提高移动最小二乘近似无网格方法计算效率的技术和理论

Meshless (or meshfree) methods, which are approximations based on nodes, can overcome the disadvantage that the traditional numerical methods depend on the mesh of the solution domain. The basis of meshless methods is shape functions. The moving least square (MLS) approximation is an important approximate method to form shape functions in mshless mthods. But it may form an ill-conditioned or singular system of algebra equations. Besides, the matrix of the system must be inverted, which leads to the decrease in stability and the increase in computational cost. Consequently, the computational efficiency of the associated meshless methods is reduced. The aim of this project is to develop some techniques to overcome these drawbacks. First, some improved algorithms of the MLS will be established using orthogonal basis functions and complex variable theory. In these improved algorithms, the system of algebra equations neither ill-conditioned nor singular, and there is no need to inverse the system matrix. Then, incorporating these improved algorithms into the element-free Galerkin method and the Galerkin boundary node method, the computational efficiency of these meshless methods will be improved. To further improve the computational efficiency, adaptive algorithms of these meshless methods and their coupling with the finite element and boundary element methods will also be researched. Since the absent mathematically theoretical foundation embarrassed the in-depth development of meshless methods to a certain extent, this project will also research the stability and approximate error of these aforementioned improved algorithms of the MLS, and the error estimates of the associated meshless methods. This project will provide efficient meshless algorithms for numerical solutions of boundary value problems in science and engineering and will enrich the mathematical theory of meshless methods. Therefore, this project will advance the in-depth development of meshless methods.

无网格法采用基于点的近似，能克服传统数值方法对网格单元的依赖。形函数是无网格法的基石，移动最小二乘近似是构造形函数的重要方法，但形成的法方程矩阵可能病态或奇异，且需求逆，稳定性差和计算量大，导致相应无网格法计算效率低。 本项目旨在发展一些技术来克服这些缺点，首先利用正交基函数和复数等理论建立移动最小二乘近似的改进算法，使得法方程矩阵不病态奇异，也不用求逆，然后将改进算法融入无单元Galerkin法和Galerkin边界点法等无网格法中来提高计算效率，还将研究这些无网格法的自适应算法以及与有限元和边界元法的耦合算法来进一步提高效率。 由于误差估计等数学理论的缺乏在一定程度上限制了无网格法的发展应用，本项目还研究移动最小二乘近似的改进算法的稳定性和逼近误差、以及相应无网格法的误差等理论。 本项目将为数值求解科学工程问题提供高效的无网格算法，同时将完善无网格法的数学理论，促进无网格法的深入发展。

移动最小二乘近似是一种构造无网格方法形函数的重要方法，为了提高基于移动最小二乘近似的无网格方法的计算效率，本项目开展了如下研究工作：任意维移动最小二乘近似和改进移动最小二乘近似的逼近误差和稳定性，二维复变量移动最小二乘近似的性质、逼近误差和稳定性，三维空间区域边界曲面上的复变量移动最小二乘近似及其逼近误差，改进无单元Galerkin法和复变量无单元Galerkin法的误差和收敛性，改进移动最小二乘近似、复变量移动最小二乘近似和插值型移动最小二乘近似在Galerkin边界点法中的应用和相应的误差理论，改进无单元Galerkin法和复变量无单元Galerkin法的自适应算法及其与有限元和边界元方法的耦合算法，求解二维Stokes方程和广义Oseen方程等不可压缩流体力学方程的无单元Galerkin法等等。本项目现已发表或录用学术论文32篇，其中SCI论文29篇。本项目建立了一些较为高效的移动最小二乘近似无网格算法及相关误差、收敛性和稳定性等数学理论，丰富了现有无网格方法的算法和理论，有助于促进无网格方法的研究进展，具有重要的理论和应用价值。