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…     

On the block element method in nonstationary problems

The block element method is used to study and solve the boundary-value problems of continuum mechanics for materials with time-varying characteristics. An example of constructing a block element is given for a boundary-value problem related to nonstationary behavior of a continuum in a four-dimensional space, with time taken into account. Pseudodifferential equations describing the block element parameters are derived. It is shown that, in the theory of block elements, the difference between the boundary and initial conditions in a nonstationary boundary-value problem disappears.     

Double porous screen element for gas–liquid phase separation

We consider an assembly of two parallel porous screens suspended in a tube at a distance L. The screens are connected by wicking aids. If one screen is brought into contact with a wetting liquid, the other screen will be wetted as well enclosing gas in between. Due to surface tension in the screen pores, the gas can only be removed from the chamber when the pressure difference across one screen exceeds the bubble point. With such a double porous screen element it is therefore possible to block liquid flow using trapped gas as plug. We present a model approach, experiments and numerical calculations on the performance of such a screen element. The model is based on capillary transport in vertical and radial capillaries and allows to predict how fast the element will trap the gas to become operational. For the experiments, we have built such an element using Dutch Twilled weaves made of stainless steel. Placed in a vertical tube and initially dry, it is wetted from below or above and submitted to an increasing pressure difference until breakthrough occurs where the element fails. Corresponding numerical calculations elucidate what happens within the element when it fails. Our results confirm the concept of the double porous screen element and encourage its application as liquid management device.     

Finite element analysis of lift build-up due to aerofoil indicial motion

In this paper the problem of impulsively started aerofoil or suden change of incidence of an aerofoil in incompressible potential flow is investigated. The essence of solution lies in the representation of a timely and spatially varying wake in a largely irrotational potential flow field. This is achieved by representing the wake through velocity potential difference, which seems to be the only way of imposing a velocity difference condition in the finite element context with velocity potentials as the basic unknowns. Superposition is employed to meet various boundary conditions, which is justified by the linearity of the problem. The finite element solutions are compared with those from singularity method.     

求解结构动应力的振型函数差分法

本文利用常规有限元方法的计算结果，结合数值计算方法对振型函数进行［Ｌ］算子的微分计算，从而可方便迅速获得到复杂结构动应力响应，并对梁和板进行了计算，计算结果表明该方法具有较高的精度，较一般的动态有限元具有通用性强，计算简单等特点。     

A Dorodnitsyn finite element formulation for laminar boundary layer flow

The Dorodnitsyn boundary later formulation is given a finite element interpretation and found to generate very accurate and economical solutions when combined with an implicit, non-iterative marching scheme in the downstream direction. The algorithm is of order (Δ²u, Δx) whether linear or quadratic elements are used across the boundary layer. Solutions are compared with a Dorodnitsyn spectral formulation and a conventional finite difference formulation for three Falkner-Skan pressure gradient cases and the flow over a circular cylinder. With quadratic elements the Dorodnitsyn finite element formulation is approximately five times more efficient than the conventional finite difference formulation.     

On Taylor weak statement finite element methods for computational fluid dynamics

A Taylor series augmentation of a weak statement (a ‘Taylor weak statement’ or ‘Taylor-Galerkin’ method) is used to systematically reduce the dispersion error in a finite element approximation of the one-dimensional transient advection equation. A frequency analysis is applied to determine the phase velocity of semi-implicit linear, quadratic and cubic basis one-dimensional finite element methods and of several comparative finite difference/finite volume algorithms. The finite element methods analysed include both Galerkin and Taylor weak statements. The frequency analysis is used to obtain an improved linear basis Taylor weak statement finite element algorithm. Solutions are reported for verification problems in one and two dimensions and are compared with finite volume solutions. The improved finite element algorithms have sufficient phase accuracy to achieve highly accurate linear transient solutions with little or no artificial diffusion.     

Berechnung instationärer Temperaturfelder mit finiten Elementen in Raum- und Zeitdimension

A programme is demonstrated which apart from linear finite elements in time also includes elements with shape functions of the second and third degree. The algorithm for discretization in the time dimension is described and, using the example of a parabolic time element, the coefficients required to form the element matrices are given. A parabolic alteration within a time element is taken into consideration for the boundary conditions, this being of special importance in reproducing periodic processes. On the basis of certain test examples the efficiency of the process is examined by comparison with the customary mean difference method.     

