ADAPTIVE PARALLEL MULTIGRID SOLUTION OF 2D INCOMPRESSIBLE NAVIER–STOKES EQUATIONS

In this paper an adaptive parallel multigrid method and an application example for the 2D incompressible Navier–Stokes equations are described. The strategy of the adaptivity in the sense of local grid refinement in the multigrid context is the multilevel adaptive technique (MLAT) suggested by Brandt. The parallelization of this method on scalable parallel systems is based on the portable communication library CLIC and the message-passing standards: PARMACS, PVM and MPI. The specific problem considered in this work is a two-dimensional hole pressure problem in which a Poiseuille channel flow is disturbed by a cavity on one side of the channel. Near geometric singularities a very fine grid is needed for obtaining an accurate solution of the pressure value. Two important issues of the efficiency of adaptive parallel multigrid algorithms, namely the data redistribution strategy and the refinement criterion, are discussed here. For approximate dynamic load balancing, new data in the adaptive steps are redistributed into distributed memories in different processors of the parallel system by block remapping. Among several refinement criteria tested in this work, the most suitable one for the specific problem is that based on finite-element residuals from the point of view of self-adaptivity and computational efficiency, since it is a kind of error indicator and can stop refinement algorithms in a natural way for a given tolerance. Comparisons between different global grids without and with local refinement have shown the advantages of the self-adaptive technique, as this can save computer memory and speed up the computing time several times without impairing the numerical accuracy. © 1997 By John Wiley & Sons, Ltd. Int. J. Numer. Methods Fluids 24, 875–892, 1997.     

A PARALLEL BLOCK-STRUCTURED MULTIGRID METHOD FOR THE PREDICTION OF INCOMPRESSIBLE FLOWS

In this paper a parallel multigrid finite volume solver for the prediction of steady and unsteady flows in complex geometries is presented. For the handling of the complexity of the geometry and for the parallelization a unified approach connected with the concept of block-structured grids is employed. The parallel implementation is based on grid partitioning with automatic load balancing and follows the message-passing concept, ensuring a high degree of portability. A high numerical efficiency is obtained by a non-linear multigrid method with a pressure correction scheme as smoother. By a number of numerical experiments on various parallel computers the method is investigated with respect to its numerical and parallel efficiency. The results illustrate that the high performance of the underlying sequential multigrid algorithm can largely be retained in the parallel implementation and that the proposed method is well suited for solving complex flow problems on parallel computers with high efficiency.     

利用多小波自适应格式求解流体力学方程

高阶计算格式的高精度、高分辨率对提高复杂流场的计算水平有重要的意义,为了提高AUSMPW格式对流场计算中激波等间断的分辨率,减小数值振荡,在原有AUSMPW格式的基础之上,利用多小波对函数进行多尺度分解,并采取阈值的方法生成自适应网格,提出了一种新的基于多小波自适应算法的AUSMPW格式,理论上可以达到任意阶精度. 将所得的压强、密度与原格式、TVD格式及WENO格式的计算结果进行了比较分析. 结果表明改进后的AUSMPW格式较原格式具有更高的分辨率、更强的捕捉间断的能力及更低的数值耗散.      

A NON-LINEAR ADAPTIVE FULL TRI-TREE MULTIGRID METHOD FOR THE MIXED FINITE ELEMENT FORMULATION OF THE NAVIER–STOKES EQUATIONS

The full adaptive multigrid method is based on the tri-tree grid generator. The solution of the Navier–Stokes equations is first found for a low Reynolds number. The velocity boundary conditions are then increased and the grid is adapted to the scaled solution. The scaled solution is then used as a start vector for the multigrid iterations. During the multigrid iterations the grid is first recoarsed a specified number of grid levels. The solution of the Navier–Stokes equations with the multigrid residual as right-hand side is smoothed in a fixed number of Newton iterations. The linear equation system in the Newton algorithm is solved iteratively by CGSTAB preconditioned by ILU factorization with coupled node fill-in. The full adaptive multigrid algorithm is demonstr ated for cavity flow. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1037ndash;1047, 1997.     

