Segregated finite element algorithms for the numerical solution of large-scale incompressible flow problems

This paper presents results of an ongoing research program directed towards developing fast and efficient finite element solution algorithms for the simulation of large-scale flow problems. Two main steps were taken towards achieving this goal. The first step was to employ segregated solution schemes as opposed to the fully coupled solution approach traditionally used in many finite element solution algorithms. The second step was to replace the direct Gaussian elimination linear equation solvers used in the first step with iterative solvers of the conjugate gradient and conjugate residual type. The three segregated solution algorithms developed in step one are first presented and their integrity and relative performance demonstrated by way of a few examples. Next, the four types of iterative solvers (i.e. two options for solving the symmetric pressure type equations and two options for solving the non-symmetric advection–diffusion type equations resulting from the segregated algorithms) together with the two preconditioning strategies employed in our study are presented. Finally, using examples of practical relevance the paper documents the large gains which result in computational efficiency, over fully coupled solution algorithms, as each of the above two main steps are introduced. It is shown that these gains become increasingly more dramatic as the complexity and size of the problem is increased.     

Stress–Strain Analysis of Thin-Walled Shells with Massive Ribs

A multifrontal method is proposed for solving large-scale finite-element problems with a symmetric stiffness matrix. This approach is based on node ordering rather than element ordering as in the traditional frontal or multifrontal methods. This allows us to employ the efficient well-known ordering techniques such as the minimum-degree algorithm or the nested dissection method. A thin-walled cylindrical shell with massive ribs is considered as an example. The efficiency of the method proposed is illustrated against the classical skyline and aggregation multilevel conjugate-gradient iterative solvers     

A 3D FEM–BEM rolling contact formulation for unstructured meshes

This work presents a new formulation for solving 3D steady-state rolling contact problems. The convective terms for computing the tangential slip velocities involved in the rolling problem, are evaluated using a new approximation inspired in numerical fluid dynamics techniques for unstructured meshes. Moreover, the elastic influence coefficients of the surface points in contact are approached by means of the finite element method (FEM) and/or the boundary element method (BEM). The contact problem is based on an Augmented Lagrangian Formulation and the use of projection functions to establish the contact restrictions. Finally, the resulting nonlinear equations set is solved using the generalized Newton method with line search (GNMls), presenting some acceleration strategies as: a new and more simplified projection operator, which makes it possible to obtain a quasi-complementarity of the contact variables, reducing the number of contact problem unknowns, and using iterative solvers. The presented methodology is validated solving some rolling contact problems and analyzed for some unstructured mesh examples.     

Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier–Stokes equations

This paper presents a numerical study of the 3D flow around a cylinder which was defined as a benchmark problem for the steady state Navier–Stokes equations within the DFG high‐priority research program flow simulation with high‐performance computers by Schafer and Turek (Vol. 52, Vieweg: Braunschweig, 1996). The first part of the study is a comparison of several finite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary fitted higher order finite element methods are in general most accurate. Our numerical study improves the hitherto existing reference values for the benchmark parameters considerably. The second part of the study deals with efficient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The flexible GMRES method with a multiple discretization multigrid method proves to be the best solver. Copyright © 2002 John Wiley & Sons, Ltd.     

Performance of iterative methods for Newtonian and generalized Newtonian flows

We consider the use of accelerated gradient-type iterative methods for solution of Newtonian and certain non-Newtonian (power-law and Bingham models) viscous flow problems. The formulations are based on penalty and mixed finite element methods, and such factors as the effect of the penalty parameter, asymmetry, continuation and preconditioning are examined.     

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.     

Computation of the stabilization parameter for the finite element solution of advective–diffusive problems

In a previous paper a general procedure for deriving stabilized finite element schemes for advective type problems based on invoking higher order balance laws over finite size domains was presented. This provides an expression for the element stabilization parameter in terms of the solution residual and its first derivatives in a kind of iterative or adaptative manner. Details of the application of this procedure to 1D and 2D advective–diffusive problems are given. Some examples of applications showing the potential of the new approach are presented. © 1997 John Wiley & Sons, Ltd.     

Stability of convective flows in cavities: solution of benchmark problems by a low‐order finite volume method

A problem of stability of steady convective flows in rectangular cavities is revisited and studied by a second‐order finite volume method. The study is motivated by further applications of the finite volume‐based stability solver to more complicated applied problems, which needs an estimate of convergence of critical parameters. It is shown that for low‐order methods the quantitatively correct stability results for the problems considered can be obtained only on grids having more than 100 nodes in the shortest direction, and that the results of calculations using uniform grids can be significantly improved by the Richardson's extrapolation. It is shown also that grid stretching can significantly improve the convergence, however sometimes can lead to its slowdown. It is argued that due to the sparseness of the Jacobian matrix and its large dimension it can be effective to combine Arnoldi iteration with direct sparse solvers instead of traditional Krylov‐subspace‐based iteration techniques. The same replacement in the Newton steady‐state solver also yields a robust numerical process, however, it cannot be as effective as modern preconditioned Krylov‐subspace‐based iterative solvers. Copyright © 2006 John Wiley & Sons, Ltd.     

