Unified analysis for stabilized methods of low-order mixed finite elements for stationary Navier-Stokes equations

A unified analysis is presented for the stabilized methods including the pres- sure projection method and the pressure gradient local projection method of conforming and nonconforming low-order mixed finite elements for the stationary Navier-Stokes equa- tions. The existence and uniqueness of the solution and the optimal error estimates are proved.     

Superconvergence of nonconforming finite element penalty scheme for Stokes problem using L 2 projection method

A modified penalty scheme is discussed for solving the Stokes problem with the Crouzeix-Raviart type nonconforming linear triangular finite element. By the L ² projection method, the superconvergence results for the velocity and pressure are obtained with a penalty parameter larger than that of the classical penalty scheme. The numerical experiments are carried out to confirm the theoretical results.     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

New nonconforming finite element method for solving transient Naiver-Stokes equations

For transient Naiver-Stokes problems, a stabilized nonconforming finite element method is presented, focusing on two pairs inf-sup unstable finite element spaces, i.e., pNC/pNC triangular and pNQ/pNQ quadrilateral finite element spaces. The semi- and full-discrete schemes of the stabilized method are studied based on the pressure projection and a variational multi-scale method. It has some attractive features： avoiding higher-order derivatives and edge-based data structures, adding a discrete velocity term only on the fine scale, being effective for high Reynolds number fluid flows, and avoiding increased computation cost. For the full-discrete scheme, it has second-order estimations of time and is unconditionally stable. The presented numerical results agree well with the theoretical results.     

A class of Wilson arbitrary quadrilateral elements for an axisymmetric problem

A class of modified Wilson arbitrary quadrilateral nonconforming elements for an axisymmetric problem is proposed. Their convergence is proven by means of the strong patch test. The structure of this finite element class is investigated. Thus a general method of axisymmetric nonconforming elements with convergence properties is presented. Project supported by the Foundation of the Chinese Ministry of Machine-Building Industry     

A nonconforming spectral element ocean model

A nonconforming spectral element ocean model, which allows a combination of higher‐ and lower‐order elements in a single formulation, is presented. The choice between the order of interpolating polynomials and the number of elements can be adjusted locally in a subregion of a domain, based on the geometric and dynamic properties of a solution. High‐order elements are applied in regions with smooth properties and achieve high‐order convergence rates. In the regions where smoothness of the solution is limited and/or geometric requirements are complex, low‐order elements are used. This paper presents a nonconforming spectral element method based on mortar elements. Convergence of the method is analyzed using several elliptic and hyperbolic test problems in two and three dimensions. To test the method, a study of wave propagation through a nonconforming interface for two problems in a realistic geometry is also presented. Copyright © 2000 John Wiley & Sons, Ltd.     

精化直接刚度法与不协调模式

基于不协调元的收敛条件，建立了精化直接刚度法（ＲＤＳＭ）所对应的不协调位移模式，从而使ＲＤＳＭ成为建立单变量不协调元的有实用性的一般方法。     

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。     

Stabilized Crouzeix-Raviart element for the coupled Stokes and Darcy problem

This paper introduces a new stabilized finite element method for the coupled Stokes and Darcy problem based on the nonconforming Crouzeix-Raviart element. Optimal error estimates for the fluid velocity and pressure are derived. A numerical example is presented to verify the theoretical predictions.     

Nonconforming stabilized combined finite element method for Reissner-Mindlin plate

Based on combination of two variational principles, a nonconforming stabilized finite element method is presented for the Reissner-Mindlin plates. The method is convergent when the finite element space is energy-compatible. Error estimates are derived. In particular, three finite element spaces are applied in the computation. Numerical results show that the method is insensitive to the mesh distortion and has better performence than the MITC4 and DKQ methods. With properly chosen parameters, high accuracy can be obtained at coarse meshes.     

Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations

This paper proposes a new nonconforming finite difference streamline diffusion method to solve incompressible time-dependent Navier-Stokes equations with a high Reynolds number. The backwards difference in time and the Crouzeix-Raviart （CR） element combined with the P0 element in space are used. The result shows that this scheme has good stabilities and error estimates independent of the viscosity coefficient.     

PEGASE: A NAVIER–STOKES SOLVER FOR DIRECT NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS

A hybrid conservative finite difference/finite element scheme is proposed for the solution of the unsteady incompressible Navier–Stokes equations. Using velocity–pressure variables on a non-staggeredgrid system, the solution is obtained with a projection method basedon the resolution of a pressure Poisson equation. The new proposed scheme is derived from the finite element spatial discretization using the Galerkin method with piecewise bilinear polynomial basis functions defined on quadrilateral elements. It is applied to the pressure gradient term and to the non-linear convection term as in the so-called group finite element method. It ensures strong coupling between spatial directions, inhibiting the development of oscillations during long-term computations, as demonstrated by the validation studies. Two- and three-dimensional unsteady separated flows with open boundaries have been simulated with the proposed method using Cartesian uniform mesh grids. Several examples of calculations on the backward-facing step configuration are reported and the results obtained are compared with those given by other methods. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 833–861, 1997.     

几何精确的非协调等几何NURBS有限元

本文尝试将传统的非协调有限元技术推广到等几何有限元领域,建立了基于精确几何的非协调等几何分析方法,旨在拓展等几何分析应用范围,以便于等几何分析技术能真正实现CAD和FEA的融合,从而真正实现了无需划分网格的目的。我们定义了非协调的NURBS几何（类似非协调元）,给出了NURBS曲面之间几何弱连续的充分条件,进而定义了非协调的等几何分析,将之归纳为带约束驻值问题,并用拉格朗日方法进行求解。两个算例证明这种方法的有效性。未来的工作主要是证明这种方法在不同几何连续性条件下的收敛性以及将之应用到更广的领域。     

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.     

轴对称非协调元分片检验检验函数的结构

导出了轴对称非协调元分片检验中检验函数的形式,其位移函数是一种不完全的线性函数。从而给出了轴对称非协调调元分析检验更准确,完全的描述。     

A method for solving the factorized vorticity-stream function equations by finite elements

A new finite element method for solving the time-dependent incompressible Navier-Stokes equations with general boundary conditions is presented. The two second-order partial differential equations for the vorticity and the stream function are factorized, apart from the non-linear advection term, by eliminating the coupling due to the double specification on the stream function at (a part of) the boundary. This is achieved by reducing the no-slip boundary conditions to projection integral conditions for the vorticity field and by evaluating the relevant quantities involved according to an extension of the method of Glowinski and Pironneau for the biharmonic problem. Time integration schemes and iterative algorithms are introduced which require the solution only of banded linear systems of symmetric type. The proposed finite element formulation is compared with its finite difference equivalent by means of a few numerical examples. The results obtained using 4-noded bilinear elements provide an illustration of the superiority of the finite element based spatial discretization.     

非线性广义杂交元

选择新的三类变量建立了放松单元间连续性条件的用于非线性有限元分析的泛函，并由此建立了基于不协调模式的非线性广义杂交元方法。     

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier- Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.     

discontinuous galerkin method for discontinuous temperature field problems

针对不连续温度场问题建立了一种间断Galerkin有限元方法,该方法的主要特点是允 许插值函数在单元边界上存在跳变. 在建立有限元方程时,通过在单元边界上引入数值通量 项和稳定性项来处理间断效应,并且数值通量可以直接由接触热阻的定义式导出. 数值算例 表明该方法可以很方便且准确地捕捉到结构内部由于接触热阻而引起的温度跳变,同时在局 部高梯度温度场的模拟方面也比常规连续Galerkin有限元方法效率明显要高. 该方法也为研 究由接触热阻引起的温度场与应力场之间的耦合问题提供了一种新的数值模拟手段.