跨音叶栅流场中Euler方程的对角化隐式算法

本文在非因式分解隐式通量分裂有限面积法的基础81提出了一种单向隐式对角化算法,并给出了相应的固壁边界条件的一种精确的隐式处理方法.数值结果表明,这种方法是高效率的。     

一种改进的分块隐式数值方法

提出了一种改进的分块隐式数值方法，在贴体坐标和交错网格下以逆变速度分量和压力Ｕ，Ｖ，Ｗ，ｐ为基本求解变量，由此克服了原分块隐式数值方法求解复杂边界流动时的困难．９０°弯管流动数值计算初步表明，本文提出的方法合理、可行     

ADAPTIVE LINEARIZATION AND GRID ITERATIONS WITH THE TRI-TREE MULTIGRID REFINEMENT–RECOARSEMENT ALGORITHM FOR THE NAVIER–STOKES EQUATIONS

The tri-tree algorithm for refinements and recoarsements of finite element grids is explored. The refinement–recoarsement algorithm not only provides an accurate solution in certain parts of the grid but also has a major influence on the finite element equation system itself. The refinements of the grid lead to a more symmetric and linear equation matrix. The recoarsements will ensure that the grid is not finer than is necessary for preventing divergence in an iterative solution procedure. The refinement–recoarsement algorithm is a dynamic procedure and the grid is adapted to the instant solution. In the tri-tree multigrid algorithm the solution from a coarser grid is scaled relatively to the increase in velocity boundary condition for the finer grid. In order to have a good start vector for the solution of the finer grid, the global Reynolds number or velocity boundary condition should not be subject to large changes. For each grid and velocity solution the element Reynolds number is computed and used as the grid adaption indicator during the refinement–recoarsement procedure. The iterative tri-tree multigrid method includes iterations with respect to the grid. At each Reynolds number the same boundary condition s are applied and the grid is adapted to the solution iteratively until the number of unknowns and elements in the grid becomes constant. In the present paper the following properties of the tri-tree algorithm are explored: the influence of the increase in boundary velocities and the size of the grid adaption indicator on the amount of work for solving the equations, the number of linear iterations and the solution error estimate between grid levels. The present work indicates that in addition to the linear and non-linear iterations, attention should also be given to grid adaption iterations. © 1997 by John Wiley & Sons, Ltd.     

动力学平衡方程的Euler中点辛差分求解格式

给出了动力学方程${\pmb M}\ddot {\pmb x} + {\pmb C}\dot {\pmb x} + {\pmb K \pmb x} = {\pmb R}$的二阶Euler中点隐式差分求解格式,分保守系统、无 阻尼受迫振动系统和阻尼系统3种情况, 讨论了算法中Jacobi矩阵${\pmb A}$的性质,譬 如${\pmb A}$是否为辛矩阵以及谱半径等. 对于无阻尼系统,证明了无论是否存在外 载荷,Jacobi 矩阵都是辛矩阵. 证明了辛矩阵的所有本征值的模为1,其谱半径永远 为1, 以及$\delta = 0.5$和$\alpha = 0.25$的Newmark算法就是Euler中点隐式差 分格式,对保守系统它们都是辛算法. 严格证 明了Euler中点辛格式是严格保持系统能量的. 通过算例详细讨论了保辛算法用于求解非保 守系统动态特性的优越性,如广义保结构特性等;分析了保辛算法的相位误差以及由其引起 的系统的附加能量特性;分析了保辛算法和$\delta \ne 0.5$的Newmark算法的精度随着激励频率与系统固有频率比的变化情况等     

线化欧拉方程的高阶间断有限元数值解法研究

采用高阶间断有限元法于非结构网格上针对复杂外形数值求解声学控制方程------线化欧拉方程. 背景流场采用有限体积法于结构网格求得, 一种高精度数据传递方法将基于有限体积法的背景流场数据传递到声场计算所采用的较为稀疏的非结构网格上, 保证了背景流场信息的完整和精确. 为提高计算效率, 采用了一种更为直接的Quadrature-FreeImplementation技术以及网格分区并行技术. 数值结果表明采用高阶的情况下即使在稀疏的网格上也可以捕捉到细微的声场结构.      

