In density-based topology optimization of flow problems, flow in the solid domain is generally inhibited using a penalization approach. Setting an appropriate maximum magnitude for the penalization traditionally requires manual tuning to find an acceptable compromise between flow solution accuracy and design convergence. In this work, three penalization approaches are examined, the Darcy (D), the Darcy with Forchheimer (DF), and the newly proposed Darcy with filtered Forchheimer (DFF) approach. Parameter tuning is reduced by analytically deriving an appropriate penalization magnitude for accuracy of the flow solution. The Forchheimer penalization is found to be required to reliably predict the accuracy of the flow solution. The state-of-the-art D and DF approaches are improved by developing the novel DFF approach, based on a spatial average of the velocity magnitude. In comparison, the parameter selection in the DFF approach is more reliable, as convergence of the flow solution and objective convexity are more predictable. Moreover, a continuation approach on the maximum penalization magnitude is derived by numerical inspection of the convexity of the pressure drop response. Using two-dimensional optimization benchmarks, the DFF approach reliably finds accurate flow solutions and is less prone to converge to inferior local optima.

中文翻译：

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使用 Darcy 和 Forchheimer 惩罚减少流动问题拓扑优化中的参数调整

在基于密度的流动问题拓扑优化中，通常使用惩罚方法抑制固体域中的流动。传统上，为罚分设置适当的最大幅度需要手动调整，以在流动求解精度和设计收敛性之间找到可接受的折衷方案。在这项工作中，研究了三种惩罚方法，即 Darcy （D）、Darcy with Forchheimer （DF） 和新提出的 Darcy with filtered Forchheimer （DFF） 方法。通过解析推导出流解准确度的适当惩罚幅度，可以减少参数调整。发现需要 Forchheimer 惩罚来可靠地预测流动解的准确性。通过开发基于速度大小的空间平均值的新型 DFF 方法，改进了最先进的 D 和 DF 方法。相比之下，DFF 方法中的参数选择更可靠，因为流动解和目标凸性的收敛性更可预测。此外，通过对压降响应的凸性进行数值检查，得出了最大惩罚幅度的连续方法。使用二维优化基准，DFF 方法可以可靠地找到准确的流动解，并且不易收敛到较差的局部最优值。