Isothermal extrusion of non-dilute fiber suspensions

The extrusion of a rod-like fiber suspension is a Newtonian solvent, as a first step to the fast and inexpensive production of composite materials, is investigated. The analysis is carried out by means of an integral constitutive equation for a non-dilute suspension, streamlined finite element for liquid with memory, and Newton iteration of nonlinear integro-differential equations. The predictions show substantial differences between dilute and nondilute fiber suspension regarding the processing conditions (pressure drop, velocity distribution, die-swell) and the resulting fiber orientation. Nondilute fiber suspensions exhibit substantial shear-thinning and negligible elasticity as evidenced by the small die-swell, and fiber concentration viscosity-thickening as evidenced by the large pressure drop. The fiber orientation is computed by solving the orientation distribution function along selected streamlines of the complex velocity field. It is shown that the fiber orientation far downstream can be made independent of the random fiber orientation at the inlet.     

高阶谱元区域分解算法求解定常方腔驱动流

主要利用Jacobian-free的Newton-Krylov方法求解定常不可压缩Navier-Stokes方程,将基于高阶谱元法的区域分解Stokes算法的非定常时间推进步作为Newton迭代的预处理,回避了传统Newton方法Jacobian矩阵的显式装配,节省了程序内存,同时降低了Newton迭代线性系统的条件数,且没有非线性对流项的隐式求解,大大加快了收敛速度。对有分析解的Kovasznay流动的计算结果表明,本高阶谱元法在空间上有指数收敛的谱精度,且对定常解的Newton迭代是二次收敛的。本文模拟了二维方腔顶盖一致速度驱动流,同基准解符合得很好,表明本文方法是准确可靠的。本文还考虑了Re=800时方腔顶盖正弦速度驱动流,除得到已知的一个稳定对称解和一对稳定非对称解外,还获得了一对新的不稳定的非对称解。     

Singular finite elements for the sudden-expansion and the die-swell problems

The singular finite element method is used to solve the sudden-expansion and the die-swell problems in order to improve the accuracy of the solution in the vicinity of the singularity and to speed up the convergence. The method requires minor modifications to standard finite element schemes, and even coarse meshes give more accurate results than refined ordinary finite element meshes. Improved normal stress results for the sudden-expansion problem have been obtained for various Reynolds numbers up to 100 using the singular elements constructed for the creeping flow problem. In addition, the normal stresses at the walls appear to be insensitive to the singularity powers used in the construction of the singular basis functions. The die-swell problem is solved using the singular elements constructed for the stick–slip problem. The singular elements accelerate the convergence of the free surface dramatically.     

Olver迭代与Newton迭代的比较

Olver迭代是一个立方收敛的求根公式,而Newton迭代仅是平方收敛,但前者却不如后者为人们所熟知,以至于近来有作者其推导了一个新的高阶迭代公式,而实际就是Olver迭代公式却浑然不如。那么,到底是什么原因导致Olver迭代没有被广大的计算方法教科书介绍呢？本文对Newton迭代与Olver迭代做了详尽的分析,给出了两者各自的精度表达式,并对两者进行了比较,结论是：从计算效率及精度方面综合考虑,Olver迭代公式不如Newton迭代公式实用。     

Comparison of Iterative Methods for Improved Solutions of the Fluid Flow Equation in Partially Saturated Porous Media

Abstract. The Picard and modified Picard iteration schemes are often used to numerically solve the nonlinear Richards equation governing water flow in variably saturated porous media. While these methods are easy to implement, they are only linearly convergent. Another approach to solve the Richards equation is to use Newton's iterative method. This method, also known as Newton–Raphson iteration, is quadratically convergent and requires the computation of first derivatives. We implemented Newton's scheme into the mixed form of the Richards equation. As compared to the modified Picard scheme, Newton's scheme requires two additional matrices when the mixed form of the Richards equation is used and requires three additional matrices, when the pressure head-based form is used. The modified Picard scheme may actually be viewed as a simplified Newton scheme.Two examples are used to investigate the numerical performance of different forms of the 1D vertical Richards equation and the different iterative solution schemes. In the first example, we simulate infiltration in a homogeneous dry porous medium by solving both, the h based and mixed forms of Richards equation using the modified Picard and Newton schemes. Results shows that, very small time steps are required to obtain an accurate mass balance. These small times steps make the Newton method less attractive.In a second test problem, we simulate variable inflows and outflows in a heterogeneous dry porous medium by solving the mixed form of the Richards equation, using the modified Picard and Newton schemes. Analytical computation of the Jacobian required less CPU time than its computation by perturbation. A combination of the modified Picard and Newton scheme was found to be more efficient than the modified Picard or Newton scheme.     

