参数拓扑优化与多分辨率有限元模型
Parametric Topology Optimization with Multi-Resolution Finite Element Models
AI总结：
本文提出了一种在不确定性下使用多分辨率有限元模型进行拓扑优化的方法，通过双保真设置结合粗细网格模型，减少计算成本并生成与单分辨率网格相似的设计，同时验证了双保真有限元近似的误差界。
摘要：
我们提出了一种在不确定性下使用多分辨率有限元模型进行拓扑优化的方法。我们使用我们的框架在双保真设置中，其中可用低分辨率和高分辨率的网格模型。廉价的低分辨率模型用于探索参数空间并近似参数化高分辨率模型及其灵敏度，其中参数同时考虑结构载荷和刚度。我们为双保真有限元（FE）近似及其灵敏度提供了误差界，并进行了数值研究以验证这些理论估计。我们通过基准合规性最小化问题展示了我们的方法，其中我们显示在昂贵的问题如制造变异性下的拓扑优化中计算成本显著降低，同时生成与单分辨率网格获得的设计几乎相同。我们还通过我们的双保真有限元近似计算生成设计的参数化 Von-Mises 应力，并将其与标准蒙特卡洛模拟进行比较。我们提供的算法实现扩展了众所周知的 MATLAB 中 88 行拓扑优化代码。
英文摘要：
We present a methodical procedure for topology optimization under uncertainty with multi-resolution finite element models. We use our framework in a bi-fidelity setting where a coarse and a fine mesh corresponding to low- and high-resolution models are available. The inexpensive low-resolution model is used to explore the parameter space and approximate the parameterized high-resolution model and its sensitivity where parameters are considered in both structural load and stiffness. We provide error bounds for bi-fidelity finite element (FE) approximations and their sensitivities and conduct numerical studies to verify these theoretical estimates. We demonstrate our approach on benchmark compliance minimization problems where we show significant reduction in computational cost for expensive problems such as topology optimization under manufacturing variability while generating almost identical designs to those obtained with single resolution mesh. We also compute the parametric Von-Mises stress for the generated designs via our bi-fidelity FE approximation and compare them with standard Monte Carlo simulations. The implementation of our algorithm which extends the well-known 88-line topology optimization code in MATLAB is provided.
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