非线性有限元的若干基本问题

本文介绍了非线性有限元中的若干基本问题。其中包括有关应变、应力和非线性平衡方程的一些基本概念,基于不同非线性广义变分原理的位移模式、杂交模式和拟协调模式几何非线性有限元及其在壳体屈曲问题中的应用等。      

The displacement function of quasi-conforming element and its node error

Based on the strain formulation of the quasi-conforming finite element, displacement functions are constructed which have definite physical meaning, and a conclusion can be obtained that the coefficients of the constant and the linear strain are uniquely determined, and the quasi-conforming finite element method is convergent to constant strain. There are different methods for constructing the rigid displacement items, and different methods correspond to different order node errors, and this is different from ordinary displacement method finite element. Contributed by TANG Li-min Biography: HE Dong-sheng (1964-)     

压电层合梁静动力学分析的高精度梁单元

给出了一个对复合材料压电层合梁进行数值分析的高精度压电层合梁单元。基于Shi三阶剪切变形板理论的位移场和Layer-wise理论的电势场,用力-电耦合的变分原理及Hamilton原理推导了压电层合梁单元列式。采用拟协调元方法推导了一个可显式给出单元刚度矩阵的两节点压电层合梁单元,并应用于压电层合梁的力-电耦合弯曲和自由振动分析。计算结果表明,该梁单元给出的梁挠度和固有频率与解析解吻合良好,并优于其它梁单元的计算结果,说明了本文所给压电层合梁单元的可靠性和准确性。研究结果可为力-电耦合作用下压电层合梁的力学分析提供一个简单、精确且高效的压电层合梁单元。     

A triangular quasi-conforming finite element for transient dynamic analysis

A type of 3 node triangular element is constructed by the Quasi-conforming method, which may be used to solve the equation of a type of inverse problem of wave propagation after Laplace transformation Δu−A ² u=0. The strains in the element are approximated by an exponential function and the string-net function between neighbouring elements is approximated by one dimensional general solution of the equation. Furthermore the strain field satisfies the equation, and therefore in the derivation of the element formulation, no shape function is needed. In this sense, it is a kind of hybrid element. Compared with the ordinary linear triangular element, the new one features higher precision with coarse meshes. Some numerical tests are presented. The project is supported by the National Natural Science Foundation of China.     

QUASI-CONFORMING FINITE ELEMENT APPROXIMATION FOR A FOURTH ORDER VARIATIONAL INEQUALITY WITH DISPLACEMENT OBSTACLE

In this paper, unconventional quasi-conforming finite element approximation for a fourth order variational inequality with displacement obstacle is considered, the optimal order of error estimate O(h) is obtained which is as same as that of the conventional finite elements.     

Exact geometry based quasi-conforming analysis for Euler-Bernoulli beam

The problem of quick analysis using exact geometry data was proposed by Hughes et al. and the isogeometric analysis framework was introduced as a solution. In this letter, the exact geometry concept is combined into the quasi-conforming framework and a novel method, i.e., the exact geometry based quasi-conforming analysis is proposed. In present method the geometry is exactly described by non-uniform rational B-spline bases, while the solution space by traditional polynomial bases. Present method combines the merits of both isogeometric analysis and quasi-conforming finite element method. In this letter Euler-Bernoulli beam problem is solved as an example and the results show that the present method is effective and promising.     

Nonlinear quasi-conforming finite element method

The nonlinear quasi-conforming FEM is presented based on the basic concept of the quasiconforming finite element. First, the incremental principle of stationary potential energy is discussed. Then, the formulation process of the nonlinear quasi-conforming FEM is given. Lastly, two computational examples of shells are given.     

非线性多类变量有限元的进展

本文介绍了非线性多类变量有限元,即非线性的杂交元和拟协调元近几年来的发展及其变分基础.     

拟协调等腰梯形壳元显式几何刚度阵及屈曲分析

采用拟协调元方法推导了等腰梯形薄壳元的显式几何刚度阵，用于组合结构屈曲分析的计算，结果表明，这种单元的几何刚度阵收敛快、精度好。     

用拟协调元直接构造平面任意四边形单元——进入有限元的禁区

利用拟协调元方法,在直角坐标系下直接构造了一族平面任意四边形单元,对其收敛性进行了分析,并与平面等参元进行了对比研究.结果证明平面任意四边形单元可采用多项式基函数直接列式,并可以保障单元的收敛性;拟协调元列式可以使平面问题的有限元方法得到统一.与平面等参元相比,单元列式简单,性能稳定,具有显式的刚度阵,计算量小,这说明对于有限元平面问题拟协调元是一个更正确、有效的做法.