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.      

拟协调九参数三角形板单元的性态分析

本文利用九参数三角形板单元,对拟协调元方法作了一系列的性态分析。文中着重讨论了单元内和单元边界上位移插值函数对收敛性的影响;论述了选择假设应变场的原则。对于引入场变量平衡条件时单元的性质及拟协调元和杂交元的联系和区别进行了分析,同时并给出了用拟协调元方法构造杂交应力模型的一般列式。     

论九参数拟协调元的扩展

九参数拟协调有限元是唐立民等人于1980年提出的一种计算简便的薄板单元。该单元具有较好的收敛性。本文利用新的插值观重新分析九参数拟协调元,得出拟协调元的形函数空间。同时,构造出一类具有较好收敛性的新有限元,其形函数空间有明显的表达方式。     

论九参数拟协调元的扩展

九参数拟协调有限元是唐立民等人于１９８０年提出的一种计算简便的薄板单元。该单元具有较好的收敛性。本文利用新的插值观重新分析九参数拟协调元，得出拟协调元的形函数空间，同时，构造出一类具有较好收敛性的新有元，其形函数空间有明显的表达方法。     

板的多变量有限元分析方法

本文论述了修正的 Hu-Washizu广义变分原理和拟协调列式对板的有限元分析,以及修正 Hu-Washizu广义变分原理与拟协调列式的关系,并以此方法构造一个Mindlin类型板单元。数值结果表明,似乎此法更适合Mindlin类型厚-簿通用板弯曲单元的构造。     

平面理性元的收敛性证明

理性元直接在物理面内列式，并用微分方程的解插值，不用等参技术而在计算面内用多项式插值．由于其解析的特性，即使是不协调元也可证明其收敛性．本文的证明采用力学方法，故易于为力学工作者所接受，且可用于多种单元的结构．收敛性证明可给理性有限元以坚实的理论基础．     

用于几何非线性分析的内参型非协调元法

本文把内参型非协调元的理论和方法推广应用于几何非线性分析。对方法的合理性、可行性、单元收敛条件及实用简化方案等诸方面作了初步探讨。文章以大位移平面问题为例进行了详细讨论,给出了有限元非线性列式的全过程和数值结果。     

带旋转自由度的精化非协调平面四边形等参元

对于C~1类不协调元文[1]提出了一种精比直接刚度法。本文进一步将其应用到C~0类问题,建立一种带旋转自由度的不协调平面四边形单元,其协调部分是用人Allman插值法建立的单元函数,不协调部分用了四个内部自由度。该单元能保证通过分片试验,保持了单变量有限元列式简单、性能可靠(无多余零能模式及坐标不变性)等长处,同时,还具备多变量有限元(杂交/拟协调元)高精度的优点。算例表明,本文提出的单元收敛、可靠、高精度且高效率。     

三维电磁场辛有限元列式及电磁共振腔的计算

针对三维共振腔的电磁场分析,利用Maxwell方程的对偶方程体系形式,从其相应的对偶变量变分原理出发,导出了三维电磁场辛有限单元的详细列式。为了有限元列式的保辛,变分原理被积函数可导向对于对偶变量为对称的形式。变分原理的边界积分项对于相邻单元相互抵消。由于采用了对偶变量的插值函数,使得电磁场单元构造可以在层面上进行,从而避免了所谓的连续性问题。无物理意义的零本征解可采用奇异值分解加以排除。文末分别对矩形及圆柱形的共振腔做了数值计算并与解析解和棱边元计算结果进行对比,算例表明了列式及算法的有效性。     

弱连续条件下的九参三角形板元

首先对薄板弯曲平衡方程的弱形式进行了推导,导出保证单元收敛的弱协调条件,即三角形顶点函数值连续和三边的法向导数积分连续这两个条件;对比拟协调元、广义协调元和双参数法中所使用的3个积分连续条件,本条件更弱;再对这3个积分协调条件的构成方法进行了总结和分析,现有采用积分连续条件构造的有限元大都采用了这些构成方法.采用弱协调条件构造有限元,比原来的构造范围更广,井以此构造出几种单元作为算例.采用这种构成法还可构造多种单元,它们都具有采用最小势能原理法构成有限元的简便的优点,并在任意网格下收敛到真解.     

