偏微分方程与微分代数方程的一致求解方法

中国机械工程 ›› 2015, Vol. 26 ›› Issue (4): 441-445.

偏微分方程与微分代数方程的一致求解方法

李志华;喻军;杨红光

杭州电子科技大学，杭州，310018

出版日期:2015-02-25 发布日期:2015-02-25
基金资助:
国家自然科学基金资助项目（51275141）；浙江省自然科学基金资助项目（Y1100901）

Consistent Solving Method of PDE and DAE

Li Zhihua;Yu Jun;Yang Hongguang

Hangzhou Dianzi University,Hangzhou,310018

Online:2015-02-25 Published:2015-02-25
Supported by:
National Natural Science Foundation of China（No. 51275141）;Zhejiang Provincial Natural Science Foundation of China（No. Y1100901）

摘要/Abstract

摘要：

Modelica 语言是一种复杂物理系统多领域统一建模语言，但目前该语言只能解决由微分代数方程（DAE）描述的问题，而不能解决由偏微分方程（PDE）表达的问题。为此，提出一种偏微分方程与微分代数方程的一致求解方法，利用所构建的径向基函数配点无网格法直接将偏微分方程在空间上离散成一系列的微分代数方程，然后采用成熟的微分代数方程求解器进行求解。实例结果表明，该方法在不改变 Modelica 语法的前提下，能较好地实现偏微分方程与微分代数方程的一致求解，且求解精度高、边界条件处理简单，有利于Modelica直接求解复杂工程系统中多领域耦合、时间域与空间域耦合的复杂问题。

关键词: 多领域统一建模, Modelica, 偏微分方程（PDE）, 微分代数方程（DAE）

Abstract:

Modelica is a multi-domain unified modeling language for modeling and simulation of large and complex physical systems. However, it dealt only with DAE but not with PDE. A consistent solving method of PDE and DAE was proposed. The PDE was transferred into a series of DAEs with the meshless method of radial basis function collocation, and was solved by the mature DAE solver in MWorks platform based on Modelica. Results show that this consistent solving method realizes the consistent solution of PDE and DAE under the premise of not changing Modelica grammar, and has high accuracy and the convenience of dealing with boundary conditions, which is conducive to solve complex engineering systems with multi-domain coupling and time domain and space domain coupling.

Key words: multi-domain unified modeling, Modelica, partial differential equation (PDE), differential-algebraic equation (DAE)

中图分类号:

TH122
TP391

李志华, 喻军, 杨红光. 偏微分方程与微分代数方程的一致求解方法[J]. 中国机械工程, 2015, 26(4): 441-445.

Li Zhihua, Yu Jun, Yang Hongguang. Consistent Solving Method of PDE and DAE[J]. China Mechanical Engineering, 2015, 26(4): 441-445.

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