An ILU preconditioner with coupled node fill-in for iterative solution of the mixed finite element formulation of the 2D and 3D Navier-Stokes equations

In the present paper, preconditioning of iterative equation solvers for the Navier-Stokes equations is investigated. The Navier-Stokes equations are solved for the mixed finite element formulation. The linear equation solvers used are the orthomin and the Bi-CGSTAB algorithms. The storage structure of the equation matrix is given special attention in order to avoid swapping and thereby increase the speed of the preconditioner. The preconditioners considered are Jacobian, SSOR and incomplete LU preconditioning of the matrix associated with the velocities. A new incomplete LU preconditioning with fill-in for the pressure matrix at locations in the matrix where the corner nodes are coupled is designed. For all preconditioners, inner iterations are investigated for possible improvement of the preconditioning. Numerical experiments are executed both in two and three dimensions.     

AN ADAPTIVE UNSTRUCTURED TRI-TREE ITERATIVE SOLVER FOR MIXED FINITE ELEMENT FORMULATION OF THE STOKES EQUATIONS

An iterative adaptive equation solver for solving the implicit Stokes equations simultaneously with tri-tree grid generation is developed. The tri-tree grid generator builds a hierarchical grid structure which is mapped to a finite element grid at each hierarchical level. For each hierarchical finite element grid the Stokes equations are solved. The approximate 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. In order to reduce the overall work, the element matrices are integrated analytically beforehand. 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. The efficiency of the incomplete coupled node fill-in preconditioner is shown to be largely dependent on the global node numbering. The preconditioner is therefore tested for the natural node ordering of the tri-tree grid generator and for different ways of sorting the nodes.     

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.     

The three-dimensional prolonged adaptive unstructured finite element multigrid method for the Navier–Stokes equations

This paper presents the development of the three- dimensional prolonged adaptive finite element equation solver for the Navier–Stokes equations. The finite element used is the tetrahedron with quadratic approximation of the velocities and linear approximation of the pressure. The equation system is formulated in the basic variables. The grid is adapted to the solution by the element Reynolds number. An element in the grid is refined when the Reynolds number of the element exceeds a preset limit. The global Reynolds number in the investigation is increased by scaling the solution for a lower Reynolds number. The grid is refined according to the scaled solution and the prolonged solution for the lower Reynolds number constitutes the start vector for the higher Reynolds number. Since the Reynolds number is the ratio of convection to diffusion, the grid refinements act as linearization and symmetrization of the equation system. The linear equation system of the Newton formulation is solved by CGSTAB with coupled node fill-in preconditioner. The test problem considered is the three-dimensional driven cavity flow. © 1997 John Wiley & Sons, Ltd.     

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.     

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.     

FAST ITERATIVE SOLVERS FOR THE DISCRETIZED INCOMPRESSIBLE NAVIER–STOKES EQUATIONS

In this paper some iterative solution methods of the GMRES type for the discretized Navier–Stokes equations are treated. The discretization combined with a pressure correction scheme leads to two different types of systems of linear equations: the momentum system and the pressure system. These systems may be coupled to one or more transport equations. For every system we specify a particular ILU-type preconditioner and show how to vectorize these preconditions. Finally, some numerical experiments to show the efficiency of the proposed methods are presented.     

A method for finite element parallel viscous compressible flow calculations

This paper presents the parallelization aspects of a solution method for the fully coupled 3D compressible Navier-Stokes equations. The algorithmic thrust of the approach, embedded in a finite element code NS3D, is the linearization of the governing equations through Newton methods, followed by a fully coupled solution of velocities and pressure at each non-linear iteration by preconditioned conjugate gradient-like iterative algorithms. For the matrix assembly, as well as for the linear equation solver, efficient coarse-grain parallel schemes have been developed for shared memory machines, as well as for networks of workstations, with a moderate number of processors. The parallel iterative schemes, in particular, circumvent some of the difficulties associated with domain decomposition methods, such as geometry bookkeeping and the sometimes drastic convergence slow-down of partitioned non-linear problems.     

Finite element solution of the incompressible Navier-Stokes equations by a Helmholtz velocity decomposition

Finite element solution methods for the incompressible Navier-Stokes equations in primitive variables form are presented. To provide the necessary coupling and enhance stability, a dissipation in the form of a pressure Laplacian is introduced into the continuity equation. The recasting of the problem in terms of pressure and an auxiliary velocity demonstrates how the error introduced by the pressure dissipation can be totally eliminated while retaining its stabilizing properties. The method can also be formally interpreted as a Helmholtz decomposition of the velocity vector. The governing equations are discretized by a Galerkin weighted residual method and, because of the modification to the continuity equation, equal interpolations for all the unknowns are permitted. Newton linearization is used and at each iteration the linear algebraic system is solved by a direct solver. Convergence of the algorithm is shown to be very rapid. Results are presented for two-dimensional flows in various geometries.     

