多目标拓扑优化中柔度与体积的局部非支配曲线映射

中文摘要 AI 辅助

本文研究了柔度与体积的拓扑优化中的非支配集。在多目标拓扑优化文献中，标量化是逼近该集合（更广为人知的名称为帕累托前沿）的一种流行方法。然而，先前的工作表明，通过标量化获得的逐点逼近掩盖了其潜在的数学结构：在施加长度尺度下，该集合由局部非支配曲线段组成。为了研究这一结构，本文采用了一种简单的方法：(i) 生成具有严格控制拓扑的局部最优解，以及 (ii) 使用延拓将这些点扩展为局部非支配曲线。随后，本文进行了两轮数值实验。第一轮将该方法应用于多个悬臂梁示例，得出了关于拓扑复杂性和对称性的新见解，并确认了先前关于连续性、光滑性和凸性的结论。第二轮研究了两种数值不稳定性：(i) 局部非支配曲线缺乏唯一性，以及 (ii) 其不连续性。前者与观察到的优化景观平坦性相关，而后者则归因于分支分裂并与分岔理论相关联。具体而言，我们将这些不连续性分类为亚临界、对称破缺叉形分岔的高维类比。本文最后提出了未来研究建议，以解决多目标优化景观的这些特征。

英文摘要

This paper studies the nondominated set for topology optimization of compliance and volume. In the multiobjective topology optimization literature, scalarization is a popular method to approximate this set, which is better known as the Pareto frontier. However, previous work indicates that the pointwise approximation obtained by scalarization obfuscates its underlying mathematical structure: under an imposed length scale, it consists of locally nondominated curve segments. To investigate this structure, the paper uses a simple methodology to (i) generate local optima with a tightly controlled topology and (ii) use continuation to extend these points to locally nondominated curves. The paper then performs two rounds of numerical experiments. The first round applies the methodology to several cantilever examples. This leads to novel insights regarding topological complexity and symmetry and confirms prior conclusions regarding continuity, smoothness and convexity. The second round investigates two numerical instabilities: (i) a lack of uniqueness and (ii) discontinuities of the locally nondominated curves. The former is linked to an observed flatness of the optimization landscape, whereas the latter is attributed to branch splitting and linked to bifurcation theory. Namely, we classify these discontinuities as higher-dimensional analogues of the subcritical, symmetry-breaking pitchfork bifurcation. The paper concludes with suggestions for future research to address these features of the multiobjective optimization landscape.

发表机构

• KU Leuven（荷语鲁汶大学）

机构由 AI 辅助整理，请以论文原文为准。