基于凸模型的多学科不确定性优化设计方法
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摘要
复杂工程系统通常涉及到相互耦合的多个学科,而且其中往往存在不确定性因素,所以研究多学科不确定性优化设计问题有一定的工程意义。然而由于多学科系统涉及的变量众多且变量之间的关系复杂,要得到不确定变量的分布规律有一定困难,凸模型在处理不确定性信息较少的问题中有其优势,有望应用于工程问题。凸模型理论和多学科优化理论都是面向复杂工程系统的新的方法学,因而进行两者交叉研究,可以相互促进其理论体系的进一步完善。因此,本文基于凸模型理论,研究多学科不确定性优化设计方法,为多学科设计优化的工程应用提供必要的基础。本文的主要研究工作如下：
(1)本文首先研究了基于凸模型多学科系统的变差分析,采用高斯-塞德尔型迭代方法来求得耦合变量变差范围,进而求得其它状态变量变差范围。利用一次二阶矩方法中的验算点法求解非概率可靠性指标,采用多学科可行方法进行多学科分析,得到了求解基于凸模型的多学科系统可靠性分析的方法。
(2)提出了一种针对多学科优化中目标函数和约束存在不确定性问题的求解方法。该方法基于区间数的序关系方法将不确定性目标函数转化为两目标的确定性目标函数,并对两目标进行线性加权转化为单目标问题；通过改进的满意度方法将不确定性不等式约束转换为确定性不等式约束,等式约束中的状态变量通过高斯—塞得尔迭代求得；采用罚函数法将约束问题转化为无约束问题进行求解并将多学科可行方法作为优化方法。
(3)针对约束存在不确定性的多学科问题,建立了一种新的多学科稳健性优化设计方法。该方法中采用凸模型描述不确定参量,通过遗传算法求得约束的最大值,根据实际情况选取满意度水平后再求得相应的约束值,多学科优化过程依靠两级集成系统合成法实现。该方法对于处理多学科不确定性优化设计问题有一定的指导意义。
(4)将序列优化与可靠性评价方法应用于基于凸模型的多学科不确定优化之中以提高计算效率。常规的基于可靠性的多学科设计优化问题是一个三层循环嵌套优化问题,最外层为确定性优化,中间层为可靠性分析,最内层为多学科分析。序列优化与可靠性评价方法的核心思想是将可靠性分析从多学科优化中分离出来,可靠性分析与确定性多学科优化过程构成一个循环。在该方法中,可靠性分析采用功能度量法,多学科优化方法采用多学科可行方法或者二级系统一体化合成优化方法。
(5)通过一种灵敏度信息更新方法来提高多学科不确定性优化设计的计算效率,该方法的基本思想是减少在灵敏度分析中调用学科分析次数。首先在灵敏度信息中找出非灵敏项及线性项,然后在一定的多学科优化循环次数中不对非灵敏项及线性项的灵敏度值进行更新,采用近似的灵敏度信息来代替真实的灵敏度信息。将该方法嵌入到多学科不确定性优化设计之中得到了基于灵敏度更新策略的多学科不确定性优化设计方法。
Complex engineering system usually involves coupled multidisciplinaries, and it usually contains uncertainties. Therefore, it is necessary to investigate multidisciplinary uncertain design optimization methods to meet engineering requirement. However, it is very difficult to obtain the distribution of uncertain variables because of the large number of variables and the complex relationship that exists between variables. Convex model theory has advantages in handling non-deterministic problems with a few uncertainties information, and it's likely applied in engineering problems. Both convex model theory and multidisciplinary optimization theory are newly developed methodology for complex engineering system. Therefore, study on the integration of the two theories can improve their theoretical system. Based on convex model theory, this dissertation mainly focuses on investigating multidisciplinary uncertain design optimization method and providing fundamental understanding for multidisciplinary design optimization in engineering applications. As a result, the following studies are carried out in this dissertation:
(1) First, the technique for calculating the variation ranges of multidisciplinary system is investigated based on convex model theory. Gauss-Sidel iteration method is utilized to calculate the variation intervals of coupling variables, and then the variation intervals of the other state variables can be obtained. A reliability analysis method of multidisciplinary system is proposed based on convex model. First order reliability analysis is used to calculate the non-probabilistic reliability index, and the multidisciplinary feasible method is applied to perform multidisciplinary analysis.
