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.     

Global convergence of inexact Newton methods for transonic flow

In computational fluid dynamics, non-linear differential equations are essential to represent important effects such as shock waves in transonic flow. Discretized versions of these non-linear equations are solved using iterative methods. In this paper an inexact Newton method using the GMRES algorithm of Saad and Schultz is examined in the context of the full potential equation of aerodynamics. In this setting, reliable and efficient convergence of Newton methods is difficult to achieve. A poor initial solution guess often leads to divergence or very slow convergence. This paper examines several possible solutions to these problems, including a standard local damping strategy for Newton's method and two continuation methods, one of which utilizes interpolation from a coarse grid solution to obtain the initial guess on a finer grid. It is shown that the continuation methods can be used to augment the local damping strategy to achieve convergence for difficult transonic flow problems. These include simple wings with shock waves as well as problems involving engine power effects. These latter cases are modelled using the assumption that each exhaust plume is isentropic but has a different total pressure and/or temperature than the freestream.     

A convergence accelerator of a linear system of equations based upon the power method

This paper considers the convergence rate of an iterative numerical scheme as a method for accelerating at the post‐processor stage. The methodology adapted here is: (1) residual eigenmodes included in the origin of the convex hull are eliminated; (2) remaining residual terms are smoothed away by the main convergence algorithm. For this purpose, the polynomial matrix approach is employed for deriving the characteristic equation by two different methods. The first method is based on vector scaling and the second is based on the normal equations approach. The input for both methods is the solution difference between two consecutive iteration/cycle levels obtained from the main program. The singular value decomposition was employed for both methods due to the ill‐conditioned structure of the matrices. The use of the explicit form of the Richardson extrapolation in the present work overrules the need to employ the Richardson iteration with a Leja ordering. The performance of these methods was compared with the GMRES algorithm for three representative problems: two‐dimensional boundary value problem using the Laplace equation, three‐dimensional multi‐grid, potential solution over a sphere and the one‐dimensional steady state Burger equation. In all three examples both methods have the same rate of convergence, or better, as that of the GMRES method in terms of computer operational count. However, in terms of storage requirements, the method based upon vector scaling has a significant advantage over the normal equations approach as well as the GMRES method, in which only one vector of the N grid‐points is required. Copyright © 2001 John Wiley & Sons, Ltd.     

A SOLVER USING NEWTON‘S METHOD FOR UNSTEADY VISCOUS FLOWS

A solver is developed for time-accurate computations of viscous flows based on the conception of Newton‘s method. A set of pseudo-time derivatives are added into governing equations and the discretized system is solved using GMRES algorithm. Due to some special properties of GMRES algorithm, the solution procedure for unsteady flows could be regarded as a kind of Newton iteration. The physical-time derivatives of governing equations are discretized using two different approaches, I.e., 3-point Euler backward, and Crank-Nicolson formulas, both with 2nd-order accuracy in time but with different truncation errors. The turbulent eddy viscosity is calculated by using a version of Spalart~Allmaras one-equation model modified by authors for turbulent flows. Two cases of unsteady viscous flow are investigated to validate and assess the solver, I.e., low Reynolds number flow around a row of cylinders and transonic bi-circular-arc airfoil flow featuring the vortex shedding and shock buffeting problems, respectively. Meanwhile, comparisons between the two schemes of timederivative discretizations are carefully made. It is illustrated that the developed unsteady flow solver shows a considerable efficiency and the Crank-Nicolson scheme gives better results compared with Euler method.     

二维定常湍流计算中的GMRES算法

在以前工作的基础上将广义极小残差（Generalized Minimum RESidual (GMRES)算法发展到用于求解二维可压Favier平均Navier-Stokes方程组。控制方程经Newton线化处理后构成近似的线性系统,然后采用分别耦合了LUSGS和ILU两种预处理矩阵的GMRES算法求解。Spalart-Allmaras湍流模型被用来封闭流体控制方程组,采用与流体控制方程非耦合的方式,使用LUSGS方法求解。对GMRES算法中矩阵－向量的乘积采用了有限插分方法,从而避免了精确的左端系数矩阵的计算和存储。对预处理矩阵的两种使用方法（左预处理和右预处理）进行了分析和讨论。用两个算例对LUSGS和ILU两各预处理矩阵进行了比较,同时探讨了左预处理和右预处理各自的优缺点。通过对Sajben扩压器和NACA0012有攻角流动的计算,表明带有预处理的GMRES算法在二维定常跨音黏性流动计算中相比于得到广泛应用的DDADI方法具有很大优势,左预处理要优于右预处理。     

