论文标题
在模型降低中使用光谱子手机进行最佳模式选择
Using Spectral Submanifolds for Optimal Mode Selection in Model Reduction
论文作者
论文摘要
大型非线性系统的模型降低通常涉及控制方程式在通过精心选择的模式跨越的线性子空间上的投影。选择与减少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，我们提出了一个基于近期光谱亚策略（SSM）理论的严格模式选择标准，该标准有助于对模态子空间的管理非线性方程的可靠投影。 SSM是相位空间中确切的不变歧管，它充当线性正常模式的非线性连续性。我们的标准确定了关联的临界线性正常模式，其相关SSM的局部曲率最大。然后，这些模式应包括在任何基于投影的模型降低中，因为它们对非线性最敏感。为了使此模式选择自动，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过程的开源数值实现。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过程的功能。
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.
