The energy orthogonal relation between conforming and non-conforming displacements of triangular element

Based on the variational principle of combinative stability, combined hybrid methods posed by Zhou Tianxiao are absolutely convergent and stabilized. Zhou has advocated a systematic approach to enhanced stress/strain schemes and has designed a family of lower-order elements which are affine-equivalent ton-cube (n=2,3). The energy orthogonal relation between the conforming part and the non-conforming part of displacements interpolation functions in triangular element is given, in which the stress is interpolated by linear polynomials on each element, but the displacements are interpolated by the sum of conforming linear and non-conforming quadratic polynomials. Furthermore, this element is equivalent to the conforming triangular linear element, that is, the non-conforming parts have no contribution to enhanced strains.     

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.     

An algebraic factorisation scheme for spectral element solution of incompressible flow and scalar transport

A spectral element algorithm for solution of the unsteady incompressible Navier–Stokes and scalar (species/heat) transport equations is developed using the algebraic factorisation scheme. The new algorithm utilises Nth order Gauss–Lobatto–Legendre points for velocity and the scalar, while (N-2)th order Gauss–Legendre points are used for pressure. As a result, the algorithm does not require inter-element continuity for pressure and pressure boundary conditions on solid surfaces. Implementations of the algorithm are performed for conforming and non-conforming grids. The latter is accomplished using both the point-wise matching and integral projection methods, and applied for grids with both polynomial and geometric non-conformities. Code validation cases include the unsteady scalar convection equation, and Kovasznay flow in two- and three-dimensional domains. Using cases with analytical solutions, the algorithm is shown to achieve spectral accuracy in space and second-order accuracy in time. The results for the Boussinesq approximation for buoyancy-driven flows, and the species mixing in a continuous flow micro-mixer are also included as examples of applications that require long-time integration of the scalar transport equations.     

拟协调元研究综述

拟协调元是有限元中十分重要的、具有特色的一种列式体系. 拟协调元列式简单、灵活, 统一了协 调元、非协调元等列式方法. 在列式中, 拟协调元将几何方程和平衡方程同时弱化, 并强调基函数在有限元空 间中的重要作用; 借助对位移和应变离散精度的控制, 拟协调元保障了单元的收敛性, 并可以利用泰勒展开校 核进行简便直接的收敛性分析. 研究者们利用拟协调元已经构造了大量的优秀的单元, 并广泛地应用到结构问 题、流体流动问题、非线性分析、稳定性和破坏分析等方面. 这些工作集中体现了拟协调元的理论价值和工程 应用价值. 对拟协调列式方法、列式理论和已发表文献中的主要拟协调单元进行了总结. 最后对拟协调的研究 工作进行了展望.      

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。     

The solid variational principles of the discrete from—The variational principles of the discrete analysis by the finite element method

This paper suggests a new solid variational principle of discrete form. Basing on the true case of the discrete analysis by the finite element method and considering the variable boundaries of the elements and the unknown functions of piecewise approximation, the unknown functions have various discontinuities at the interfaces between successive element.Thus, we have used mathematical technique of variable boundary with discontinuity of the unknown functions, based on the conditions that the first variation vanishes immediately, to establish the solid variation principles of discrete form. It generalizes the classical and non-classical variational principles. Successive equations that have to be satisfied by the unknown functions are the convergency necessary conditions for the finite elements method (including conforming and non-conforming). They expand that convergency necessary conditions of the compatibility conditions in the internal interfaces.     

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.     

A generalized variational formulation for convective heat transfer

The Scope of this paper is to develop the basic equations for a variational formulation which can be used to solve problems related to convection and/or diffusion dominated flows. The formulation is based on the introduction of a generalized quantity defined as the hear displacement. The governing equation is expressed in terms of this quantity and a variational formulation is developed which leads to a system of equations similar in form to Lagrange's equations of mechanics. These equations can be used for obtaining approximate solutions, though they are of particular interest for application of the finite element method. As an example of the formulation two finite element models are derived for solving convectiondiffusion boundary value problems. The performance of the two models is investigated and numerical results are given for different cases of convection and diffusion with two types of boundary conditions. The applications of the developed formulations are not limited to convection-diffusion problems but can also be applied to other types of problems such as mass transfer, hydrodynamics and wave propagation.     

基于非协调元特征法的奇异性问题分析

提出了一个基于位移的、分析平面尖劈尖端奇性应力场和位移场问题的非协调FE特征分析法．该方法与过去原有求解裂纹尖端近似场的有限元特征分析方法导出公式的出发点不同，并且采用的单元形式为非协调元，尖劈尖端邻域内的位移场假定没有采用奇异变换技术，运用该方法处理了若干尖劈和接头的算例，所有的计算结果表明，该方法较原有方法使用的单元少而且精度高，具有应用广泛性。     

An efficient numerical model for contact-induced crack propagation analysis

This paper presents an efficient boundary element formulation for the simulation of contact-induced crack propagation problems. The formulation combines a load-scaling boundary element technique for frictional contact analysis and a dual boundary element method for axisymmetric crack propagation. These previously developed contact and fracture mechanics algorithms are revisited and adapted for the proposed coupled formulation. The novelty of the present work is the development of an efficient algorithm capable of simulating contact-induced crack propagation, while keeping proper equilibrium conditions in the frictional contact area. An efficient combination of the two separate formulations is not straightforward, and possible approaches are explained. The algorithm is validated by comparing the results of several indentation tests with available experimental data and analytical expressions. Its efficiency is also verified in frictional, conforming and non-conforming, indentation simulations.     

