基于密度的结构拓扑optimization的Heaviside投影再引入单元删除
Revisiting element removal for density-based structural topology optimization with reintroduction by Heaviside projection
摘要：
我们提出了一种基于Bruns和Tortorelli [1]的元素移除思想的策略，旨在降低计算成本并规避基于密度的拓扑优化的潜在数值不稳定性。设计变量和相对密度都表示在一个固定的，均匀的有限元网格，并通过过滤和Heaviside投影连接。分析域中相对密度低于指定阈值的区域从正演分析中删除，并由虚拟节点边界条件代替。随着优化的进行，这带来了计算成本的逐步削减，并有助于减轻与低密度区域相关的数值不稳定性。由于所有设计变量都保持有效，并且在正式的灵敏度分析中建模，因此可以很容易地重新引入删除的区域。所提出的方法的一个关键特征是，赫维赛德函数通过放大滤波器范围内的灵敏度的大小来促进材料沿结构边界的重新引入。几个二维和三维的结构拓扑优化的例子，包括线性和非线性柔度最小化，设计的力逆变器，频率和屈曲载荷最大化。该方法被证明是有效的，在生产优化的设计等同于或几乎等同于那些没有元素删除，同时提供显着的计算节省。
英文摘要：
We present a strategy grounded in the element removal idea of Bruns and Tortorelli [1] and aimed at reducing computational cost and circumventing potential numerical instabilities of density-based topology optimization. The design variables and the relative densities are both represented on a fixed, uniform finite element grid, and linked through filtering and Heaviside projection. The regions in the analysis domain where the relative density is below a specified threshold are removed from the forward analysis and replaced by fictitious nodal boundary conditions. This brings a progressive cut of the computational cost as the optimization proceeds and helps to mitigate numerical instabilities associated with low-density regions. Removed regions can be readily reintroduced since all the design variables remain active and are modeled in the formal sensitivity analysis. A key feature of the proposed approach is that the Heaviside functions promote material reintroduction along the structural boundaries by amplifying the magnitude of the sensitivities inside the filter reach. Several 2D and 3D structural topology optimization examples are presented, including linear and nonlinear compliance minimization, the design of a force inverter, and frequency and buckling load maximization. The approach is shown to be effective at producing optimized designs equivalent or nearly equivalent to those obtained without the element removal, while providing remarkable computational savings.
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