具有几何对称性的12参数矩形板元

1 引言 三角形板元中,形式最简单的是九参数元,节点参数是单元三个顶点上的函数值和两个一阶偏导数值,非协调九参三角形板元的研究取得了丰硕成果,根据不同方法已构造出多种收敛性能好的单元.相比之下,矩形板元的研究较少见报道.矩形板元中形式最简单的是12参元,节点参数是单元4个顶点上的函数值和两个一阶偏导数值,这类似于九参三角形板元.常见的12参矩形板元是ACM元,其形函数空间是完整3次多项式空间加上两个4次多项式的基函数,ACM元是C°元,但位移形函数的外法向导数平均值在单元间不连续,这类似于Zienkiewicz九参三角形板元,但由于矩形单元的特殊形状,ACM元是收敛的.龙驭球教授等在[1]中提出一种12参矩形广义协调元,其位移形函数的外法向导数平均值在     

关于几种三角形板元等价性的注记

1．引言求解薄板弯曲问题的广义协调元法近年来有了发展,本文讨论几种三角形板元之间的等价关系．我们证明,[2]中构造的GCⅢ－T9元等价于Specht元问．同一作者在[4]中构造的GCⅡ－T9元实际上也等价于Specht元,因此,它们彼此等价．另外,[7]中用对称列式构造的一个广义协调元是和作者在问中所引进的单元等价．例已指出,[7]中的那个元等价于问中的VZ1元,所以[7]和[6]中的两个广义协调元均等价于已知的VZ1元．2．广义协调GCⅢ－T9单元设三角形单元K的三个顶点依反时针方向分别为p i（xi,yi）,节点参数是pi点处的函数值及其一阶导…     

位移障碍下四阶变分不等式问题的非协调有限元一般误差估计式

1.引 言 关于二阶变分不等式问题的非协调有限元逼近已有大量研究[1-5].但是,对于四阶变分不等式的研究相对而言较少[6-7].[8,9,10]给出了位移障碍问题的非协调有限元,包括C0元(如Zienkiewicz元及Adini元)和非C0元(如Morley元及De Veubeke元)逼近的理论分析及最优误差估计.经过仔细分析发现,其成功的关键技巧是充分利用上述单元的一个     

Specht九参数板元的分析

Zienkiewicz 九参数的不协调三角形板元以单元顶点上的函数值和两个一阶偏导数值作为节点参数,形式简单,总体自由度少,但它只对特殊剖分才收敛,不便于实际应用.后来相继出现一些对 Zienkiewicz元的改进形式,如拟协调元,TRUNC元,广义协调元等.这是一类非常规的有限元,均为三结点九参数三角形元.从常规有限元     

MITC元的分析

1 引言 有限元求解厚薄板通用的R-M(Reissner-Mindlin)模型板问题,单元只需具有C°连续性,这一点优于需具有C~1连续性的Kirchhoff模型薄板单元.但是当板厚趋于零时,通常的低阶C°元却不收敛,这就是所谓的Locking现象.Bathe和Brezzi等将R-M板模型转化成2阶椭园问题与Stokes问题的耦合形式,据此提出求解R-M板问题的混合扦值单元MITC~([1]、[2]、[3]):设挠度ω的形函数空间是W,转角β=(βx,βy)的形空间是B,在计算剪切应变时,分别将βx,βy按不同方式投影到空间和.数值结果表明这类单元具有很好的收敛性.本文分析MITC元,导出投影算子的显表达式,根据[5]关于Locking现象的一个数学分析,证明当板厚趋于零时,投影算子的选取方式使剪切应变部分对应于特定点上的Kirchhoff条件,引起Locking现象的因素被消除,从而显式证明MITC元避免了Locking 现象. 2 MITC元的整体性质 考虑R-M板弯曲问题,求挠度,转角,使下列板的能量泛函达极小: (1) (2) 其中E是杨氏模量,υ是Possion比,0<υ<1/2,t是板厚,k是剪力校正因子,Ω是板的中面占有的平面区域,f是横向荷载.(1)的第一项是弯曲应变能,第二项是剪切应变能. 设有限元空间是W_h×B_h,W_hH_0~1(Ω),B_h[H_0~1(Ω)]~2,J_h是Ω的单元部分,Ω=K,K是单元,对(1)的直接离散是求(     