LINEAR AND GEOMETRICALLY NONLINEAR ANALYSIS WITH 4-NODE PLANE QUASI-CONFORMING ELEMENT WITH INTERNAL PARAMETERS

A linear 4-node quadrilateral quasi-conforming plane element with internal parameters is proposed. The element preserves advantages of the quasi-conforming technique, including an explicit stiffness matrix, which can be applied to nonlinear problems. The weak patch test guarantees the convergence of the element. Then the linear element is extended to the geometrically nonlinear analysis in the framework of Total Lagrangian(TL) formulation. The numerical tests indicate that the present element is accurate and insensitive to mesh distortion.     

基于最佳质量网格的薄板问题的非协调流形方法

对于大部分非协调板单元,使用规则网格能得到很好的效果。但是,当网格不规则时,非协调元的数值特性将变得很差,甚至收敛性得不到保证。为解决网格依赖性问题,许多专家学者提出了改造单元,如拟协调元法和广义协调元法,这些方法能解决收敛性问题,但是数值实践证明没有一种单元能在所有情况下都具有良好的数值特性。考虑到流形方法采用两套完全独立的覆盖系统,可以用规则的数学网格来作为数学覆盖进行插值,取得最佳的插值效果,单元收敛性便能得到保证。再结合适用于流形方法的变分提法,建立起流形方法处理非规则物理边界非协调板单元的一般格式。以ACM薄板单元为例,与ANSYS、拟协调元法和广义协调元法进行了对比,证明本文方法在处理具有曲线边界的薄板弯曲问题时具有收敛快和精度高等优势。     

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-)     

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。     

The curved beam element and its convergence rate

The quasi-conforming element of the curved beam and shallow curved beam is given in this paper. Numerical examples illustrate that the quasi-conforming elements of the curved beam and shallow curved beam which is used to approximate the curved beam have better accuracy than the straight beam element. The curved beam element constructed by displacement method can not satisfy rigid body motion condition and the very fine grids have to be used in order to satisfy rigid body motion condition approximately.In this paper it is proved that the straight beam element and the quasi-conforming element of the curved beam and shallow curved beam, when element size is reduced infinitely, have convergence rate with the same order O(l²) and when regular elements are used l is the element length.The Project Supported by National Natural Science Foundation of China.     

Characterization of macroscopic tensile strength of polycrystalline metals with two-scale finite element analysis

The objective of this contribution is to develop an elastic-plastic-damage constitutive model for crystal grain and to incorporate it with two-scale finite element analyses based on mathematical homogenization method, in order to characterize the macroscopic tensile strength of polycrystalline metals. More specifically, the constitutive model for single crystal is obtained by combining hyperelasticity, a rate-independent single crystal plasticity and a continuum damage model. The evolution equations, stress update algorithm and consistent tangent are derived within the framework of standard elastoplasticity at finite strain. By employing two-scale finite element analysis, the ductile behaviour of polycrystalline metals and corresponding tensile strength are evaluated. The importance of finite element formulation is examined by comparing performance of several finite elements and their convergence behaviour is assessed with mesh refinement. Finally, the grain size effect on yield and tensile strength is analysed in order to illustrate the versatility of the proposed two-scale model.     

Anisotropic rectangular nonconforming finite element analysis for Sobolev equations

An anisotropic rectangular nonconforming finite element method for solving the Sobolev equations is discussed under semi-discrete and full discrete schemes. The corresponding optimal convergence error estimates and superclose property are derived, which are the same as the traditional conforming finite elements. Furthermore, the global superconvergence is obtained using a post-processing technique. The numerical results show the validity of the theoretical analysis.     

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.