基于DC规划的非概率结构可置信性极值响应分析与优化设计

When considering the problem of the extreme responses of structures with non-probabilistic parameters, because of the absence of guarantee in mathematical theory, the analysis methods based on the approximation would fail for the structures with fairly large uncertainties in parameters. To circumvent this drawback, the concept of the confidence extreme response has been proposed, where a group of extreme responses strictly contain the real region of the response of uncertain structures. Usually, the problem of the confidence extreme responses of structures with non-probabilistic parameters are formulated as finding the global optimal solutions of indefinite quadratic programming problems with a convex feasible region. Due to the limitation of solution techniques for the confidence extreme response, previous analyses are mostly restricted to the trusses with ellipsoidal convex model parameters subjected to static load. DC(difference of convex functions) programming is an effectively approach for solving the above quadratic programming problems globally. The project presents a study on the modeling methodologies and their associated algorithms based on DC programming to find the confidence extreme static response, the confidence extreme dynamic response, and the confidence extreme modal values for general structures, for different types of non-probabilistic parameters. The confidence optimization technique based on DC programming for structures with non-probabilistic parameters will be developed. The research in this project will provide great significance theoretical basis for confidence response analysis and confidence optimization of structures with non-probabilistic parameters.

在处理非概率(有界不确定)参数结构极值响应分析问题时，由于没有任何数学理论保证，现有基于近似的分析方法对于较大不确定性参数问题常常会失效。因此，有关学者提出了可置信性极值响应概念：在严格数学意义下，一组结构极值响应完全包络住结构响应范围。通常非概率参数结构可置信性极值响应问题被建模为可行域为凸域，目标函数为不定二次函数的全局最优化问题。由于求解手段的限制，前人研究局限于具有椭球型不确定性参数的桁架结构可置信性极值静力响应的分析。DC(difference of convex functions)规划是求解此类二次规划问题全局最优解的新兴高效方法。本项目基于DC规划，针对不同类型非概率参数，提出适合任意结构形式的广义非概率参数结构可置信性静力、动力极值响应、模态极值建模理论与求解方法；发展基于DC规划的高效非概率参数结构可置信性优化设计方法。预期成果对非概率结构分析与优化提供重要理论依据。

在处理非概率(有界不确定)参数结构极值响应分析问题时，现有分析方法大多基于对响应函数的一阶近似。当参数的不确定性较大时，由于没有任何数学理论保证，采用一阶近似方法很可能会失效。针对此问题，有关学者提出了可置信性极值响应概念：在严格数学证明意义下，一组结构极值响应完全包络住结构的响应范围。通常非概率参数结构可置信性极值响应问题被建模为可行域为凸域，目标函数为不定二次函数的全局最优化问题。由于求解手段的限制，前人研究局限于具有椭球型不确定性参数的桁架结构可置信性静力极值响应的分析。DC(difference of convex functions)规划是求解此类二次规划问题全局最优解的新兴高效方法。本项目基于DC规划，针对不同类型非概率参数，提出适合任意结构形式的广义非概率参数结构可置信性静力、动力极值响应、模态极值建模理论与求解方法；发展基于DC规划的高效非概率参数结构可置信性优化设计方法。课题预期成果对于非概率结构分析与优化设计提供重要理论依据。