区间非概率多目标优化设计方法及其在车身设计中的应用
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摘要
现代经济、工业的发展使优化技术在各个领域得到了广泛应用,很多工程问题常常涉及到多目标优化问题,多个目标之间的相互竞争和相互冲突常常使得最优解不存在,因此,不能简单地应用单目标优化方法来解决。此外,不确定性也广泛存在于实际工程问题中,这使得传统的优化理论和方法难以直接得到应用,给求解带来了困难和挑战。随机和模糊优化方法是两类传统的不确定优化方法,然而要获得不确定量的精确概率分布和模糊隶属度函数较为困难。区间数优化是一类相对较新的不确定优化方法,它利用区间对不确定参数建模,只需要较少的信息即可获得变量的上下界,有较好的经济性和方便性。当前区间不确定优化问题大多是单目标优化问题,然而在实际工程中常常涉及到区间多目标优化问题,特别是非线性区间数多目标优化问题。目前为止,还没有发展出一种有效的算法来处理该类问题。为此,本论文对非线性区间不确定多目标优化问题进行了研究。
本文的研究工作按以下几个方面展开:首先,提出了一种非线性区间不确定性多目标优化转换模型,实现了不确定多目标优化问题向确定性优化问题的转换。其次,基于该转换模型,将自适应近似模型技术引入非线性区间多目标优化,构造出一种具有一定工程实用性的高效非线性区间不确定优化算法,主要解决两层嵌套优化造成的效率低下问题。再次,基于该转换模型,拓展到不确定多学科优化问题中,构造出一种适用于多学科多目标的不确定优化算法。最后,将算法应用于车身设计领域中的一些实际问题。基于此思路,本文主要研究内容如下:
(1)提出了一种基于非线性区间的不确定多目标转化模型。基于区间序关系和区间可能度,将不确定多目标的目标函数和约束转化为确定性的目标函数和约束。通过转换模型,得到确定性的两层嵌套优化问题。基于多目标遗传算法和序列二次规划算法的两层嵌套优化算法来求解转换后的确定多目标优化问题。对车辆耐撞性和薄板冲压成形两类不确定优化问题进行求解。结果表明该算法具有较好的工程实用性。
(2)提出了一种基于自适应近似模型技术的不确定多目标优化算法。整个优化过程由一系列的近似不确定优化问题迭代完成,通过拉丁方试验设计,在设计空间和不确定空间进行采样,建立目标函数和约束的Kriging近似模型。自适应方法的每一迭代步,通过非线性区间数优化求解转换后的确定优化问题,获得Pareto解集和不确定集,组成设计点集,根据空间填充设计标准,从设计点集选择几个设计点对近似模型更新直到收敛为止。该算法不仅更新了设计空间和不确定空间,提高了近似模型的精度,而且减少了样本的数量,提高了优化效率。
(3)在区间非概率可靠性指标的基础上,建立了具有可靠性指标约束的多目标优化模型。针对该内层优化为极小极大问题的嵌套优化模型,转换为易于处理的等效形式,同时通过区间序关系给出了目标函数稳健性的求解方法。
(4)提出了一种基于多学科可行方法的区间不确定多目标优化算法。该多学科多目标算法是三层循环求解,最内层通过学科分析求得状态变量;中间层搜索不确定量,求得目标和约束函数的区间;基于区间序关系和区间可能度,转换为确定性多学科多目标优化问题,外层多目标优化算法求解该转换后的确定性优化问题。并通过数值算例验证了该方法的有效性。
The rapid development with modern industry and economy has made the optimization technique be greatly used in many kinds of fields.Many engineering problems often involve multi-objective optimization problem. Multiple objectives are always competitive and conflicting, which often result in inexistence of optimal solution. Therefore, multi-objective optimization problem (MOOP) can’t be efficiently solved by using the single objective optimization method.Moreover, the uncertainties also cause many difficulties to directly use the conventional optimization approaches and optimization theories,and bring challenges for MOOP. Stochastic and fuzzy multi-objective optimizations are two types of traditional uncertain multi-objective optimization methodologies. Unfortunately, and it is difficult to construct the precise probability distributions or fuzzy membership functions. The interval number optimization is a relatively newly-developed uncertain optimization method, in which interval is used to model the uncertainty of variables. Thus the variation bounds of the uncertain variables are only required, which can be obtained through only a small amount of uncertainty information. The interval number optimization method mostly focuses on the single objective optimization in dealing with uncertainties, while uncertain multi-objective optimization more often involves in engineering problems. It is still at preliminary stage for the nonlinear interval number optimization research. So far, a nonlinear interval number algorithm with fine efficiency and precision has still not been developed to deal with this class of problems. So it is necessary to study the nonlinear interval uncertain multi-objective optimization (NIUMO).
