A new finite element technique for the analysis of steady viscous flow problems

A new finite element technique for two-dimensional viscous incompressible fluid flow problems is presented in this paper. The vorticity transport equation is integrated in a small control volume, which results in the conservation law of vorticity. The finite element technique is applied to this equation together with the continuity equation, where simple linear triangular elements with three nodes are used for the formulation. Resulting sets of algebraic equations are solved by the use of a kind of relaxation method. Numerical results for viscous flow past a cavity show good agreement with experimental results.     

The convergence of a mixed finite element method for steady convective incompressible viscous flow

Some results on the convergence of the assumed deviatoric stress-pressure-velocity mixed finite element method for steady, convective, incompressible, viscous flow are given. An abstract error estimate is proved, which shows that the same LBB conditions for hybrid finite element method for Stokes flow are also applicable to the present method. An unusual term appears in the estimate, the rate of convergence for this term is examined. To make our idea clear, the same finite element method is applied to single elliptic equations first.This work was supported by the Science Foundation of Academia Sinica, No. (84)-103     

A large-scale three-dimensional finite element analysis of incompressible viscous fluid flow

An efficient finite element scheme for large-scale three-dimensional flow analysis is proposed. Focus of attention is placed on the time integration algorithm and some techniques for the reduction of the storage requirement, including the one-point quadrature technique and an iterative matrix solver. Application to large-scale three-dimensional problems is given to demonstrate the feasibility of the proposed scheme.     

A boundary element method for steady incompressible thermoviscous flow

A boundary element formulation is presented for moderate Reynolds number, steady, incompressible, thermoviscous flows. The governing integral equations are written exclusively in terms of velocities and temperatures, thus eliminating the need for the computation of any gradients. Furthermore, with the introduction of reference velocities and temperatures, volume modelling can often be confined to only a small portion of the problem domain, typically near obstacles or walls. The numerical implementation includes higher order elements, adaptive integration and multiregion capability. Both the integral formulation and implementation are discussed in detail. Several examples illustrate the high level of accuracy that is obtainable with the current method.     

A semi-implicit fractional step finite element method for viscous incompressible flows

This paper describes a new semi-implicit finite element algorithm for time-dependent viscous incompressible flows. The algorithm is of a general type and can handle both low and high Reynolds number flows, although the emphasis is on convection dominated flows. An explicit three-step method is used for the convection term and an implicit trapezoid method for the diffusion term. The consistent mass matrix is only used in the momentum phase of the fractional step algorithm while the lumped mass matrix is used in the pressure phase and in the pressure Poisson equation. An accuracy and stability analysis of the algorithm is provided for the pure convection equation. Two different types of boundary conditions for the end-of-step velocity of the fractional step algorithm have been investigated. Numerical tests for the lid-driven cavity at Re=1 and Re=7500 and flow past a circular cylinder at Re=100 are presented to demonstrate the usefulness of the method.     

Steady and unsteady finite element analysis of incompressible viscous fluid

Finite element procedures and related illustrative numerical examples for incompressible viscous fluid motion are discussed in this paper. The steady flow problem is solved by the Newton–Raphson method and the perturbation method. By numerical examples, it can be shown that the combined use of the Newton–Raphson method and perturbation method is suitable. For the analysis of unsteady flow, the perturbation method is employed. Assuming that the basic flow is known, unsteady flow is calculated by accumulating the solution of the linearized equation in which the boundary values are varied by small amounts. Steady flows of temperature dependent free convection are also discretized and analyzed by the same procedure as the conventional finite element Galerkin method. For shape functions, quadratic polynomials are used for velocity and temperature, and linear polynomials for pressure. It is to be noted that the selections of shape functions and solution method are the keys to the analysis of highly non-linear fluid flow problems such as those discussed in this paper.     

基于有限元的能量流分析方法研究

区别于传统的统计能量分析,通过对有限元方法计算结果的后处理也可进行能量流分析。为提高计算效率,分析过程中对某些与频率相关的积分进行了推导简化。在此基础上,以Matlab为平台编写了相关计算程序。而后,对具体算例进行基于有限元的能量流分析并与统计能量分析的结果进行比较。最后,基于此算例进一步对基于有限元的能量流分析方法进行改进并对相应的计算程序进行优化提高计算效率。     

Finite element analysis of incompressible viscous flow in a helical pipe

This paper presents an accurate finite element procedure to deal with steady state, fully developed and incompressible viscous flow in helical pipes with arbitrary curvatures and torsions. The full Navier-Stokes equations and continuity equation have been explicitly derived using a non-orthogonal helical coordinate system. To obtain the final simultaneous non-linear algebraic equations, a pressure-velocity finite element formulation is formulated based on the Galerkin Method.The combined influence of finite curvature and finite torsion on the helical flow is studied. The secondary flow patterns and contours of axial velocity of helical flows show the significant distinction with those of toroidal flows. Further, the effect of torsion on flow rates can be neglected.Several numerical examples are presented. Excellent correlations between the computed results and available referenced solutions can be drawn.     

A three-step finite element method for unsteady incompressible flows

This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. The stability analysis of the one-dimensional purely convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax-Wendroff finite element method. The method is cost effective for incompressible flows, because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present three-step finite element method does not contain any new higher-order derivatives, and is suitable for solving non-linear multi-dimensional problems and flows with complicated outlet boundary conditions. The three-step finite element method has been used to simulate unsteady incompressible flows, such as the vortex pairing in mixing layer. The properties of the flow fields are displayed by the marker and cell technique. The obtained numerical results are in good agreement with the literature.     

