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 引用本文: 铁军, 隋允康, 彭细荣. 互逆规划的拓宽和深化及其在结构拓扑优化的应用[J]. 力学学报, 2020, 52(6): 1822-1837.
Tie Jun, Sui Yunkang, Peng Xirong. WIDENING AND DEEPENING OF RECIPROCAL PROGRAMMING AND ITS APPLICATION TO STRUCTURAL TOPOLOGY OPTIMIZATION[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(6): 1822-1837.
 Citation: Tie Jun, Sui Yunkang, Peng Xirong. WIDENING AND DEEPENING OF RECIPROCAL PROGRAMMING AND ITS APPLICATION TO STRUCTURAL TOPOLOGY OPTIMIZATION[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(6): 1822-1837.

WIDENING AND DEEPENING OF RECIPROCAL PROGRAMMING AND ITS APPLICATION TO STRUCTURAL TOPOLOGY OPTIMIZATION

• 摘要: 本文的工作涉及数学与力学两方面,数学方面：(1) 将数学规划论中新提出的互逆规划,从 s-m 型 (或称为 m-s 型) 发展出 s-s 型和 m-m 型互逆规划 (其中 s 意为单目标,m 意为多目标),从而使互逆规划的定义完备成为 3 种;(2) 从 KKT 条件审视互逆规划的两方面,得到了互逆规划双方求解涉及拟同构和拟同解的 3 个定理,并且予以证明,提供了在求解中对于互逆规划双方在求解中相互借鉴的理论基础;(3) 对一对互逆规划双方皆合理的情况和某一方不合理的情况,皆给出了求解策略和具体解法. 力学方面：(1) 给出结构优化设计模型合理与否的诠释;(2) 在互逆规划对结构拓扑优化的应用中,提出了不合理结构拓扑优化模型的求解策略;(3) 给出了借助 MVCC 模型 (多个柔顺度约束下的体积最小化) 解决 MCVC 模型 (对于给定体积下的多个柔顺度的最小化) 的途径,其中的建模基于 ICM (独立连续映射) 方法. 用 Matlab 编程给出了数值算例. 其中两个数学问题图示了互逆规划的双方关系;其中,结构拓扑优化问题是众多结构拓扑优化中的两个个案;数值结果均支持了本文提出的互逆规划理论.

Abstract: The work of this paper involves two aspects of mathematics and mechanics. In terms of mathematics: (1) The reciprocal programming newly proposed in the mathematical programming theory is developed from the s-m type (or called m-s type) to the s-s type and m-m type reciprocal programming (among which s means single goal, m means multiple goals), so that the definition of reciprocal programming is complete into three types; (2) From the KKT condition to examine the two aspects of reciprocal programming, it is obtained that the three theorems of two sides of reciprocal programming which involves quasi-isomorphism theorem and quasi-simultaneous solution theorems. Moreover, the proofs of the three theorems provide a theoretical basis for the two parties to reference and compare from each other in the solution of reciprocal programming; (3) Respectively, the solution strategies and detailed solutions are given for the case where both sides of a pair of reciprocal programming are reasonable ,and the case where one aspect is unreasonable while the other is reasonable. In terms of mechanics: (1) This paper gives the explanation of whether the structural optimization design model is reasonable or not; (2) In the application of reciprocal programming to structural topology optimization, a solution strategy for unreasonable structural topology optimization models is proposed; (3) A way to solve the MCVC model (the minimization of multiple compliances under a given volume) with the help of the MVCC model (Minimization of the volume under multiple compliance constraints), where the modeling is based on the ICM (Independent Continuous Mapping) method. At the end of this article, four numerical examples are given by Matlab codes, where two of the mathematical programming problems illustrate the relationship between the two sides of reciprocal programming, and two of structural topology optimization problems is two cases in many structural topology optimization problems. The numerical results support the reciprocal programming theory proposed in this paper.

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