无穷域势流问题的有限元／差分线法混合求解

提出了一种将有限元和差分线法相结合求解无穷域势流问题的算法。用两同心圆将求解域划分为存在重叠的有限和无限两个区域,在有限和无限域上分别用有限元和差分线法求解Laplace方程边值问题。用差分线法推导出的关系式修正有限元方程,求解该方程组从而得到原问题的解。本算法将求解无穷域问题转化为代数特征值问题和有限域内线性方程组的...     

Finite difference and finite element methods for mhd channel flows

A Galerkin finite element method and two finite difference techniques of the control volume variety have been used to study magnetohydrodynamic channel flows as a function of the Reynolds number, interaction parameter, electrode length and wall conductivity. The finite element and finite difference formulations use unequally spaced grids to accurately resolve the flow field near the channel wall and electrode edges where steep flow gradients are expected. It is shown that the axial velocity profiles are distorted into M-shapes by the applied electromagnetic field and that the distortion increases as the Reynolds number, interaction parameter and electrode length are increased. It is also shown that the finite element method predicts larger electromagnetic pinch effects at the electrode entrance and exit and larger pressure rises along the electrodes than the primitive-variable and streamfunction–vorticity finite difference formulations. However, the primitive-variable formulation predicts steeper axial velocity gradients at the channel walls and lower axial velocities at the channel centreline than the streamfunction–vorticity finite difference and the finite element methods. The differences between the results of the finite difference and finite element methods are attributed to the different grids used in the calculations and to the methods used to evaluate the pressure field. In particular, the computation of the velocity field from the streamfunction–vorticity formulation introduces computational noise, which is somewhat smoothed out when the pressure field is calculated by integrating the Navier–Stokes equations. It is also shown that the wall electric potential increases as the wall conductivity increases and that, at sufficiently high interaction parameters, recirculation zones may be created at the channel centreline, whereas the flow near the wall may show jet-like characteristics.     

Numerical solution of axisymmetric cavity flows using the boundary element method

This paper presents a formulation of the boundary element method (BEM) for solution of axisymmetric cavity flow problems. The governing equation is written in terms of Stokes' stream function, requiring a new fundamental solution to be found. The iterative procedure for adjusting the free-surface position is similar to that used for planar cavity flows. Numerical results are compared with finite difference and finite element solutions, showing the robustness of the BEM model.     

关于无振荡、无自由参数有限元格式的研究

利用双曲守恒律方程的Ｔａｙｌｏｒ弱解表达式，建立了有限元法修正方程，选择合适的展开式系数能得到一系列数值格式．通过稳定性分析研究了格式的稳定性、色散误差与有限元修正方程导数项系数之间的关系，该关系与差分法的ＮＮＤ格式一致．在选定格式下，通过ＣＦＬ数可控制有限元离散解的振荡而使格式不含自由参数．最后，用数值算例验证了这一关系，并在二、三维欧拉方程作了推广应用．     

Element functions of discrete operator difference method

ＩｎｔｒｏｄｕｃｔｉｏｎＤｉｓｃｒｅｔｅｏｐｅｒａｔｏｒｗａｓｐｕｓｈｅｄｆｏｒｗａｒｄｉｎｐａｐｅｒｓ [1 ,2 ] ,ｗｈｉｃｈｔｒｉｅｄｔｏｕｎｉｆｙｆｉｎｉｔｅｅｌｅｍｅｎｔｍｅｔｈｏｄａｎｄｄｉｆｆｅｒｅｎｃｅｍｅｔｈｏｄｉｎｔｏｏｎｅｕｎｉｆｏｒｍｆｒａｍｅａｎｄｂｅｎｅｆｉｔｕｓｆｏｒｆｉｎｄｉｎｇｎｅｗｍｅｔｈｏｄｓ.ＰｒｏｆｅｓｓｏｒＬＩＲｏｎｇ＿ｈｕａｅｔａｌ.ｇａｖｅａｍｅｔｈｏｄ‘ｇｅｎｅｒａｔｅｄｉｆｆｅｒｅｎｃｅｍｅｔｈｏｄ’[3,4 ]ｉｓａｋｉｎｄｏｆｉｎｎｏｖａｔｉｏｎａｎｄｄ…     

