Two 8-node quadrilateral spline elements by B-net method

Isoparametric quadrilateral elements are widely used in the finite element method, but the accuracy of the isoparametric quadrilateral elements will drop obviously deteriorate due to mesh distortions. Spline functions have some properties of simplicity and conformality. Two 8-node quadrilateral elements have been developed using the triangular area coordinates and the B-net method, which can exactly model the quadratic field for both convex and concave quadrangles. Some appropriate examples are employed to evaluate the performance of the proposed elements. The numerical results show that the two spline elements can obtain solutions which are highly accurate and insensitive to mesh distortions.     

A cubic quadrilateral spline element with concave shapes

Basic requirement for applying isoparametric element is that the element has to be convex and no violent distortion is allowed. In this paper, a cubic quadrilateral spline element with 12 nodes has been developed using the triangular area coordinates and the B-net method, which can exactly model the cubic field for quadrilateral element with both convex and concave shapes. Neither mapping nor coordinate transformation is required and the spline element can obtain high accuracy solutions and insensitive to mesh distortions.     

基于面积坐标与B网方法的四边形样条单元

传统等参元方法中, S型等参元完备阶较低,对网格畸变敏感, L型等参元具有高阶完备性但需要使用内部节点. 另外,由于引入等参变换, 采用数值积分可能导致总刚度矩阵出现奇异性.利用三角形面积坐标与B网方法建立了一类平面四边形的样条单元函数,它们的特点是满足协调条件, 克服网格畸变敏感性.其中8节点和12节点单元分别为2次和3次样条函数,对直角坐标分别具有二阶和三阶完备性, 高于相同节点的S型等参元.通过算例测试了这些样条单元, 并与等参元和其它四边形单元比较,数值结果显示了它们的高精度和有效性.      

一个不闭锁和抗畸变的四边形厚板元

构造一个彻底消除剪切闭锁现象并且对网格畸变不敏感的四边形厚薄板通用单元RPAQ。在方法上有三个特点：第一,在厚板挠度和转角的试函数中,采用了合理匹配方案,从而在源头上彻底消除了剪切闭锁现象;第二,采用四边形面积坐标,以代替通常的等参坐标,从而使网格畸变时仍然保持高精度;第三,采用广义协调元做法,使协调条件的采用灵活多样,并保证单元的收敛性。进行了一系列数值例题测试,表明单元RPAQ能自动消除闭锁现象,在由薄板到厚板的不同情况下,在各种网格畸变的情况下,都能体现出良好的精度和数值稳定性。     

A 3D pyramid spline element

In this paper,a 13-node pyramid spline element is derived by using the tetrahedron volume coordinates and the B-net method,which achieves the second order completeness in Cartesian coordinates.Some appropriate examples were employed to evaluate the performance of the proposed element.The numerical results show that the spline element has much better performance compared with the isoparametric serendipity element Q20 and its degenerate pyramid element P13 especially when mesh is distorted,and it is comparable to the Lagrange element Q27.It has been demonstrated that the spline finite element method is an efficient tool for developing high accuracy elements.     

Application of the quadrilateral area coordinate method： a new element for laminated composite plate bending problems

Recently, some new quadrilateral finite elements were successfully developed by the Quadrilateral Area Coordinate (QAC) method. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. In this paper, a new displacement-based, 4-node 20-DOF (5-DOF per node) quadrilateral bending element based on the first-order shear deformation theory for analysis of arbitrary laminated composite plates is presented. Its bending part is based on the element AC-MQ4, a recent-developed high-performance Mindlin-Reissner plate element formulated by QAC method and the generalized conforming condition method; and its in-plane displacement fields are interpolated by bilinear shape functions in isoparametric coordinates. Furthermore, the hybrid post-processing procedure, which was firstly proposed by the authors, is employed again to improve the stress solutions, especially for the transverse shear stresses. The resulting element, denoted as AC-MQ4-LC, exhibits excellent performance in all linear static and dynamic numerical examples. It demonstrates again that the QAC method, the generalized conforming condition method, and the hybrid post-processing procedure are efficient tools for developing simple, effective and reliable finite element models. The project is supported by the National Natural Science Foundation of China (10502028), the Special Foundation for the Authors of the Nationwide (China) Excellent Doctoral Dissertation (200242), and the Science Research Foundation of China Agricultural University (2004016).     

Construction of n-sided polygonal spline element using area coordinates and B-net method

In general, triangular and quadrilateral elements are commonly applied in two-dimensional finite element methods. If they are used to compute polycrystalline materials, the cost of computation can be quite significant. Polygonal elements can do well in simulation of the materials behavior and provide greater flexibility for the meshing of complex geometries. Hence, the study on the polygonal element is a very useful and necessary part in the finite element method. In this paper, an n-sided polygonal element based on quadratic spline interpolant, denoted by PS2 element, is presented using the triangular area coordinates and the B-net method. The PS2 element is conforming and can exactly model the quadratic field. It is valid for both convex and non-convex polygonal element, and insensitive to mesh distortions. In addition, no mapping or coordinate transformation is required and thus no Jacobian matrix and its inverse are evaluated. Some appropriate examples are employed to evaluate the performance of the proposed element.     

