三维不可压缩粘性Navier-Stokes方程的维数分裂-插值型无网格方法

Solving the Navier-Stokes equation numerically is an important research topic in computational fluid dynamics. Meshless method is another important numerical method after finite element method in science and engineering computing. For there is no good accuracy when the anisotropic nodes distribution is used to construct the shape functions in the moving least-squares method, an improved interpolating moving least-squares method adapted for the anisotropic nodes distribution will be presented, and the corresponding theories of the error estimates will also be discussed. Then, considering the difficulties of mesh generation in vicious fluid field and the anisotropic nodes distribution arisen from the boundary layer phenomena, the interpolating element-free Galerkin method adapted to the anisotropic nodes distribution will be presented for the 2-D steady and unsteady incompressible vicious Navier-Stokes equations, and the corresponding error estimates also will be discussed. At last, since the element-free Galerkin method for 3-D problems has no high efficiency, by coupling the dimension splitting method, the dimension splitting – interpolating element-free Galerkin method adapted for the anisotropic nodes will be presented for the 3-D steady and unsteady incompressible vicious Navier-Stokes equations, and the corresponding error estimates theories of the coupled method for 3-D Navier-Stokes equations will be studied. This project will provide a high efficient numerical method for the nonlinear analysis of vicious incompressible flow, and promote the development of meshless method.

寻求Navier-Stokes方程的数值解是计算流体力学研究的重要课题之一，而无网格方法是继有限元法之后科学和工程计算中又一重要的数值方法。本项目首先针对移动最小二乘法在处理各向异性节点时的不适应性，建立适合各向异性节点分布的改进的插值型移动最小二乘法，并研究其误差估计；然后，针对粘性流体问题中碰到的网格剖分困难以及因边界层现象而经常采用各向异性节点的特点，研究定常和非定常二维不可压缩粘性Navier-Stokes方程的各向异性节点分布的插值型无单元Galerkin方法，并研究其误差估计；最后，针对无单元Galerkin方法在处理三维流体问题时计算效率较低的问题，研究定常和非定常的三维不可压缩粘性Navier-Stokes方程各向异性节点分布的维数分裂-插值型无单元Galerkin方法，并研究其误差估计。本项目将为不可压粘性流体非线性分析提供高效的数值方法，并促进无网格方法的研究进展。

通过引入虚拟节点，本项目首先建立了适合各向异性节点分布的改进的插值型移动最小二乘法，并研究其误差估计。这些虚拟节点不增加最后离散代数方程的自由量个数，它们只影响形函数的构造，从而有利于降低流体问题最后离散方程组的自由度和求解效率。对于流体问题，由于传统的无单元Galerkin方法不满足Babuska–Brezzi（BB）条件，可能导致解的不稳定性，本项目基于提出的改进的插值型移动最小二乘法和Galerkin变分弱形式，通过引入变分多尺度方法和局部细化多项式基函数，研究了二维不可压缩粘性流体问题的各向异性节点分布的插值型无单元Galerkin方法，并研究其误差估计；最后，针对无单元Galerkin方法在处理三维问题时计算效率较低的问题，通过引入维数分裂法，研究三维不可压缩粘性流体问题的各向异性节点分布的维数分裂-插值型无单元Galerkin方法，并研究其误差估计。对本项目提出的方法编写了Matlab计算程序，并进行了数值验证，结果显示改进的无网格方法对于流体问题具有很好的计算精度和计算效率。