非协调元在复合材料安定下限分析中的应用

采用应力函数法,结合均匀化理论和应变法,在细观层次上研究了复合材料的极限和安定分析,获取复合材料代表性体积元在载荷加载历史未知下的容许承载域。利用8节点非协调等参元离散结构,获取弹性应力场和自平衡残余应力场,建立复合材料在细观层次上安定下限的优化格式。在满足计算精度的同时,大大降低了优化规模。以周期性纤维增强金属基复合材料为例,验证了该单元在安定下限分析中的有效性和可靠性。     

Taylor-Galerkin-based spectral element methods for convection-diffusion problems

Several explicit Taylor-Galerkin-based time integration schemes are proposed for the solution of both linear and non-linear convection problems with divergence-free velocity. These schemes are based on second-order Taylor series of the time derivative. The spatial discretization is performed by a high-order Galerkin spectral element method. For convection-diffusion problems an operator-splitting technique is given that decouples the treatment of the convective and diffusive terms. Both problems are then solved using a suitable time scheme. The Taylor-Galerkin methods and the operator-splitting scheme are tested numerically for both convection and convection-diffusion problems.     

Solution of free-boundary problems using finite-element/Newton methods and locally refined grids: Application to analysis of solidification microstructure

A new method is presented for the solution of free-boundary problems using Lagrangian finite element approximations defined on locally refined grids. The formulation allows for direct transition from coarse to fine grids without introducing non-conforming basis functions. The calculation of elemental stiffness matrices and residual vectors are unaffected by changes in the refinement level, which are accounted for in the loading of elemental data to the global stiffness matrix and residual vector. This technique for local mesh refinement is combined with recently developed mapping methods and Newton's method to form an efficient algorithm for the solution of free-boundary problems, as demonstrated here by sample calculations of cellular interfacial microstructure during directional solidification of a binary alloy.     

A regular boundary element method for fluid flow

The Boundary Element Method is now well established as a valid numerical technique for the solution of field problems, equal to the Finite Element Method in generality and surpassing it in computational efficiency in some cases.¹ In this paper is presented a 'Regular Boundary Element Method' as applied to inviscid laminar fluid flow problems. It involves the formation of a system of regular integral equations obtained by moving the singularity outside the domain of the given problem. It is also shown that non-conforming elements may be used whereby freedoms are not defined at the geometric nodes under the boundary element discretization. A linear element is developed here; higher order variants could easily be defined. Satisfactory numerical results have been obtained using the proposed regular method with both conventional (continuous across the boundary) and non-conforming boundary elements for two-dimensional inviscid laminar fluid flow problems having regular and singular solutions.     

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。     

线性定常对流占优对流扩散问题的无网格解法

应用无网格Galerkin方法求解对流占优对流扩散问题时会出现非物理现象的数值伪振荡,本文将SUPG方法、GLS方法、SGS方法与无网格Galerkin方法相耦合,成功解决了对流扩散方程中对流项占优时的数值伪振荡问题。运用本文构造的方法,采用线性基和具有C2连续的权函数,应用移动最小二乘法可容易地构造高阶导数连续的形函数,从而避免了有限元方法中当采用线性元插值时,因忽略稳定项中二阶导数项而降低计算精度和稳定性的问题。数值实验表明:本文构造的方法具有计算精度高、稳定性好、计算算法实施简单、前后处理方便的优点,这些方法不仅能适用于对流项占优问题,而且也能很好地消除反应项占优时的数值伪振荡问题。     

New type of dual bem and Green's-function-library strategy for fracture analysis in complex structures

A new type of dual boundary integral equations (DBIE) is presented first, through which, a smaller system of equations needs to be solved in fracture analysis. Then a non-conforming crack tip element in two-dimensional problems is proposed. The exact formula for the hypersingular integral over the non-conforming crack tip element is given next. By virtue of Green's-function-library strategy, a series of stress intensity factors (SIF) of different crack orientations, locations and/or sizes in a complicated structure can be obtained easily and efficiently. Finally, several examples of fracture analysis in two dimensions are given to demonstrate the accuracy and efficiency of the method proposed. Partially supported by the Aeronautical Science Foundation of China (No. 99C53026)     

基于Cosserat理论的广义协调元法

首次将广义协调元理论与Cosserat理论相结合,构造了一个基于Cosserat理论的平面四节点广义协调等参元。依据常应力与线性应力下的广义协调条件,推导了广义协调位移,进而得到有限元列式。利用广义协调元,一方面克服了协调元过于刚硬的缺点,另一方面消除了非协调元不一定收敛的弱点。分析了带有圆孔的应力集中问题,可以看出,...     

两个新型的八结点块体单元—三维离散算子差分法

给出了弹性力学三维问题的离散算子差分法 ,讨论离散算子差分法在三维问题中的特点 ,意在为该方法的进一步发展提供依据 ,为应用弱形式进行数值求解的研究提供参考。本文从弹性力学平衡方程更为一般的弱形式出发 ,给出了含边界参数的弱形式方程。由该方程不仅可以得到有限元法 ,还可得到离散算子差分法。给出了两个八结点块体单元 ,虽然单元中位移函数是非协调的 ,不需特殊处理便可保证离散格式收敛 ,并对单元位移有十分好的反映能力。