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中文摘要：
常规有限元方法的插值函数通常仅仅从数学层面上考虑单元的几何性质，忽视了与物理问题相关的物性参数，因此可能降低数值分析的效果．理性有限元的构造方法与常规有限元法不同，其插值函数使用的是控制微分方程解析解的线性组合，求解过程是在物理域内直接列式，对单元的应变场和应力场同时进行插值，并在单元级别考虑分片实验的要求并直接进行修正，最终形成与问题物性参数紧密相关的单元刚度阵．该方法避免了传统方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．利用空间各向异性问题的基本解，从最小势能原理出发，构造出两种满足分片实验要求的二十节点理性块体单元．数值算例表明，所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．
英文摘要：
For the conventional finite element method, only the geometry and node locations of elements were considered in the interpolation functions, while the physical parameters which reflect the key features of the physical problems were ignored, so its numerical performance may be not satisfying in some cases. The construction of the rational finite element method was different from that of the conventional finite element method. The linear combinations of the fundamental solutions to the problem＇ s controlling differential equations were used as the inter- polation functions, so the stress and strain fields were interpolated directly in the physical do- main at the same time. The transfer matrix was modified at the element level to pass the patch test, and the resulting element stiffness matrix was related closely to the physical parameters of the problem. The rational finite element avoids the separation between the mathematical and physical aspects of a problem, so the stability and accuracy of numerical analysis could be im- proved significantly. Two kinds of 20-node rational brick elements based on the principle of minimum potential energy and satisfying the requirements of the patch test, were constructed according to the fundamental solutions to general 3D anisotropic problems. Numerical examples show that the rational elements give numerical results with not only high accuracy, but also good numerical stability.
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