.(in Chinese)) [8] 邹丽, 王振, 宗智, 王喜军, 张朔. 指数同伦法对Cauchy条件下变系数Burgers方程的解析与数值分析[J]. 应用数学和力学, 2014,35(7): 777-789.(ZOU Li, WANG Zhen, ZONG Zhi, WANG Xi-jun, ZHANG Shuo. Analytical and numerical investigation of the variable coefficient Burgers equation under Cauchy condition with the exponential homotopy method[J]. Applied Mathematics and Mechanics,2014,35(7): 777-789.(in Chinese)) [9] LIN Zheng-hua, LI Yong, YU Bo. A combined homotopy interior point method for general nonlinear programming problems[J]. Applied Mathematics and Computation,1996,80(2/3): 209-224. [10] 薛齐文, 杨海天. 二阶非定常热传导反问题的多宗量辨识[J]. 计算力学学报, 2007,24(4): 425-429.(XUE Qi-wen, YANG Hai-tian. Identification of multi-variables of inverse two-order transient heat conduction problems[J]. Chinese Journal of Computational Mechanics,2007,24(4): 425-429.(in Chinese)) [11] 吉猛, 姜潮, 韩硕. 一种基于同伦分析的结构可靠性功能度量法[J]. 计算力学学报, 2015,32(2): 149-153.(JI Meng, JIANG Chao, HAN Shuo. A performance measure approach of structural reliability based on homotopy analysis[J]. Chinese Journal of Computational Mechanics,2015,32(2): 149-153.(in Chinese)) [12] 薛齐文, 张军, 魏伟. 基于同伦技术的偶应力反问题求解[J]. 计算力学学报, 2011,28(2): 243-247.(XUE Qi-wen, ZHANG Jun, WEI Wei. Solving inverse couple stress problem via homotopy method[J]. Chinese Journal of Computational Mechanics,2011,28(2): 243-247.(in Chinese)) [13] 韩硕. 基于同伦算法的结构可靠性分析[D]. 硕士学位论文. 长沙: 湖南大学, 2014.(HAN Shuo. Structural reliability analysis based on homotopy algorithm[D]. Master Thesis. Changsha: Hunan University, 2014.(in Chinese)) [14] Cidade G A G, Anteneodo C, Roberty N C, Neto A J S. A generalized approach for atomic force microscopy image restoration with Bregman distances as Tikhonov regularization terms[J]. Inverse Problem in Engineering,2000,8(5): 457-472. [15] WANG Hong, WANG Xi-cheng. Parameter estimation for metabolic networks with Bregman regularization homotopy inversion algorithm[J]. Journal of Theoretical Biology,2014,343(2): 199-207.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142