由欧拉角确定一次回转轴的简单解法

欧拉-达朗贝尔定理表明,定点运动刚体的有限位移可以通过绕一固定轴的一次转动实现。本文介绍了一种利用欧拉角求解刚体从任意一个姿态到另一个姿态作一次转动的转轴位置及回转角度的计算方法,给出了以欧拉角表示的一次回转轴公式。和求解坐标变换矩阵特征向量的方法比较,该算法计算量小,在应用上更为简便。     

MULTIGRID AND KRYLOV SUBSPACE METHODS FOR THE DISCRETE STOKES EQUATIONS

Discretization of the Stokes equations produces a symmetric indefinite system of linear equations. For stable discretizations a variety of numerical methods have been proposed that have rates of convergence independent of the mesh size used in the discretization. In this paper we compare the performance of four such methods, namely variants of the Uzawa, preconditioned conjugate gradient, preconditioned conjugate residual and multigrid methods, for solving several two-dimensional model problems. The results indicate that multigrid with smoothing based on incomplete factorization is more efficient than the other methods, but typically by no more than a factor of two. The conjugate residual method has the advantage of being independent of iteration parameters.     

AN UNFACTORED IMPLICIT MOVING MESH METHOD FOR THE TWO-DIMENSIONAL UNSTEADY N–S EQUATIONS

An unfactored implicit time-marching method for the solution of the unsteady two-dimensional Reynolds-averaged thin layer Navier–Stokes equations is presented. The linear system arising from each implicit step is solved by the conjugate gradient squared (CGS) method with preconditioning based on an ADI factorization. The time-marching procedure has been used with a fast transfinite interpolation method to regenerate the mesh at each time step in response to the motion of the aerofoil. The main test cases examined are from the AGARD aeroelastic configurations and involve aerofoils oscillating rigidly in pitch. These test cases have been used to investigate the effect of various parameters, such as CGS tolerance and laminar/turbulent transition location, on the accuracy and efficiency of the method. Comparisons with available experimental data have been made for these cases. In order to illustrate the application of the mesh generator and flow solver to more general flows where the aerofoil deforms, results for an NACA 0012 aerofoil with an oscillating trailing edge flap are also shown.     

广义对流扩散方程的隐式特征线混合元方法

提出了一个广义对流扩散方程的混合有限元方法,方程的基本变量及其空间梯度和流量在单 元内均作为独立变量分别插值. 基于胡海昌-Washizu三变量广义变分原理结合特征线法给 出了控制方程的单元弱形式. 混合元方法采用基于一点积分方案并结合可以滤掉虚假的 数值震荡的隐式特征线法. 数值结果证明了所提出的方法可以提供和四点积分同样的数 值计算结果,并能够提高计算效率.     

求解气动方程的混合反扩散方法

本文综述了反扩散格式的发展,指出利用混合反扩散方法,理论上既可导出Beam—Warming以及Jameson等发展的含参数的格式,又可导出不含参数的TVD格式.研究了含参数的混合反扩散格式和不含参数的反扩散格式,并介绍了格式的应用情况.      

