高维有限元模型中不变流形及其约化动力学的计算
How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models
摘要：
不变流形是定量和定性理解动力系统中非线性现象的重要构造。例如，在非线性阻尼机械系统中，谱子流形已经成为通过精确的降阶模型计算强迫响应曲线、骨干曲线、分离共振曲线（孤立点）的有用工具。对于保守的非线性力学系统，李雅普诺夫次中心流形和他们的约化动力学提供了一种方法来识别非线性的幅频关系的形式保守的骨干曲线。尽管不变流形提供了这些强大的预测，但它们的使用主要限于低维的学术例子。这是因为一些挑战使得它们的计算对于由有限元模型描述的现实工程结构不可行。在这项工作中，我们解决这些计算的挑战和发展的方法计算不变的流形和减少动力学在非常高维的非线性系统所产生的空间离散化的偏微分方程。我们说明我们的计算算法的机械结构的有限元模型，范围从一个简单的梁包含几十个自由度的飞机机翼包含超过十万个自由度。
英文摘要：
Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude-frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite-element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite-element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred-thousand degrees of freedom.
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。
