将特征映射设计对齐到给定密度场：一阶与二阶方法
Aligning feature-mapping designs to given density fields, first and second order
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中文总结 AI 辅助
本文提出一种基于梯度的分阶段方法，将特征映射配置对齐到给定密度场，并首次引入特征映射的Hessian公式，在一阶和二阶优化器上验证了其有效性。
中文摘要 AI 辅助
特征映射方法通过在固定分析网格上使用显式几何基元来表示结构设计，这是基于密度的拓扑优化（SIMP）中密度场的一种可解释且参数化的替代方案。然而，与SIMP不同，特征映射优化对初始设计敏感。我们提出了一种基于梯度的方法，将特征映射配置对齐到给定的（伪）密度场，例如，用于初始化后续的特征映射优化。该分阶段方法基于最小二乘跟踪公式，并先采用一种仅奖励对齐而不惩罚不匹配的变体。在没有底层有限元模拟的情况下，迭代成本低廉，但孤立特征获得的灵敏度信息很少。作为补救措施，我们提出了一种基于自动参数化贝塞尔曲线的不对称过渡函数。我们在二维悬臂梁和五杆设计上演示了该方法；对于后者，我们展示了基于几何投影方法的特征缩放变量的可选特征最小化。该方法与一阶优化器配合良好。然而，由于没有状态问题和变量数量较少，二阶公式也具有吸引力。据作者所知，我们首次提出了特征映射的Hessian公式，并报告了在我们的基准问题上的一阶和二阶优化器的行为。为完整起见，我们还给出了基于状态柔度的精确Hessian，这需要对每个特征变量额外求解一次有限元系统。
英文摘要
Feature-mapping methods represent structural designs by explicit geometric primitives on fixed analysis grids, an interpretable and parametric alternative to the density fields of density-based topology optimization (SIMP). In contrast to SIMP, however, feature-mapping optimization is sensitive to the initial design. We present a gradient-based approach to align a feature-mapping configuration to a given (pseudo) density field, e.g., to initialize a subsequent feature-mapping optimization. The staged approach is based on a least-squares tracking formulation, preceded by a variant that only rewards alignment and does not penalize mismatch. Without an underlying finite element simulation, iterations are cheap, but isolated features receive little sensitivity information. As a remedy, we propose an asymmetric transition function based on automatically parametrized Bézier curves. We demonstrate the approach for a 2D cantilever and the five-bar design; for the latter, we show optional feature minimization based on the feature scaling variable of the geometry projection method. The approach works well with first-order optimizers. The absence of a state problem and the small number of variables, however, also make a second-order formulation attractive. To the best of the authors' knowledge, we present the first Hessian formulation for feature mapping and report the behavior of first- and second-order optimizers on our benchmark problems. For completeness, we also give the exact Hessian of the state-based compliance, which requires one additional solution of the FEM system per feature variable.
发表机构
- Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）
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