弱有限元方法在线弹性问题中的应用

本文考虑弱有限元（简称WG）方法在线弹性问题中的应用.WG方法是传统有限元方法的推广，用于偏微分方程的数值求解.和传统有限元一样，它的基本思想源于变分原理.WG方法的特点是使用在剖分单元内部和剖分单元边界上分别有定义的分片多项式函数（即弱函数）作为近似函数来逼近真解，并针对弱函数定义相应的弱微分算子代入数值格式进行计算.除此之外，WG方法允许在数值格式中引进稳定子以实现近似函数的弱连续性.WG方法具有允许使用任意多边形或多面体剖分，数值格式与逼近函数构造简单，易于满足相应的稳定性条件等优点.本文考虑WG方法在求解线弹性问题中的应用.围绕线弹性问题数值求解中常见的三个问题，即：数值格式的强制性，闭锁性，应力张量的对称性介绍WG方法在线弹性问题求解中的应用.

WEAK GALERKIN FINITE ELEMENT METHOD FOR LINEAR ELASTICITY PROBLEMS

This article considers the application of the weak Galerkin finite element (WG) method to linear elasticity problems. The WG method is a generalization of the traditional finite element method, which is used to solve numerical solutions of partial differential equations. In WG, the weak function, a piecewise polynomial function that is defined both inside the element and on the boundary of the element, is used as an approximate function and weak differential operators are given correspondingly. Moreover, stabilizers are introduced to keep the weak continuity of the approximate function. In the WG method, partitions could be arbitrary polygons or polyhedrons that satisfies the shape regular conditions. In addition the numerical format and the approximate function are easy to construct. In this paper, we introduce the application of the WG method in solving linear elasticity problems by solving three common problems in the numerical methods for linear elasticity problems, namely: the coerciveness, locking property, and the symmetry of stress tensor.