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.     

A spectral streamline diffusion finite element coupling method of unsteady transport equation in the field of neutron logging

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对流_扩散问题的6节点三角形单元流线迎风有限元法和自适应网格重分技术

A streamline upwind finite element method using 6-node triangular element is presented. The method is applied to the convection term of the governing transport equation directly along local streamlines. Several convective-diffusion examples are used to evaluate efficiency of the method. Results show that the method is monotonic and does not produce any oscillation. In addition, an adaptive meshing technique is combined with the method to further increase accuracy of the solution, and at the same time, to minimize computational time and computer memory requirement.     

Fractional four-step finite element method for analysis of thermally coupled fluid-solid interaction problems

An integrated fluid-thermal-structural analysis approach is presented. In this approach, the heat conduction in a solid is coupled with the heat convection in the viscous flow of the fluid resulting in the thermal stress in the solid. The fractional four-step finite element method and the streamline upwind Petrov-Galerkin (SUPG) method are used to analyze the viscous thermal flow in the fluid. Analyses of the heat transfer and the thermal stress in the solid are performed by the Galerkin method. The second-order semiimplicit Crank-Nicolson scheme is used for the time integration. The resulting nonlinear equations are linearized to improve the computational efficiency. The integrated analysis method uses a three-node triangular element with equal-order interpolation functions for the fluid velocity components, the pressure, the temperature, and the solid displacements to simplify the overall finite element formulation. The main advantage of the present method is to consistently couple the heat transfer along the fluid-solid interface. Results of several tested problems show effiectiveness of the present finite element method, which provides insight into the integrated fluid-thermal-structural interaction phenomena.     

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。     

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.     

Anisotropic adaptive stabilized finite element solver for RANS models

Aerodynamic characteristics of various geometries are predicted using a finite element formulation coupled with several numerical techniques to ensure stability and accuracy of the method. First, an edge‐based error estimator and anisotropic mesh adaptation are used to detect automatically all flow features under the constraint of a fixed number of elements, thus controlling the computational cost. A variational multiscale‐stabilized finite element method is used to solve the incompressible Navier‐Stokes equations. Finally, the Spalart‐Allmaras turbulence model is solved using the streamline upwind Petrov‐Galerkin method. This paper is meant to show that the combination of anisotropic unsteady mesh adaptation with stabilized finite element methods provides an adequate framework for solving turbulent flows at high Reynolds numbers. The proposed method was validated on several test cases by confrontation with literature of both numerical and experimental results, in terms of accuracy on the prediction of the drag and lift coefficients as well as their evolution in time for unsteady cases.     

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.     

Numerical modelling of shear-dependent mass transfer in large arteries

A numerical scheme for the simulation of blood flow and transport processes in large arteries is presented. Blood flow is described by the unsteady 3D incompressible Navier–Stokes equations for Newtonian fluids; solute transport is modelled by the advection–diffusion equation. The resistance of the arterial wall to transmural transport is described by a shear-dependent wall permeability model. The finite element formulation of the Navier–Stokes equations is based on an operator-splitting method and implicit time discretization. The streamline upwind/Petrov–Galerkin (SUPG) method is applied for stabilization of the advective terms in the transport equation and in the flow equations. A numerical simulation is carried out for pulsatile mass transport in a 3D arterial bend to demonstrate the influence of arterial flow patterns on wall permeability characteristics and transmural mass transfer. The main result is a substantial wall flux reduction at the inner side of the curved region. © 1997 John Wiley & Sons, Ltd.     

SOLUTION OF THE ADVECTION–DIFFUSION EQUATION USING A COMBINATION OF DISCONTINUOUS AND MIXED FINITE ELEMENTS

When transport is advection-dominated, classical numerical methods introduce excessive artificial diffusion and spurious oscillations. Special methods are required to overcome these phenomena. To solve the advection‒diffusion equation, a numerical method is developed using a discontinuous finite element method for the discretization of the advective terms. At the discontinuities of the approximate solution, numerical advective fluxes are calculated using one-dimensional approximate Riemann solvers. The method is stabilized with a multidimensional slope limiter which introduces small amounts of numerical diffusion when sharp concentration fronts occur. In addition, the diffusive term is discretized using a mixed hybrid finite element method. With this approach, numerical oscillations are completely avoided for a full range of cell Peclet numbers. The combination of discontinuous and mixed finite elements can be easily applied to 2D and 3D models using various types of elements in regular and irregular meshes. Numerical tests show good agreement with 1D and 2D analytical solutions. This approach is compared at the same time with two different numerical methods, a standard mixed finite method and a finite volume approach with high-resolution upwind terms. Regular and irregular meshes are used for the numerical tests to study the mesh effects on the numerical results. Our data show that in all cases this approach performs well. © 1997 by John Wiley & Sons, Ltd.     

