An AMG Preconditioner for Solving the Navier-Stokes Equations with a Moving Mesh Finite Element Method

AMG preconditioners are typically designed for partial differential equation solvers and divergence-interpolation in a moving mesh strategy. Here we introduce an AMG preconditioner to solve the unsteady Navier-Stokes equations by a moving mesh finite element method. A $4P1$ − $P1$ element pair is selected based on the data structure of the hierarchy geometry tree and two-layer nested meshes in the velocity and pressure. Numerical experiments show the efficiency of our approach.     

A fast and robust iterative solver for nonlinear contact problems using a primal‐dual active set strategy and algebraic multigrid

For extending the usability of implicit FE codes for large‐scale forming simulations, the computation time has to be decreased dramatically. In principle this can be achieved by using iterative solvers. In order to facilitate the use of this kind of solvers, one needs a contact algorithm which does not deteriorate the condition number of the system matrix and therefore does not slow down the convergence of iterative solvers like penalty formulations do. Additionally, an algorithm is desirable which does not blow up the size of the system matrix like methods using standard Lagrange multipliers. The work detailed in this paper shows that a contact algorithm based on a primal‐dual active set strategy provides these advantages and therefore is highly efficient with respect to computation time in combination with fast iterative solvers, especially algebraic multigrid methods. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical Methods for Accurate Computation of Design Sensitivities for Two-Dimensional Navier-Stokes Problems

Gradient computations can be a limiting factor in algorithm efficiency and accuracy for optimization based design. In this paper, we present three parameterized flow problems and consider the evaluation of state sensitivities both theoretically and numerically. Existence and uniqueness results are given for the sensitivities of a specific group of two-dimensional Navier-Stokes problems. We then turn our attention to obtaining numerical approximations to state sensitivities. We show convergence of our numerical sensitivities using a problem having an exact solution. Next, two problems, flow around a cylinder and flow over a bump, are used to evaluate several computational schemes. In particular, a local projection scheme for improved state derivative approximations and the use of an adaptive finite element scheme are shown to be important techniques for obtaining accurate sensitivity approximations. Lastly, we evaluate the impact of these computational techniques on cost function and gradient calculation.     

求解二维Navier-Stokes方程的移动网格方法

为了减少解在较小的局部区域内有着很强的奇异性、剧烈变化等的偏微分方程求解问题的计算量,提出了一种基于方程求解的移动网格方法,并将其应用于二维不可压缩Navier-Stokes方程的求解．与已有的大部分移动网格方法不同,网格节点的移动距离是通过求解一个变系数扩散方程得到的,避免了做区域映射,也不需要对控制函数进行磨光处理,所以算法很容易编程实现．数值算例表明所提算法能够在解梯度较大的位置加密网格,从而在保证提高数值解的分辨率的前提下,可以很好地节省了计算量．由于Navier-Stokes 的典型性,所得算法能够推广到求解很大一类偏微分方程数值问题．     

Navier-Stokes方程自由面数值模拟

A Crank-Nicolson finite difference method for incompressible Navier-Stokes equa-tions is developed and is applied to find the numerical solution of wave elevation on the free surface in a two dimensional tank. The numerical simulation of wave eleva-tion on the free surface is studied with different excited accelerations and Reynolds number. From the benchmarks, the proposed finite difference method agrees well with the previous published works. As shown in the numerical results, we draw a conclusion that the wave elevations decay gradually with time. The number of the beating periods diminish gradually when the Reynolds number decreases. Finally, we find that the wave elevations on the free surface keep a related stable height, when the Reynolds number decreases.     

非定常N-S方程的双重网格数值模拟

本文采用全离散双重网格算法（时间变量采用Eular全隐式格式离散,空间变量采用混合有限元离散）,对非定常Navier-Stokes（N-S）方程进行数值模拟.双重网格算法的基本思想是,首先在粗网格有限元空间X^H上求解一个非线性问题,然后在细网格有限元空间Xh（h＜＜H）上求解一个线性问题.数值实验结果表明：在保持几乎相同精度的前提下,双重网格算法比标准有限元算法节省近一半的计算时间,说明了新算法求解非定常N-S方程的可行性和高效性.     

Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

We propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.     

