状态空间框架用于平凡与拓扑超材料随机分析

中文摘要 AI 辅助

拓扑声子晶体和弹性超材料支持由全局拓扑不变量定义的边缘态，为在存在缺陷时仍能正常工作的振动控制和波导器件提供了一条途径。然而，制造引起的空间变异性可能扰动这些不变量并损害其鲁棒性，因此在设计过程中对其量化至关重要。在本工作中，我们首先证明先前提出的线性时变（LTV）公式在数学上等价于基于传递矩阵的谱元方法，适用于基本杆、圣维南轴和欧拉-伯努利梁理论。然后，确定性公式通过随机线性时变（SLTV）框架扩展到随机分析。所提出的方法结合了蒙特卡洛模拟、随机傅里叶级数和解析的Karhunen-Loève展开，提供了LTV公式所需的闭式随机场及其导数。SLTV方法能够计算具有任意变化几何和机械性能的一维波导的随机色散图和强迫响应。由于基于LTV的转移矩阵能够隔离单个波模，而无需传统基于特征问题公式所需的模式跟踪，该框架特别适用于评估拓扑不变量，特别是Zak相位，并评估在空间变异性下拓扑带的鲁棒性。针对杆、轴和欧拉-伯努利梁的数值结果证明了所提出框架作为在连续空间不确定性下平凡和拓扑周期波导的确定性和随机分析统一方法的适用性。

英文摘要

Topological phononic crystals and elastic metamaterials support edge states defined by global topological invariants, offering a route toward vibration-control and wave-guiding devices that remain functional in the presence of defects. However, manufacturing-induced spatial variability can perturb these invariants and compromise their robustness, making its quantification essential during design. In this work, we first demonstrate that a previously proposed linear time-varying (LTV) formulation is mathematically equivalent to the spectral element method based on transfer matrices for elementary rod, Saint-Venant shaft, and Euler-Bernoulli beam theories. The deterministic formulation is then extended to stochastic analyses through a stochastic linear time-varying (SLTV) framework. The proposed methodology combines Monte Carlo simulations with stochastic Fourier series and an analytical Karhunen--Loève expansion, providing closed-form stochastic fields and their derivatives required by the LTV formulation. The SLTV approach enables the computation of stochastic dispersion diagrams and forced responses of one-dimensional waveguides with arbitrarily varying geometry and mechanical properties. Because the LTV-based transition matrix isolates individual wavemodes without requiring the mode tracking needed by conventional eigenproblem-based formulations, the framework is particularly suited for evaluating topological invariants, specifically the Zak phase, and assessing the robustness of topological bands under spatial variability. Numerical results for rods, shafts, and Euler-Bernoulli beams demonstrate the applicability of the proposed framework as a unified methodology for deterministic and stochastic analyses of trivial and topological periodic waveguides under continuous spatial uncertainty.

发表机构

• University of Campinas, School of Mechanical Engineering（坎皮纳斯大学机械工程学院）
• University of São Paulo, São Carlos School of Engineering, Department of Aeronautical Engineering（圣保罗大学圣卡洛斯工程学院航空工程系）
• KU Leuven, Faculty of Engineering Technology, Department of Mechanical Engineering, LMSD Division, Ghent Campus（荷语鲁汶大学工程技术学院机械工程系LMSD分部根特校区）
• Instituto Universitario de Matemática Pura y Aplicada, Departamento de Matemática Aplicada, Universitat Politècnica de València（瓦伦西亚理工大学应用数学研究所应用数学系）

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