This dissertation mainly focuses the NIUMO problem. Firstly, a transformation model is proposed to deal with the NIUMO problem, through which the uncertain optimization can be transformed into a deterministic optimization problem. Secondly, an adaptive approximation method is presented to solve the low efficiency problem, which is caused by the nesting optimization. Thirdly, the transformation model is extended to uncertain multidisciplinary design optimization problem. Thus, an algorithm is constructed to solve the interval multidisciplinary multi-objective problem. Finally, the algorithm is applied to the design problem of vehicle body. The main contents are given as follows:
(1) A new uncertain multi-objective optimization method is developed based on a nonlinear interval number programming. Based on the order relation of interval number and possibility degree of interval number, the uncertain multi-objective optimization is transformed into a deterministic non-constraint multi-objective optimization in terms of a penalty function. The multi-objective genetic algorithm and sequential quadratic programming algorithm are used to solve two layers nesting optimization problems, which are based on the deterministic multi-objective model of transformation. Thus, a new hybrid algorithm is proposed to solve the nonlinear interval number optimization problem. The optimization algorithm is successfully applied in complicated engineering problems, including the crashworthiness and sheet metal forming optimization. The application of the engineering problem demonstrates the effectiveness of the present method.
(2) An adaptive approximation method is suggested to deal with an nonlinear interval uncertain multi-objective optimization (NIUMO) problem. The whole optimization process consists of a sequence of approximate optimization problems. The approximation models of uncertain objective functions and constrains are constructed by the Kriging models with the Latin Hypercube Design (LHD) samples in design space and uncertain field. In each iteration step, the approximate optimization can be created, and it can be solved by NIUMO. Then a Pareto set predicted by the approximations is identified through the NIUMO method, and an uncertainty set is generated by the Pareto set. Then design points set can be obtained, which consists of the Pareto set and uncertainty set. According to a space-filling design criterion, the approximation models are updated using a few design points belonging to design points set of the current iteration until convergence. This algorithm can improve the precision of approximation models by updating the design space and uncertain space. Furthermore, the algorithm can reduce the sample points at a certain extent and improve the optimization efficiency.
(3) Based on the definition of non-probability reliability index, a mathematical model for nonlinear interval multi-objective optimization design considering reliability constraints is formulated as a nested optimization problem, in which the inner layer is a min-max problem associated with evaluation of the reliability index. An approach is proposed to transform the optimization problem into its equivalent form. Based on the order relation of interval number, the objective function robustness can be ensured.
(4) A method is suggested to solve uncertain multidisciplinary multi-objective optimization problem based on the multidisciplinary feasible method (MDF) and interval. It is three-loop optimization procedure. The state variables are solved by the iteration of the multidisciplinary system analysis in the inner loop. The interval of objective functions and constraints are obtained in the middle loop. Based on the order relation of interval number and possibility degree of interval number, the uncertain multidisciplinary optimization can be transformed into the deterministic multidisciplinary optimization. The multi-objective optimization algorithm is used to solve the deterministic multidisciplinary multi-objective optimization problem in the outer loop. Numerical examples are presented to demonstrate the effectiveness and practicability of the present method.
本文的研究工作按以下几个方面展开:首先,提出了一种非线性区间不确定性多目标优化转换模型,实现了不确定多目标优化问题向确定性优化问题的转换。其次,基于该转换模型,将自适应近似模型技术引入非线性区间多目标优化,构造出一种具有一定工程实用性的高效非线性区间不确定优化算法,主要解决两层嵌套优化造成的效率低下问题。再次,基于该转换模型,拓展到不确定多学科优化问题中,构造出一种适用于多学科多目标的不确定优化算法。最后,将算法应用于车身设计领域中的一些实际问题。基于此思路,本文主要研究内容如下:
(1)提出了一种基于非线性区间的不确定多目标转化模型。基于区间序关系和区间可能度,将不确定多目标的目标函数和约束转化为确定性的目标函数和约束。通过转换模型,得到确定性的两层嵌套优化问题。基于多目标遗传算法和序列二次规划算法的两层嵌套优化算法来求解转换后的确定多目标优化问题。对车辆耐撞性和薄板冲压成形两类不确定优化问题进行求解。结果表明该算法具有较好的工程实用性。
(2)提出了一种基于自适应近似模型技术的不确定多目标优化算法。整个优化过程由一系列的近似不确定优化问题迭代完成,通过拉丁方试验设计,在设计空间和不确定空间进行采样,建立目标函数和约束的Kriging近似模型。自适应方法的每一迭代步,通过非线性区间数优化求解转换后的确定优化问题,获得Pareto解集和不确定集,组成设计点集,根据空间填充设计标准,从设计点集选择几个设计点对近似模型更新直到收敛为止。该算法不仅更新了设计空间和不确定空间,提高了近似模型的精度,而且减少了样本的数量,提高了优化效率。
(3)在区间非概率可靠性指标的基础上,建立了具有可靠性指标约束的多目标优化模型。针对该内层优化为极小极大问题的嵌套优化模型,转换为易于处理的等效形式,同时通过区间序关系给出了目标函数稳健性的求解方法。
(4)提出了一种基于多学科可行方法的区间不确定多目标优化算法。该多学科多目标算法是三层循环求解,最内层通过学科分析求得状态变量;中间层搜索不确定量,求得目标和约束函数的区间;基于区间序关系和区间可能度,转换为确定性多学科多目标优化问题,外层多目标优化算法求解该转换后的确定性优化问题。并通过数值算例验证了该方法的有效性。
