陈述式仿真模型相容性分析与约简方法研究
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摘要
数学建模与仿真已经成为分析产品技术性能的一项重要技术。随着科学技术的高速发展,产品结构和功能日趋复杂与异构。复杂产品通常是机械、电子、液压、控制等多领域子系统的综合体。为优化复杂产品的设计,必然需要将不同领域的子系统模型集成到一起以实现复杂产品整体性能的协同仿真。Modelica语言是一种基于方程的物理系统建模语言,它继承和统一了先前多种面向对象建模语言的诸多优点,非常适合于现代多领域复杂系统建模。面向方程的分析优化器是基于Modelica语言的建模仿真平台的一项核心技术,本文对其进行了深入研究。
模型相容是模型可以求解的前提。本文分析了造成仿真模型奇异的原因,提出了基于图论算法的相容性分析方法,显著地提高了奇异模型的识别和修正效率。该方法通过在二部图上计算最大匹配实现了模型的相容性判定。引入虚构方程替代组件的外部连接方程,通过识别奇异组件细化判别范围。在此基础上,使用结构过滤规则和语义过滤规则进一步缩小判别范围,最终给出有效的修正信息。
陈述式建模形成的DAE系统规模很大,求解时间长。为支持复杂多领域系统的高效率仿真,本文研究了DAE系统的符号约简技术。结合Modelica模型的特点,通过定义规范转换规则,基于方程的二叉树表示实现了方程表达式的规范转换。运用图论算法消除等值方程,分解方程系统,简化代数环,有效地降低了模型的规模和耦合度,显著地加快了模型的求解速度。
数值求解高指标DAE系统是十分困难的。本文讨论了DAE系统的指标分析策略,阐明了指标约简的基本原理。通过定义最小结构奇异子集和加权二部图,提出了基于加权二部图的指标约简方法,该方法可以找出所有需要微分的方程。基于方程的二叉树表示实现了方程的符号微分。结合符号约简技术,给出了大规模DAE系统的指标分析方法,提高了分析效率。在此基础上讨论了DAE模型的相容初始化问题,给出了初始条件的相容性判定标准和初始化方程系统求解策略。
基于上述研究成果,设计实现了面向Modelica模型的分析优化器MWOptimizer,为开发基于Modelica语言的建模仿真环境奠定了技术基础,并成功地将其集成到了多领域物理系统混合建模与仿真平台MWorks中。
Mathematical modeling and simulation has become an important technology to analyze product performance. Along with advance in science and technology, modern products are increasingly complex and heterogeneous. Complex products generally are composed of subsystems from multiple engineering domains such as mechanic, electric, hydraulic, and control system components. To optimize complex products, it is necessary to integrate simulation models from several engineering domains to cooperatively simulate the total behavior of complex products. Modelica is an equation based language for physical systems modeling. The language unifies and generalizes previous modeling languages, and is appropriate to model modern complex multi-domain systems very well. The optimizer for equations is a kernel of a Modelica based modeling and simulation tool, its key issues are studied in this dissertation.
The consistency of a model is the precondition for that the model can be solved. The causes of inconsistencies of simulation models are analyzed, and a consistency analysis method based on graph theoretical algorithms is proposed, which can enhance the error detecting and correcting process. The method determines whether a model is consistent or not by computing a maximum matching in the bipartite graph associated with the model. In order to find out singular components, fictitious equations are used to replace the equations generated from the connections of the component. The equations and variables, which are responsible for the inconsistencies, are reduced by singular components, and then are further reduced by the structural filtering rule and the semantic filtering rule. Finally, efficient correcting messages for the user are elaborated.
The DAE system resulted from declarative model is of very large dimension, and its numerical solution requires excessively long computation times. Techniques for the symbolic manipulation of general DAE systems are presented, and used for model reduction purpose, to support efficient simulation of complex multi-domain systems. Canonical transformation rules are used to simplify and transform equations, which are represented as binary trees, into canonical forms. Some graph theoretical algorithms are employed to address the specific problems, which are elimination of trivial equations by means of substitution, and partitioning the whole DAE system into subsystems which can be solved in sequence, and reducing algebraic loops by tearing algorithm. These symbolic manipulations can considerably reduce the dimension and coupling of a DAE system to make the simulation of the DAE system be executed much more efficiently.
It is numerically difficult to solve a high-index DAE system. The strategy for index analysis of DAE system is discussed, and the basic principle of index reduction is elaborated. The minimally structurally singular subset and the weighted bipartite graph are introduced, and then an index reduction algorithm based on weighted bipartite graph is developed, which can locate those subsets of the system equations which need to be differentiated. By representing equation as binary tree, a symbolic method for calculating the derivative of equation is presented. On the basis of the techniques proposed for model reduction, an index analysis method for large scale DAE system is proposed, which can improve index reduction efficiency. Finally, the consistent initialization of DAE model is discussed, the criterion for determining whether an initial condition is consistent or not is given, and the strategy for solving the initialization equation system is presented.
