摘要
有限单元法是目前最为广泛使用的数值分析方法,能够有效求解各类工程和科学问题。但在一些需要求单元内部场变量的情况下,传统形函数插值方法往往不能保证其结果精度。针对此问题,提出了一种高精度求解方案,基本思路是将有限单元与Taylor展开相结合,建立并求解单元内部未知场量关于已知单元结点场量的线性方程组,各方程中系数取决于点的相对位置关系。研究发现:所提方案算法简单,可适用于多维度、多阶次单元,在非线性场和高阶场中性能卓越,是一种高效高精度的新型求解算法,可服务于非线性计算的中间步骤,及满足高精度后处理的需求。该方法并不局限于有限元方法,可推广用于包括无网格方法在内的各类离散数值方法。
Finite element method(FEM) is the most widely used numerical analysis method, which is an effective solution to all kinds of engineering and scientific problems. However, the interpolation based on shape function often does not have desired accuracy in cases where the internal field variables of the finite element need to be solved. To deal with this kind of problems, a high-precision solution is proposed, and the basic idea of the solution is the combination of finite element and Taylor expansion. The value of internal field is obtained by establishing and solving linear equations of unknown field variables in the element with respect to known field values at element nodes. The coefficients of each equation depend on the relative position of the points. The results illustrate that the proposed method, which implement a relatively simple algorithm, is applicable to multi-dimension and multi-order elements. It also shows an excellent performance in the application of nonlinear and high-order field functions. This new solution algorithm with high computational efficiency and high precision is expected to serve the intermediate steps of nonlinear computation and satisfy the requirements of high precision post-processing. The method is not limited to FEM and can be extended to all kinds of discrete numerical methods including meshless method.
作者
次惠岭
白冰
雷宏武
崔银祥
CI Hui-ling;BAI Bing;LEI Hong-wu;CUI Yin-xiang(State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences,Wuhan,Hubei 430071,China;University of Chinese Academy of Sciences,Beijing 100049,China;Hubei Key Laboratory of Environmental Geotechnology,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences,Wuhan,Hubei 430071,China)
出处
《岩土力学》
EI
CAS
CSCD
北大核心
2021年第11期3137-3146,共10页
Rock and Soil Mechanics
基金
国家自然科学基金项目(No.41972316)。
关键词
有限单元法
场量
TAYLOR展开
高精度
finite element method
field quantity
Taylor expansion
high precision
