粘性流体动力学有限元变分原理

本文把建立有限元变分原理的一种新方法“N>2直接方法”从固体力学推广到流体力学,并用该方法把粘性流体动力学的广义功率消耗原理和广义变分原理发展成为有限元变分原理。还在论证中发现,相邻有限元交界面上的应力协调条件会自然地满足而无需引进任何拉民乘子。本文还建立了混合杂交非协调元的变分原理和广义变分原理,它解除了全部协调性约束条件和其它的边界性约束条件,但是并不增加待定的拉氏乘子,因此使有限元计算得到简化。本文结果可以作为粘性流体动力学有限元计算的基础定理。     

多参量断裂力学广义杂交/混合裂纹有限元法

通过研究广为人知的断裂力学单变量八节点位移裂纹QPE元和Akin族奇异单元法，本文运用经典局部裂纹解析解，与非协调假设应力杂交-混合元列式方法相结合，提出用于分层各向异性材料的多变量半解析假设应力奇异广义杂交／混合裂纹有限元法，能克服现有位移裂纹元法的域应力分布精度低和高次单元所需计算容量大的局限性，互为补充，更有利于结构裂纹扩展分析和应用研究。文中设计了一个半解析奇异裂纹平面单元，各向同性材料板算例验证了退化二次八节点协调位移裂纹元及六节点非协调奇异应力裂纹元，说明采用稀疏及加密单元网格，两类裂纹单元分别从上下逼近收敛于实验和理论参考解，可得到吻合程度较好的1／√r奇异应变和应力分量以及应力强度因子值，表明了本文奇异裂纹单元理论的优越性。     

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.     

分区界面元-有限元-无限元混合模型

利用界面元良好的相容性,引入过渡界面元的概念．实现了界面元与有限元二种数值计算方法的结合,并提出了一种界面元-有限元-无限元混合模型。这种混合模型既可以发挥界面元计算精度高、适用于不连续变形等优点．又能够充分利用有限元的计算效率和无限元方便处理无限域介质的特点,较为和谐地解决了计算精度和计算效率的矛盾。数值算例表明,本文所建立的混合模型的有效性,揭示此类混合模型具有广阔的工程应用前景。     

近场动力学与有限元的混合建模方法

综述了近场动力学与有限元混合建模方法的研究进展,阐明了各种混合建模方法的基本原理与特点,并重点介绍本课题组在近场动力学与有限元方法混合建模方面的研究工作。现有近场动力学与有限元混合建模方法包括位移协调约束、力耦合、混合函数方法以及子模型方法等,除子模型方法外,都可归结为并行式多尺度分析方法,其基本思想是将计算结构划分为近场动力学子域、有限元子域以及两者的交界区域(或重叠区域、或界面单元、或过渡区域)。子模型方法可归结为显-显分析方法,先采用显式有限元进行整体分析,后采用近场动力学方法对重点区域进行分析。混合建模方法需要着重提高交界区域的计算精度,并且消除虚假力和虚假应力波问题。提出了通过力耦合的近场动力学与有限元混合建模的隐式分析方法,该方法不再设置重叠区,通过杆单元连接近场动力学子域与有限元子域,其中界面上的有限元结点不仅与其所在单元的其他结点发生作用,还通过杆单元与以其为圆心、一定半径的圆域内的其他物质点相互作用。研究表明,本文提出的混合模型和求解方法既能有效解决裂纹扩展等不连续问题,又可提高计算效率,为工程结构破坏问题的计算分析提供一种有效方法。     