(2) A method is suggested to solve the multidisciplinary optimization problem with uncertainties both in objective function and constraints. Based on the method of order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, and then the deterministic objective functions are linear weighted to get a single-objective function. The uncertain inequality constrains are converted into deterministic inequality constrains through modified possibility degree. The state variables are obtained through Gauss-Sidel iteration. The constraint optimization problem is changed into non-constraint optimization problem by penalty function method, and multidisciplinary feasible method is used as optimization strategy.
(3) A multidisciplinary robust design optimization method is presented to solve the optimization problem of multidisciplinary design with uncertainties in constraints. Uncertain variables are assumed as convex model and genetic algorithm is applied to solve the maximum values of constraint intervals. The values of constraints can be obtained after possibility degree level is predetermined. Bi-level integrated system synthetic method is used as multidisciplinary optimization solver. Numerical example is investigated to demonstrate the efficiency of the method. The method can provide the necessary theoretical basis for multidisciplinary uncertain design optimization.
(4) A sequential multidisciplinary optimization and reliability assessment method based on convex model is proposed to improve computational efficiency. The conventional reliability-based multidisciplinary design optimization strategy has tri-level loops:the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. The cental idea of sequential optimization and reliability assessment method is to decouple the reliability analysis from multidisciplinary optimization with sequential cycles of reliability analysis and multidisciplinary optimization. In the method, the reliability analysis method is performance measure approach, and multidisciplinary feasible method or bi-level integrated system synthesis is used to optimize the multidisciplinary system.
(5) A sensitivity information updating method is developed to improve computational efficiency in multidisciplinary uncertain design optimization. This method aimed at reducing the number of disciplinary analysis called by sensitivity analysis. At first, the linear and insensitive terms in sensitivity information are identified, and then the values of the linear and insensitive terms are temporarily kept invariable for multiple multidisciplinary design optimization cycles. The true sensitivity information is replaced by approximated sensitivity information. The sensitivity information updating method is integrated into multidisciplinary uncertain design optimization method.
(1)本文首先研究了基于凸模型多学科系统的变差分析,采用高斯-塞德尔型迭代方法来求得耦合变量变差范围,进而求得其它状态变量变差范围。利用一次二阶矩方法中的验算点法求解非概率可靠性指标,采用多学科可行方法进行多学科分析,得到了求解基于凸模型的多学科系统可靠性分析的方法。
(2)提出了一种针对多学科优化中目标函数和约束存在不确定性问题的求解方法。该方法基于区间数的序关系方法将不确定性目标函数转化为两目标的确定性目标函数,并对两目标进行线性加权转化为单目标问题；通过改进的满意度方法将不确定性不等式约束转换为确定性不等式约束,等式约束中的状态变量通过高斯—塞得尔迭代求得；采用罚函数法将约束问题转化为无约束问题进行求解并将多学科可行方法作为优化方法。
(3)针对约束存在不确定性的多学科问题,建立了一种新的多学科稳健性优化设计方法。该方法中采用凸模型描述不确定参量,通过遗传算法求得约束的最大值,根据实际情况选取满意度水平后再求得相应的约束值,多学科优化过程依靠两级集成系统合成法实现。该方法对于处理多学科不确定性优化设计问题有一定的指导意义。
(4)将序列优化与可靠性评价方法应用于基于凸模型的多学科不确定优化之中以提高计算效率。常规的基于可靠性的多学科设计优化问题是一个三层循环嵌套优化问题,最外层为确定性优化,中间层为可靠性分析,最内层为多学科分析。序列优化与可靠性评价方法的核心思想是将可靠性分析从多学科优化中分离出来,可靠性分析与确定性多学科优化过程构成一个循环。在该方法中,可靠性分析采用功能度量法,多学科优化方法采用多学科可行方法或者二级系统一体化合成优化方法。
(5)通过一种灵敏度信息更新方法来提高多学科不确定性优化设计的计算效率,该方法的基本思想是减少在灵敏度分析中调用学科分析次数。首先在灵敏度信息中找出非灵敏项及线性项,然后在一定的多学科优化循环次数中不对非灵敏项及线性项的灵敏度值进行更新,采用近似的灵敏度信息来代替真实的灵敏度信息。将该方法嵌入到多学科不确定性优化设计之中得到了基于灵敏度更新策略的多学科不确定性优化设计方法。