跨音速大迎角绕流的Euler方程数值模拟

采用保角变换与代数方法相结合，生成全场统一的贴体、正交Ｏ－Ｈ型网格．采用有限体积法求解Ｅｕｌｅｒ方程，模拟具有歼击机外形的全机及翼身组合体大迎角跨音速绕流．计算表明，法向力系数、气动中心位置及压力分布的计算结果与实验结果吻合良好     

比例边界坐标插值方法在谱元法中的应用——无穷域Euler方程的数值模拟

将比例边界坐标插值方法引入谱元法, 构成比例边界谱单元, 对无穷域Euler方程进行数值模拟.阐述了比例边界谱单元的基本使用方法以及基于比例边界谱元的Runge-Kutta间断Galerkin方法求解Euler方程的过程;计算了无穷域圆柱和NACA0012翼型绕流问题, 并与已有结果进行了比较, 显示了计算结果的正确性.用基于比例边界谱元的间断Galerkin方法求解无穷域Euler方程时, 最多只需将求解域划分为2个子域, 避免了一般谱方法将求解域划分为9个或者27个子域的麻烦. 比例边界谱单元为无穷域Euler方程的直接求解提供了一个可供参考的方法.     

A PRESSURE-CORRECTION METHOD FOR THE SOLUTION OF INCOMPRESSIBLE VISCOUS FLOWS ON UNSTRUCTURED GRIDS

An unstructured grid, finite volume method is presented for the solution of two-dimensional viscous, incompressible flow. The method is based on the pressure-correction concept implemented on a semi-staggered grid. The computational procedure can handle cells of arbitrary shape, although solutions presented herein have been obtained only with meshes of triangular and quadrilateral cells. The discretization of the momentum equations is effected on dual cells surrounding the vertices of primary cells, while the pressure-correction equation applies to the primary-cell centroids and represents the conservation of mass across the primary cells. A special interpolation scheme s used to suppress pressure and velocity oscillations in cases where the semi-staggered arrangement does not ensure a sufficiently strong coupling between pressure and velocity to avoid such oscillations. Computational results presented for several viscous flows are shown to be in good agreement with analytical and experimental data reported in the open literature.     

A NON–LINEAR ADAPTED TRI–TREE MULTIGRID SOLVER FOR FINITE ELEMENT FORMULATION OF THE NAVIER–STOKES EQUATIONS

An iterative adaptive equation multigrid solver for solving the implicit Navier–Stokes equations simultaneously with tri-tree grid generation is developed. The tri-tree grid generator builds a hierarchical grid structur e which is mapped to a finite element grid at each hierarchical level. For each hierarchical finite element multigrid the Navier–Stokes equations are solved approximately. The solution at each level is projected onto the next finer grid and used as a start vector for the iterative equation solver at the finer level. When the finest grid is reached, the equation solver is iterated until a tolerated solution is reached. The iterative multigrid equation solver is preconditioned by incomplete LU factorization with coupled node fill-in. The non-linear Navier–Stokes equations are linearized by both the Newton method and grid adaption. The efficiency and behaviour of the present adaptive method are compared with those of the previously developed iterative equation solver which is preconditioned by incomplete LU factorization with coupled node fill-in.     

气体动理学统一算法的隐式方法研究

目前的气体动理学统一算法(unified gas kinetic scheme, 简称UGKS) 在求解高速流动问题时的计算效率,难以满足求解复杂工程问题的需求. 为了提高该算法的计算效率, 本文对模型方程的对流项和碰撞项进行了隐式处理, 并针对UGKS 界面通量与演化时间相关的特点, 引入了演化时间平均界面通量, 通过对控制方程矩阵进行近似LU 分解(lower-upper decomposition), 实现了隐式UGKS. 不同来流马赫数的圆柱绕流算例测试表明, 只要演化时间选取得当, 隐式方法可以得到与显式方法完全相同的结果, 且计算效率可以提高1~2 个量级.      

非结构隐式方法在空心弹丸流场模拟中的应用

采用非结构网格的LU-SGS隐式算法计算三维Euler方程,数值模拟了不同马赫数以及不同攻角下某空心弹丸绕流流场,分析了流场的波系结构及其升阻力特性,计算结果表明空心弹丸的阻力系数比同口径的普通弹丸的阻力系数大约小30％,空心弹丸的阻力系数以及升力系数随攻角的变化规律与普通弹丸一致．