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.
翻译:我们提出了一个基于布鲁斯和托托雷利[1]要素删除概念的战略,目的是降低计算成本,绕过基于密度的地形优化的潜在数字不稳定性。设计变量和相对密度都体现在固定的、统一的有限元素网格上,并通过过滤和 Heaviside 投影连接。分析领域的相对密度低于特定阈值的区域从远端分析中删除,代之以虚构的节点边界条件。这导致计算成本随着优化的收益而逐步削减,有助于减少与低密度区域相关的数字不稳定性。去除的区域可以随时重新出现,因为所有设计变量仍然活跃,并在正式敏感度分析中建模。拟议方法的一个关键特征是,海维赛德函数通过扩大过滤所达到的敏感度,促进沿着结构界限重新进行材料的重新引入。提出了几个2D和3D结构表层优化实例,包括最大限度地减少线性和非线性合规性,设计了与低密度区域有关的力量,频率和重负负负负载的频率,可以随时重新引入。该方法的主要特征是,在不作等同的优化的计算时,将产生有效的节约。