Boundary element analysis of Navier-Stokes equations

As arrangements, the fundamental solutions of anisotropic convective diffusion equations of transient incompressible viscous fluid flow and boundary elements analysis of the diffusion equation are presented. Secondly, by considering that convective diffusion equations and Navier-Stokes equations are mathematical formulations of mass and momentum conservation law respectively, and that consequently, both physical contents and equation styles are analogous, boundary integral formulations for Navier-Stokes equations are proposed on the basis of formulation of diffusion equations.     

Some numerical schemes using curvilinear coordinate grids for incompressible and compressible Navier-Stokes equations

In this review paper some numerical schemes recently developed by the authors and their coworkers for analysing the cascade flows of turbomachinery are described. These schemes use the curvilinear coordinate grid and solve the momentum equations of contravariant velocities (volume flux). The compressible flow schemes are based on the delta-form approximate-factorization finite-difference scheme, and are improved by using the diagonalization, the flux difference splitting and thetvd schemes to save computational effort and to increase stability and resolvability. Furthermore, using higher-order compacttvd muscl schemes, we can capture not only shock waves but also contact surfaces very sharply. On the other hand, the incompressible flow schemes are based on the well-knownSMAC scheme, and are extended to the curvilinear coordinate grid and further to the implicit scheme to reduce computations. These schemes, like thesmac scheme, satisfy the continuity condition identically, and suppress the occurrence of spurious errors. In both the compressible and incompressible schemes, for the turbulent flow thek-ɛ turbulence model with the law of the wall or considering the low Reynolds number effects is employed, and for the unsteady flow the Crank-Nicholson method is employed and the solution at each time step is obtained by the Newton iteration. Use of the volume flux instead of the physical velocity is inevitable for theMAC type schemes, and makes it easy to impose boundary conditions. Finally, some calculated results using the present schemes are shown.     

FLEXMG: A new library of multigrid preconditioners for a spectral/finite element incompressible flow solver

A new library called FLEX MG has been developed for a spectral/finite element incompressible flow solver called SFELES. FLEX MG allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods. Two families of algebraic multigrid preconditioners have been implemented, namely smooth aggregation‐type and non‐nested finite element‐type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non‐nested finite element‐type multigrid so that our approaches can be considered as hybrid. Our aggregation‐type multigrid is smoothed with either a constant or a linear least‐square fitting function, whereas the non‐nested finite element‐type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand‐alone solvers or coupled with a GMRES method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics, their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in the study. Copyright © 2010 John Wiley & Sons, Ltd.     

A parallel frontal solver for finite element applications

In finite element simulations, the overall computing time is dominated by the time needed to solve large sparse linear systems of equations. We report on the design and development of a parallel frontal code that can significantly reduce the wallclock time needed for the solution of these systems. The algorithm used is based on dividing the finite element domain into subdomains and applying the frontal method to each subdomain in parallel. The so‐called multiple front approach is shown to reduce the amount of work and memory required compared with the frontal method and, when run on a small number of processes, achieves good speedups. The code, HSL_MP42, has been developed for the Harwell Subroutine Library (http://www.numerical.rl.ac.uk/hsl). It is written in Fortran 90 and, by using MPI for message passing, achieves portability across a wide range of modern computer architectures. Copyright © 2001 John Wiley & Sons, Ltd.     

Navier-Stokes方程的集中质量非协调有限元法

本文通过所谓的速度-压力型公式讨论了Navier-Stokes方程的集中质量非协调有限元法(半离散情形)。首先给出了所讨论方程的集中质量非协调有限元逼近格式,其次对所讨论方程的真解与逼近格式的解之间的误差进行了分析,最后利用Navier-Stokes投影算子及其性质,得到了在确定模意义下的速度、压力误差估计,且某些误差估计能达到最优。     

求解非定常N-S方程的稳定化分数步长法研究

本文研究了求解非定常Navier-Stokes方程的稳定化分数步长法.首先,通过一阶精度的算子分裂,将非线性项和不可压缩条件分裂到两个不同的子问题中,并对非线性项采用Oseen迭代.格式分为两步:第一步求解一个线性椭圆问题;第二步求解一个广义的Stokes问题.这两个子问题关于速度都满足齐次Dilichlet边界条件.同时,在格式的第二步添加了局部稳定化项,使用等阶序对来加强数值解的稳定性.通过能量估计方法,对速度与压力做了收敛性分析和误差估计.最后,数值实验验证了方法的有效性.     

