基于局部Petrov-Galerkin过程的自然单元法及其面向对象的设计与实现

NATURAL ELEMENT METHOD BASED ON LOCAL PETROV-GALERKIN PROCEDURE AND ITS OBJECT-ORIENTED DESIGN AND IMPLEMENTATION

提出了一种能够用于实际岩土及地下工程全自动分析计算的数值计算新方法——局部自然单元法。该方法利用求解点的自然邻结点和Voronoi结构构造整体近似位移函数,且在求解区域的局部Delaunay三角形子域上使用局部Petrov-Galerkin过程建立整体求解的系统平衡方程,其积分可在背景三角形网格的形心上采用一点数值积分得到。同时,利用对象设计思想建立了局部自然单元法的自动建模类、分析求解类、施工过程控制类及其相关类和方法,为进一步研究和系统开发工作提供了一个易于维护和扩展的基本系统框架。数值算例结果表明,理论正确,方法有效。

The local natural element method (LNEM) which can be used in the automatic analysis of geotechnical engineering problems is implemented for solving the elasticity problem. In LNEM, the whole interpolations are constructed with respect to the natural neighbour nodes and Voronoi tessellation of the given point, and a local weak form over the local Delaunay triangular sub-domain is used to obtain the discretized system of equilibrium equation. As a result, the numerical integration of the equilibrium equation can be analytically calculated at the center of the background triangular quadrature meshes. The design and management of the object about LNEM are also studied by abstracting the data and procedure of LNEM as auto-model building class, analysis class, control class, and other relevant classes and methods. Numerical results indicate that the theory and programs are accurate and effective.