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| Variable density topology optimization method considering structural stability |
| ZHANG Ri-cheng1, ZHAO Jiong1, WU Qing-long1, XIONG Xiao-lei1, ZHOU Qi-cai1, JIAO Hong-yu1,2 |
1. School of Mechanical Engineering, Tongji University, Shanghai, 201804, China;
2. School of Automotive Engineering, Changshu Institute of Technology, Suzhou 215500, China |
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Abstract To achieve the topology optimization of plane model with stability constraints, the stability problem is introduced into the traditional variable density method. The mathematical model of optimization problem was established by using the relative density of element as design variable, structural volume and instability load factor as constraints and structural flexibility as objective, and a variable density topology optimization method considering structural stability was proposed. Based on the sensitivity analysis of the flexibility, volume and instability load factor to design variables, the optimization criteria were derived according to the Lagrange multiplier method and Kuhn-Tucker condition. At the same time, the Lagrange multiplier of the criteria was solved through Taylor expansion of the constraints. The geometric stiffness matrix of the plane four-node quadrilateral element was deduced to calculate the geometry strain energy of the optimization criteria. Finally, the proposed method was verified by an example. Through the comparison with the traditional variable density method, it was illustrated that the proposed method could notably improve the stability of the optimization result. The research results have important guiding significance for the optimal design of the slender compression structure, and have certain reference value for the stability design of the structure.
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Received: 30 November 2017
Published: 28 August 2018
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考虑结构稳定性的变密度拓扑优化方法
将稳定性问题引入传统变密度法中，可实现包含稳定性约束的平面模型结构拓扑优化。以单元相对密度为设计变量，结构柔度最小为目标函数，结构体积和失稳载荷因子为约束条件建立优化问题数学模型，提出了一种考虑结构稳定性的变密度拓扑优化方法。通过分析结构柔度、体积、失稳载荷因子对设计变量的灵敏度，并基于拉格朗日乘子法和Kuhn-Tucker条件，推导了优化问题的迭代准则。同时，利用基于约束条件的泰勒展开式求解优化准则中的拉格朗日乘子。通过推导平面四节点四边形单元几何刚度矩阵的显式表达式，得到了优化准则中的几何应变能。最后，通过算例对提出的方法进行了验证，并与不考虑稳定性的传统变密度拓扑优化方法进行对比，结果表明该方法能显著提高拓扑优化结果的稳定性。研究结果对细长受压结构的优化设计有重要指导意义，对结构的稳定性设计有一定参考价值。
关键词：
稳定性,
拓扑优化,
变密度法,
几何刚度,
灵敏度
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