悬索桥结构分析中的索-鞍座单元

为在悬索桥结构的有限元分析中真实、简洁、高效地模拟索鞍,本文建立了一类新的单元。新单元包括索段的一端固定在与其接触部分为单一半径圆弧的索鞍上,另一端分别位于索鞍两侧的两节点“左索-鞍座单元”和“右索-鞍座单元”,以及索段两端点分别位于索鞍两侧,中间一点固定于鞍座上的三节点“索-鞍座单元”,后者的鞍槽可为两不同半径圆弧的组合。根据要求的成桥状态几何参数确定结构的无应力状态时,可利用前二者进行悬索桥的单跨分析。新单元通过自动调整索与鞍座的脱离点而处于平衡状态,从而简化了计算。单元算法的推导基于有限元分析的基本原理和弹性悬链线的精确解,并利用了处于平衡状态时索与鞍座之间的内力关系。新单元可象常规单元一样直接用于成桥状态或施工过程中悬索桥结构的有限元分析。设计的算例验证了新单元的正确性,并举例说明了新单元在悬索桥结构分析中的应用。     

局部有限元模型边界条件的直接传递方法

提出一种将整体分析得到的节点力或节点位移直接传递到精细化局部有限元模型的方法,即部分混合单元法。沿精细化局部有限元模型周边建立一组过渡单元,该组过渡单元采用与整体模型一致的单元类型和模拟方式,其外侧边界上的节点与整体模型节点的相对坐标对应,内侧边界与精细化局部有限元模型采用基于面约束的方式连接。在外侧边界上根据节点坐标对应施加整体分析获得的节点力或节点位移,过渡单元就可直接将边界条件传递到精细化局部有限元模型。通过贵州红水河特大桥钢-混结合段的精细化有限元分析,验证了本文方法的实用性和有效性。     

无网格Galerkin法与非协调元方法的耦合

无网格Galerkin法(EFGM)处理不可压缩问题时不存在自锁现象，有限元方法(FEM)也常被用来与其耦合以方便地施加边界条件和提高计算效率。在有限元方法中使用等参元，EFGM与FEM的耦合方法在处理不可压缩问题时仍然存在自锁现象。本文在有限元方法中，采用非协调元，将无网格kGalerkin法与非协调元耦合，保留了耦合方法的优点，且避免了求解不可压缩问题时的自锁现象。算例显示本文方法在分析平面应变不可压缩问题时能得到合理的结果。     

Theoretical basis and general optimal formulations of isoparametric generalized hybrid/mixed element model for improved stress analysis

By the modified three-field Hu-Washizu principle, this paper establishes a theoretical foundation and general convenient formulations to generate convergent stable generalized hybrid/mixed element (GH/ME) model which is invariant with respect to coordinate, insensitive to geometric distortion and suitable for improved stress computation. In the two proposed formulations, the stress equilibrium and orthogonality constraints are imposed through incompatible displacement and internal strain modes respectively. The proposed model by the general formulations in this paper is characterized by including assumed stress/strain, assumed stress, variable-node, singular, compatible and incompatible GH/ME models. When using regular meshes or the constant values of the isoparametric Jacobian Det in the assumed strain interpolation, the incompatible GH/ME model degenerates to the hybrid/mixed element model. Both general and concrete guidelines for the optimal selection of element shape functions are suggested. By means of the GH/ME theory in this paper, a family of new GH/ME can be and have been easily constructed. The software can also be developed conveniently because all the standard subroutines for the corresponding isoparametric displacement elements can be utilized directly. Modified version of a conference paper presented at Int. Conf. on EPMESC IV, July 29–Aug. 3, 1992, Dalian, China     

金沙江虎跳峡河段岸坡变形破坏的相关动力因子研究

石墨层和单臂碳纳米管都是以C---C共价键结合的. 在小变形条件下C---C键的势能可用谐和函 数来描述，这与梁单元的变形能具有相同的形式，因此可以用梁单元等效C---C键的作用. 提出了一种C---C键的等效梁单元有限元模型，该模型能够完备地替代谐和势描述C---C键的 伸长、面内键角变化、离面键角变化和扭转. 通过分析石墨层的典型受载情况得到了等效梁 单元的参数，以及等效梁单元参数与谐和势参数的关系，并用该模型计算了单臂碳纳米管的 杨氏模量和泊松比，计算结果为相关文献所验证.     

