Bidirectional Evolutionary Topology Optimization for Stress Minimization Based on the Modified Couple Stress Elasticity
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摘要: 研究并探讨了基于应力的双向渐进结构拓扑优化(BESO)法在修正偶应力弹性理论中的应用. 该方法允许对尺寸问题相关的微观结构均质连续体进行拓扑优化. 其通过引入一种与尺寸相关的，基于修正偶应力理论的，非经典等效应力的新颖公式，对经典的BESO技术进行了扩展，并在体积约束的条件下进行应力最小化设计. 设计变量的迭代更新依赖于灵敏度分析，其涉及对目标函数p范数全局应力的直接求导. 理论中涉及高阶弹性，因此为了满足有限元实现时需要的C1节点连续性，在插值中将传统的Lagrange插值与一个含待定系数的插值函数相结合. 通过三个不同的数值算例，分析了尺寸效应对应力优化设计过程及结果的影响. 同时探讨了其他参数包括范数p值和材料体积分数的作用. 获得的研究结果证明了所提出的基于应力的BESO方法在涉及尺寸效应相关的拓扑优化设计方向的潜力.Abstract: The application of stress-based bidirectional evolutionary structural optimization (BESO) in the context of the modified couple stress elasticity theory was investigated. This methodology allows for structure optimization of homogenized continuum with a microstructural composition of size effects. The classical BESO technique was extended through the introduction of a novel formulation of couple stress based non-classical equivalent stress, and the minimization design was conducted under the constraint of volume criterion. The iterative update of design variables relies on the sensitivity analysis involving direct derivation of the enriched p-norm global stress with couple stress contributions. Since the high-order elasticity is involved, the FEM implementation requires at least the C1 nodal continuity. Thus, a Lagrangian finite element complemented by additional integration functions was implemented. The method was validated with 3 distinct cases through investigation of the size effects on the stress optimization and the subsequent structure design. The impacts of other parameters including the norm p value and the material volume fraction, were explored. The results demonstrate the potential of the proposed stress-based BESO method in addressing structural optimization of problems involving size effects.
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Key words:
- topology optimization /
- BESO /
- stress-based /
- modified couple stress /
- size effect
edited-byedited-by1) （我刊编委冯志强来稿） -
图 1 修正偶应力理论下单位体积δxδy上的平面应力分布
Figure 1. The plane stress distribution on a unit volume δxδy according to the modified couple stress elasticity
图 4 矩形梁结构：设计域以360×60网格离散
Figure 4. The rectangular beam structure with the design domain is discretized by a 360×60 mesh
图 5 基于经典弹性理论的应力最优拓扑结构
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 5. Stress optimization topology structures based on the classical elasticity theory
图 6 基于修正偶应力理论的应力最优拓扑结构及应力分量云图(H/l=107，103)
Figure 6. Stress optimization topology structures and stress component contours based on the couple stress theory (H/l=107, 103)
图 7 基于修正偶应力理论的应力最优拓扑结构及应力分量云图(H/l=100，10)
Figure 7. Stress optimization topology structures and stress component contours based on the couple stress theory (H/l=100, 10)
图 8 基于修正偶应力理论的应力最优拓扑结构及应力分量云图(H/l=5，2.5)
Figure 8. Stress optimization topology structures and stress component contours based on the couple stress theory (H/l=5, 2.5)
图 9 σPN值随尺寸效应的演化：偶应力优化模型与经典弹性优化模型的比较
Figure 9. Evolution of σPN vs. the size effect: comparison between the couple stress-based optimization model and the classical elasticity-based model
图 11 应力范数p值的影响：应力优化结构设计和应力σe分布云图的比较
Figure 11. Influences of stress norm p values: comparison of stress-optimized structural designs and stress σe distribution contours
图 13 体积限制分数的影响：应力优化结构设计和应力σe分布云图的比较
Figure 13. Influences of the volume fractions constraint: comparison of stress-optimized structural designs and stress σe distribution contours
表 1 不同体积分数下的全局应力水平对比
Table 1. Comparison of global stress levels under different volume fractions
volume limitation V* classical theory-based σPN/MPa modified couple stress theory-based σPN/MPa 60% 1.872 2.060 40% 2.384 2.665 20% 4.588 4.908 
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