A preconditioned semi‐staggered dilation‐free finite volume method for the incompressible Navier–Stokes equations on all‐hexahedral elements

A new semi‐staggered finite volume method is presented for the solution of the incompressible Navier–Stokes equations on all‐quadrilateral (2D)/hexahedral (3D) meshes. The velocity components are defined at element node points while the pressure term is defined at element centroids. The continuity equation is satisfied exactly within each elements. The checkerboard pressure oscillations are prevented using a special filtering matrix as a preconditioner for the saddle‐point problem resulting from second‐order discretization of the incompressible Navier–Stokes equations. The preconditioned saddle‐point problem is solved using block preconditioners with GMRES solver. In order to achieve higher performance FORTRAN source code is based on highly efficient PETSc and HYPRE libraries. As test cases the 2D/3D lid‐driven cavity flow problem and the 3D flow past array of circular cylinders are solved in order to verify the accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

ℋ︁‐LU factorization in preconditioners for augmented Lagrangian and grad‐div stabilized saddle point systems

The (mixed finite element) discretization of the linearized Navier–Stokes equations leads to a linear system of equations of saddle point type. The iterative solution of this linear system requires the construction of suitable preconditioners, especially in the case of high Reynolds numbers. In the past, a stabilizing approach has been suggested which does not change the exact solution but influences the accuracy of the discrete solution as well as the effectiveness of iterative solvers. This stabilization technique can be performed on the continuous side before the discretization, where it is known as ‘grad‐div’ (GD) stabilization, as well as on the discrete side where it is known as an ‘augmented Lagrangian’ (AL) technique (and does not change the discrete solution). In this paper, we study the applicability of ??‐LU factorizations to solve the arising subproblems in the different variants of stabilized saddle point systems. We consider both the saddle point systems that arise from the stabilization in the continuous as well as on the discrete setting. Recently, a modified AL preconditioner has been proposed for the system resulting from the discrete stabilization. We provide a straightforward generalization of this approach to the GD stabilization. We conclude the paper with numerical tests for a variety of problems to illustrate the behavior of the considered preconditioners as well as the suitability of ??‐LU factorization in the preconditioners. Copyright © 2010 John Wiley & Sons, Ltd.     

A comparison of preconditioners for incompressible Navier–Stokes solvers

We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the so‐called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium‐sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright © 2007 John Wiley & Sons, Ltd.     

含J2摄动的有限时间航天器追逃博弈问题研究

航天器追逃博弈是航天器在轨捕获任务的一个重要问题,具有极高的军民两用双重价值．针对有限时间且考虑J2摄动的航天器追逃博弈问题,本文提出了一种精确的求解方法．该方法的核心思想是将有限时间的航天器追逃博弈问题建模为有限时间二人零和对策,则博弈中两航天器的最优控制策略可以转化为有限时间二人零和对策的鞍点解．在鞍点解的求解过程中,本文首先基于考虑J2摄动的非线性动力学方程,将两航天器动力学方程和始末边值条件与鞍点解必要条件结合得到两点边值问题,然后提出一种结合遗传算法和配点法的混合算法求解该两点边值问题以得到精确的鞍点解．本文利用数值仿真对所提方法的有效性进行了验证．结果表明：(i) 在航天器追逃博弈过程中,J2摄动对两航天器的最优控制策略具有较大影响;(ii) 所提方法能够精确求解出两航天器在有限时间的追逃博弈过程中的最优控制策略．     

Stability of transonic two-phase flow

The nature of a singular point in the stability of one-dimensional transonic flow of a vapor-drop mixture in a channel of variable cross section is considered within the framework of a two-lquid hydrodynamical model. It is shown that the singular point in the case of any lags of the drops preserves the nature of a saddle inherent to homogeneous gas flow, shifting only towards the divergent part of the channel if the content of condensed phase is not too high. Here the transition of subsonic two-phase flow into supersonic flow is stable and the predominance of drop agglomeration over fragmentation and the positive curvature of the channel profile are stabilizing factors. The saddle nature of the singularity is possible only if the lag of the drops is not too high in the case of flows with a higher content of condensed phase. In the opposite case, the point at which the speed of sound is attained loses the nature of a saddle point.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 163–171, January–February, 1976.     

A least-squares finite element method for incompressible Navier-Stokes problems

A least-squares finite element method based on the velocity–pressure–vorticity formulation was proposed for solving steady incompressible Navier-Stokes problems. This method leads to a minimization problem rather than to the saddle point problem of the classic mixed method and can thus accommodate equal-order interpolations. The method has no parameter to tune. The associated algebraic system is symmetric and positive definite. In order to show the validity of the method for high-Reynolds-number problems, this paper provides numerical results for cavity flow at Reynolds number up to 10 000 and backward-facing step flow at Reynolds number up to 900.     