Coupled Navier–Stokes/molecular dynamics simulations in nonperiodic domains based on particle forcing

This paper focuses on coupling methods for hybrid Navier–Stokes/molecular dynamics (MD) simulations. The computational domain is split in a continuum flow region, where a finite‐volume discretisation of the Navier–Stokes equations is used, and one or more particle domains, where molecular level modelling of the flow is employed. The domains are defined with a partial overlap, in which the flow states are coupled through an exchange of the velocity components. For the steady flows considered, an under‐relaxed Newton iteration method is used to drive the coupled system to convergence. The main focus of the present work is on methods to impose nonperiodic boundary conditions on the particle domain(s). A particle forcing is applied in the direction normal to the particle domain boundary to impose the boundary normal velocity component. A novel aspect of the present work is the extension of this method to more general nonplanar particle domain boundaries. The main contribution of the paper is the development of a particle forcing method in the direction tangential to the domain boundary, which is based on the equivalent continuum‐flow boundary shear stresses along with an iterative forcing strength adjustment based on the extrapolated particle boundary velocity. Furthermore, an adaptation scheme is presented, which uses the finite‐volume flux residuals of the particle bin averaged velocity field as a truncation criterion for the iterative force‐update scheme. It is demonstrated that by comparing the residual reduction for the momentum equation in the nonhomogeneous directions during the molecular dynamics simulations with that for a homogeneous direction, the forcing iteration at which the statistical noise in the velocity field dominates the uncertainty in the forcing strength can be determined. At this point the iteration can be truncated. It is shown that with adaptive schemes of this type, the total number of MD evaluations required in a coupled Navier–Stokes/MD simulation can be reduced relative to a hybrid scheme with a fixed number of forcing‐strength updates. Copyright © 2011 John Wiley & Sons, Ltd.     

非常应变模式数字相关方法的初值估计

本文给出了基于高精度非常应变子区位移模式数字相关方法的Newton-Raphson迭代法求解的新通用公式,对相关迭代算法中的初值估计问题进行了研究,提出两种初值估计方法：（1）利用“实时相减”和“精密调节”相结合的方法而获得零初值;（2）快速迭代初值估计方法,从而有效地解决了Newton-Raphson迭代算法中的初值估计问题,并提高了迭代的收敛速度。     

A matrix‐free preconditioned Newton/GMRES method for unsteady Navier–Stokes solutions

The unsteady compressible Reynolds‐averaged Navier–Stokes equations are discretized using the Osher approximate Riemann solver with fully implicit time stepping. The resulting non‐linear system at each time step is solved iteratively using a Newton/GMRES method. In the solution process, the Jacobian matrix–vector products are replaced by directional derivatives so that the evaluation and storage of the Jacobian matrix is removed from the procedure. An effective matrix‐free preconditioner is proposed to fully avoid matrix storage. Convergence rates, computational costs and computer memory requirements of the present method are compared with those of a matrix Newton/GMRES method, a four stage Runge–Kutta explicit method, and an approximate factorization sub‐iteration method. Effects of convergence tolerances for the GMRES linear solver on the convergence and the efficiency of the Newton iteration for the non‐linear system at each time step are analysed for both matrix‐free and matrix methods. Differences in the performance of the matrix‐free method for laminar and turbulent flows are highlighted and analysed. Unsteady turbulent Navier–Stokes solutions of pitching and combined translation–pitching aerofoil oscillations are presented for unsteady shock‐induced separation problems associated with the rotor blade flows of forward flying helicopters. Copyright © 2000 John Wiley & Sons, Ltd.     

Symmetry and ordinary differential constraints

The representative generalized symmetries of any ordinary differential equation are described in terms of its invariants. This identifies the evolution equations compatible with a given constraint. The restriction of the flow of a compatible equation to the solution space of the constraint is generated by the corresponding internal symmetry. This reduces the evolution equation to a finite dimensional system of first-order ordinary differential equations. The Euler–Lagrange equation of any conserved density of a given evolution equation yields such a reduction. Other examples include the generalized method of separation of variables, the characterization of separable evolution equations, and the characterization of equations with complete families of wave solutions. A Newton equation is compatible with an ordinary differential constraint if and only if the constraint is affine, with force field symmetry, in which case the equation reduces to a finite-dimensional dynamical system. Newton equations with complete families of characteristic solutions reduce to central force problems on solution spaces of linear constraints.     

A Newton multigrid method for steady-state shallow water equations with topography and dry areas

A Newton multigrid method is developed for one-dimensional (1D) and two-dimensional (2D) steady-state shallow water equations (SWEs) with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.     

