多学科设计优化的分解、协同及不确定性研究
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摘要
多学科设计优化方法是从现代复杂系统设计的实际应用背景出发而提出的。在解决模型复杂、设计约束与设计变量多、设计过程中信息量大的优化设计问题时优化效率较高。本文对多学科设计优化理论与方法进行了深入而系统的研究,归纳而言,工作主要集中在以下几个方面。
本文首先对灵敏度分析技术进行了研究。在多学科设计优化过程中,灵敏度分析技术起着重要的作用,它不仅在优化时能提供导数信息,而且其中的一种特殊灵敏度分析技术——全局灵敏度方程可用来进行多学科系统的分解。本文系统地研究与比较了适合多学科设计优化的各种灵敏度分析方法。在全局灵敏度方程基础上,导出二阶全局灵敏度方程,并利用二阶全局灵敏度方程对多学科系统进行了基于学科的分解。利用二阶全局灵敏度方程进行分解使系统分解后与原多学科系统维持较好的一致性。
在多学科设计优化中,由于一些工程仿真模型较复杂且计算费时,因此常用近似模型来代替。本文从两方面对近似模型进行研究。首先,对多学科设计优化中常使用的五种近似技术——响应面模型、Kriging模型、BP神经网络模型、多元自适应回归样条、径向基函数进行了研究与比较,对其适用情况进行了分析。其次,在Kriging模型的基础上,提出了一种增强型Kriging近似模型,其理论基础是贝叶斯推论。增强型Kriging模型不需要对模型参数进行重新计算,只需用新的样本数据对Kriging模型进行更新,从而提高模型的近似精度。由于模型建立过程中,样本信息来源不同,因此可灵活地处理模型的精度,并且模型可重复使用。
本文在基于并行子空间优化方法的基础上,提出了一种基于二阶全局灵敏度方程的系统分解与基于增强型Kriging近似技术进行协同的多学科设计优化方法。方法首先利用二阶全局灵敏度方程对多学科系统进行分解,整个多学科系统的耦合由二阶全局灵敏度方程来维持。各个多学科子系统进行独立优化,得到各个学科子系统的优化解。利用增强型Kriging近似技术对分解后的多学科子系统优化解进行协同,得到一个全局优化解。如此循环,最终收敛到原系统的最优解。相比全局灵敏度方程,二阶全局灵敏度方程能更为精确的对多学科系统进行分解,更好地维持系统的一致性,从而减少了整个多学科系统进行分解与协同的优化迭代次数。由于增强型Kriging近似技术使用各个子系统优化后得到的新样本点对模型进行更新,提高了近似模型的精度,因此使用增强型Kriging技术对分解后的系统协调能力较强,多学科设计优化迭代中全局优化解能更快地收敛到最优解。
在多学科设计优化过程中,设计变量的不确定性将影响优化结果的不确定性,因此,本文最后对多学科设计优化过程中的不确定性进行了研究。文章探讨了多学科设计优化中不确定性的各种来源,并将不确定性的来源分为输入量的不确定性和模型的不确定性。本文利用概率方法定量地处理多学科设计优化中的不确定性,研究了多学科设计优化中的不确定传播。由于不确定性计算的复杂性,尽管不确定性已被认识,但在多学科设计优化中较少被考虑。本文采用Kriging方法建立不确定性计算的代理模型,并进行了多学科不确定性设计优化。
In today’s engineering design, the number of design variables involved and the amount of data handled becomes so large that attempting to optimize the design by traditionally sequential approach is both intractable and costly. Multidisciplinary design optimization (MDO) emerges as an enabling methodology for the design of complex systems. The increasing economic competition of industrial markets and growing complexity of engineering problems has sparked increasing interest to MDO in recently years.
Comprehensively and systemically the present thesis studies the theories and methods of MDO. Broadly speaking, the investigation focuses on the following four aspects.
The sensitivity analysis (SA) is an important element of MDO research field. It is used not only for derivatives computation in single disciple but also for system decomposition. The first research area of thesis focuses on the sensitivity analysis. Firstly, The SA methods that fit MDO are studied systemically. Several individual disciplinary SA methods are introduced, also the details of each method. And comparisons of these methods are presented. On this basis of global sensitivity equation, the second order global sensitivity equation is derived. Based on the second order global sensitivity equation, the multidisciplinary system could be decomposed into several independent subsystems.