High‐order ILU preconditioners for CFD problems

This paper tests a number of incomplete lower–upper (ILU)‐type preconditioners for solving indefinite linear systems, which arise from complex applications such as computational fluid dynamics (CFD). Both point and block preconditioners are considered. The paper focuses on ILU factorization that can be computed with high accuracy by allowing liberal amounts of fill‐in. A number of strategies for enhancing the stability of the factorizations are examined. Copyright © 2000 John Wiley & Sons, Ltd.     

Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations

We study different variants of the augmented Lagrangian (AL)‐based block‐triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput. 2006; 28 : 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual method (GMRES) applied to various finite element and Marker‐and‐Cell discretizations of the Oseen problem in two and three space dimensions. Both steady and unsteady problems are considered. Numerical experiments show the effectiveness of the proposed preconditioners for a wide range of problem parameters. Implementation on parallel architectures is also considered. The AL‐based approach is further generalized to deal with linear systems from stabilized finite element discretizations. Copyright © 2010 John Wiley & Sons, Ltd.     

关于Orr—Sommerfeld方程的Chebyshev谱方法的讨论

本文讨论了Ｏｒｒ－Ｓｏｍｍｅｒｅｌｄ方程的各种Ｃｈｅｂｙｓｈｅｖ谱离散方法，数值证明了Ｃｈｅｂｙｓｈｅｖ配置法离散Ｏｒｒ－Ｓｏｍｍｅｒｆｅｌｄ方程没有伪谱，并以此构造了适于任意平面平行速度剖面情形，对时间和时空稳定性模式一致有效的无伪谱的离离散方法，其中，对时空稳定性问题本文给出了一种新的迭代法可以快速有效地求出复频率的鞍点，对平面Ｐｏｉｓｅｕｉｌｌｅ流，Ｂｌａｓｉｕｓ边界层流和Ｇａｕｓｓ模型尾迹     

A regularization method for a boundary variational inequality of the second kind associated to a friction problem

A scalar contact problem with friction is formulated as a boundary variational inequality of the second kind. The presence of the non-differentiable friction functional causes difficulties when approximating it. We present two approaches to overcome these difficulties: A regularization procedure leading to a non-linear boundary variational equation, for which we propose an iterative process and the second one is a boundary mixed variational formulation involving Lagrange multiplier. We reformulate our problem in terms of a saddle point problem for the corresponding boundary Lagrangian and describe Uzawa's algorithm to compute it.     

Towards the numerical simulation of the horizontal slug front

Steps towards the numerical simulation of the flow behind the slug front in horizontal slug flow performed with a streamfunction-vorticity representation of the mean flow and an energy dissipation model for the turbulence are discussed. The flow field consists of two vortices, one saddle point and four stagnation regions. Attention is focused on the following boundary conditions: moving wall jet, moving wall, free jet velocity discontinuity and vertical liquid-gas open surface. A dissipation flux boundary condition is suggested to simulate the interaction of the turbulent eddies with the open surface. A method to assess the necessity to use a transport model equation for the dissipation rather than a geometric specification of a length is suggested. Three different ways to characterize the mixing zone length are proposed.     

Some special purpose preconditioners for conjugate gradient-like methods applied to CFD

Standard preconditioners such as incomplete LU decomposition perform well when used with conjugate gradient-like iterative solvers such as GMRES for the solution of elliptic problems. However, efficient computation of convection-dominated problems requires, in general, the use of preconditioners tuned to the particular class of fluid-flow problems at hand. This paper presents three such preconditioners. The first is applied to the finite element computation of inviscid (Euler equations) transonic and supersonic flows with shocks and uses incomplete LU decomposition applied to a matrix with extra artificial dissipation. The second preconditioner is applied to the finite difference computation of unsteady incompressible viscous flow; it uses incomplete LU decomposition applied to a matrix to which a pseudo-compressible term has been added. The third method and application are similar to the second, only the LU decomposition is replaced by Beam-warming approximate factorization. In all cases, the results are in very good agreement with other published results and the new algorithms are found to be competitive with others; it is anticipated that the efficiency and robustness of conjugate-gradient-like methods will render them the method of choice as the difficulty of the problems that they are applied to is increased.     