非协调四结点平面等参位移元新列式方法

本文导出一种等参协调元位移函数的新的表示方法,在此基础上建立起了构造等参非协调元的新方法。作为实例,构造出两个可以给出单刚显式的四结点平面非协调新单元。     

平面理性元的收敛性证明

理性元直接在物理面内列式，并用微分方程的解插值，不用等参技术而在计算面内用多项式插值．由于其解析的特性，即使是不协调元也可证明其收敛性．本文的证明采用力学方法，故易于为力学工作者所接受，且可用于多种单元的结构．收敛性证明可给理性有限元以坚实的理论基础．     

A hybrid-stress element based on Hamilton principle

A novel hybrid-stress finite element method is proposed for constructing simple 4-node quadrilateral plane elements, and the new element is denoted as HH4-3fl here. Firstly, the theoretical basis of the traditional hybrid-stress elements, i.e., the Hellinger-Reissner variational principle, is replaced by the Hamilton variational principle, in which the number of the stress variables is reduced from 3 to 2. Secondly, three stress parameters and corresponding trial functions are introduced into the system equations. Thirdly, the displacement fields of the conventional bilinear isoparametric element are employed in the new models. Finally, from the stationary condition, the stress parameters can be expressed in terms of the displacement parameters, and thus the new element stiffness matrices can be obtained. Since the required number of stress variables in the Hamilton variational principle is less than that in the Hellinger-Reissner variational principle, and no additional incompatible displacement modes are considered, the new hybrid-stress element is simpler than the traditional ones. Furthermore, in order to improve the accuracy of the stress solutions, two enhanced post-processing schemes are also proposed for element HH4-3β. Numerical examples show that the proposed model exhibits great improvements in both displacement and stress solutions, implying that the proposed technique is an effective way for developing simple finite element models with high performance.     

A Hybrid Boundary/Finite Element Method for Simulating Viscous Flows and Shapes of Droplets in Electric Fields

A mixed boundary element and finite element numerical algorithm for the simultaneous prediction of the electric fields, viscous flow fields, thermal fields and surface deformation of electrically conducting droplets in an electrostatic field is described in this paper. The boundary element method is used for the computation of the electric potential distribution. This allows the boundary conditions at infinity to be directly incorporated into the boundary integral formulation, thereby obviating the need for discretization at infinity. The surface deformation is determined by solving the normal stress balance equation using the weighted residuals method. The fluid flow and thermal fields are calculated using the mixed finite element method. The computational algorithm for the simultaneous prediction of surface deformation and fluid flow involves two iterative loops, one for the electric field and surface deformation and the other for the surface tension driven viscous flows. The two loops are coupled through the droplet surface shapes for viscous fluid flow calculations and viscous stresses for updating the droplet shapes. Computing the surface deformation in a separate loop permits the freedom of applying different types of elements without complicating procedures for the internal flow and thermal calculations. Tests indicate that the quadratic, cubic spline and spectral boundary elements all give approximately the same accuracy for free surface calculations; however, the quadratic elements are preferred as they are easier to implement and also require less computing time. Linear elements, however, are less accurate. Numerical simulations are carried out for the simultaneous solution of free surface shapes and internal fluid flow and temperature distributions in droplets in electric fields under both microgravity and earthbound conditions. Results show that laser heating may induce a non-uniform temperature distribution in the droplets. This non-uniform thermal field results in a variation of surface tension along the surface of the droplet, which in turn produces a recirculating fluid flow in the droplet. The viscous stresses cause additional surface deformation by squeezing the surface areas above and below the equator plane.     

Compatible dynamic finite element with diagonalized consistent mass matrix

In this paper, the compatible dynamic finite elements with diagonalized consistent mass matrix are studied. In previous papers^[1,2], the author studied the dynamic finite elements with diagonalized consistent mass matrix, but all of them are incompatible elements. In this paper, the compatible form functions are obtained not only for the tetrahedron elements, but also for the triangular ring elements, with diagonalized consistent mass matrices. This kind of finite elements can be used for the treatment of impact problems, vibration problems, and problems involving time coordinates, including the linear and nonlinear problems.     

