Model reduction of large nonlinear systems often involves the projection of the governing equations onto linear subspaces spanned by carefully-selected modes. The criteria to select the modes relevant for reduction are usually problem-specific and heuristic. In this work, we propose a rigorous mode-selection criterion based on the recent theory of Spectral Submanifolds (SSM), which facilitates a reliable projection of the governing nonlinear equations onto modal subspaces. SSMs are exact invariant manifolds in the phase space that act as nonlinear continuations of linear normal modes. Our criterion identifies critical linear normal modes whose associated SSMs have locally the largest curvature. These modes should then be included in any projection-based model reduction as they are the most sensitive to nonlinearities. To make this mode selection automatic, we develop explicit formulas for the scalar curvature of an SSM and provide an open-source numerical implementation of our mode-selection procedure. We illustrate the power of this procedure by accurately reproducing the forced-response curves on three examples of varying complexity, including high-dimensional finite element models.

中文翻译：

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在模型简化中使用光谱子流形进行最优模式选择

大型非线性系统的模型简化通常涉及将控制方程投影到由精心选择的模式跨越的线性子空间。选择与归约相关的模式的标准通常是特定于问题的和启发式的。在这项工作中，我们基于最近的谱子流形 (SSM) 理论提出了一个严格的模式选择标准，这有助于将控制非线性方程可靠地投影到模态子空间上。SSM 是相空间中的精确不变流形，充当线性正态模式的非线性延续。我们的标准确定了临界线性正常模式，其相关的 SSM 具有局部最大的曲率。这些模式应包含在任何基于投影的模型简化中，因为它们对非线性最敏感。为了使这种模式选择自动化，我们为 SSM 的标量曲率开发了明确的公式，并提供了我们模式选择程序的开源数值实现。我们通过在三个不同复杂性的示例（包括高维有限元模型）上准确再现强制响应曲线来说明此程序的强大功能。