一种平面问题的加权Nitsche间断伽辽金有限元法

以经典间断伽辽金有限元法求解弹性力学界面问题,存在着由于稳定系数取值不当引起的数值不稳定问题,而加权Nitsche间断伽辽金有限元法可以缓解这种问题,但仅应用于常量单元离散的情况。为解决上述问题,基于加权Nitsche间断伽辽金有限元法,针对平面弹性力学问题,推导了四节点四边形单元离散情况下的加权系数和稳定参数的计算公式,建立了权重与稳定参数间的定性依赖关系。通过建立和求解广义特征值问题,实现了加权系数和稳定参数的自动计算,使得高阶单元的使用成为可能。通过数值试验检验了方法的收敛性和稳定性。结果表明:在求解均匀或材料分区不均匀介质问题时,加权Nitsche间断伽辽金有限元法均表现出良好的稳定性,且计算结果具有较高的精度。所提出的方法在一定程度上无须人工干预,具有高效率、高精度和良好的稳定性,可以应用于复杂界面问题。

A weighted Nitsche discontinuous Galerkin finite element method for plane problems

The classical discontinuous Galerkin finite element method has the unstable numerical problem resulting from the inappropriate stability parameter for elasticity problem with interfaces. This problem can be released by the weighted Nitsche discontinuous Galerkin finite element method, but only for constant elements. To solve the above problems, the weights and the stabilization parameters of the weighted Nitsche discontinuous Galerkin finite element method are derived with four-node quadrilateral elements discretization for plane elasticity problems, and a qualitative dependence between the weights and the stabilization parameters is established. The weights and the stabilization parameters are evaluated automatically by setting up and solving generalized eigenvalue problems. This makes the use of high-order elements possible. The convergence and stability of the proposed method are verified through numerical examples. The results show that the weighted Nitsche discontinuous Galerkin finite element method has good stability and high accuracy for both homogeneous and heterogeneous problems in the material partition. In some extent, the method needs less manual work, and has high efficiency, high stability and better accuracy, making it suitable for solution of complicated interface problems.