标题： 基于傅里叶级数展开的射击法求解有限元问题的准周期解的并行计算 显示英文标题

标题： Parallelized computation of quasi-periodic solutions for finite element problems: A Fourier series expansion-based shooting method

摘要： 高维非线性机械系统存在准周期解，这对于理解动力系统至关重要。 这些准周期解位于由超时间中的复杂偏微分方程（PDE）所控制的不变环面上。 在此，我们提出了一种基于傅里叶级数展开的射击方法（FSE-Shooting），用于并行计算具有$d$个基础频率（$d \ge 2$）的准周期解。 我们将相关的$d$-环面表示为在（$d-1$）-环面上初始化的轨迹集合。 我们推导出一组适用于这些轨迹的常微分方程（ODE）。 我们还推导出一组耦合这些轨迹初始状态和终端状态的边界条件，然后通过耦合条件形成一组非线性代数方程。 我们使用傅里叶级数展开来参数化（$d-1$）-环面，并使用射击方法迭代与初始环面相关的傅里叶系数，使得耦合条件得到满足。 特别是，这些轨迹的终端点通过 Newmark 积分并行计算，其中时间点和傅里叶系数通过交替频域-时域方法相互转换。 设计了一个直接的相位条件，以跟踪具有先验未知基础频率的准周期解。 此外，FSE-Shooting 的副产品也可直接用于计算 Lyapunov 指数，以评估准周期解的稳定性。 三个有限元系统的实验结果表明，FSE-Shooting 在高维非线性动力系统中具有高效性和通用性，包括一个具有$1872$个自由度的三维壳体结构。 显示英文摘要

摘要： High-dimensional nonlinear mechanical systems admit quasi-periodic solutions that are essential for the understanding of the dynamical systems. These quasi-periodic solutions stay on some invariant tori governed by complex PDEs in hyper-time. Here, we propose a Fourier series expansion-based shooting method (FSE-Shooting) for the parallelized computation of quasi-periodic solution with $d$ base frequencies ($d \ge 2$). We represent the associated $d$-torus as a collection of trajectories initialized at a ($d-1$)-torus. We drive a set of ODEs that hold for any of these trajectories. We also derive a set of boundary conditions that couple the initial and terminal states of these trajectories and then formulate a set of nonlinear algebraic equations via the coupling conditions. We use Fourier series expansion to parameterize the ($d-1$)-torus and shooting method to iterate the Fourier coefficients associated with initial torus such that the coupling conditions are satisfied. In particular, the terminal points of these trajectories are parallelized computed via Newmark integration, where the time points and Fourier coefficients are transformed to each other by alternating Frequency-Time method. A straightforward phase condition is devised to track the quasi-periodic solutions with priori unknown base frequencies. Additionally, the by-product of the FSE-Shooting can be also directly used to compute the Lyapunov exponents to assess the stabilities of quasi-periodic solutions. The results of three finite element systems show the efficiency and versatility of FSE-Shooting in high-dimensional nonlinear dynamical systems, including a three-dimensional shell structure with $1872$ DOFs.