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。     

Comparison of three solvers for viscoelastic fluid flow problems

The computational efficiency of three numerical schemes has been examined for the solution of a linearized system of equations resulting from the finite element discretization of a viscoelastic fluid flow problem. The first scheme is a modified frontal solver, which solves the linear system of equations directly. The other two, one based on a biconjugate gradient stabilized (BiCGStab) method and another based on a generalized minimal residual (GMRES) method, are iterative schemes. The stick-slip problem and the four-to-one contraction problem were analyzed and the viscoelastic fluid was assumed to obey the Oldroyd-B model. The two iterative schemes are superior to the direct scheme in terms of CPU time consumed and the BiCGStab scheme is even faster than the GMRES scheme. The range of convergence for both iterative schemes is compatible with that of the direct scheme.     

EFFECTS OF IMPLICIT PRECONDITIONERS ON SOLUTION ACCELERATION SCHEMES IN CFD

Several solution acceleration techniques, used to obtain steady state CFD solutions as quickly as possible, are applied to an implicit, upwind Euler solver to evaluate their effectiveness. The implicit system is solved using either ADI or ILU and the solution acceleration techniques evaluated are quasi-Newton iteration, Jacobian freezing, multigrid and GMRES. ILU is a better preconditioner than ADI because it can use larger time steps. Adding GMRES does not always improve the convergence. However, GMRES preconditioned with ILU and multigrid can take advantage of Jacobian freezing to produce an efficient scheme that is relatively independent of grid size and grid quality.     

Assessment of some iterative methods for non-symmetric linear systems arising in computational fluid dynamics

Various tests have been carried out in order to compare the performances of several methods used to solve the non-symmetric linear systems of equations arising from implicit discretizations of CFD problems, namely the scalar advection-diffusion equation and the compressible Euler equations. The iterative schemes under consideration belong to three families of algorithms: relaxation (Jacobi and Gauss-Seidel), gradient and Newton methods. Two gradient methods have been selected: a Krylov subspace iteration method (GMRES) and a non-symmetric extension of the conjugate gradient method (CGS). Finally, a quasi-Newton method has also been considered (Broyden). The aim of this paper is to provide indications of which appears to be the most adequate method according to the particular circumstances as well as to discuss the implementation aspects of each scheme.     

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。     

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.     

Accurate iteration-free mixed-stabilised formulation for laminar incompressible Navier–Stokes: Applications to fluid–structure interaction

Stabilised mixed velocity–pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier–Stokes. In these formulations, the Newton–Raphson scheme is employed to solve the nonlinearity in the convection term. One fundamental issue with this approach is the computational cost incurred in the Newton–Raphson iterations at every load/time step. In this paper, we present an iteration-free mixed finite element formulation for incompressible Navier–Stokes that preserves second-order temporal accuracy of the generalised-alpha and related schemes for both velocity and pressure fields. First, we demonstrate the second-order temporal accuracy using numerical convergence studies for an example with a manufactured solution. Later, we assess the accuracy and the computational benefits of the proposed scheme by studying the benchmark example of flow past a fixed circular cylinder. Towards showcasing the applicability of the proposed technique in a wider context, the inf–sup stable P2–P1 pair for the formulation without stabilisation is also considered. Finally, the resulting benefits of using the proposed scheme for fluid–structure interaction problems are illustrated using two benchmark examples in fluid-flexible structure interaction.     

MODELLING OF WAVE PROPAGATION IN THE NEARSHORE REGION USING THE MILD SLOPE EQUATION WITH GMRES-BASED ITERATIVE SOLVERS

The mild slope equation in its linear and non-linear forms is used for the modelling of nearshore wave propagation. The finite difference method is used to descretize the governing elliptic equations and the resulting system of equations is solved using GMRES-based iterative method. The original GMRES solution technique of Saad and Schultz is not directly applicable to the present case owing to the complex coefficient matrix. The simpler GMRES algorithm of Walker and Zhou is used as the core solver, making the upper Hessenberg factorization unneccessary when solving the least squares problem. Several preconditioning-based acceleration strategies are tested and the results show that the GMRES-based iteration scheme performs very well and leads to monotonic convergence for all the test-cases considered.     