Vibration analysis of two-dimensional structures using micropolar elements

Based on the micropolar theory(MPT), a two-dimensional(2 D) element is proposed to describe the free vibration response of structures. In the context of the MPT, a 2 D formulation is developed within the ABAQUS finite element software. The user-defined element(UEL) subroutine is used to implement a micropolar element. The micropolar effects on the vibration behavior of 2 D structures with arbitrary shapes are studied. The effect of micro-inertia becomes dominant, and by considering the micropolar effects, the frequencies decrease. Also, there is a considerable discrepancy between the predicted micropolar and classical frequencies at small scales, and this difference decreases when the side length-to-length scale ratio becomes large.     

Finite element analysis of density flow using the velocity correction method

A finite element method is proposed for the analysis of density flow which is induced by a difference of density. The method employs the idea that density variation can be pursued by using markers distributed in the flow field. For the numerical integration scheme, the velocity correction method is successfully used, introducing a potential for the correction of velocity. This method is useful because one can use linear interpolation functions for velocity, pressure and potential based on the triangular finite element. The final equations can be formulated using the quasi-explicit finite element method. A flume in a tank with sloping bottom has been analysed by the present method. The computed results show extremely good agreement with the experimental observations.     

动压推力轴承的边界元分析

本文利用Green第二公式,将Reynolds方程转化为沿边界的积分方程,并将非线性项作为自由项的一部分处理,采用常单元离散边界Γ,用迭代技术求出油膜压力分布,与有限差分法和有限元法比较,边界元法的结果更接近解析解.     

Improving the accuracy of discontinuous Galerkin schemes at boundary layers

The resolution of boundary layers typically requires fine grids in the wall normal direction, which leads to anisotropic elements being refined towards the wall. Best practice guidelines for the mesh generation of stretched boundary layer grids exist for the finite volume or finite difference discretizations. A similar resolution of boundary layers with DG schemes can be achieved with a coarser grid because of the subgrid resolution of the DG scheme. High order schemes incorporate the possibility of high order element mappings, resulting in different resolution properties inside the element. In this paper, we show that the use of an internal element mapping in combination with a stretched grid can be used to reduce the error of the boundary layer approximation by an order of magnitude in comparison with the classical linear internal element mapping. The boundary layer is modeled by a one‐dimensional singular perturbation problem. In addition, we discuss the construction of the element mappings by interpolation and investigate the limits of the stretching function such that the resulting element Jacobian remains positive. A parameter study shows the influence of the element mapping for different polynomial degrees on the solution. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Numerical solution of a first-order conservation equation by a least square method

Least square methods have been frequently used to solve fluid mechanics problems. Their specific usefulness is emphasized for the solution of a first-order conservation equation. On the one hand, the least square formulation embeds the first-order problem into equivalent second-order problem, better adapted to discretization techniques due to symmetry and positive-definiteness of the associated matrix. On the other hand, the introduction of a least square functional is convenient for finite element applications. This approach is applied to the model problem of the conservation of mass (the unknown is the density ρ) in a nozzle with a specified velocity field (u, v), possibly including jumps along lines simulating shock waves. This represent a preliminary study towards the solution of the steady Euler equations. A finite difference and a finite element method are presented. The choice of the finite difference scheme and of a continuous finite element representation for the groups of variables (ρu, ρv) is discussed in terms of conservation of mass flux. Results obtained with both methods are compared in two numerical tests with the same mesh system.     

A simple and efficient BEM algorithm for planar cavity flows

This paper presents a boundary element formulation for solution of planar Riabouchinsky cavity flow problems. An iterative procedure for adjusting the free surface position is developed and shown to be stable and convergent. Numerical results are compared with finite difference and finite element solutions, showing the superior accuracy of the BEM models.