Non-linear steady state vibration of frames by finite element method

A finite element method based on the virtual work principle to determine the steady state response of frams in free or forced periodic vibration is introduced. The axial and flexural deformations are coupled by mean of the induced axial force along the element. The spatial discretization of the deformations is achieved by the usual finite element method and the time discretization by Fourier coefficients of the nodal displacements. No unconventional element matrices are needed. After applying the harmonic balance method, a set of non-linear algebraic equations of the Fourier coefficients is obtained. These equations are solved by the Newtonian iteration method in terms of the Fourier coefficient increments. Nodal damping can easily be included by a diagonal damping matrix. The direct numerical determination of the Fourier coefficient increments is difficult owing to the presence of peaks, loops and discontinuities of slope along the amplitude-frequency response curves. Parametric construction of the response curves using the phase difference between the response and excitation is recommended to provide more points during the rapid change of the phase (i.e. at resonance). For undamped natural vibration, the method of selective coefficients adopted. Numerical examples on the Duffing equation, a hinged–hinged beam, a clamped–hinged beam, a ring and a frame are given. For reasonably accurate results, it is shown that the number of finite elements must be sufficient to predict at least the linear mode at the frequency of interest and the number of harmones considered must satisfy the conditions of completeness and balanceability, which are discussed in detail.     

A high order finite element for completely incompressible creeping flow

A finite element method for simulating the creeping flow of an incompressible material is presented. The method allows for (1) a quadratic approximation of the velocity field, (2) material incompressibility everywhere within an element and (3) the ability to follow the flow through large changes of the material boundaries. A candle slowly bending under its own weight is simulated for illustrative purposes.     

On the superiority of the mixed element free Galerkin method for solving the steady incompressible flow problems

The element free Galerkin (EFG) method is a promising method for solving flow problems, but it meets the difficulty of volumetric locking for solving the incompressible flow problems. In this paper, a mixed EFG method is proposed for solving the steady incompressible flow problems, which avoids the volumetric locking and inherits the meshfree properties. The method employs two sets of nodes, one for the velocity approximation and the other for the pressure approximation. Specially, the ratio between the velocity node number and the pressure node number is taken as the only indicator for the locking behavior of the mixed EFG method. And inf–sup tests are carried out to investigate the relationship between the ratio and locking behavior. By two numerical examples, the accuracy, rate of convergence and efficiency of the mixed EFG method are also carefully studied. The results show that the accuracy, convergence and efficiency of the mixed EFG method are superior to that of the time-related fractional step methods.     

Integral equation formulations with the primitive variables for incompressible viscous fluid flow problems

New integral equation formulations for steady and unsteady flow problems of an incompressible viscous fluid are presented. The so-called direct approach in which the velocity vector and the pressure are inclued as unknowns is employed in this paper. The nonlinear boundary value, and the initial-boundary value problems described with the Navier-Stokes equations are transformed into integral equations by the method of weighted residuals. Fundamental solutions of the Stokes approximate equations are used as the weight function. The fundamental solution tensors are presented for the steady-state and unsteady-state problems. For the unsteady-state problem, we derive not only the time-dependent fundamental solution tensor but also the one using the finite difference approximation for the time derivative. A numerical example of the two-dimensional driven cavity flow is given to show the validity and effectiveness of the method.     

A new fast finite element method for dislocations based on interior discontinuities

A new technique for the modelling of multiple dislocations based on introducing interior discontinuities is presented. In contrast to existing methods, the superposition of infinite domain solutions is avoided; interior discontinuities are specified on the dislocation slip surfaces and the resulting boundary value problem is solved by a finite element method. The accuracy of the proposed method is verified and its efficiency for multi‐dislocation problems is illustrated. Bounded core energies are incorporated into the method through regularization of the discontinuities at their edges. Though the method is applied to edge dislocations here, its extension to other types of dislocations is straightforward. Copyright © 2006 John Wiley & Sons, Ltd.     

ODDLS: A new unstructured mesh finite element method for the analysis of free surface flow problems

This paper introduces a new stabilized finite element method based on the finite calculus (Comput. Methods Appl. Mech. Eng. 1998; 151 :233–267) and arbitrary Lagrangian–Eulerian techniques (Comput. Methods Appl. Mech. Eng. 1998; 155 :235–249) for the solution to free surface problems. The main innovation of this method is the application of an overlapping domain decomposition concept in the statement of the problem. The aim is to increase the accuracy in the capture of the free surface as well as in the resolution of the governing equations in the interface between the two fluids. Free surface capturing is based on the solution to a level set equation. The Navier–Stokes equations are solved using an iterative monolithic predictor–corrector algorithm (Encyclopedia of Computational Mechanics. Wiley: New York, 2004), where the correction step is based on imposing the divergence‐free condition in the velocity field by means of the solution to a scalar equation for the pressure. Examples of application of the ODDLS formulation (for overlapping domain decomposition level set) to the analysis of different free surface flow problems are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Comments on “A conjugate-residual—FEM for incompressible viscous flow analysis”

Communicated by S. N. Atluri, April 1, 1989     

SUPG finite element computation of viscous compressible flows based on the conservation and entropy variables formulations

In this article, we present our investigation and comparison of the SUPG-stabilized finite element formulations for computation of viscous compressible flows based on the conservation and entropy variables. This article is a sequel to the one on inviscid compressible flows by Le Beau et al. (1992). For the conservation variables formulation, we use the SUPG stabilization technique introduced in Aliabadi and Tezduyar (1992), which is a modified version of the one described in Le Beau et al. (1992). The formulation based on the entropy variables is same as the one introduced in Hughes et al. (1986).The two formulations are tested on three different problems: adiabatic flat plate at Mach 2.5, Reynolds number 20,000; Mach 3 compression corner at Reynolds number 16,800; and Mach 6 NACA 0012 airfoil at Reynolds number 10,000. In all cases, we show that the results obtained with the two formulations are very close. This observation is the same as the one we had in Le Beau et al. (1992) for inviscid flows.