A net inflow method for incompressible viscous flow with moving free surface

A new finite element procedure called the net inflow method has been developed to simulate time-dependent incompressible viscous flow including moving free surfaces and inertial effects. As a fixed mesh approach with triangular element, the net inflow method can be used to analyse the free surface flow in both regular and irregular domains. Most of the empty elements are excluded from the computational domain, which is adjusted successively to cover the entire region occupied by the liquid. The volume of liquid in a control volume is updated by integrating the net inflow of liquid during each iteration. No additional kinetic equation or material marker needs to be considered. The pressure on the free surface and in the liquid region can be solved explicitly with the continuity equation or implicitly by using the penalty function method. The radial planar free surface flow near a 2D point source and the dam-breaking problem on either a dry bed or a still liquid have been analysed and presented in this paper. The predictions agree very well with available analytical solutions, experimental measurements and/or other numerical results.     

An efficient algorithm for strain history tracking in finite element computations of non-Newtonian fluids with integral constitutive equations

A new finite element technique has been developed for employing integral-type constitutive equations in non-Newtonian flow simulations. The present method uses conventional quadrilateral elements for the interpolation of velocity components, so that it can conveniently handle viscoelastic flows with both open and closed streamlines (recirculating regions). A Picard iteration scheme with either flow rate or elasticity increment is used to treat the non-Newtonian stresses as pseudo-body forces, and an efficient and consistent predictor-corrector scheme is adopted for both the particle-tracking and strain tensor calculations. The new method has been used to simulate entry flows of polymer melts in circular abrupt contractions using the K-BKZ integral constitutive model. Results are in very good agreement with existing numerical data. The important question of mesh refinement and convergence for integral models in complex flow at high flow rate has also been addressed, and satisfactory convergence and mesh-independent results are obtained. In addition, the present method is relatively inexpensive and in the meantime can reach higher elasticity levels without numerical instability, compared with the best available similar calculations in the literature.     

Fully coupled finite volume solutions of the incompressible Navier–Stokes and energy equations using an inexact Newton method

An inexact Newton method is used to solve the steady, incompressible Navier–Stokes and energy equation. Finite volume differencing is employed on a staggered grid using the power law scheme of Patankar. Natural convection in an enclosed cavity is studied as the model problem. Two conjugate-gradient -like algorithms based upon the Lanczos biorthogonalization procedure are used to solve the linear systems arising on each Newton iteration. The first conjugate-gradient-like algorithm is the transpose-free quasi-minimal residual algorithm (TFQMR) and the second is the conjugate gradients squared algorithm (CGS). Incomplete lower-upper (ILU) factorization of the Jacobian matrix is used as a right preconditioner. The performance of the Newton- TFQMR algorithm is studied with regard to different choices for the TFQMR convergence criteria and the amount of fill-in allowed in the ILU factorization. Performance data are compared with results using the Newton-CGS algorithm and previous results using LINPACK banded Gaussian elimination (direct-Newton). The inexact Newton algorithms were found to be CPU competetive with the direct-Newton algorithm for the model problem considered. Among the inexact Newton algorithms, Newton-CGS outperformed Newton- TFQMR with regard to CPU time but was less robust because of the sometimes erratic CGS convergence behaviour.     

Numerical study of flows of complex fluids between eccentric cylinders using transformation functions

In this paper, we investigate fluid flows between eccentric cylinders by means of two stream‐tube analyses. The first method considers a one‐to‐one global transformation function that allows the physical domain to be transformed into a mapped domain, used as computational domain, that involves concentric streamlines. The second approach uses local transformations and domain decomposition techniques to deal with mixed flow regimes. Both formulations are particularly adapted for handling time‐dependent constitutive equations, since particle‐tracking problems are avoided. Mass conservation is verified in both formulations and the relevant numerical procedure can be carried out using simple meshes built on the mapped streamlines. Fluids obeying anelastic and viscoelastic constitutive equations are considered in the calculations. The numerical results are consistent with those in the literature for the flow rates tested. Application of the method to the K‐BKZ memory‐integral constitutive equation highlights significant differences between the model predictions and those provided by more simple rheological models. Copyright © 2002 John Wiley & Sons, Ltd.     

