互逆规划理论及其用于建立结构拓扑优化的合理模型

隋允康; 彭细荣; 叶红玲; 铁军

doi:10.6052/0459-1879-19-259

互逆规划理论及其用于建立结构拓扑优化的合理模型

doi: 10.6052/0459-1879-19-259

隋允康^*,,
彭细荣^†3),,
叶红玲^*,
铁军^**

^* 北京工业大学机电学院工程力学系,北京 100022
^† 湖南城市学院土木工程学院,湖南益阳 413000
^** 天津财经大学理工学院,天津 300222

基金项目: 1) 国家自然科学基金资助项目(11672103)

详细信息

作者简介:
2) 隋允康,教授,主要研究方向：结构优化. E-mail:ysui@bjut.edu.cn

通讯作者:
彭细荣

中图分类号: O343.1
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- 被引次数: 0
出版历程
- 收稿日期: 2019-09-12
- 刊出日期: 2019-11-18

RECIPROCAL PROGRAMMING THEORY AND ITS APPLICATION TO ESTABLISH A REASONABLE MODEL OF STRUCTURAL TOPOLOGY OPTIMIZATION

Sui Yunkang^*
,,
Peng Xirong^{†3)
,},
Ye Hongling^*,
Tie Jun^**

^* Numerical Simulation Center for Engineering, Beijing University of Technology, Beijing 100022, China
^† School of Civil Engineering, Hunan City University, Yiyang 413000, Hunan,China
^** Tianjin University of Finance and Economics, Institute of Technology, Tianjin 300222, China

摘要

摘要: 在数学规划的领域里定义了互逆规划——各自目标函数与约束条件位置相互交换的一对规划. 接着指出,尽管互逆规划与对偶规划在表面上似乎类似,但是二者存在 5 点不同：(1) 是否为同一个问题的不同;(2) 存在``对偶间隙''与否的不同;(3) 设计变量数目的不同;(4) 是否单目标与多目标问题的不同;(5) 问题合理与否的不同. 然后,基于互逆规划的定义,用以审视结构拓扑优化模型,给出如下结果：(1) 从这个角度洞悉,在结构优化中,确实有不合理的模型一直被沿用着;(2) 找到了修正不合理模型使之合理化的方法;(3) 对于给定体积下的柔顺度最小化 (MCVC) 模型,指出了其不合理的原因;(4) MCVC 模型实际是互逆规划的 m 方,由此建立起其对应的 s 方, 即给出了多个柔顺度约束的体积最小化 (MVCC) 模型;(5)给出了MVCC模型中的结构柔顺度约束的物理解释和算法,论证了 ICM (independent continuous and mapping) 方法以往关于全局化应力约束的概念和方法;(6)数值算例表明了 MCVC 与 MVCC 模型作为互逆规划的差异,且印证了 MVCC 模型的合理性.MCVC 模型在不同体积约束及多工况下不同的权系数时,得到最优拓扑不同;但 MVCC 模型在多工况柔顺度约束下可得到唯一的最优拓扑.
- 结构拓扑优化设计 /
- 互逆规划 /
- MCVC 模型 /
- MVCC 模型 /
- ICM 方法
Abstract: Based on mathematical programming theory, the reciprocal programming is defined as a pair of programming which objective and constraint functions are changed each other. After that, it is pointed out that reciprocal programming and dual programming seem similar but there are five differences between them: (1) the difference in whether they are the same problem or not; (2) the difference in whether there exists a dual gap or not; (3) the difference in the number of design variables; (4) the difference between single-objective and multi-objective problems; (5) the difference between reasonable and unreasonable problems. Finally, based on the definition of the reciprocal programming, the structural topology optimization model is examined; and following results are obtained: (1) from this perspective, it is clear that there exists indeed unreasonable models that have been used in structural optimization; (2) a way to correct the unreasonable model and make it reasonable is put forward; (3) the reasons that the minimizing compliance model with volume constraint (MCVC for short) is unreasonable are presented; (4) according to the theory presented in this paper, the MCVC model is actually the m-aspect of reciprocal programming, so its corresponding s-aspect is established, that is, the minimizing volume model with multiple compliance constraints (MVCC for short); (5) the physical interpretation and algorithm of structural compliance constraints in the MVCC model are presented; and the concepts and methods of global stress constraint in the ICM (Independent continuous and mapping ) method are demonstrated; (6) numerical examples show the differences between the MCVC and MVCC model as a pair of reciprocal programming and verify the rationality of the MVCC model. Different optimized topologies are obtained for the MCVC model with different volume constraints and different weighting coefficients for multiple load cases. But a unique optimized topology can be achieved by the MVCC model with compliance constraints under multiple load case.
- structural topology optimization /
- reciprocal programming /
- MCVC model /
- MVCC model /
- ICM method

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参考文献(1)

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