三维弹性问题高次有限元方程的代数多层网格法

Algebraic multigrid method for higher-order finite element equations in three dimensional linear elasticity

有限元法是数值求解三维弹性问题的一类重要的离散化方法,高次有限元又是其中的一类常用有限元。由于高次元对问题具有更好的逼近效果及具有某些特殊的优点,如能解决弹性问题的闭锁现象(Poisson’s ratiolocking),使得它们在实际计算中被广泛使用。但与线性元相比,它具有更高的计算复杂性。通过分析高次有限元空间与线性有限元空间之间的关系,提出了一种求解三维弹性问题高次有限元方程的两水平方法,然后,通过调用现有的代数多层网格法求解粗水平方程,建立了求解高次有限元方程的AMG法。数值实验表明,本文设计的AMG法对求解三维弹性问题高次有限元方程具有很好的计算效率和鲁棒性。

As for finite element method, the higher-order elements are often used in that they are superior and necessary under certain conditions over the low-order ones, for example, they can overcome the Poisson＇s ratio locking. However, they have much higher computational complexity than the linear elements. In this paper, we firstly introduce this method for elliptic problems, to the solution of three dimensional elasticity problems discretized using higher-order elements and propose a two-level method by algebraic approaches. With the existing algebraic multigrid（L_AMG） method used as a solver on the first coarse level, an AMG method is then designed for high-order discretizations. The results of various numerical experiments show that the resulting AMG method is more robust and efficient for the solution of higher-order finite element equations in three dimensional linear elasticity.