A new fully coupled solution of the Navier-Stokes equations

A fully coupled method for the solution of incompressible Navier-Stokes equations is investigated here. It uses a fully implicit time discretization of momentum equations, the standard linearization of convective terms, a cell-centred colocated grid approach and a block-nanodiagonal structure of the matrix of nodal unknowns. The Method is specific in the interpolation used for the flux reconstruction problem, in the basis iterative method for the fully coupled system and in the acceleration means that control the global efficiency of the procedure. The performance of the method is discussed using lid-driven cavity problems, both for two and three-dimensional geometries, for steady and unsteady flows.     

A control volume finite‐element method for numerical simulating incompressible fluid flows without pressure correction

This paper presents a numerical model to study the laminar flows induced in confined spaces by natural convection. A control volume finite‐element method (CVFEM) with equal‐order meshing is employed to discretize the governing equations in the pressure–velocity formulation. In the proposed model, unknown variables are calculated in the same grid system using different specific interpolation functions without pressure correction. To manage memory storage requirements, a data storage format is developed for generated sparse banded matrices. The performance of various Krylov techniques, including Bi‐CGSTAB (Bi‐Conjugate Gradient STABilized) with an incomplete LU (ILU) factorization preconditioner is verified by applying it to three well‐known test problems. The results are compared to those of independent numerical or theoretical solutions in literature. The iterative computer procedure is improved by using a coupled strategy, which consists of solving simultaneously the momentum and the continuity equation transformed in a pressure equation. Results show that the strategy provides useful benefits with respect to both reduction of storage requirements and central processing unit runtime. Copyright © 2006 John Wiley & Sons, Ltd.     

Preconditioned conjugate gradient methods for the incompressible Navier-Stokes equations

A robust technique for solving primitive variable formulations of the incompressible Navier-Stokes equations is to use Newton iteration for the fully implicit non-linear equations. A direct sparse matrix method can be used to solve the Jacobian but is costly for large problems; an alternative is to use an iterative matrix method. This paper investigates effective ways of using a conjugate-gradient-type method with an incomplete LU factorization preconditioner for two-dimensional incompressible viscous flow problems. Special attention is paid to the ordering of unknowns, with emphasis on a minimum updating matrix (MUM) ordering. Numerical results are given for several test problems.     

Using a segregated finite element scheme to solve the incompressible Navier-Stokes equations

In this paper, a segregated finite element scheme for the solution of the incompressible Navier-Stokes equations is proposed which is simpler in form than previously reported formulations. A pressure correction equation is derived from the momentum and continuity equations, and equal-order interpolation is used for both the velocity components and pressure. Algorithms such as this have been known to lead to checkerboard pressure oscillations; however, the pressure correction equation of this scheme should not produce these oscillations. The method is applied to several laminar flow situations, and details of the methods used to achieve converged solutions are given.     

Simultaneous solution of the Navier-Stokes and elastic membrane equations by a finite element method

Fluid flow through a significantly compressed elastic tube occurs in a variety of physiological situations. Laboratory experiments investigating such flows through finite lengths of tube mounted between rigid supports have demonstrated that the system is one of great dynamical complexity, displaying a rich variety of self-excited oscillations. The physical mechanisms responsible for the onset of such oscillations are not yet fully understood, but simplified models indicate that energy loss by flow separation, variation in longitudinal wall tension and propagation of fluid elastic pressure waves may all be important. Direct numerical solution of the highly non-linear equations governing even the most simplified two-dimensional models aimed at capturing these basic features requires that both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. To accomplish this, previous algorithms have decoupled the solid and fluid mechanics, solving for each separately and converging iteratively on a solution which satisfies both. This paper describes a finite element technique which solves the incompressible Navier-Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary. The elastic boundary position is parametized in terms of distances along spines in a manner similar to that which has been used successfully in studies of viscous free surface flows, but here the membrane curvature equation rather than the kinematic boundary condition of vanishing normal velocity is used to determine these diatances and the membrane tension varies with the shear stresses exerted on it by the fluid motions. Bothy the grid and the spine positions adjust in response to membrane deformation, and the coupled fluid and elastic equations are solved by a Newton-Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse. Solutions to the steady problem are discussed, along with an indication of how the time-dependent problem might be approached.     

A composite multigrid method for calculating unsteady incompressible flows in geometrically complex domains

A time-accurate, finite volume method for solving the three-dimensional, incompressible Navier-Stokes equations on a composite grid with arbitrary subgrid overlapping is presented. The governing equations are written in a non-orthogonal curvilinear co-ordinate system and are discretized on a non-staggered grid. A semi-implicit, fractional step method with approximate factorization is employed for time advancement. Multigrid combined with intergrid iteration is used to solve the pressure Poisson equation. Inter-grid communication is facilitated by an iterative boundary velocity scheme which ensures that the governing equations are well-posed on each subdomain. Mass conservation on each subdomain is preserved by using a mass imbalance correction scheme which is secondorder-accurate. Three test cases are used to demonstrate the method's consistency, accuracy and efficiency.     

