Numerical study around the corotational Maxwell model for the viscoelastic fluid flows

We present numerical results for the FEM (finite element method) presented in [Comput. Methods Appl. Mech. Engrg. 191 (2002) 5045–5065]. This method is devoted to the approximation of fluid flows obeying the Oldroyd model. A particularity of this method, is to take into account the purely viscoelastic case, the so-called Maxwell model, important in practice. Numerical results are given for a fluid flowing in an abrupt plane 4 to 1 contraction. We use the corotational Maxwell model as benchmark in the choice of our computations. Results are also given for the upper convected Maxwell model. Interesting effects appear on the velocity profile: a phenomenon of quasi slip at the downstream wall.     

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J‐G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher‐order) finite elements. This method can achieve high‐order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Numerical investigations of the XFEM for solving two-phase incompressible flows

This paper discusses the application of the extended finite element method (XFEM) to solve two-phase incompressible flows. The Navier–Stokes equations are discretised using the Taylor–Hood finite element. To capture the different discontinuities across the interface, kink or jump enrichments are used for the velocity and/or pressure fields. However, these enrichments may lead to an inappropriate combination of interpolations. Different polynomial enrichment orders and different enrichment functions are investigated; only the stable combination will be used afterward.

In cases with a surface tension force, the accuracy mainly relies on the precise computation of the normal and curvature. A novel method for computing normal vectors to the interface is proposed. This method employs successive mesh refinements inside the cut elements. Comparisons with analytical and numerical solutions demonstrate that the method is effective. Moreover, the mesh refinement improves the sub-integration in the XFEM and allows for a precise re-initialisation procedure.     

生物芯片微通道周期性电渗流特性

以双电层的Poisson-Boltzmann方程和黏性不可压缩流体运动的Navier-Stokes方程为 基础,提出二维均匀微通道周期电渗流的解析解. 分析结果表明,周期电渗流速度大 小不但与双电层特性和外电场有关, 而且与流动雷诺数(Re = \omega h^2/\nu )密切相关. 随雷诺数增加,双电层滑移速度下降. 当离开固壁距离增加时,双电层以外区域流动速度快 速衰减,速度滞后相位角明显增加. 研究发现在微通道有波浪状速度剖面. 给出在低雷 诺数时的周期电渗流渐近解,它的速度振幅与定常电渗流速度相同,并具有柱栓式速度分布 形态. 还得到在微通道宽对双电层厚的比值(\kappa h)很小时,Debye-H\"{u}ckel近似 的周期电渗流解, 并与解析解进行分析比较 微通道,双电层,周期电渗流,雷诺数     

Numerical simulation of natural and mixed convection flows by Galerkin‐characteristic method

A numerical investigation is performed to study the solution of natural and mixed convection flows by Galerkin‐characteristic method. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization in primitive variables. It can be interpreted as a fractional step technique where convective part and Stokes/Boussinesq part are treated separately. The main feature of the proposed method is that, due to the Lagrangian treatment of convection, the Courant–Friedrichs–Lewy (CFL) restriction is relaxed and the time truncation errors are reduced in the Stokes/Boussinesq part. Numerical simulations are carried out for a natural convection in squared cavity and for a mixed convection flow past a circular cylinder. The computed results are compared with those obtained using other Eulerian‐based Galerkin finite element solvers, which are used for solving many convective flow models. The Galerkin‐characteristic method has been found to be feasible and satisfactory. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical simulation method for viscoelastic flows with free surfaces—fringe element generation method

To simulate filling flow in injection moulding for viscoelastic fluids, a numerical method, based on a finite element method and a finite volume method, has been developed for incompressible isothermal viscoelastic flow with moving free surfaces. The advantages of this method are, first, good applicability to arbitrarily shaped mould geometries and, second, accurate treatment for boundary conditions on the free surface. Typical filling flows are simulated, namely filling flow into a 1:4 expansion cavity with and without an obstacle. Numerical results predict the position of weld lines and air-traps. The method also indicates the effects of elongational flow on molecular orientation.     