A variational multiscale Newton–Schur approach for the incompressible Navier–Stokes equations

In the following paper, we present a consistent Newton–Schur (NS) solution approach for variational multiscale formulations of the time‐dependent Navier–Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three‐dimensional problems and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non‐symmetric matrices. In addition to the quadratic convergence characteristics of a Newton–Raphson‐based scheme, the NS approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two‐level approach to stabilizing the incompressible Navier–Stokes equations based on a coarse and fine‐scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three‐dimensional problems for Reynolds number up to 1000 including steady and time‐dependent flows. Copyright © 2009 John Wiley & Sons, Ltd.     

多重网格法与Newton-Raphson法耦合求线弹流数值解

利用多重网格算法,在最粗网格上,采用Newton-Raphson迭代法模式精解线接触弹流非线性方程组,充分利用了多重网格法与Newton-Raphson法各自的优点,计算实践表明,求解弹流问题的数值解过程在收敛与稳定性方面均有较大改善,且有相当宽的载荷参数适用范围。     

HIGH-ORDER DISCONTINUOUS GALERKIN SOLUTION OF N-S EQUATIONS ON HYBRID MESH

针对层流NS方程发展了混合网格上的高阶间断有限元方法,给出了物面边界高阶近似的具体步骤以及近物面弯曲单元的处理方法。对数值离散产生的非线性方程组采用牛顿迭代进行求解,每个牛顿循环采用预处理广义最小余量法求解产生的大型稀疏线性系统。使用该方法得到了典型算例的数值结果,并跟前人的计算结果进行了比较。计算结果表明,混合网格上应用高阶间断有限元方法求解黏性流动具有很好的应用前景。     

Solution of the discretized incompressible Navier-Stokes equations with the GMRES method

We describe some experiences using interative solution methods of GMRES type to solve the discretized Navier-Stokes equations. The discretization combined with a pressure correction scheme leads to two different systems of equations: the momentum equations and the pressure equation. It appears that a fast solution method for the pressure equation is obtained by applying the recently proposed GMRESR method, or GMRES combined with a MILU preconditioner. The diagonally scaled momentum equations are solved by GMRES(m), a restarted version of GMRES.     

Analysis of preconditioned iterative solvers for incompressible flow problems

Solving efficiently the incompressible Navier–Stokes equations is a major challenge, especially in the three‐dimensional case. The approach investigated by Elman et al. (Finite Elements and Fast Iterative Solvers. Oxford University Press: Oxford, 2005) consists in applying a preconditioned GMRES method to the linearized problem at each iteration of a nonlinear scheme. The preconditioner is built as an approximation of an ideal block‐preconditioner that guarantees convergence in 2 or 3 iterations. In this paper, we investigate the numerical behavior for the three‐dimensional lid‐driven cavity problem with wedge elements; the ultimate motivation of this analysis is indeed the development of a preconditioned Krylov solver for stratified oceanic flows which can be efficiently tackled using such meshes. Numerical results for steady‐state solutions of both the Stokes and the Navier–Stokes problems are presented. Theoretical bounds on the spectrum and the rate of convergence appear to be in agreement with the numerical experiments. Sensitivity analysis on different aspects of the structure of the preconditioner and the block decomposition strategies are also discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

预条件GMRES(m)法迭代求解大规模边界元弹性问题

提出了一种基于单元节点的块雅可比预条件方法,扩大了边界元法的计算规模,使之可用于大规模工程问题的求解.数值实验说明了这种预条件技术的有效性,表明预条件GMRES(m)算法具有较好的收敛特性,适合于求解大规模问题边界元弹性问题所形成的稠密非对称线性方程组.     

Compressible viscous flow calculations using compatible finite element approximations

We discuss in this paper the numerical simulation of compressible viscous flows by a combination of finite element methods for the space approximation, an implicit second-order multistep scheme for the time discretization and GMRES iterative methods for solving the non-linear problems encountered at each time step. Numerical results corresponding to flows around aerofoils and aerospace vehicles illustrate the possibilities of these methods.