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.     

On iterative methods for the incompressible Stokes problem

In this paper, we discuss various techniques for solving the system of linear equations that arise from the discretization of the incompressible Stokes equations by the finite‐element method. The proposed solution methods, based on a suitable approximation of the Schur‐complement matrix, are shown to be very effective for a variety of problems. In this paper, we discuss three types of iterative methods. Two of these approaches use the pressure mass matrix as preconditioner (or an approximation) to the Schur complement, whereas the third uses an approximation based on the ideas of least‐squares commutators (LSC). We observe that the approximation based on the pressure mass matrix gives h‐independent convergence, for both constant and variable viscosity. Copyright © 2010 John Wiley & Sons, Ltd.     

Improved preconditioned conjugate gradient method and its application in F. E. A. for engineering

In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method(PCCG).The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix.The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix.This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems,and simultaneously contrasted with other methods.The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations,It is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.     

Accuracy and convergence of element-by-element iterative solvers for incompressible fluid flows using penalty finite element model

The ability of two types of Conjugate Gradient like iterative solvers (GMRES and ORTHOMIN) to resolve large-scale phenomena as a function of mesh density and convergence tolerance limit is investigated. The flow of an incompressible fluid inside a sudden expansion channel is analysed using three meshes of 400, 1600 and 6400 bilinear elements. The iterative solvers utilize the element-by-element data structure of the finite element technique to store and maintain the data at the element level. Both the mesh density and the penalty parameter are found to influence the choice of the convergence tolerance limit needed to obtain accurate results. An empirical relationship between the element size, the penalty parameter, and the convergence tolerance is presented. This relationship can be used to predict the proper choice of the convergence tolerance for a given penalty parameter and element size.     

SOLUTION OF THE ADVECTION–DIFFUSION EQUATION USING A COMBINATION OF DISCONTINUOUS AND MIXED FINITE ELEMENTS

When transport is advection-dominated, classical numerical methods introduce excessive artificial diffusion and spurious oscillations. Special methods are required to overcome these phenomena. To solve the advection‒diffusion equation, a numerical method is developed using a discontinuous finite element method for the discretization of the advective terms. At the discontinuities of the approximate solution, numerical advective fluxes are calculated using one-dimensional approximate Riemann solvers. The method is stabilized with a multidimensional slope limiter which introduces small amounts of numerical diffusion when sharp concentration fronts occur. In addition, the diffusive term is discretized using a mixed hybrid finite element method. With this approach, numerical oscillations are completely avoided for a full range of cell Peclet numbers. The combination of discontinuous and mixed finite elements can be easily applied to 2D and 3D models using various types of elements in regular and irregular meshes. Numerical tests show good agreement with 1D and 2D analytical solutions. This approach is compared at the same time with two different numerical methods, a standard mixed finite method and a finite volume approach with high-resolution upwind terms. Regular and irregular meshes are used for the numerical tests to study the mesh effects on the numerical results. Our data show that in all cases this approach performs well. © 1997 by John Wiley & Sons, Ltd.     

US-FE-LSPIM四边形单元及其在几何非线性问题中的应用

为了提高在网格畸变时的数值计算精度,基于非对称有限单元的概念,提出US-FE-LSPIM四边形单元。该单元是利用传统的四节点等参元形函数集和FE—LSPIM四边形单元形函数集分别作为检验函数和试函数而构成。前者用于满足单元间和单元内的位移连续性要求,后者用于满足位移完备性要求。该单元结合了有限单元法和无网格法的优点,能...     

Robust semidirect finite difference methods for solving the Navier–Stokes and energy equations

Semidirect solution techniques can be an effective alternative to the more conventional iterative approaches used in many finite difference methods. This paper summarizes several semidirect techniques which generally have not been applied to the Navier–Stokes and energy equations in finite difference form. The methods presented use both successive substitution and Jacobian-based updates as well as two variations of Broyden's full matrix update. A hybrid method is also presented, as is a norm-reducing search technique that can be used to enhance the convergence characteristics of any semidirect approach. These methods have been compared with the well known iterative methods SIMPLE and SIMPLER. The comparison was performed on the natural convection and driven cavity problems. The semidirect methods proved to be reliably convergent without the need for a priori specification of variable under-relaxation factors, which was necessary with the iterative methods. Natural convection and driven cavity solutions have been readily obtained with the proposed methods for Rayleigh and Reynolds numbers up to 10⁹ and 10⁶ respectively. Of the semidirect techniques, the hybrid approach was the most robust. From an arbitrary zero initial guess this method was able to obtain a solution to the natural convection problem for Rayleigh numbers three orders of magnitude larger than was possible with the Newton-Raphson update. The computational effort required by the semidirect methods is comparable to that required by the iterative methods; however, the memory requirements can be significantly greater.     