Numerical study for influences of material parameters on hydrostatic bulging of metal sheet

In this paper, the influences of various material parameters, the hardening exponent (n), the rate sensitivity (m). the thickness anisotropy parameter (R) and the index M in the Hosford and Hill yield function, on the hydrostatic bulging of a circular clamped sheet of ductile metal materials are analysed by introducting a rigid-viscoplastic finite element method. By numerical studies, an empirical relationship within the average limit thickness strain – ₃ ^* and the material parameters (n andm) is obtained. Besides, it has been found that the influences of surface shapes of the yield function on the average limit thickness strain can be reflected by the Barlat'sP value which represents the effects ofR andM values.This work is supported by the National Natural Science Foundation and Natural Science Foundation of the Youth of China.     

Parallelization of a vorticity formulation for the analysis of incompressible viscous fluid flows

A parallel computer implementation of a vorticity formulation for the analysis of incompressible viscous fluid flow problems is presented. The vorticity formulation involves a three‐step process, two kinematic steps followed by a kinetic step. The first kinematic step determines vortex sheet strengths along the boundary of the domain from a Galerkin implementation of the generalized Helmholtz decomposition. The vortex sheet strengths are related to the vorticity flux boundary conditions. The second kinematic step determines the interior velocity field from the regular form of the generalized Helmholtz decomposition. The third kinetic step solves the vorticity equation using a Galerkin finite element method with boundary conditions determined in the first step and velocities determined in the second step. The accuracy of the numerical algorithm is demonstrated through the driven‐cavity problem and the 2‐D cylinder in a free‐stream problem, which represent both internal and external flows. Each of the three steps requires a unique parallelization effort, which are evaluated in terms of parallel efficiency. Copyright © 2002 John Wiley & Sons, Ltd.     

Two-scale finite element method for piezoelectric problem in periodicstructure

The prediction of the mechanical and electric properties of piezoelectric fibre composites has become an active research area in recent years. By means of introducing a boundary layer problem, some new kinds of two-scale finite element methods for solutions to the electric potential and the displacement for composite material in periodic struc- ture under the coupled piezoelectricity are derived. The coupled two-scale relation of the electric potential and the displacement is set up, and some finite element...     

一种新的金属压缩试验方法的有限元分析

对一种新的金属压缩试验方法进行了有限元数值分析，论证了这种方法的可行性和有性，并且对不同尺寸的试件进行了有限元计算．     

亚、跨、超音速及不可压流动的数值分析方法的研究

为了对亚、跨、超音速及不可压无粘流动进行数值模拟,将LU－SGS方法与预处理方法结合,给出了PLU－SGS方法。方程离散基于有限体积法,采用高阶精度AUSMPW格式。方程求解采用了特征边界条件。通过典型算例的数值试验对比分析,表明PLU－SGS方法可以有效地对亚、跨、超音速及不可压流动进行数值模拟,并具有较高的计算精度和收敛速度。     

A modeling and vibration analysis of a piezoelectric micro-pump diaphragm

The vibration analysis of a micro-pump diaphragm is presented. A piezoelectric micro-pump is studied. For this purpose, a dynamic model of the micro-pump is derived. The micro-pump diaphragm is modeled as circular double membranes, a piezoelectric one as actuator and a silicon one for representing the membrane for pumping action. The damping effect of the fluid is introduced into the equations. Vibration analysis is established by explicitly solving the dynamic model. The natural frequencies and mode shapes are calculated. The orthogonality conditions of the system are discussed. To verify the results, the finite-element micro-pump model is developed in ANSYS software package. The results show that the two methods are well comparable.     

层状压电陶瓷致动器中力电耦合场奇异性的数值分析

首先推导了不同压电材料界面裂纹尖端处的扇形区域内包含基本方程、裂纹面D-P边界条件和交界面处边界条件的弱形式。通过假设力电耦合位移场(位移和电势)与到裂纹尖端距离的(λ 1)次方成正比,可以得到一个分析压电材料裂纹尖端处力电耦合场奇异性的特殊的一维有限元列式。该一维有限元列式只需对扇形区域在角度方向上离散,最后的总体方程为一个关于λ的二次特征根方程。探讨了层状压电陶瓷致动器中可能出现奇异力电耦合场的部位的裂纹面边界条件及交界面处边界条件,进而将该一维有限元法进行推广,用于研究了这些部位的力电耦合场的奇异性。通过数值算例与相应的精确解的比较表明该方法是正确的,而且仅用很少单元就可以得到非常精确的结果。     

