Hou Songming, Li Zhilin, Wang Liqun, Wang Wei
Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA, 71272, USA.
Commun Comput Phys. 2012;12(2):595-612. doi: 10.4208/cicp.160910.130711s. Epub 2012 Feb 20.
Solving elasticity equations with interfaces is a challenging problem for most existing methods. Nonetheless, it has wide applications in engineering and science. An accurate and efficient method is desired. In this paper, an efficient non-traditional finite element method with non-body-fitting grids is proposed to solve elasticity equations with interfaces. The main idea is to choose the test function basis to be the standard finite element basis independent of the interface and to choose the solution basis to be piecewise linear satisfying the jump conditions across the interface. The resulting linear system of equations is shown to be positive definite under certain assumptions. Numerical experiments show that this method is second order accurate in the L(∞) norm for piecewise smooth solutions. More than 1.5th order accuracy is observed for solution with singularity (second derivative blows up) on the sharp-edged interface corner.
对于大多数现有方法而言，求解带有界面的弹性方程是一个具有挑战性的问题。尽管如此，它在工程和科学领域有着广泛的应用。人们期望有一种准确且高效的方法。在本文中，提出了一种采用非拟合网格的高效非传统有限元方法来求解带有界面的弹性方程。其主要思想是选择与界面无关的标准有限元基作为检验函数基，并选择满足跨界面跳跃条件的分段线性函数作为解基。在某些假设下，所得的线性方程组被证明是正定的。数值实验表明，对于分段光滑解，该方法在(L(\infty))范数下具有二阶精度。对于在尖锐边缘界面角处具有奇异性（二阶导数发散）的解，观察到其精度超过(1.5)阶。
