几何非线性结构的模型降阶方法：非线性技术综述
Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques
摘要：
本文综述了几何非线性结构模型降阶的非线性方法，重点介绍了基于不变流形理论的降阶方法。非线性方法与基于线性的技术的不同之处在于它们使用非线性映射而不是添加新的向量来扩大投影基。不变流形首先被引入到振动理论中的非线性模态（NNM）的背景下，并已初步计算从模态基础，使用图形表示或范式的方法来计算映射和约化动力学。这些发展首先回顾了历史的角度来看，其中的主要应用首先面向结构模型，可以表示感谢偏微分方程（PDE）。然后，它们被替换在更一般的上下文中的参数化的不变流形，允许统一的方法。然后讨论了用有限元法离散结构的具体情况。隐式凝聚，产生到应力流形上的投影，和模态导数，在二次流形的框架中使用，首先进行审查。最后，最近的发展允许直接计算的降阶模型（ROM）依赖于不变流形理论进行了详细说明。应用的例子和扩展的方法来处理进一步的并发症进行审查。最后，突出了开放的问题和未来的方向。
英文摘要：
This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes (NNMs) and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations (PDE). They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then the specific case of structures discretized with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models (ROMs) relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.
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