Direct numerical simulations of mechanical metamaterials are often computationally prohibitive due to multi-scale geometric complexity. This work develops a non-locality homogenization method (NLHM) integrating multi-scale asymptotic homogenization with generalized finite element principles for static and dynamic analysis. The method converts heterogeneous materials with inclusions or voids into mechanically equivalent homogeneous continua, bypassing resource intensive microstructure modeling. Three representative mechanical metamaterials (conventional, auxetic, and zero Poisson’s ratio metamaterials) are constructed to validate NLHM performance. A comprehensive comparative analysis of static mechanical responses (specifically tensile and shear behaviors) and dynamic characteristics (including modal frequencies, band structures, and vibration transmission properties) is conducted and rigorously benchmarked against conventional finite element method (FEM) simulation results. Numerical results demonstrate NLHM achieves comparable accuracy while requiring merely 0.17% of traditional FEM computational time, equivalent to hundreds of times efficiency improvement, with maximum errors below 5%. The approach reduces mesh complexity by over 90% through eliminating explicit microstructure discretization. This non-locality homogenization framework enables efficient prediction of critical deformation mechanisms in metamaterials while maintaining multi-scale mechanical fidelity. Its computational tractability addresses a fundamental challenge in analyzing architecturally complex metamaterial systems, particularly beneficial for design optimization workflows requiring iterative simulations. The proposed methodology establishes a paradigm for accelerating multi-scale metamaterial analysis without sacrificing accuracy, offering practical value for advancing functional metamaterial development.

中文翻译：

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一种用于预测机械超材料静态和动态行为的宏观非局域均质化方法

由于多尺度几何复杂性，机械超材料的直接数值模拟通常在计算上令人望而却步。这项工作开发了一种非局域均质化方法 （NLHM），该方法将多尺度渐近均质化与广义有限元原理相结合，用于静态和动态分析。该方法将具有夹杂物或空隙的非均质材料转换为机械等效的均质连续体，绕过了资源密集型微观结构建模。构建了三种具有代表性的机械超材料 （传统超材料、增补超材料 和零泊松比超材料） 来验证 NLHM 性能。对静态机械响应（特别是拉伸和剪切行为）和动态特性（包括模态频率、频带结构和振动传输特性）进行了全面的比较分析，并与传统的有限元方法 （FEM） 仿真结果进行了严格的基准测试。数值结果表明，NLHM 实现了相当的精度，而只需要传统 FEM 计算时间的 0.17%，相当于效率提高了数百倍，最大误差低于 5%。该方法通过消除显式微结构离散化，将网格复杂性降低了 90% 以上。这种非局域均质化框架能够有效预测超材料中的关键变形机制，同时保持多尺度机械保真度。它的计算可处理性解决了分析架构复杂的超材料系统的基本挑战，特别有利于需要迭代仿真的设计优化工作流程。 所提出的方法建立了一种在不牺牲准确性的情况下加速多尺度超材料分析的范式，为推进功能超材料开发提供了实用价值。