The rapid development with modern industry and economy has made the optimization technique be greatly used in many kinds of fields.Many engineering problems often involve multi-objective optimization problem. Multiple objectives are always competitive and conflicting, which often result in inexistence of optimal solution. Therefore, multi-objective optimization problem (MOOP) can’t be efficiently solved by using the single objective optimization method.Moreover, the uncertainties also cause many difficulties to directly use the conventional optimization approaches and optimization theories,and bring challenges for MOOP. Stochastic and fuzzy multi-objective optimizations are two types of traditional uncertain multi-objective optimization methodologies. Unfortunately, and it is difficult to construct the precise probability distributions or fuzzy membership functions. The interval number optimization is a relatively newly-developed uncertain optimization method, in which interval is used to model the uncertainty of variables. Thus the variation bounds of the uncertain variables are only required, which can be obtained through only a small amount of uncertainty information. The interval number optimization method mostly focuses on the single objective optimization in dealing with uncertainties, while uncertain multi-objective optimization more often involves in engineering problems. It is still at preliminary stage for the nonlinear interval number optimization research. So far, a nonlinear interval number algorithm with fine efficiency and precision has still not been developed to deal with this class of problems. So it is necessary to study the nonlinear interval uncertain multi-objective optimization (NIUMO).
This dissertation mainly focuses the NIUMO problem. Firstly, a transformation model is proposed to deal with the NIUMO problem, through which the uncertain optimization can be transformed into a deterministic optimization problem. Secondly, an adaptive approximation method is presented to solve the low efficiency problem, which is caused by the nesting optimization. Thirdly, the transformation model is extended to uncertain multidisciplinary design optimization problem. Thus, an algorithm is constructed to solve the interval multidisciplinary multi-objective problem. Finally, the algorithm is applied to the design problem of vehicle body. The main contents are given as follows:
(1) A new uncertain multi-objective optimization method is developed based on a nonlinear interval number programming. Based on the order relation of interval number and possibility degree of interval number, the uncertain multi-objective optimization is transformed into a deterministic non-constraint multi-objective optimization in terms of a penalty function. The multi-objective genetic algorithm and sequential quadratic programming algorithm are used to solve two layers nesting optimization problems, which are based on the deterministic multi-objective model of transformation. Thus, a new hybrid algorithm is proposed to solve the nonlinear interval number optimization problem. The optimization algorithm is successfully applied in complicated engineering problems, including the crashworthiness and sheet metal forming optimization. The application of the engineering problem demonstrates the effectiveness of the present method.
(2) An adaptive approximation method is suggested to deal with an nonlinear interval uncertain multi-objective optimization (NIUMO) problem. The whole optimization process consists of a sequence of approximate optimization problems. The approximation models of uncertain objective functions and constrains are constructed by the Kriging models with the Latin Hypercube Design (LHD) samples in design space and uncertain field. In each iteration step, the approximate optimization can be created, and it can be solved by NIUMO. Then a Pareto set predicted by the approximations is identified through the NIUMO method, and an uncertainty set is generated by the Pareto set. Then design points set can be obtained, which consists of the Pareto set and uncertainty set. According to a space-filling design criterion, the approximation models are updated using a few design points belonging to design points set of the current iteration until convergence. This algorithm can improve the precision of approximation models by updating the design space and uncertain space. Furthermore, the algorithm can reduce the sample points at a certain extent and improve the optimization efficiency.
(3) Based on the definition of non-probability reliability index, a mathematical model for nonlinear interval multi-objective optimization design considering reliability constraints is formulated as a nested optimization problem, in which the inner layer is a min-max problem associated with evaluation of the reliability index. An approach is proposed to transform the optimization problem into its equivalent form. Based on the order relation of interval number, the objective function robustness can be ensured.
(4) A method is suggested to solve uncertain multidisciplinary multi-objective optimization problem based on the multidisciplinary feasible method (MDF) and interval. It is three-loop optimization procedure. The state variables are solved by the iteration of the multidisciplinary system analysis in the inner loop. The interval of objective functions and constraints are obtained in the middle loop. Based on the order relation of interval number and possibility degree of interval number, the uncertain multidisciplinary optimization can be transformed into the deterministic multidisciplinary optimization. The multi-objective optimization algorithm is used to solve the deterministic multidisciplinary multi-objective optimization problem in the outer loop. Numerical examples are presented to demonstrate the effectiveness and practicability of the present method.
引文
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