压电薄板屈曲有限元分析及DKQ单元

在机电耦合本构方程基础上,利用Hamilton原理推导了压电薄板屈曲分析的有限元特征方程和机电耦合的内力计算公式,在有限元实现中选择了基于Kirchhoff薄板假定的四边形薄板单元（DKQ单元）,并给出该单元的几何刚度阵及其数值积分方法。在大型通用有限元分析和优化设计软件系统JIFEX中实现了该方法。给出的数值验证了DKQ单元在屈曲分析和压电薄板静力分析中具有较高精度和收敛性,通过机械荷载和电荷载联合作用下的临界荷载计算,表明压电耦合效应能够影响结构的稳定性,可以通过改变外加电压对结构稳定性进行控制。     

QUADRILATERAL 4-NODE QUASI-CONFORMING PLANE ELEMENT WITH INTERNAL PARAMETERS

在拟协调框架之下,利用新的内参形函数构造了一个四边形四节点拟协调平面单元. 新的内参位移函数也可以添加到等参单元Q4 中来构造新的内参型等参单元. 新构造的拟协调单元QC6N 具有显式刚度矩阵,因而效率更高. 数值例子表明相比于四节点等参单元,新构造的单元可以提高计算精度和抗网格畸变的能力.     

内参型四边形四节点拟协调平面单元

在拟协调框架之下,利用新的内参形函数构造了一个四边形四节点拟协调平面单元. 新的内参位移函数也可以添加到等参单元Q4 中来构造新的内参型等参单元. 新构造的拟协调单元QC6N 具有显式刚度矩阵,因而效率更高. 数值例子表明相比于四节点等参单元,新构造的单元可以提高计算精度和抗网格畸变的能力.      

薄板小波有限元理论及其应用

利用样条小波尺度函数构造了常用的三角形和矩形薄板单元的位移函数,得到了利用小波函数表示的形函数。采用合理的局部坐标,对单元进行压缩,使单元在局部坐标区间上有其值,成功地推导出了分域的三角形和矩形薄板小波有限元列式。在此基础上,提出了弹性地基薄板的小波有限元求解方法。通过两个算例对薄板的挠度和弯矩进行了计算,数值结果表明,求解结果具有收敛快、精度高的特点。     

A 17-node quadrilateral spline finite element using the triangular area coordinates

Isopaxametric quadrilateral elements are widely used in the finite element method. However, they have a disadvantage of accuracy loss when elements are distorted. Spline functions have properties of simpleness and conformality. A 17onode quadrilateral element has been developed using the bivaxiate quaxtic spline interpolation basis and the triangular area coordinates, which can exactly model the quartic displacement fields. Some appropriate examples are employed to illustrate that the element possesses high precision and is insensitive to mesh distortions.     

一种提高四边形四节点平面壳单元计算精度的新方法

平面壳单元是由平面应力单元和平板弯曲单元叠加组合而成,具有简单的理论表达,但是它在计算曲面壳体结构时误差较大。为了进一步提高平面壳单元的计算精度,本文提出了一种计算平面壳单元刚度矩阵的新方法。通过该方法在高斯积分点建立多个单元局部坐标系,并保证每个局部坐标系都位于单元在高斯点处的切平面上,从而可以有效适应曲面壳体形状,达到进一步提高平面壳单元计算精度的目的。为了在这种新坐标系下计算单元刚度矩阵,给出了求解形函数对局部坐标的导数、局部到自然坐标系积分转换的雅可比、以及局部到整体坐标系的转换矩阵的新型计算方法。通过将这些新坐标系以及新计算方法运用到平面壳单元DKQ24中,可以有效提高平面壳单元尤其是在计算曲面壳体时的精度。计算结果表明,本文方法和平面壳单元相结合,不仅具有平面壳单元简单的理论表达式,还能得到满意的精度。另外,本文方法还可以应用到其他类型的平面壳单元,为提高其他类型平面壳单元的计算精度提供了一种新的途径和思路,具有广阔的应用前景。     

一种精确积分的四结点板弯曲单元

本文从Ｍｉｎｄｌｉｎ／Ｒｅｉｓｓｎｅｒ理论出发,采用一种新的平行四边形母单元和相应的形函数推导四结点板弯曲单元刚度矩阵的精确积分解。弯曲应变和横向剪切应变分别采用不同的插值公式构成单元刚度矩阵。理论和算例分析表明本文方法克服了“闭锁”现象并能应用于很薄的板,单元刚度矩阵计算速度比采用数值积分计算的同类单元的快四倍。     

