求解由点云离散化的定向曲面上的不可压缩Navier-Stokes方程

中文摘要 AI 辅助

我们提出了一种无网格数值求解器，用于求解由表面点云表示的定向曲面上的不可压缩Navier-Stokes方程。在曲面上，与刚度和压力-速度耦合相关的数值挑战会加剧，此外，曲面上的向量微积分与欧几里得空间中的对应内容存在差异。所提出的方法在欧拉参考系下直接作用于表面点云，无需计算网格或网格，即可实现空间和时间上的一致近似，且精度可达六阶。不可压缩约束通过弱人工可压缩性近似在局部施加，避免了全局矩阵求逆。我们证明该方法能为表面向量场和微分算子提供一致且收敛的近似。研究了误差、空间分辨率与人工马赫数之间的关系，并表征了人工振荡的频谱。我们给出了不可压缩Navier-Stokes方程在对称曲面（如球面和环面）以及参数化和非参数化非对称曲面上的数值解。由于该方法可直接作用于非结构化表面点云，为基于图像的几何结构模拟提供了一种有前景的方法，例如用于从显微视频中研究生物形态发生。

英文摘要

We present a meshfree numerical solver for the incompressible Navier-Stokes equations on oriented curved surfaces that are represented by surface point clouds. On curved surfaces, numerical challenges pertaining to stiffness and pressure-velocity coupling are exacerbated. Moreover, vector calculus on curved surfaces differs from its Euclidean counterpart. The presented method operates on surface point clouds in an Eulerian frame of reference without requiring a computational grid or mesh. It achieves consistent approximation in space and time with high order of accuracy; we demonstrate up to order six. The incompressibility constraint is locally imposed as a weak artificial compressibility approximation, avoiding global matrix inversion. We show that the method provides consistent and convergent approximations of surface vector fields and differential operators. We study the relationship between error, spatial resolution, and artificial Mach number and characterize the frequency spectrum of the artificial oscillations. We provide numerical solutions of the incompressible Navier-Stokes equations on symmetric surfaces, such as the sphere and torus, and on parametric and non-parametric asymmetric surfaces. Since the proposed method works directly on unstructured surface point clouds, it provides a promising approach for simulations on image-derived geometries, such as in biological morphogenesis from microscopy videos.

发表机构

• Dresden University of Technology（德累斯顿工业大学）
• Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）
• Center for Systems Biology Dresden（德累斯顿系统生物学中心）

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