关于不完全双二次非协调板元的收敛性

多年来,工程界普遍认为Irons的分片检验准则是检验非协调元收敛性的一个充要条件。作者在[3,4]中曾对三类四边形无证明了非协调元可以不通过分片检验仍然收敛,可见分片检验并非必要。最近,吴茂庆在[5]中给出了一个八个自由度的不完全双二次矩形板元,其形状函数由矩形四个角点上的函数值与四边中点上的法向导数值确定.这是一个非协调元,形状函数及其一阶偏导数在相邻单元的共同边界上不连续,有点象Morley元.[5]称此非协调元不通过分片检验,但却收敛,并给出收敛速度的一个估计:     

一般组合弹性结构的有限元分析

提出一个求解一般组合弹性结构问题的有限元方法:对体件的位移、 板件的纵向位移和杆件的纵向位移用线性协调元离散, 对杆件的纵向转角用二次协调元离散, 对板件的横向位移和杆件的横向位移分别用Morley元和三次Hermite元离散. 在相应的非协调元空间上建立了广义Korn不等式, 进而证得有限元方法的唯一可解性. 最后导出方法在能量模意义下的最优误差估计.     

Zienkiewicz三角板元的极限问题

引言 非协调单元是有限元求解板的弯曲问题的重要方法之一.自从[1]提出Ziekiewicz三角板元以来,此单元成为广泛采用的一种非协调性单元.考虑到三角形的几何形状,此     

二阶问题的一个类Wilson非协调元

§1.引言 Wilson元是工程计算中常用的一种非协调元,数值计算效果很好,但是Wilson元对于任意四边形网格却不能收敛.石钟慈在[1]中限制四边形单元剖分,要求四边形单元满足对角线中点距离d_K=o(h_K~2),而[2]—[3]则修改了双线性形式,即在刚度矩阵元素的计算中采用某种数值积分,这两种方法均使得Wilson元达到收敛.另外,通过改变形状函数,[4]—[5]提出了一个六参数非协调四边形单元QP6,它是推广的Wilson元.此元对任意四边形网格能够收敛,但其单元上的形状函数非常依赖单元本身.     

二阶问题的一个类Wilson非协调元

§1.引言 Wilson元是工程计算中常用的一种非协调元,数值计算效果很好,但是Wilson元对于任意四边形网格却不能收敛.石钟慈在[1]中限制四边形单元剖分,要求四边形单元满足对角线中点距离d_K=o(h_K~2),而[2]—[3]则修改了双线性形式,即在刚度矩阵元素的计算中采用某种数值积分,这两种方法均使得Wilson元达到收敛.另外,通过改变形状函数,[4]—[5]提出了一个六参数非协调四边形单元QP6,它是推广的Wilson元.此元对任意四边形网格能够收敛,但其单元上的形状函数非常依赖单元本身.     

A stabilized mixed formulation for finite strain deformation for low-order tetrahedral solid elements

A stabilized mixed finite element formulation for four-noded tetrahedral elements is introduced for robustly solving small and large deformation problems. The uniqueness of the formulation lies within the fact that it is general in that it can be applied to any type of material model without requiring specialized geometric or material parameters. To overcome the problem of volumetric locking, a mixed element formulation that utilizes linear displacement and pressure fields was implemented. The stabilization is provided by enhancing the rate of deformation tensor with a term derived using a bubble function approach. The element was implemented through a user-programmable element of the commercial finite element software ANSYS. Using the ANSYS platform, the performance of the element was evaluated by comparing the predicted results with those obtained using mixed quadratic tetrahedral elements and hexahedral elements with a B-bar formulation. Based on the quality of the results, the new element formulation shows significant potential for use in simulating complex engineering processes.     

非协调曲边四边形十二自由度平板弯曲单元

本文利用精确元法,给出一个十二自由度曲边四边形板弯曲单元.该方法不需要变分原理,适用于任意正定和非正定偏微分方程.利用这个方法,单元之间的协调条件很容易满足,仅须位移和内力在单元节点上连续,即可保证所得到的解收敛于精确解.利用本文方法所获得的解,无论是位移还是内力可同时有二阶收敛精度.文末给出数值算例.表明了本文所得到的单元有非常好的精度.     

Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems

Natural superconvergence of the least-squares finite element method is surveyed for the one-and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method. The second author was supported in part by the US National Science Foundation under Grant DMS-0612908.     