基于S-A湍流模型的Runge-Kutta有限元算法

采用一方程S-A模型(Spalart-Allmaras模型)封闭雷诺时均N-S方程(RANS方程)进行湍流数值计算,可以减少方程求解数量,节约计算时间。本文对其进行了有限元数值算法研究,首先通过沿流线坐标变换,得到无对流项RANS方程,并引入三阶Runge-Kutta法对其进行时间离散;然后利用沿流线的Taylor展开解决坐标变换带来的网格更新的困难;最后采用Galerkin法进行空间离散,得到湍流模型的有限元算法。基于方柱绕流和覆冰输电线绕流模型,与试验结果进行对比,验证了该算法的有效性,与一阶数值算法相比,该算法在精度和收敛性方面更具优势。     

计算离心泵叶轮流场的扫描流束有限元方法

本文提出一种求解离心式叶轮流场的数值方法，将流动求解区域离散为有限个由流线构成其边界的单元，采用伽辽金法建立的单元方程在一条流束上集合为方程组，流线上的节点坐标亦作为未知量包含在有限元方程中，通过扫描计算，逐步解得流线位置及流动参数。本文应用叶轮的通流理论流动模型，采用扫描流速有限元方法对离心泵叶轮流场进行了计算，并与有关文献作了比较。     

基于Newton/Gauss-Seidel迭代的DGM隐式方法

在Newton迭代方法的基础上,对高阶精度间断Galerkin有限元方法(DGM)的时间隐式格式进行了研究. Newton迭代 法的优势在于收敛效率高效,并且定常和非定常问题能够统一处理,对于非定常问题无需引入双时间步策略. 为了避免大型矩阵的求逆,采用一步Gauss-Seidel迭代和Matrix-free技术消去残值Jacobi矩阵的上、下三角矩阵,从而只需计算和存储对角(块)矩阵. 对角(块)矩阵采用数值方法计算. 空间离散采用Taylor基,其优势在于对于任意形状的网格,基函数的形式是一致的,有利于在混合网格上推广. 利用该方法,数值模拟了Bump绕流和NACA0012翼型绕流. 计算结果表明,与显式的Runge-Kutta时间格式相比,隐式格式所需的迭代步数和CPU时间均在很大程度上得到减少,计算效率能够提高1～ 2个量级.     

Iteration schemes for rapid post‐stall aerodynamic prediction of wings using a decambering approach

Nonlinear aerodynamics of wings may be evaluated using an iterative decambering approach. In this approach, the effect of flow separation due to stall at any wing section is modeled as an effective reduction in section camber. The approach uses a wing analysis method for potential‐flow calculations and viscous airfoil lift curves for the sections as input. The calculation procedure is implemented using a Newton–Raphson iteration to simultaneously satisfy the boundary condition, which comes from potential‐flow wing theory, and drive the sectional operating points toward their respective viscous lift curves, as required for convergence. Of particular interest in this research is the calculation of the residuals during the Newton iteration. Unlike a typical implementation of the Newton iteration, the residual calculation is not performed via a straightforward function evaluation, but rather by estimating the target operating points on the input viscous lift curves. Estimation of these target operating points depends on the assumptions made in the cross‐coupling of the decambering at the different sections. This paper presents four residual calculation schemes for the decambering approach. The residual calculation schemes are compared against each other to assess computational speed and robustness. Decambering results are also compared with higher‐order computational fluid dynamics (CFD) solutions for rectangular and swept wings. Results from the best scheme compare well with the CFD solutions for the rectangular wing, motivating further development of the method. Poor predictions for the swept wings are traced to spanwise propagation of separated flow at stall, highlighting the limitations of the current approach. Copyright © 2014 John Wiley & Sons, Ltd.     

The integrated singular basis function method for the stick-slip and the die-swell problems

We further develop a new singular finite element method, the integrated singular basis function method (ISBFM), for the solution of Newtonian flow problems with stress singularities. The ISBFM is based on the direct subtraction of the leading local solution terms from the governing equations and boundary conditions of the original problem, followed by a double integration by parts applied to those integrals with singular contributions. The method is applied to the stick-slip and the die-swell problems and improves the accuracy of the numerical results in both cases. In the case of the die-swell problem it considerably accelerates the convergence of the free surface profile with mesh refinement. The advantages and disadvantages of the ISBFM when compared to other singular methods are also discussed.     

Variational iteration method for two-dimensional steady slip flow in micro-channels

In this paper, the two-dimensional steady slip flow in microchannels is investigated. Research on micro flow, especially on micro slip flow, is very important for designing and optimizing the micro electromechanical system (MEMS). The Navier-Stokes equations for two-dimensional steady slip flow in microchannels are reduced to a nonlinear third-order differential equation by using similarity solution. The variational iteration method (VIM) is used to solve this nonlinear equation analytically. Comparison of the result obtained by the present method with numerical solution reveals that the accuracy and fast convergence of the new method.