A simple method for simulating viscoelastic fluid flows in slit channel

A low-cost semi-analysis finite element technique, named the finite piece method (FPM) is presented in this article. It aims to solve three-dimensional (3D) viscoelastic slit flows. The viscoelastic stress of the fluid is modelled using an K-BKZ integral constitutive equation of the Wagner type. Picard iteration is used to solve non-linear equations. The FPM is tested on flow problems in both planar and contraction channels. The accuracy of the method is assessed by comparing flow distributions and pressure with results obtained by 3D finite element method (FEM). It shows that the solution accuracy is excellent and a substantial amount of computing time and memory requirement can be saved.     

非结构混合网格高超声速绕流与磁场干扰数值模拟

对均匀磁场干扰下的二维钝头体无粘高超声速流场进行了基于非结构混合网格的数值模拟.受磁流体力学方程组高度非线性的影响及考虑到数值模拟格式的精度,目前在此类流场的数值模拟中大多使用结构网格及有限差分方法,因而在三维复杂外形及复杂流场方面的研究受到限制.本文主要探索使用非结构网格(含混合网格)技术时的数值模拟方法.控制方程为耦合了Maxwell方程及无粘流体力学方程的磁流体力学方程组,数值离散格式采用Jameson有限体积格心格式,5步Runge-Kutta显式时间推进.计算模型为二维钝头体,初始磁场均匀分布.对不同磁感应强度影响下的高超声速流场进行了数值模拟,并与有限的资料进行了对比,得到了较符合的结果.     

Solving the fully nonlinear weakly dispersive Serre equations for flows over dry beds

We describe a numerical method for solving the Serre equations that can simulate flows over dry bathymetry. The method solves the Serre equations in conservation law form with a finite volume method. A finite element method is used to solve the auxiliary elliptic equation for the depth‐averaged horizontal velocity. The numerical method is validated against the lake at rest analytic solution, demonstrating that it is well‐balanced. Since there are currently no known nonstationary analytical solutions to the Serre equation that involve bathymetry, a nonstationary forced solution, involving bathymetry was developed. The method was further validated and its convergence rate established using the developed nonstationary forced solution containing the wetting and drying of bathymetry. Finally, the method is also validated against experimental results for the run‐up of a solitary wave on a sloped beach. The finite‐volume finite‐element approach to solving the Serre equation was found to be accurate and robust.     

Numerical residual elimination method of finite differential equations

In this paper,a new elimination of finite differential equations has been discussed.It applies the numerical direct iteration to obtain the residual equations,in which the number of unknowns has been reduced greatly.The solution process is simple and efficient,and the solution is exact     

Numerical simulation of flows over 2D and 3D backward-facing inclined steps

The work deals with the numerical solution of incompressible turbulent flow in a channel with a backward-facing step having various inclination angles. Also, the inclination of upper wall is considered. The mathematical model is based on the Reynolds averaged Navier–Stokes equations. The governing equations are closed by the explicit algebraic Reynolds stress (EARSM) model according to Wallin and Johansson or by linear eddy viscosity models (SST, TNT k–ω). The numerical solution is carried out by the implicit finite-volume method based on the artificial compressibility and by the finite-element method amd both approaches compared. The numerical simulations use as reference the experimental data by Makiola and Driver and Seegmiller in large aspect ratio channels. In these cases, the results are obtained by 2D and 3D simulations. Further narrow channel PIV experimental data are used as reference for 3D simulations.     

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered‐mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix‐free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical well test for well with finite conductivity vertical fracture in coalbed

A new well test model is developed for the hydraulic fractured well in coalbed by considering the following aspects： methane desorption phenomena, finite conductivity vertical fractures, and asymmetry of the fracture about the well. A new parameter is introduced to describe the storage of the fracture, which is named as a combined fracture storage. Another new concept called the fracture asymmetry coefficient is used to define the asymmetry of the fracture about the well. Finite element method （FEM） is used to solve the new mathematical model. The well test type curves and pressure fields are obtained and analyzed. The effects of the combined fracture storage, desorption factor, fracture conductivity, and fracture asymmetry coefficient on the well test type curves are discussed in detail. In order to verify the new model, a set of field well test data is analyzed.