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     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

Generalized Fourier analyses of the advection–diffusion equation—Part I: one‐dimensional domains

This paper presents a detailed multi‐methods comparison of the spatial errors associated with finite difference, finite element and finite volume semi‐discretizations of the scalar advection–diffusion equation. The errors are reported in terms of non‐dimensional phase and group speed, discrete diffusivity, artificial diffusivity, and grid‐induced anisotropy. It is demonstrated that Fourier analysis provides an automatic process for separating the discrete advective operator into its symmetric and skew‐symmetric components and characterizing the spectral behaviour of each operator. For each of the numerical methods considered, asymptotic truncation error and resolution estimates are presented for the limiting cases of pure advection and pure diffusion. It is demonstrated that streamline upwind Petrov–Galerkin and its control‐volume finite element analogue, the streamline upwind control‐volume method, produce both an artificial diffusivity and a concomitant phase speed adjustment in addition to the usual semi‐discrete artifacts observed in the phase speed, group speed and diffusivity. The Galerkin finite element method and its streamline upwind derivatives are shown to exhibit super‐convergent behaviour in terms of phase and group speed when a consistent mass matrix is used in the formulation. In contrast, the CVFEM method and its streamline upwind derivatives yield strictly second‐order behaviour. In Part II of this paper, we consider two‐dimensional semi‐discretizations of the advection–diffusion equation and also assess the affects of grid‐induced anisotropy observed in the non‐dimensional phase speed, and the discrete and artificial diffusivities. Although this work can only be considered a first step in a comprehensive multi‐methods analysis and comparison, it serves to identify some of the relative strengths and weaknesses of multiple numerical methods in a common analysis framework. Published in 2004 by John Wiley & Sons, Ltd.     

基于滑移界面耦合技术的旋转电机磁场仿真方法

提出了一种基于滑移界面耦合技术的旋转电机磁场仿真方法。首先,对旋转电机问题建立等效弱形式,用Lagrange乘子法施加Coulomb规范条件和滑移界面处的磁矢势连续性条件;然后,采用混合单元方法离散整个求解域中的未知量,采用棱边单元法离散滑移界面处的Lagrange矢量乘子,并采用多点约束法耦合滑移界面处的Lagrange标量乘子自由度,该方法无须在旋转电机模型的非匹配网格中构建生成树,即可自动保证磁矢势解的唯一性;最后,采用旋转线圈案例和简化的永磁同步电机案例验证了本文方法的有效性。     

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.     

Optimal control of material concentration using Fourier series

This paper presents an optimal control of the material concentration using Fourier series and finite element method. It is assumed that the optimal control value can be expanded into a Fourier series. The Fourier coefficient is identified to minimize the performance function and the optimal control value is determined. The Sakawa–Shindo algorithm is used for the minimization algorithm. The advection–diffusion equation and shallow water equation are used for the analysis of material concentration and water flow. The Crank–Nicolson scheme and finite element method using bubble function element with stabilized control parameter are employed as temporal and special discretization. Copyright © 2004 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。     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

Godunov–mixed methods for immiscible displacement

The immiscible displacement problem in reservoir engineering can be formulated as a system of partial differential equations which includes an elliptic pressure–velocity equation and a degenerate parabolic saturation equation. We apply a sequential numerical scheme to this problem where time splitting is used to solve the saturation equation. In this procedure one approximates advection by a higher-order Godunov method and diffusion by a mixed finite element method. Numerical results for this scheme applied to gas–oil centrifuge experiments are given.     

间断Galerkin有限元和有限体积混合计算方法研究

通过局部坐标变换而建立的非正交单元间断Galerkin(DG)有限元计算方法计算精度高， 但计算量大、内存需求大；而非结构网格有限体积方法虽然准确计算热流的问题目 前还没有完全解决，但其具有计算速度快和内存需求小的优点. 该研究是将有 限元和有限体积方法的优点结合，发展有限元和有限体积的混合方法. 在物面 附近黏性占主导作用的区域内采用有限元方法进行计算，在远离物面的区域采用快速的有限 体积方法进行计算，在有限元和有限体积方法结合处要保证通量守恒. 通过算例说明有 限元和有限体积混合方法既能保证黏性区域的热流计算精度和流场结构的分辨率，又能 降低内存需求和提高计算效率，使有限元方法应用于复杂外形(实际工程问题)的计 算成为可能.