Properties of Chaotic and Regular Boundary Crisis in Dissipative Driven Nonlinear Oscillators

The phenomenon of the chaotic boundary crisis and the related concept of the chaotic destroyer saddle has become recently a new problem in the studies of the destruction of chaotic attractors in nonlinear oscillators. As it is known, in the case of regular boundary crisis, the homoclinic bifurcation of the destroyer saddle defines the parameters of the annihilation of the chaotic attractor. In contrast, at the chaotic boundary crisis, the outset of the destroyer saddle which branches away from the chaotic attractor is tangled prior to the crisis. In our paper, the main point of interest is the problem of a relation, if any, between the homoclinic tangling of the destroyer saddle and the other properties of the system which may accompany the chaotic as well as the regular boundary crisis. In particular, the question if the phenomena of fractal basin boundary, indeterminate outcome, and a period of the destroyer saddle, are directly implied by the structure of the destroyer saddle invariant manifolds, is examined for some examples of the boundary crisis that occur in the mathematical models of the twin-well and the single-well potential nonlinear oscillators.     

On the dynamics of compressor surge

Investigations concerned with the stability of stationary states and the possibility of self-excited oscillation (surge) occurring in systems with a centrifugal compressor (or a centrifugal pump) lead, for a simplified model, to an analysis of a set of two first-order differential equations. The paper presents such an analysis for the case when the machine characteristic can be expressed by a continuous unique curve as well as for that when the characteristic is neither a unique nor even a smooth curve. It is shown which of the singular points is the saddle point and in the case of the latter type of characteristic, which point can be taken for the saddle; this approach is believed to make practical analyses more straightforward.     

Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation

Based on a new global variational formulation, a spectral element approximation of the incompressible Navier–Stokes/Euler coupled problem gives rise to a global discrete saddle problem. The classical Uzawa algorithm decouples the original saddle problem into two positive definite symmetric systems. Iterative solutions of such systems are feasible and attractive for large problems. It is shown that, provided an appropriate pre‐conditioner is chosen for the pressure system, the nested conjugate gradient methods can be applied to obtain rapid convergence rates. Detailed numerical examples are given to prove the quality of the pre‐conditioner. Thanks to the rapid iterative convergence, the global Uzawa algorithm takes advantage of this as compared with the classical iteration by sub‐domain procedures. Furthermore, a generalization of the pre‐conditioned iterative algorithm to flow simulation is carried out. Comparisons of computational complexity between the Navier–Stokes/Euler coupled solution and the full Navier–Stokes solution are made. It is shown that the gain obtained by using the Navier–Stokes/Euler coupled solution is generally considerable. Copyright © 2000 John Wiley & Sons, Ltd.     

Surface flow pattern and local mass transfer on the free-end surface of a finite circular cylinder

 Relation between the surface flow pattern and the local mass transfer characteristic on the free end surface of a finite circular cylinder mounted on a flat plate was investigated using the oil surface flow visualization and the naphthalene sublimation technique. The aspect ratio [the ratio of the cylinder length to the cylinder diameter (H/D)] was 1.25 and the Reynolds number based on the diameter was 1.48 × 10⁵. Several kinds of critical points on the free-end surface were identified by the visualization and the mass transfer measurement around those critical points through the naphthalene sublimation technique was then performed to investigate the relation between the flow and mass transfer. The free-end surface flow was topologically characterized by the presence of a saddle point and two focal points in the fore-half portion of the surface and two nodes and a saddle point in the rear-half portion. It was found that the rate of mass transfer was largest in the vicinity of the nodes in the rear-half of the free-end surface. Received on 26 April 2000     

On Approximate Representation of Periodic Motions at a Saddle Point

The results of a qualitative analysis of a bistable oscillator are applied to the approximate integration of a system with a closed trajectory about a saddle point.     

On the efficiency of linearization schemes and coupled multigrid methods in the simulation of a 3D flow around a cylinder

This paper studies the efficiency of two ways to treat the non‐linear convective term in the time‐dependent incompressible Navier–Stokes equations and of two multigrid approaches for solving the arising linear algebraic saddle point problems. The Navier–Stokes equations are discretized by a second‐order implicit time stepping scheme and by inf–sup stable, higher order finite elements in space. The numerical studies are performed at a 3D flow around a cylinder. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical method for optimum motion of undulatory swimming plate in fluid flow

A numerical method for the optimum motion of an undulatory swimming plate is presented. The optimum problem is stated as minimizing the power input under the condition of fixed thrust. The problem is singular for the invisible modes, and therefore the commonly used Lagrange multiplier method cannot predict an optimum solution but just a saddle point. To eliminate the singularity, an additional amplitude inequality constraint is added to the problem. A numerical optimization code with a sequential quadratic programming method is used to solve the problem. The method is applied to several cases of the motion of two-dimensional and three-dimensional undulatory plates, and the optimum results are obtained.