Two-Grid Finite Element Methods for the Steady Navier-Stokes/Darcy Model

Two-grid finite element methods for the steady Navier-Stokes/Darcy model are considered. Stability and optimal error estimates in the $H^1$-norm for velocity and piezometric approximations and the $L^2$-norm for pressure are established under mesh sizes satisfying $h=H^2$. A modified decoupled and linearised two-grid algorithm is developed, together with some associated optimal error estimates. Our method and results extend and improve an earlier investigation, and some numerical computations illustrate the efficiency and effectiveness of the new algorithm.     

非定常不可压Navier-Stokes方程基于Crank-Nicolson格式的两水平变分多尺度方法

不可压缩粘性流是密度不发生变化的流体运动．它们被用来描述许多重要的物理现象,例如：天气、洋流、绕翼型流动和动脉内的血液流动．Navier-Stokes方程是不可压缩粘性流的基本方程．因此,求解Navier-Stokes方程的数值方法在近几十年得到了广泛的关注．本文主要给出非定常不可压Navier-Stokes方程基于Crank-Nicolson格式的两水平变分多尺度方法．该方法分为两步：第一步,在粗网格上求解稳定的非线性Navier-Stokes系统;第二步,在细网格上求解稳定的线性问题去校正粗网格上的解．通过该方法推导的速度的误差估计关于时间是二阶收敛的．数值实验验证了在粗细网格匹配合理的情形下,本文的方法与直接在细网格上使用单网格的变分多尺度方法相比,可以节约大量的计算时间．     

Numerical modelling of tidal flows in complex estuaries including turbulence: an unstructured finite volume solver and experimental validation

In this paper an unstructured finite volume model for quasi‐2D tidal flow with wet–dry fronts and turbulence modelling is presented, and applied to the Crouch–Roach estuarine system (Essex, U.K.). Two depth averaged turbulence models, a mixing length model and a k–ε model, are used in the numerical computations. An additional limiter to the production of turbulence due to bed friction is introduced in order to improve the performance and numerical stability of the model near wet–dry fronts. In addition to a first‐order and a second‐order schemes, an hybrid second‐order/first‐order upwind scheme which improves the accuracy of the first‐order scheme while maintaining a good numerical stability is used to discretize the convective flux. Numerical results are compared with observed current speed and water level data, with particular reference to the ability of the model to reproduce shallow water tidal harmonics. Copyright © 2006 John Wiley & Sons, Ltd.     

非定常不可压缩Navier-Stokes方程的特征稳定化非协调有限元方法

数值求解非定常不可压缩Navier-Stokes方程的难点之一在于强烈的非线性容易引发非物理震荡,本文结合可以有效减弱此种震荡的特征线离散方法,基于局部Gauss积分之差的稳定化格式,采用最低等阶非协调混合有限元对NCP1-P1,构造出求解非定常不可压缩Navier-Stokes方程的特征稳定化非协调混合有限元方法。证明了该方法的全离散格式是无条件稳定的,并给出逼近解的相应误差估计。     

Interpolation functions in control volume finite element method

 The main contribution of this paper is the study of interpolation functions in control volume finite element method used in equal order and applied to an incompressible two-dimensional fluid flow. Especially, the exponential interpolation function expressed in the elemental local coordinate system is compared to the classic linear interpolation function expressed in the global coordinate system. A quantitative comparison is achieved by the application of these two schemes to four flows that we know the analytical solutions. These flows are classified in two groups: flows with privileged direction and flows without. The two interpolation functions are applied to a triangular element of the domain then; a direct comparison of the results given by each interpolation function to the exact value is easily realized. The two functions are also compared when used to solve the discretized equations over the entire domain. Stability of the numerical process and accuracy of solutions are compared. Received: 20 October 2002 / Accepted: 2 December 2002     

A robust and efficient numerical finite element method for cables

Numerical simulation of cable systems remain delicate due to their geometrical nonlinearity and also to their intrinsic unilateral constitutive law. Indeed Finite Element approaches (if not implemented carefully) fail to predict accurate equilibrium for cable structures. The major issue to be addressed is the ill-conditioning, starting configuration and wrong choice of descent direction during iterative methods. An iterative scheme based on Finite Element Method is presented to overcome this issue, even with large number of elements.