Surrogate models are an effective approach for alleviating some of the problems associated with the direct use of modern computerized analysis techniques in MDO environment. Surrogate models shift the computational burden from the optimization problem to the problem of building the surrogate models. Additionally, surrogate models filter out the noise inherent to most numerical analysis procedures by providing smooth approximate functions. The second research area of thesis is two-fold. One, a large-scale comparison of five popular means for building approximation models is presented. Second, Based on Kriging surrogate of computer simulations and experiments, a new means enables the creation of surrogate models that are advanced Kriging surrogate fitted with data obtained from deterministic numerical models and/or experimental data using optimal sampling. The statistical theory of Bayesian inference is used to support the building of surrogates. Information from computer simulations of different levels of accuracy and detail is integrated, updating surrogates sequentially to improve their accuracy. Surrogates are updated in sequential stages. They are flexible in their accuracy level and can be reusable because they can be updated as information is gathered.
The primary goal of MDO is to decompose a large multidisciplinary system into a related grouping of smaller, more tractable, coupled subsystems. The third research area involves on the decomposition of the MDO. Based on the second order global sensitivity equations and advanced Kriging approximation, a new MDO method is proposed. The method optimizes decomposed subspaces concurrently, followed by a coordination procedure of global approximation. The global approximation used in the coordination procedure is formed using the advanced Kriging approximation strategy. Optimization of a global approximation problem is used as a coordination procedure for directing system convergence and resolving subspace conflicts. The global approximation problem is formed using information generated during concurrent subspace optimization. At each subspace, non-local analyses are approximated at the subspaces using second order global sensitivity equations. Local analyses are performed using analyses packages available to the subspace design team. System coupling is maintained and updated using second order global sensitivity equations approach.
Finally, the fourth area of research involves the uncertainty in multidisciplinary design optimization. This study examines sources of uncertainty and classifies these errors into two categories, that is, the bias error associated with the disciplinary design tool and the precision error of the input data. For the formulation presented in this thesis, we employ probability methods to uncertainty. In this research an investigation of how uncertainty propagates through a multidisciplinary system analysis subject to the bias errors and the precision errors is undertaken. Though the usefulness of multidisciplinary design optimization formulation to evaluate uncertainty encountered in the design process is widely acknowledged, its implementation is rare. One of the reasons is due to the complexity and computational burden when evaluating the uncertainty in MDO. The approach proposed to this problem is to employ Kriging approximation to create a uncertain surrogate, and then perform the evaluation of uncertainty in MDO.
本文首先对灵敏度分析技术进行了研究。在多学科设计优化过程中,灵敏度分析技术起着重要的作用,它不仅在优化时能提供导数信息,而且其中的一种特殊灵敏度分析技术——全局灵敏度方程可用来进行多学科系统的分解。本文系统地研究与比较了适合多学科设计优化的各种灵敏度分析方法。在全局灵敏度方程基础上,导出二阶全局灵敏度方程,并利用二阶全局灵敏度方程对多学科系统进行了基于学科的分解。利用二阶全局灵敏度方程进行分解使系统分解后与原多学科系统维持较好的一致性。
在多学科设计优化中,由于一些工程仿真模型较复杂且计算费时,因此常用近似模型来代替。本文从两方面对近似模型进行研究。首先,对多学科设计优化中常使用的五种近似技术——响应面模型、Kriging模型、BP神经网络模型、多元自适应回归样条、径向基函数进行了研究与比较,对其适用情况进行了分析。其次,在Kriging模型的基础上,提出了一种增强型Kriging近似模型,其理论基础是贝叶斯推论。增强型Kriging模型不需要对模型参数进行重新计算,只需用新的样本数据对Kriging模型进行更新,从而提高模型的近似精度。由于模型建立过程中,样本信息来源不同,因此可灵活地处理模型的精度,并且模型可重复使用。
本文在基于并行子空间优化方法的基础上,提出了一种基于二阶全局灵敏度方程的系统分解与基于增强型Kriging近似技术进行协同的多学科设计优化方法。方法首先利用二阶全局灵敏度方程对多学科系统进行分解,整个多学科系统的耦合由二阶全局灵敏度方程来维持。各个多学科子系统进行独立优化,得到各个学科子系统的优化解。利用增强型Kriging近似技术对分解后的多学科子系统优化解进行协同,得到一个全局优化解。如此循环,最终收敛到原系统的最优解。相比全局灵敏度方程,二阶全局灵敏度方程能更为精确的对多学科系统进行分解,更好地维持系统的一致性,从而减少了整个多学科系统进行分解与协同的优化迭代次数。由于增强型Kriging近似技术使用各个子系统优化后得到的新样本点对模型进行更新,提高了近似模型的精度,因此使用增强型Kriging技术对分解后的系统协调能力较强,多学科设计优化迭代中全局优化解能更快地收敛到最优解。
在多学科设计优化过程中,设计变量的不确定性将影响优化结果的不确定性,因此,本文最后对多学科设计优化过程中的不确定性进行了研究。文章探讨了多学科设计优化中不确定性的各种来源,并将不确定性的来源分为输入量的不确定性和模型的不确定性。本文利用概率方法定量地处理多学科设计优化中的不确定性,研究了多学科设计优化中的不确定传播。由于不确定性计算的复杂性,尽管不确定性已被认识,但在多学科设计优化中较少被考虑。本文采用Kriging方法建立不确定性计算的代理模型,并进行了多学科不确定性设计优化。
In today’s engineering design, the number of design variables involved and the amount of data handled becomes so large that attempting to optimize the design by traditionally sequential approach is both intractable and costly. Multidisciplinary design optimization (MDO) emerges as an enabling methodology for the design of complex systems. The increasing economic competition of industrial markets and growing complexity of engineering problems has sparked increasing interest to MDO in recently years.
