基于扩展单元插值法二维弹性问题的边界元法分析

本文把一种新型的插值方法-扩展单元插值法,用于二维弹性问题的边界元法求解。扩展单元是在原非连续单元两端添加虚节点,将非连续单元变成阶次更高的连续单元。原非连续单元的内部点被称为源节点,其形函数用来构建源节点和虚节点之间的关系,被称为RawShape。扩展单元的形函数是由源节点和虚节点构造,用于边界物理变量的插值, 称之为FineShape。扩展单元继承了连续和非连续单元的优点,同时克服了它们的缺点；既可以插值连续场,也可以插值非连续场,在不改变方程自由度的前提下(边界积分方程只在源点处配置),把插值精度提高了至少两阶,最大限度的发挥了边界积分方程试函数可以不连续的特性。最后通过数值算例来验证本文方法的精度和收敛性。

Boundary element analysis for 2D elasticity problems based on expanding element interpolation method

Boundary element analysis of 2D elasticity problems by a new expanding element interpolation method is proposed in this paper. The expanding element is made up based on a traditional discontinuous element by adding virtual nodes along the perimeter of the element. Its shape functions constructed on both source nodes and virtual nodes are referred as fine shape functions and boundary variables are interpolated by the fine shape functions. The internal nodes of the original discontinuous element are referred as source nodes and its shape function as raw shape function. The raw shape functions are used to provide additional constraint equations between variables on virtual nodes and source nodes. The expanding element inherits the advantages of both the continuous and discontinuous elements while overcomes their disadvantages. With the expanding element, both continuous and discontinuous fields on the domain boundary can be accurately approximated, and the interpolation accuracy increases by two orders compared with the original discontinuous element. While the boundary integral equations are collocated at source nodes, the size of the final system of linear equations has not change. In addition, the expanding elements take full advantages of the characteristic that the trial function of the boundary integral equation can be discontinuous. At last, a few numerical examples are presented to verify the accuracy and convergence of the proposed method.