Some numerical quadrature schemes of a non-conforming quadrilateral finite element

Numerical quadrature schemes of a non-conforming finite element method for general second order elliptic problems in two dimensional (2-D) and three dimensional (3-D) space are discussed in this paper. We present and analyze some optimal numerical quadrature schemes. One of the schemes contains only three sampling points, which greatly improves the efficiency of numerical computations. The optimal error estimates are derived by using some traditional approaches and techniques. Lastly, some numerical results are provided to verify our theoretical analysis.     

Non‐self‐overlapping structured grid generation on an n‐sided surface

Most existing meshing algorithms for a 2D or shell figure requires the figure to have exactly four sides. Generating structured grids in the n‐sided parametric region of a trimmed surface thus usually requires to first partition the region into four‐sided sub‐regions. We address the automatic structured grid generation problem in an n‐sided region by fitting a planar Gregory patch so that the partition requirement is naturally avoided. However, self‐overlapping may occur in some portions of the algebraically generated grid; this severely limits its usage in most of engineering and scientific applications where a grid system with no self‐intersecting is strictly required. To solve the problem, we use a functional optimization approach to move grid nodes in the u?v domain of the trimmed surface to eliminate the self‐overlapping. The derivatives of a Gregory patch, which are extremely difficult to compute analytically, are not required in our method. Thus, our optimization algorithm compares favourably at least in terms of speed with some other mesh optimization algorithms, such as the elliptic PDE method. In addition, to overcome the difficulty of guessing a good initial position of every grid node for the conjugate gradient method, a progressive optimization algorithm is incorporated in our optimization. Experiment results are given to illustrate the usefulness and effectiveness of the presented method. Copyright © 2004 John Wiley & Sons, Ltd.     

A note on a lowest order divergence‐free Stokes element on quadrilaterals

This short note reports a lowest order divergence‐free Stokes element on quadrilateral meshes. The velocity space is based on a P₁ spline element over the crisscross partition of a quadrilateral, and the pressure is approximated by piecewise constant. For a given quadrilateral mesh, this element is stable if and only if the well‐known Q₁‐P₀ element is also stable. Although this method is a subspace method of Qin‐Zhang's P₁‐P₀ element, their velocity solutions are precisely equal. Moreover, an explicit basis representation is also provided. These theoretical findings are verified by numerical tests.     

交叉立方体的等价定义(英文)

For the crossed cube,an equivalent definition based on the quadrilateral is given,by which we obtain the main properties of this topological architecture,and propose a procedure to find a shortest path between any two vertices in it.     

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper, we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes. Different from the former scheme [J. Comput. Phys. 285(2015), 265-279] on uniform meshes, in this paper, in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme (UGKS), we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations. We can prove that the scheme is asymptotic preserving, and especially for the distorted quadrilateral meshes, a nine-point scheme [SIAM J. SCI. COMPUT. 30(2008), 1341-1361] for the diffusion limit equations is obtained, which is naturally reduced to standard five-point scheme for the orthogonal meshes. The numerical examples on distorted meshes are included to validate the current approach.     

A lowest order divergence-free finite element on rectangular grids

It is shown that the conforming Q _2,1;1,2-Q′₁ mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q _2,1;1,2 = Q _2,1 × Q _1,2, and Q _2,1 denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. Q′₁ is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q′₁ is the divergence of the discrete velocity space Q _2,1;1,2. Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the Raviart-Thomas element.     

平面应变问题广义有限元法

结合广义有限元和理性有限元,针对平面应变问题提出一种广义四边形单元.该单元考虑泊松效应,以节点位移自由度约束弹性力学平面应变方程的半解析解,构造单元位移模式的附加项,较准确地反映真实位移场,提高了单元的计算精度.首先推导广义单元及其等参单元的形函数,之后设计分片试验和数值算例来验证单元的精度,数值算例表明,在规则和非规则网格下新单元计算精度均优于传统有限元和广义有限元.新单元精度高且易于程序实现,可应用于实际工程的结构分析.     

采用三角形面积坐标的四边形17节点样条单元

利用二元4次样条插值基和三角形面积坐标构造17节点四边形单元．这个新单元具有4次完备阶,通过一些算例测试表明了该单元有较高精度并对网格畸变不敏感．     

解Stokes问题的P1非协调四边形元的稳定化方法

本文研究了用(~P)_1-Q_0元(其中(~P)_1表示P_1非协调四边形元)解Stokes问题的非协调混合有限元稳定化逼近方法.(~P)_1-Q_0元不满足LBB条件(见[7,14] ),因而其不能直接用来求解Stokes问题.受[3] 的启发,我们提出了一种用(~P)_1-Q_0元解Stokes问题的稳定化方法,证明了这种方法的稳定性和离散问题解的存在唯一性,得到了最优误差估计.文章最后给出的数值算例验证了我们的理论结果.     

基于高阶剪切变形理论的四边形求积元板单元及其应用

近年来由各类新型复合材料或功能梯度材料构成的板结构在工程领域得到了广泛应用,其显著特点是材料性能沿板厚变化.为合理考虑横向剪切应变,许多学者基于Reddy高阶剪切变形理论,构建了不同的有限元单元对该类板结构进行分析,但其中满足$C^{1}$连续条件的单元相对较少.本文基于Reddy高阶剪切变形理论,采用求积元方法,建立了$C^{1}$连续的四边形板单元.利用该单元对均质材料、复合材料、功能梯度材料构成的等厚度矩形板、变厚度矩形板及等厚度斜板的线弹性弯曲和自由振动问题进行了计算分析,并与现有文献中的相应计算结果进行了对比.研究表明:基于高阶剪切变形理论的四边形求积元板单元具有较高的计算效率和良好的适应性,文中各类材料构成的等变厚度矩形板及等厚度斜板均只需1个单元即可得到理想的计算结果.对于等/变厚度矩形板,可仅使用9$\times$9个积分点,而对于等厚度斜板,随着斜角的增大,所需积分点的数目逐渐增多至15$\times $15.该四边形求积元板单元可进一步用于新型复合材料板的非线性分析.