Vibration and stability analysis of thick orthotropic plates using hybrid-Trefftz elements

This paper discusses the vibration and stability analysis of thick orthotropic plate structures using finite elements based on the hybrid-Trefftz formulation. While the formulation can be used for elements of arbitrary geometry, the paper concentrates on the use of a simple and robust triangular element. The key feature of the formulation is to use element interpolations that are consistent for all values of the plate thickness, including the limit when it goes to zero. This eliminates the locking problem automatically and ensures a robust approximation for thick and thin plates. Results for various problems are included to demonstrate the accuracy and efficiency of the element.     

ON THE SELECTION OF SHAPE FUNCTION SPACES OF TRIANGULAR PLATE ELEMENTS

ＯＮＴＨＥＳＥＬＥＣＴＩＯＮＯＦＳＨＡＰＥＦＵＮＣＴＩＯＮＳＰＡＣＥＳＯＦＴＲＩＡＮＧＵＬＡＲＰＬＡＴＥＥＬＥＭＥＮＴＳＣｈｅｎＳｈａｏｃｈｕｎ；ＳｈｉＤｏｎｇｙａｎｇ（Ｄｅｐｔ．ｏｆＳｙｓ．Ｓｃｉ．＆Ｍａｔｈ．，ＺｈｅｎｇｚｈｏｕＵｎｉｖ．，Ｚｈｅ...     

一种新的声椭球无限单元

提出了一种新的声椭球无限单元·　这种声无限单元基于一种新的声压表达式,这种声压表达式能够更准确地代表着椭球声场的声传播模式·　这种新方法的形函数类似于Burnett方法,而权函数定义为形函数和一个附加因子的乘积·　因为仅需要一维的数值积分,这种新方法的代码生成十分容易,就像处理一维单元一样·　耦合标准的有限元程序,这种声无限单元理论上能够高效地求解任何形状的声源的声辐射和声散射现象·　简要地推导了这种新方法,并给出了这种方法详尽的推导结果·　为更有效地检验该无限元方法的可行性,文中例子仅考虑无限元求解的精度,而不包括相应的有限元·　使用这种新方法,精确地推导出了摆动球的理论计算公式·　而长旋转椭球的例子则表明了这种方法优于边界元方法和其他声椭球无限元方法·　这些例子表明了这种新方法是切实可行的·     

NON C^0 NONCONFORMING ELEMENTS FOR ELLIPTIC FOURTH ORDER SINGULAR PERTURBATION PROBLEM

In this paper we give a convergence theorem for non C^0 nonconforming finite element to solve the elliptic fourth order singular perturbation problem. Two such kind of elements, a nine parameter triangular element and a twelve parameter rectangular element both with double set parameters, are presented. The convergence and numerical results of the two elements are given.     

A finite element method for interface problems in domains with smooth boundaries and interfaces

This paper is concerned with the analysis of a finite element method for nonhomogeneous second order elliptic interface problems on smooth domains. The method consists in approximating the domains by polygonal domains, transferring the boundary data in a natural way, and then applying a finite element method to the perturbed problem on the approximate polygonal domains. It is shown that the error in the finite element approximation is of optimal order for linear elements on a quasiuniform triangulation. As such the method is robust in the regularity of the data in the original problem.     

Partial hybrid finite elements for composite laminates

Three types of partial hybrid finite elements are presented in order to set up a global/local finite element model for analysis of composite laminates. In the global/local model, a composite laminate is divided into three different regions: global, local, and transition regions. These are modeled using three different elements. In the global region, a 4-node degenerated plate/shell element is used to model the overall response of the composite laminate. In the local region, a multilayer element is used to predict detailed stress distribution. In the transition region, a multilayer transition element is used to smoothly connect the two previous elements. The global/local finite element model satisfies the compatibility of displacement at the boundary between the global region and the local region. It also satisfies the continuity of transverse stresses at interlaminar surfaces and traction conditions on the top and bottom surfaces of composite laminates. The global/local finite element model has high accuracy and efficiency for stress analysis of composite laminates. A numerical example of analysis of a laminated strip with free edge is presented to illustrate the accuracy and efficiency of the model.     

Novel weak form quadrature element method with expanded Chebyshev nodes

Based on the principle of minimum potential energy and the differential quadrature rule, novel weak form quadrature element method is proposed. Different from the existing ones, expanded Chebyshev grid points are used as the element nodes. A simple but general way is proposed to compute the strains at the integration points explicitly by using the differential quadrature rule. For illustration and verification, quadrature bar and beam elements are established. Several examples are given. Numerical results indicate that the proposed quadrature element method allows a longer time step as compared to elements with other nodes and is an accurate and efficient method for structural analysis.