广义协调元的变分基础及几何不变性

本文根据修正的势能原理推出了广义协调元列式。从而为广义协调元提供了一种变分依据。同时讨论了广义协调元的几何不变性,即单元与其节点号编序的相关性问题。在此基础上推导了一个任意四边形薄板弯曲单元,数值结果表明该单元能保证收敛,具有与单元节点号编序无关性,与现有的同类单元相比,具有较高的精度。     

Effect of boundary representation on viscous,separated flows in a discontinuous-Galerkin Navier–Stokes solver

The effect of curved-boundary representation on the physics of the separated flow over a NACA 65(1)-412 airfoil is thoroughly investigated. A method is presented to approximate curved boundaries with a high-order discontinuous-Galerkin spectral element method for the solution of the Navier–Stokes equations. Multiblock quadrilateral element meshes are constructed with the grid generation software GridPro. The boundary of a NACA 65(1)-412 airfoil, defined by a cubic natural spline, is piecewise-approximated by isoparametric polynomial interpolants that represent the edges of boundary-fitted elements. Direct numerical simulation of the airfoil is performed on a coarse mesh and fine mesh with polynomial orders ranging from four to twelve. The accuracy of the curve fitting is investigated by comparing the flows computed on curved-sided meshes with those given by straight-sided meshes. Straight-sided meshes yield irregular wakes, whereas curved-sided meshes produce a regular Karman street wake. Straight-sided meshes also produce lower lift and higher viscous drag as compared with curved-sided meshes. When the mesh is refined by reducing the sizes of the elements, the lift decrease and viscous drag increase are less pronounced. The differences in the aerodynamic performance between the straight-sided meshes and the curved-sided meshes are concluded to be the result of artificial surface roughness introduced by the piecewise-linear boundary approximation provided by the straight-sided meshes.     

A NEW HYBRID QUADRILATERAL FINITE ELEMENT FOR MINDLIN PLATE

ＡＮＥＷＨＹＢＲＩＤＱＵＡＤＲＩＬＡＴＥＲＡＬＦＩＮＩＴＥＥＬＥＭＥＮＴＦＯＲＭＩＮＤＬＩＮＰＬＡＴＥＣｈｉｎＹｉ（秦奕）（ＴｉａｎｊｉｎＡｒｃｈｉｔｅｃｔｕｒａｌＤｅｓｉｇｎＩｎｓｔｉｔｕｔｅ，Ｔｉａｎｊｉｎ）ＺｈａｎｇＪｉｎｇ－ｙｕ（张敬宇）（Ｉ...     

四节点四边形数值流形方法及其改进

本文对四节点四边形流形元提出了改进措施,将覆盖位移函数用自然坐标表示,使得在一般非规则有限数学覆盖网格下,数值积分变得比较容易,克服了现有四节点四边形流形单元数值积分困难的缺点。数值算例将其应用于复合材料数值模拟,计算结果表明,当覆盖位移函数采用完全一阶等参多项式时,计算精度较传统有限元法有很大改进。在力应集中或应力突变的区域,无需网格加密,只需提高覆盖位移函数的阶次即可。     

基于T样条的变网格等几何薄板动力学分析

具有大位移、大变形的薄板在接触碰撞等工况下, 其局部应变会产生剧烈变化. 为了保证对其进行动力学分析的精度和计算效率, 本文整合计算机辅助设计(CAD)与计算机辅助工程(CAE)系统, 提出了一种基于T样条曲面的变网格柔性系统等几何分析方法. 首先, 建立基于T样条曲面单元的基尔霍夫薄板运动学模型, 并根据非线性格林?拉格朗日应变建立由T样条曲面单元离散的薄板弹性模型. 其次, 通过在T网格中的局部区域插入节点的方式, 达到T样条曲面网格局部更新的目的. 利用T样条混合函数细化算法得到计算新广义变量的转换矩阵, 并结合广义α法创建了变自由度系统动力学方程的求解算法, 形成了系统的T样条单元局部细化算法. 最后, 静力学算例与柔性单摆模型分别验证了T样条薄板弹性模型的正确性, 以及T样条薄板单元在动力学分析上的精度和收敛性. 通过对受冲击柔性薄板的动力学分析表明, 本文所提出T样条单元及局部细化算法可以只在接触碰撞等应变剧烈变化的区域实现局部网格细化, 从而控制系统自由度数, 提高计算效率.      

A generalized conforming isoparametric element

Based on the generalized compatibility condition under constant and linear stress field, a quadrilateral generalized conforming isoparametric element, GC-Q6, for plane stress analysis, is developed. The element GC-Q6 can be regarded as an improved form of Wilson’s non-conforming isoparametric element Q6. GC-Q6 can pass the patch test for arbitrary irregular mesh while Q6 can not. GC-Q6 degenerates to Q6 when it is a parallelogram. Numerical examples show that the GC-Q6 element gives more accurate stress solution than the existing non-conforming elements and is less sensitive to geometric distortion.