| 引用本文: | 郝宝新,周志成,曲广吉,李东泽.桁架结构拓扑优化的半定规划建模与求解[J].哈尔滨工业大学学报,2019,51(10):11.DOI:10.11918/j.issn.0367-6234.201901070 |
| HAO Baoxin,ZHOU Zhicheng,QU Guangji,LI Dongze.Modeling and solving of truss topology optimization problems based on semidefinite programming[J].Journal of Harbin Institute of Technology,2019,51(10):11.DOI:10.11918/j.issn.0367-6234.201901070 |
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| 摘要: |
| 为克服桁架结构拓扑优化传统模型中优化问题非凸、多重特征值不存在常规梯度等困难,将考虑多种约束的桁架结构拓扑优化问题建模为统一的半定规划（semidefinite programming,SDP）模型. 首先给出体积、柔度、基频和全局稳定约束的等价半定形式；然后基于桁架结构刚度和质量矩阵的线性表达式,将考虑体积、柔度和基频的优化问题表述为线性半定规划对偶规划问题的标准形式；最后分别以全局稳定约束和应力约束为例,对非线性半定约束和非线性常规约束进行了近似处理,建立了一般非线性模型的近似半定模型并给出了序列求解算法.线性半定规划模型将传统的非线性非凸模型转化为凸模型,具有良好的数值特性；对非线性约束的处理方法使统一模型既能利用半定约束的良好特性,又能够考虑多种常规约束,有助于提高优化结果的工程实用性. 优化算例表明,半定规划模型和算法具有多种约束下桁架优化问题的求解能力,且能够处理包含多重特征值的基频约束和全局稳定约束,证明了所提模型和算法求解桁架结构拓扑优化问题的有效性. |
| 关键词: 桁架结构 拓扑优化 优化模型 半定规划 序列近似 |
| DOI：10.11918/j.issn.0367-6234.201901070 |
| 分类号:TU323.4;O224 |
| 文献标识码:A |
| 基金项目: |
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| Modeling and solving of truss topology optimization problems based on semidefinite programming |
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HAO Baoxin1,ZHOU Zhicheng2,QU Guangji1,LI Dongze1
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（1.Institute of Telecommunication Satellite, China Academy of Space Technology, Beijing 100094, China; 2.China Academy of Space Technology, Beijing 100094, China）
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| Abstract: |
| To overcome the non-convexity and non-differentiability of multiple eigenvalues in traditional truss topology optimization models, a unified semidefinite programming (SDP) model was established for truss topology optimization problems with various constraints. Equivalent semidefinite forms were first provided for volume, compliance, fundamental frequency, and global stability constraints. Since the stiffness matrix and the mass matrix of truss are both linear with respect to design variables, problems considering volume, compliance, and fundamental frequency constraints were reformulated as standard dual forms linear SDP. Demonstrated by the global stability constraint and the stress constraint, nonlinear semidefinite constraints and nonlinear scalar constraints were separately approximated by simpler SDP forms at the current design point, which converts the nonlinear model to a solvable approximate SDP model. An algorithm for sequentially solving the approximate problem was then introduced to deal with the nonlinear problem. When the model contains only linear semidefinite constraints, the resultant linear SDP is convex and numerically favorable. When it concerns nonlinear constraints, the sequential approximate scheme enjoys the numerical advantage of linear semidefinite forms and also maintains the ability to handle ordinary nonlinear constraints, which may contribute to a more practical design. Examples show that the proposed SDP model and algorithm could deal with various constraints in truss topology optimization, especially fundamental frequency constraints and global stability constraints with multiple eigenvalues, which verified the effectiveness of the model and the algorithm. |
| Key words: truss topology optimization optimization model semidefinite programming (SDP) sequential approximation |