Comprehensively and systemically the present thesis studies the theories and methods of MDO. Broadly speaking, the investigation focuses on the following four aspects.
The sensitivity analysis (SA) is an important element of MDO research field. It is used not only for derivatives computation in single disciple but also for system decomposition. The first research area of thesis focuses on the sensitivity analysis. Firstly, The SA methods that fit MDO are studied systemically. Several individual disciplinary SA methods are introduced, also the details of each method. And comparisons of these methods are presented. On this basis of global sensitivity equation, the second order global sensitivity equation is derived. Based on the second order global sensitivity equation, the multidisciplinary system could be decomposed into several independent subsystems.
Surrogate models are an effective approach for alleviating some of the problems associated with the direct use of modern computerized analysis techniques in MDO environment. Surrogate models shift the computational burden from the optimization problem to the problem of building the surrogate models. Additionally, surrogate models filter out the noise inherent to most numerical analysis procedures by providing smooth approximate functions. The second research area of thesis is two-fold. One, a large-scale comparison of five popular means for building approximation models is presented. Second, Based on Kriging surrogate of computer simulations and experiments, a new means enables the creation of surrogate models that are advanced Kriging surrogate fitted with data obtained from deterministic numerical models and/or experimental data using optimal sampling. The statistical theory of Bayesian inference is used to support the building of surrogates. Information from computer simulations of different levels of accuracy and detail is integrated, updating surrogates sequentially to improve their accuracy. Surrogates are updated in sequential stages. They are flexible in their accuracy level and can be reusable because they can be updated as information is gathered.
The primary goal of MDO is to decompose a large multidisciplinary system into a related grouping of smaller, more tractable, coupled subsystems. The third research area involves on the decomposition of the MDO. Based on the second order global sensitivity equations and advanced Kriging approximation, a new MDO method is proposed. The method optimizes decomposed subspaces concurrently, followed by a coordination procedure of global approximation. The global approximation used in the coordination procedure is formed using the advanced Kriging approximation strategy. Optimization of a global approximation problem is used as a coordination procedure for directing system convergence and resolving subspace conflicts. The global approximation problem is formed using information generated during concurrent subspace optimization. At each subspace, non-local analyses are approximated at the subspaces using second order global sensitivity equations. Local analyses are performed using analyses packages available to the subspace design team. System coupling is maintained and updated using second order global sensitivity equations approach.
Finally, the fourth area of research involves the uncertainty in multidisciplinary design optimization. This study examines sources of uncertainty and classifies these errors into two categories, that is, the bias error associated with the disciplinary design tool and the precision error of the input data. For the formulation presented in this thesis, we employ probability methods to uncertainty. In this research an investigation of how uncertainty propagates through a multidisciplinary system analysis subject to the bias errors and the precision errors is undertaken. Though the usefulness of multidisciplinary design optimization formulation to evaluate uncertainty encountered in the design process is widely acknowledged, its implementation is rare. One of the reasons is due to the complexity and computational burden when evaluating the uncertainty in MDO. The approach proposed to this problem is to employ Kriging approximation to create a uncertain surrogate, and then perform the evaluation of uncertainty in MDO.
引文
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