基于多边形有限元的非高斯材料场鲁棒拓扑优化
Robust topology optimization with non-Gaussian material fields using polygonal finite elements
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- PUC-Rio（天主教大学（里约热内卢））
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中文总结 AI 辅助
提出集成多边形有限元、非高斯随机场与多项式混沌代理的鲁棒拓扑优化框架，实现高效且物理一致的可靠性设计。
中文摘要 AI 辅助
我们提出了一种鲁棒拓扑优化的计算框架，该框架集成了多边形有限元离散、空间相关的非高斯材料建模和非侵入式多项式混沌代理模型。杨氏模量的空间不确定性被表示为通过截断的Karhunen-Loève展开的无记忆变换得到的齐次非高斯随机场，通过刚度的正性确保物理可行性，同时保持规定的自协方差。多边形有限元为基于密度的非结构化网格优化提供了稳定的离散化，并减轻了棋盘格伪影和网格偏差，而稀疏多项式混沌展开能够高效估计鲁棒目标所需的低阶统计矩，其成本仅为侵入式或蒙特卡洛方法的一小部分。对悬臂梁和曲梁的数值研究表明，引入非高斯材料变异性会导致系统的载荷路径重新分布和结构体积的重新分配（6-12%），同时降低柔度散布。非侵入式代理模型在减少一个数量级的完整有限元分析次数的情况下，在3%以内复现了侵入式参考结果。这些结果表明，所提出的框架为在现实材料不确定性下保持可靠的拓扑优化设计提供了一条物理一致且计算高效的途径。
英文摘要
We present a computational framework for robust topology optimization that integrates polygonal finite-element discretizations, spatially correlated non-Gaussian material modeling, and non-intrusive polynomial-chaos surrogates. Spatial uncertainty in Young's modulus is represented as a homogeneous non-Gaussian random field obtained via a memoryless transformation of a truncated Karhunen-Loève expansion, ensuring physical admissibility through positivity of stiffness while preserving the prescribed autocovariance. Polygonal finite elements provide a stable discretization for density-based optimization on unstructured meshes and mitigate checkerboard artefacts and mesh bias, while the sparse polynomial-chaos expansion enables efficient estimation of low-order statistical moments required by the robust objective at a fraction of the cost of intrusive or Monte Carlo approaches. Numerical studies on a cantilever and a curved beam show that introducing non-Gaussian material variability leads to systematic load-path redistribution and a reallocation of 6-12% of the structural volume, together with a reduction in compliance scatter. The non-intrusive surrogate reproduces intrusive reference results within 3% using an order of magnitude fewer full finite-element analyses. These results demonstrate that the proposed framework offers a physically consistent and computationally efficient route to topology-optimized designs that remain reliable under realistic material uncertainty.