Galerkin-Petrov least squares mixed element method for stationary incompressible magnetohydrodynamics

The Galerkin-Petrov least squares method is combined with the mixed finite element method to deal with the stationary, incompressible magnetohydrodynamics system of equations with viscosity. A Galerkin-Petrov least squares mixed finite element format for the stationary incompressible magnetohydrodynamics equations is presented. And the existence and error estimates of its solution are derived. Through this method, the combination among the mixed finite element spaces does not demand the discrete Babuska-Brezzi stability conditions so that the mixed finite element spaces could be chosen arbitrartily and the error estimates with optimal order could be obtained.     

Augmented mixed finite element methods for a vorticity‐based velocity–pressure–stress formulation of the Stokes problem in 2D

In this paper, we consider an augmented velocity–pressure–stress formulation of the 2D Stokes problem, in which the stress is defined in terms of the vorticity and the pressure, and then we introduce and analyze stable mixed finite element methods to solve the associated Galerkin scheme. In this way, we further extend similar procedures applied recently to linear elasticity and to other mixed formulations for incompressible fluid flows. Indeed, our approach is based on the introduction of the Galerkin least‐squares‐type terms arising from the corresponding constitutive and equilibrium equations, and from the Dirichlet boundary condition for the velocity, all of them multiplied by stabilization parameters. Then, we show that these parameters can be suitably chosen so that the resulting operator equation induces a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and rotated Raviart–Thomas elements for the stresses. Next, we derive reliable and efficient residual‐based a posteriori error estimators for the augmented mixed finite element schemes. In addition, several numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. The present work should be considered as a first step aiming finally to derive augmented mixed finite element methods for vorticity‐based formulations of the 3D Stokes problem. Copyright © 2010 John Wiley & Sons, Ltd.     

THE COMPACTION OF TIME-DEPENDENT VISCOUS GRANULAR MATERIALS CONSIDERING INERTIAL FORCES

In the present paper, compactions of time-dependent viscous granular materials are simulated step by step using the automatic adaptive mesh generation schemes. Inertial forces of the viscous incompressible aggregates are taken into account. The corresponding conservation equations, the weighted-integral formulations, and penalty finite element model are investigated. The fully discrete finite element equations for the simulation are derived. Polygonal particles of aggregates are simplified as mixed three-node and four-node elements. The automatic adaptive mesh generation schemes include contact detection algorithms, and mesh upgrade schemes. Solutions of the numerical simulation are in good agreement with some results from literatures. With minor modification, the proposed numerical model can be applied in several industries, including the pharmaceutical, ceramic, food, and household product manufacturing.     

一个新的八节点非协调曲边四边形平面单元

本文在文[1]和文[2]的基础上,提出构造非协调有限元的新方法。该方法不用一般的变分原理,可适用任意变系数正定和非正定偏微分方程。利用这一方法得到一个新的八节点四边形平面应力单元。与一般有限元相比,位移和应力可提高一阶收敛精度。形成单刚矩阵时,不需要进行数值积分。单元之间的协调条件容易满足,文中给出收敛性证明。文末给出数值算例,表明利用本文的方法,应力和位移均可获得满意的数值精度。     

Difference scheme and numerical simulation based on mixed finite element method for natural convection problem

ntroductionLetΩ R2 beaboundeddomain .Weconsiderthefollowingnon_stationarynaturalconvectionproblem :Problem (Ⅰ )　Findu =(u1,u2 ) ,p ,andTsuchthat,foranyt1>0 ,ut- μΔu +(u· )u + p=λjT　　 ((x ,y ,t) ∈Ω× (0 ,t1) ) ,divu =0　　　　　　　　　 ((x ,y,t) ∈Ω× (0 ,t1) ) ,Tt-ΔT +λu· T =0　　 ((x,y,t) ∈Ω× (0 ,t1) ) ,u =0 ,T =0　　　　　　 ((x,y,t)∈ Ω× (0 ,t1) ) ,u(x ,y ,0 ) =0 ,　T(x,y,0 ) =f(x,y)　　 ((x,y) ∈Ω) ,whereuisthefluidvelocityvectorfield ,pthepressurefield ,Tthet…