伽辽金型无网格法的数值积分方法

无网格法直接通过节点信息构造形函数,不依赖于节点之间的有序单元连接,能够建立任意高阶连续的整体协调形函数.与传统的有限元法相比,无网格法对大变形问题、移动边界问题和高阶问题的求解有比较明显的优势.伽辽金型无网格法是目前应用最为广泛的一类无网格法.虽然无网格形函数本身不依赖于单元,但伽辽金型无网格法需要采取合适的方法进行弱形式的数值积分.由于无网格形函数一般不是多项式,具有非插值性且影响域与背景积分网格通常不重合,伽辽金型无网格法通常需要采用高阶的高斯积分进行数值积分,导致了计算效率低下,难于求解大型实际问题.因此,如何通过建立高效积分方法提高无网格法的计算效率成为无网格法研究领域的一个核心问题.论文总结了伽辽金型无网格法中若干常用的数值积分方法,并对伽辽金型无网格法的数值积分方法领域存在的一些问题进行了探讨.

A review of numerical integration approaches for Galerkin meshfree methods

Meshfree methods are capable of constructing arbitrary order smoothing and compatible shape functions only through unstructured nodes and do not reply on elements with specific connectivity. Compared with the conventional finite element methods, meshfree methods show obvious advantages in modeling large deformation, moving boundary and higher order problems. Galerkin meshfree methods are one class of most widely used meshfree methods. Although no elements are required for the shape function construction, Galerkin meshfree methods do need some kind of background cells to perform the weak form integration. Due to the rational nature and overlapping characteristics of meshfree shape functions, commonly higher order Gauss quadrature is necessary for the numerical integration of Galerkin meshfree methods, which leads to low computational efficiency and considerable difficulty to model large scale practical problems. Consequently, the development of efficient and robust integration algorithms has been an important topic for Galerkin meshfree methods. This paper first briefly discusses the imposition of essential boundary conditions, and then presents a detailed review on some typical numerical integration approaches for Galerkin meshfree methods, where the characteristics for various integration algorithms are outlined.