Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour. Such a behaviour is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical macroscale predictions, a micromorphic computational homogenization scheme has recently been developed. Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. To address them, a full Newton method is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge.

中文翻译：

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用于微晶计算均质化的牛顿求解器，能够对模式转换超材料进行多尺度屈曲分析

机械超材料具有工程化的微结构，旨在展示奇异的、通常违反直觉的有效行为。这种行为通常是通过不稳定引起的潜在周期性微观结构转变为一种或多种图案化模式来实现的。由于单个重复微结构细胞的强运动学耦合，出现了非局部行为和尺寸效应，经典均质化方案无法轻易捕捉到这些效应。此外，各个图案模式可以在空间和时间上相互影响，而在工程规模上，整个结构可以全局弯曲。为了有效的宏观数值预测，最近开发了一种微形态计算均匀化方案。尽管该框架原则上能够解释各个模式模式之间的空间和时间相互作用，但其实现依赖于基于梯度的准牛顿求解技术。这个求解器是次优的，因为 (i) 它具有次二次收敛，并且 (ii) 没有 Hessians 不允许进行适当的分叉分析。鉴于机械超材料通常依赖于可控的不稳定性，这些限制非常严重。为了解决这些问题，本文详细提供了完整的牛顿法。由于与微观框架相关的潜在位移场分解的特定模型假设，宏观切线算子的构造并不简单，涉及正交性约束。给出了总势能的第一次和第二次变化的解析表达式，并列出了完整的算法。开发的方法通过两个例子来展示，其中存在局部和全局屈曲之间的竞争以及出现多种模式模式。