线弹性问题在交错网格上的有限差分算法  

THE FINITE DIFFERENCE METHOD ON STAGGERED GRIDS FOR LINEAR ELASTIC PROBLEMS

作　　者：杨博涵 Yang Bohan(School of Mathematics,Shandong University,Jinan 250100)

机构地区：[1]山东大学数学学院,济南250100

出　　处：《高等学校计算数学学报》2023年第1期1-15,共15页Numerical Mathematics A Journal of Chinese Universities

摘　　要：1引言对线弹性材料的形变分析在工程实践中有着广泛应用,而数值方法是求解该问题的一种重要方法[1].其中,低阶有限元方法常用来处理可压缩线弹性问题[2,3].但是当材料几乎不可压缩时(拉梅常数λ→∞),普通有限元方法的收敛性会被破坏[4].这种现象被称为”locking”.基于交错网格的有限差分方法是科学计算和数值分析领域中的一种有效方法,被广泛应用于流体问题、多孔介质流问题以及麦克斯韦方程组等.The finite difference method on staggered grids is an effective numerical scheme for solving linear elastic problems.Its first-order uniform convergence aboutλ has been proved.In this paper,we introduce specific discrete interpolations to analyze the convergence.We obtain the second-order convergence in the discrete l~2norm.We also obtain the second-order convergence for some terms of the discrete h~1 norm,and the other terms are second-order convergent on uniform grids.The methods and theoretical results can be extended to three-dimensional problems.Finally,numerical experiments show that the numerical results are consistent with the theoretical analysis.

关 键 词：麦克斯韦方程组 交错网格 有限元方法 线弹性问题 线弹性材料 有限差分方法 数值分析 形变分析 

分 类 号：O241.82[理学—计算数学]