Large-scale computational fluid dynamics by the finite element method

Solution methods are presented for the large systems of linear equations resulting from the implicit, coupled solution of the Navier-Stokes equations in three dimensions. Two classes of methods for such solution have been studied: direct and iterative methods. For direct methods, sparse matrix algorithms have been investigated and a Gauss elimination, optimized for vector-parallel processing, has been developed. Sparse matrix results indicate that reordering algorithms deteriorate for rectangular, i.e. M × M × N, grids in three dimensions as N gets larger than M. A new local nested dissection reordering scheme that does not suffer from these difficulties, at least in two dimensions, is presented. The vector-parallel Gauss elimination is very efficient for processing on today's supercomputers, achieving execution rates exceeding 2.3 Gflops the Cray YMP-8 and 9.2 Gflops on the NEC on SX3. For iterative methods, two approaches are developed. First, conjugate-gradient-like methods are studied and good results are achieved with a preconditioned conjugate gradient squared algorithm. Convergence of such a method being sensitive to the preconditioning, a hybrid viscosity method is adopted whereby the preconditioner has an artificial viscosity that is gradually lowered, but frozen at a level higher than the dissipation introduced in the physical equations. The second approach is a domain decomposition one in which overlapping domain and side-by-side methods are tested. For the latter, a Lagrange multiplier technique achieves reasonable rates of convergence.     

Efficient computation of two‐dimensional steady free‐surface flows

We consider a family of steady free‐surface flow problems in two dimensions, concentrating on the effect of nonlinearity on the train of gravity waves that appear downstream of a disturbance. By exploiting standard complex variable techniques, these problems are formulated in terms of a coupled system of Bernoulli equation and an integral equation. When applying a numerical collocation scheme, the Jacobian for the system is dense, as the integral equation forces each of the algebraic equations to depend on each of the unknowns. We present here a strategy for overcoming this challenge, which leads to a numerical scheme that is much more efficient than what is normally used for these types of problems, allowing for many more grid points over the free surface. In particular, we provide a simple recipe for constructing a sparse approximation to the Jacobian that is used as a preconditioner in a Jacobian‐free Newton‐Krylov method for solving the nonlinear system. We use this approach to compute numerical results for a variety of prototype problems including flows past pressure distributions, a surface‐piercing object and bottom topographies.     

A coupled solution of the vorticity–velocity formulation of the incompressible Navier–Stokes equations

A relatively novel formulation of the Navier-Stokes equations is used for obtaining solutions of two dimensional incompressible fluid flow and convective heat transfer problems. A vorticity transport equation along with two Poisson equations for the velocity components and the energy equation are solved by a finite difference scheme. A coupled solution procedure is used for solving simultaneously the dependent variables along a line, using a block tridiagonal matrix algorithm. The formulation is found to be stable and has features that may be desirable for solving a wide variety of flow and heat transfer problems.     

Development of a hybrid finite volume/element solver for incompressible flows

In this paper, we report our development of an implicit hybrid flow solver for the incompressible Navier–Stokes equations. The methodology is based on the pressure correction or projection method. A fractional step approach is used to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free implicit cell‐centred finite volume method. The Poisson equation derived from the fractional step approach is solved by the node‐based Galerkin finite element method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centres and the auxiliary variable at cell vertices, making the current solver a staggered‐mesh scheme. Numerical examples demonstrate the performance of the resulting hybrid scheme, such as the correct temporal convergence rates for both velocity and pressure, absence of unphysical pressure boundary layer, good convergence in steady‐state simulations and capability in predicting accurate drag, lift and Strouhal number in the flow around a circular cylinder. Copyright © 2007 John Wiley & Sons, Ltd.     

注塑成型三维流动的数值模拟

建立了非等温、粘性、不可压缩、非牛顿流体流动的控制方程。为了避免同时求解耦合的压力场、速度场,本文通过修改Galerkin方法的变分方程,导出了关于压力场的拟Poisson方程,用迭代法独立地求解连续性方程、动量方程,并进行速度一粘度迭代求出最终的压力场、速度场。由于直接使用Galerkin方法求解能量方程容易引起温度场的振荡,本文采用隐式格式及“上风”法离散能量方程,用超松驰迭代法求解温度场的代数方程组。比较了模拟结果与等温管道流动的解析解及法兰的实际注射结果,算例表明本文方法可以预测注射成型流动过程中的一些重要特征。与传统Galerkin方法相比,本文方法可以减少内存,提高数值方法的稳定性。