Complex engineering system usually involves coupled multidisciplinaries, and it usually contains uncertainties. Therefore, it is necessary to investigate multidisciplinary uncertain design optimization methods to meet engineering requirement. However, it is very difficult to obtain the distribution of uncertain variables because of the large number of variables and the complex relationship that exists between variables. Convex model theory has advantages in handling non-deterministic problems with a few uncertainties information, and it's likely applied in engineering problems. Both convex model theory and multidisciplinary optimization theory are newly developed methodology for complex engineering system. Therefore, study on the integration of the two theories can improve their theoretical system. Based on convex model theory, this dissertation mainly focuses on investigating multidisciplinary uncertain design optimization method and providing fundamental understanding for multidisciplinary design optimization in engineering applications. As a result, the following studies are carried out in this dissertation:
(1) First, the technique for calculating the variation ranges of multidisciplinary system is investigated based on convex model theory. Gauss-Sidel iteration method is utilized to calculate the variation intervals of coupling variables, and then the variation intervals of the other state variables can be obtained. A reliability analysis method of multidisciplinary system is proposed based on convex model. First order reliability analysis is used to calculate the non-probabilistic reliability index, and the multidisciplinary feasible method is applied to perform multidisciplinary analysis.
(2) A method is suggested to solve the multidisciplinary optimization problem with uncertainties both in objective function and constraints. Based on the method of order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, and then the deterministic objective functions are linear weighted to get a single-objective function. The uncertain inequality constrains are converted into deterministic inequality constrains through modified possibility degree. The state variables are obtained through Gauss-Sidel iteration. The constraint optimization problem is changed into non-constraint optimization problem by penalty function method, and multidisciplinary feasible method is used as optimization strategy.
(3) A multidisciplinary robust design optimization method is presented to solve the optimization problem of multidisciplinary design with uncertainties in constraints. Uncertain variables are assumed as convex model and genetic algorithm is applied to solve the maximum values of constraint intervals. The values of constraints can be obtained after possibility degree level is predetermined. Bi-level integrated system synthetic method is used as multidisciplinary optimization solver. Numerical example is investigated to demonstrate the efficiency of the method. The method can provide the necessary theoretical basis for multidisciplinary uncertain design optimization.
(4) A sequential multidisciplinary optimization and reliability assessment method based on convex model is proposed to improve computational efficiency. The conventional reliability-based multidisciplinary design optimization strategy has tri-level loops:the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. The cental idea of sequential optimization and reliability assessment method is to decouple the reliability analysis from multidisciplinary optimization with sequential cycles of reliability analysis and multidisciplinary optimization. In the method, the reliability analysis method is performance measure approach, and multidisciplinary feasible method or bi-level integrated system synthesis is used to optimize the multidisciplinary system.
(5) A sensitivity information updating method is developed to improve computational efficiency in multidisciplinary uncertain design optimization. This method aimed at reducing the number of disciplinary analysis called by sensitivity analysis. At first, the linear and insensitive terms in sensitivity information are identified, and then the values of the linear and insensitive terms are temporarily kept invariable for multiple multidisciplinary design optimization cycles. The true sensitivity information is replaced by approximated sensitivity information. The sensitivity information updating method is integrated into multidisciplinary uncertain design optimization method.
引文
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