用于索结构找形的力密度拓扑优化
Topology optimization of force densities for form finding of cable structures
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中文总结 AI 辅助
本文提出结合力密度法与拓扑优化的方法，通过SIMP插值和二值化促进项优化索网找形，引入体积约束等得到稳定设计，可随规模扩展，能实现拓扑定性转变。
中文摘要 AI 辅助
本文提出了一种基于力密度法的索网找形拓扑优化方法。该框架将力密度法与基于连续密度变量的拓扑优化相结合，以同时确定索结构的平衡几何形态与有效连接性。设计变量被分配给索网构件，并通过SIMP插值控制其力密度，目标函数中补充了显式的二值化促进项，使得索构件在优化过程中消失或保持激活状态。通过强制满足力密度法方程得到平衡构型，同时目标函数驱动优化后的网络尽可能接近规定的参考形态。为得到物理意义明确且数值稳定的设计，引入了体积约束、被动边界区域以及受载节点处的最小连接性要求。基于由水平、垂直和斜向构件构成的索网基结构的多个数值算例被用于研究所提方法。一项专门的 rounding（圆整）研究表明，松弛的密度变量在优化过程结束时收敛至接近离散的值，阈值处理后仅需微小修正。对体积分数约束的进一步研究显示，降低允许的结构体积会触发优化拓扑的定性转变，从冗余的密集支撑构型转变为最小荷载路径。最后，求解器性能评估表明，所提公式随问题规模的扩展表现合理。
英文摘要
This paper presents a topology optimization approach for the form finding of cable networks based on the force density method. The proposed framework combines the force density method with topology optimization based on continuous density variables to identify the equilibrated geometry and effective connectivity of cable structures simultaneously. Design variables are assigned to the cable network members and control their force densities through a SIMP interpolation, complemented by an explicit binary promoting term in the objective function. As a result, cable members vanish or remain active throughout the optimization process. The equilibrium configuration is obtained by enforcing the force density method equations, while the objective drives the optimized network as close as possible to a prescribed reference form. Volume constraints, passive boundary regions, and a minimum connectivity requirement at loaded joints are incorporated to yield physically meaningful and numerically stable designs. Several numerical examples based on cable net ground structures defined by horizontal, vertical, and diagonal members are used to study the proposed approach. A dedicated rounding study shows that the relaxed density variables converge to nearly discrete values at the end of the optimization process, with only a minor correction required after thresholding. A study of the volume fraction constraint further shows that reducing the permitted structural volume can trigger a qualitative transition in the optimized topology, from a redundant, densely braced configuration to a minimal load path. Finally, an assessment of solver performance shows that the proposed formulation scales reasonably well with problem size.
