一种基于微形态计算均匀化的Newton求解器实现模式转换超材料的多尺度屈曲分析
A Newton Solver for Micromorphic Computational Homogenization Enabling Multiscale Buckling Analysis of Pattern-Transforming Metamaterials
摘要：
机械超材料具有工程微结构，旨在表现出异国情调，往往违反直觉，有效的行为。这种行为通常是通过不稳定性引起的底层周期性微结构到一个或多个图案化模式的转换来实现的。由于单个重复微结构单元的强烈运动耦合，非局部行为和尺寸效应出现，这不能容易地被经典的均匀化方案捕获。此外，各个图案模式可以在空间和时间上相互作用，而在工程规模下，整个结构可以全局屈曲。为了有效的数值宏观预测，一个微观的计算均匀化计划最近已经开发出来。虽然这个框架在原则上是能够占空间和时间之间的相互作用的个人模式，其实现依赖于基于梯度的拟牛顿解决方案技术。该求解器是次优的，因为（i）它具有次二次收敛性，以及（ii）缺少Hessians不允许进行适当的分叉分析。考虑到机械超材料通常依赖于受控的不稳定性，这些限制是严重的。为了解决这些问题，本文详细介绍了一种全牛顿法。宏观切线算子的构造并不简单，因为涉及正交约束的微观框架相关的底层位移场的分解上的特定模型假设。给出了总势能一阶变分和二阶变分的解析表达式，并列出了完整的算法。所开发的方法证明了两个例子，其中存在的竞争和多个图案模式出现的地方和全球屈曲之间。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
