In the field of fluid numerical analysis, there has been a long-standing problem: lacking of a rigorous mathematical tool to map from a continuous flow field to discrete vortex particles, hurdling the Lagrangian particles from inheriting the high resolution of a large-scale Eulerian solver. To tackle this challenge, we propose a novel learning-based framework, the Neural Vortex Method (NVM), which builds a neural-network description of the Lagrangian vortex structures and their interaction dynamics to reconstruct the high-resolution Eulerian flow field in a physically-precise manner. The key components of our infrastructure consist of two networks: a vortex representation network to identify the Lagrangian vortices from a grid-based velocity field and a vortex interaction network to learn the underlying governing dynamics of these finite structures. By embedding these two networks with a vorticity-to-velocity Poisson solver and training its parameters using the high-fidelity data obtained from high-resolution direct numerical simulation, we can predict the accurate fluid dynamics on a precision level that was infeasible for all the previous conventional vortex methods (CVMs). To the best of our knowledge, our method is the first approach that can utilize motions of finite particles to learn infinite dimensional dynamic systems. We demonstrate the efficacy of our method in generating highly accurate prediction results, with low computational cost, of the leapfrogging vortex rings system, the turbulence system, and the systems governed by Euler equations with different external forces.
A high-order semi-Lagrangian method for the consistent Monte-Carlo solution of stochastic Lagrangian drift-diffusion models coupled with Eulerian discontinuous spectral element method
采用[Natarajan和Jacobs，Computer and Fluids，2020]中开发的用于拉格朗日输运方程求解的显式半拉格朗日方法来求解随机微分方程，该方法与欧拉守恒律的不连续谱元方法（DSEM）近似一致。该方法扩展了DSEM的良好性能，包括其高阶精度，其局部和边界拟合性能和并行平台上的并行Monte-Carlo，半拉格朗日和欧拉解的一类时间依赖性问题，可以描述耦合欧拉-拉格朗日公式的高性能。半拉格朗日方法种子粒子在高斯正交配置节点内的谱元素。根据漂移速度和维纳增量强迫，粒子在时间上被明确地积分，并形成平流插值的节点基础。此插值映射回在一个半拉格朗日的时尚高斯正交点，通过最小二乘拟合使用约束的元素边界值。随机Monte-Carlo样本在正交节点上逐元素平均。稳定的显式时间步Wiener增量足够小，可以防止粒子离开元素的边界。因此，半拉格朗日方法是局部的和并行的，并且不具有网格复杂性，以及用于求解欧拉-拉格朗日公式的粒子-网格方法中常用的拉格朗日粒子求解器的并行化挑战。给出了半拉格朗日算法根据欧拉福克-普朗克方程演化解的形式证明。一维和二维漂移扩散问题的数值试验表明，该方法指数收敛常数和非常数对流和扩散速度。
A single-stage high-order compact gas-kinetic scheme in arbitrary Lagrangian-Eulerian formulation
Hong Kong University of Science and Technology（香港科技大学）；Xi’an Jiaotong University（西安交通大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）
