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Sequential conservative integer programming method for multi-constrained discrete variable structure topology optimization

求解多约束离散变量结构拓扑优化的序列保守近似整数规划算法

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Abstract

The sequential approximate integer programming (SAIP) method successfully solves multiple types of large-scale topology optimization problems by solving a sequence of separable approximate integer programming subproblems. However, these subproblems must be with quadratic/linear objective function with linear constraints that can be analytically solved by the canonical relaxation algorithm (CRA). Besides, SAIP relies on the decreasing volume fraction strategy so that it owns difficulties to confront topology optimization problems without active volume constraints. The success of MMA (method of moving asymptotic) inspires us to introduce the classical sequential conservative approximate programming that can generate a sequence of steadily improving feasible designs and present a sequential conservative integer programming (SCIP) method for the development of SAIP. This new method generates a sequence of nonlinear approximate integer programming subproblems containing the reciprocal variables, whose conservation is controlled by moving asymptotes. Since CRA is invalid for nonlinear subproblems, a simple design variable update rule derived by the KKT (Karush-Kuhn-Tucker) conditions is given. Based on the above idea, SCIP obtains stable optimization processes without relying on linear constraints, such as the volume constraint, and thus complex discrete variable topology optimization problems can be efficiently solved. Various requirements of topology optimization problems are handled, including equality or inequality constraints, and inactive or active volume constraints. Numerical results show that since the conservative property has been inherited, the convergence of the optimization process can be regulated without additional structural analysis.

摘要

序列近似整数规划方法通过求解一系列可分离的近似整数规划子问题成功求解多类大规模拓扑优化问题. 然而, 这些近似子问 题的解析求解依靠正则松弛算法, 正则松弛算法要求子问题必须由二次或线性的目标函数以及线性的约束函数构成. 同时, 序列近似 整数规划方法依赖于体分比下降策略, 这使得其难以处理不包含体分比约束的拓扑优化问题. 为进一步地完善序列近似整数规划算法,我们受移动渐近线方法的启发, 引入经典的序列保守近似规划, 以获得一系列可行性稳步提升的设计, 这个新方法称作序列保守近似整数规划算法. 新方法会产生一系列包含倒数设计变量的非线性的近似整数规划子问题, 并通过调整渐近线控制子问题的保守性. 由于正则松弛算法不能求解非线性子问题, 一种基于KKT (Karush-Kuhn-Tucker)条件的简单的设计变量更新准则被给出. 基于上述思想, 序列保守近似整数规划算法不依靠体分比约束等线性约束就可以获得稳定的优化过程, 由此可以用来求解复杂的离散变量拓扑优化 问题. 多类不同需求的拓扑优化问题被解决, 包括考虑等式约束及非等式约束的、以及考虑有效的及无效的体分比约束的问题. 数值 结果证明, 由于保守性的引入, 整个优化收敛进程可以自适应地调节无需额外的结构分析.

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Acknowledgements

This work was supported by the National Key Research and Development Plan (Grant No. 2020YFB1709401), the National Natural Science Foundation of China (Grant Nos. 12202092, 12032008, and 11821202), and the China Postdoctoral Science Foundation (Grant No. ZX20220734).

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Authors and Affiliations

  1. Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116023, China

    Kai Sun  (孙凯), Gengdong Cheng  (程耿东), Kaiqing Zhang  (张凯庆) & Yuan Liang  (梁缘)

  2. State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Dalian, 116023, China

    Kai Sun  (孙凯), Gengdong Cheng  (程耿东), Kaiqing Zhang  (张凯庆) & Yuan Liang  (梁缘)

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  1. Kai Sun  (孙凯)
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  2. Gengdong Cheng  (程耿东)
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  3. Kaiqing Zhang  (张凯庆)
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  4. Yuan Liang  (梁缘)
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Contributions

Author contributions Gengdong Cheng and Yuan Liang designed the research, helped organize the manuscript, revised and edited the final version. Kai Sun wrote the first draft of the manuscript, processed the calculation data, revised and edited the final version. Kaiqing Zhang helped improve the theory.

Corresponding author

Correspondence to Yuan Liang  (梁缘).

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Sun, K., Cheng, G., Zhang, K. et al. Sequential conservative integer programming method for multi-constrained discrete variable structure topology optimization. Acta Mech. Sin. 40, 423151 (2024). https://doi.org/10.1007/s10409-023-23151-x

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