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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (2): 131-137.doi: 10.3969/j.issn.1005-3085.2016.02.003

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自适应网格方法在Stokes问题形状最优控制中的应用

段献葆,   李飞飞,   秦新强   

  1. 西安理工大学理学院，西安 710048
  • 收稿日期:2014-12-16 接受日期:2015-05-28 出版日期:2016-04-15 发布日期:2016-06-15
  • 基金资助:
    国家自然科学基金 (61273127; 11371288)；陕西省教育厅科学研究计划项目 (11JK0494).

Optimal Shape Design of Stokes Problem by Adaptive Mesh Method

DUAN Xian-bao,   LI Fei-fei,   QIN Xin-qiang   

  1. School of Sciences, Xi'an University of Technology, Xi'an 710048
  • Received:2014-12-16 Accepted:2015-05-28 Online:2016-04-15 Published:2016-06-15
  • Supported by:
    The National Natural Science Foundation of China (61273127; 11371288); the Research Foundation of Department of Education of Shaanxi Province (11JK0494).

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摘要： 为了求解流体力学中的形状最优控制问题，本文提出了一种与最优化准则方法相耦合的自适应网格方法．优化的目标是使得流体流动的能量耗散达到最小，状态方程是Stokes问题．本算法可以在减少计算量的情况下，保证流体的界面达到较高的分辨率．最优化算法采用的是非常稳定的经典最优化准则方法，自适应网格的指示函数是通过材料分布的信息得到的．虽然本文只是考虑了Stokes问题，但所得算法可以用来解决很广泛的一类流体动力学中的形状或拓扑最优化问题．

关键词: 自适应网格方法, 最优形状控制, Stokes问题；最优化准则方法

Abstract:

In order to solve optimal shape design problem arising from fluid dynamics, an optimality criteria (OC) coupled adaptive mesh refinement algorithm has been presented in this paper. The objective is to minimize the dissipated power in the fluid, subject to the Stokes problem as the state equations. By this method, higher resolution of the interface can be obtained with a minimum of additional expense. A material distribution information based indicator is adopted during the automatic local adaptive mesh refinement process. Although only considered the Stokes problem in this study, the proposed method can be applied to a wide range of optimal shape or topology design problems in fluid dynamics.

Key words: adaptive mesh method, optimal shape control, Stokes problem, optimality criteria method

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