空间各向异性弹性问题的八节点理性单元

常规单元的插值函数通常仅考虑单元的几何形状与节点位置，而忽略了反映物理问题关键特性的物性参数，从而降低了其数值分析的效果。相反，理性有限元法是取问题微分控制方程的多项式基本解作为单元内的插值函数，其所形成的刚度阵与问题的物性参数紧密相关，因此它避免了常规有限元法对物理问题和数学问题的割裂，可显著提高数值分析的稳定性和精度。本文利用空间各向异性问题的基本解，构造出满足分片实验要求的八节点理性块体单元。数值算例表明，本文给出的理性单元不仅具有较高的求解精度，而且具有良好的数值稳定性，尤其是对较为畸形的单元反应不敏感。

A8-Node rational element for 3D anisotropic elastic problem

For conventional finite element,only the geometry and node locations are considered in the interpolation functions,while the physical parameters which reflect the key features of the physical problems is ignored,so its numerical performance may not satisfied.In contrast,for rational finite element method,the fundamental solutions of the differential equations are taken as the interpolation function and so the resulting stiffness matrix is related closely to the physical parameters of the problem.Therefore,for rational finite element,the separation between the mathematical and physical problems in the conventional finite element method can be avoided,and so the stability and accuracy of numerical analysis can improve significantly.In this paper,an eight-node brick rational element,which satisfies the requirements of the patch test,is constructed using the fundamental solution of the 3D anisotropic problems.Numerical examples show that the rational element gives numerical results with not only high accuracy,but also good numerical stability,especially it is insensitive for the ill-shaped meshes.