On the basis of the above achievement, an optimizer for Modelica models has been developed, which provides technology foundation for the development of Modelica based modeling and simulation environment, and has been integrated into a hybrid modeling and simulation tool, named MWorks, for multi-domain physical system.
模型相容是模型可以求解的前提。本文分析了造成仿真模型奇异的原因,提出了基于图论算法的相容性分析方法,显著地提高了奇异模型的识别和修正效率。该方法通过在二部图上计算最大匹配实现了模型的相容性判定。引入虚构方程替代组件的外部连接方程,通过识别奇异组件细化判别范围。在此基础上,使用结构过滤规则和语义过滤规则进一步缩小判别范围,最终给出有效的修正信息。
陈述式建模形成的DAE系统规模很大,求解时间长。为支持复杂多领域系统的高效率仿真,本文研究了DAE系统的符号约简技术。结合Modelica模型的特点,通过定义规范转换规则,基于方程的二叉树表示实现了方程表达式的规范转换。运用图论算法消除等值方程,分解方程系统,简化代数环,有效地降低了模型的规模和耦合度,显著地加快了模型的求解速度。
数值求解高指标DAE系统是十分困难的。本文讨论了DAE系统的指标分析策略,阐明了指标约简的基本原理。通过定义最小结构奇异子集和加权二部图,提出了基于加权二部图的指标约简方法,该方法可以找出所有需要微分的方程。基于方程的二叉树表示实现了方程的符号微分。结合符号约简技术,给出了大规模DAE系统的指标分析方法,提高了分析效率。在此基础上讨论了DAE模型的相容初始化问题,给出了初始条件的相容性判定标准和初始化方程系统求解策略。
基于上述研究成果,设计实现了面向Modelica模型的分析优化器MWOptimizer,为开发基于Modelica语言的建模仿真环境奠定了技术基础,并成功地将其集成到了多领域物理系统混合建模与仿真平台MWorks中。
Mathematical modeling and simulation has become an important technology to analyze product performance. Along with advance in science and technology, modern products are increasingly complex and heterogeneous. Complex products generally are composed of subsystems from multiple engineering domains such as mechanic, electric, hydraulic, and control system components. To optimize complex products, it is necessary to integrate simulation models from several engineering domains to cooperatively simulate the total behavior of complex products. Modelica is an equation based language for physical systems modeling. The language unifies and generalizes previous modeling languages, and is appropriate to model modern complex multi-domain systems very well. The optimizer for equations is a kernel of a Modelica based modeling and simulation tool, its key issues are studied in this dissertation.
The consistency of a model is the precondition for that the model can be solved. The causes of inconsistencies of simulation models are analyzed, and a consistency analysis method based on graph theoretical algorithms is proposed, which can enhance the error detecting and correcting process. The method determines whether a model is consistent or not by computing a maximum matching in the bipartite graph associated with the model. In order to find out singular components, fictitious equations are used to replace the equations generated from the connections of the component. The equations and variables, which are responsible for the inconsistencies, are reduced by singular components, and then are further reduced by the structural filtering rule and the semantic filtering rule. Finally, efficient correcting messages for the user are elaborated.
The DAE system resulted from declarative model is of very large dimension, and its numerical solution requires excessively long computation times. Techniques for the symbolic manipulation of general DAE systems are presented, and used for model reduction purpose, to support efficient simulation of complex multi-domain systems. Canonical transformation rules are used to simplify and transform equations, which are represented as binary trees, into canonical forms. Some graph theoretical algorithms are employed to address the specific problems, which are elimination of trivial equations by means of substitution, and partitioning the whole DAE system into subsystems which can be solved in sequence, and reducing algebraic loops by tearing algorithm. These symbolic manipulations can considerably reduce the dimension and coupling of a DAE system to make the simulation of the DAE system be executed much more efficiently.
It is numerically difficult to solve a high-index DAE system. The strategy for index analysis of DAE system is discussed, and the basic principle of index reduction is elaborated. The minimally structurally singular subset and the weighted bipartite graph are introduced, and then an index reduction algorithm based on weighted bipartite graph is developed, which can locate those subsets of the system equations which need to be differentiated. By representing equation as binary tree, a symbolic method for calculating the derivative of equation is presented. On the basis of the techniques proposed for model reduction, an index analysis method for large scale DAE system is proposed, which can improve index reduction efficiency. Finally, the consistent initialization of DAE model is discussed, the criterion for determining whether an initial condition is consistent or not is given, and the strategy for solving the initialization equation system is presented.
On the basis of the above achievement, an optimizer for Modelica models has been developed, which provides technology foundation for the development of Modelica based modeling and simulation environment, and has been integrated into a hybrid modeling and simulation tool, named MWorks, for multi-domain physical system.
引文
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