Conjugate gradient methods and ILU preconditioning of non-symmetric matrix systems with arbitrary sparsity patterns

Preconditioning techniques based on incomplete Gaussian elimination for large, sparse, non-symmetric matrix systems are described. A certain level of fill-in may be specified in the incomplete factorizations. All methods considered may be applied to matrices with arbitrary sparsity patterns, for instance those associated with the general preprocessor algorithms or adaptive mesh techniques. The preconditioners have been combined with five conjugate gradient-like methods and tested on finite element discretized scalar convection-diffusion equations in 2D and 3D. It is found from numerical experiments that an amount of fill-in corresponding to about 50% of the number of original non-zero matrix entries is the optimal choice for this class of preconditioners. The preconditioners show almost no sensitivity to grid distortion. In problems with significantly variable coefficients or anisotropy the preconditioners stabilize the basic iterative schemes in addition to reducing the computational work substantially, mostly by more than 90%. The modified preconditioning technique, where fill-in is added on the main diagonal, performs in general better than the standard incomplete LU factorization, but is inferior to the latter in 3D problems and for matrix systems with complicated sparsity patterns.     

Iterative adaptive implicit–explicit methods for flow problems

Iterative versions of the adaptive implicit–explicit method are presented for the finite element computation of flow problems with particular reference to incompressible flows and advection–diffusion problems. The iterative techniques employed are the grouped element-by-element and generalized minimum residual methods.     

Adjoint‐based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods

We present a robust and efficient target‐based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection–diffusion problems, including the compressible Euler and Navier–Stokes equations. The hybridization of finite element discretizations has the main advantage that the resulting set of algebraic equations has globally coupled degrees of freedom (DOFs) only on the skeleton of the computational mesh. Consequently, solving for these DOFs involves the solution of a potentially much smaller system. This not only reduces storage requirements but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint‐based mesh adaptation over uniform and residual‐based mesh refinement and secondly, to investigate the efficiency of the global error estimate. Copyright © 2014 John Wiley & Sons, Ltd.     

Level set function–based immersed interface method and benchmark solutions for fluid flexible-structure interaction

Using a hybrid Lagrangian-Eulerian approach, a level set function–based immersed interface method (LS-IIM) is proposed for the interaction of a flexible body immersed in a fluid flow. The LS-IIM involves finite volume method for the fluid solver, Galerkin finite element method for the structural solver, and a block-iterative partitioned method–based fully implicit coupling between the two solvers. The novelty of the proposed method is a level set function–based direct implementation of fluid-solid interface boundary conditions in both the solvers. Another novelty is the computation of the level set function from a geometric method instead of differential equations commonly used in level set methods—the novel geometric as compared to the traditional method is found to be more accurate and less time-consuming. The LS-IIM is demonstrated as second-order accurate. Verification study is presented first separately for both the solvers and then together for four fluid-structure interaction (FSI) problems, with different levels of complexity including lid-driven flow, channel flow, and free-stream flow. Benchmark solutions are presented for two class of FSI problems: first, easy to set up and less time-consuming and, second, a reasonably challenging and complex FSI problem involving sharp edges and forced-motion of the flexible structure. The benchmark solutions are proposed at steady state for the first problem, after a verification study with two open-source solvers and, at periodic state, after a validation with published experimental results for the second problem. Our benchmark solutions may be useful for verification study in future.     

A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion

Computational fluid mechanics techniques for examining free surface problems in two‐dimensional form are now well established. Extending these methods to three dimensions requires a reconsideration of some of the difficult issues from two‐dimensional problems as well as developing new formulations to handle added geometric complexity. This paper presents a new finite element formulation for handling three‐dimensional free surface problems with a boundary‐fitted mesh and full Newton iteration, which solves for velocity, pressure, and mesh variables simultaneously. A boundary‐fitted, pseudo‐solid approach is used for moving the mesh, which treats the interior of the mesh as a fictitious elastic solid that deforms in response to boundary motion. To minimize mesh distortion near free boundary under large deformations, the mesh motion equations are rotated into normal and tangential components prior to applying boundary conditions. The Navier–Stokes equations are discretized using a Galerkin–least square/pressure stabilization formulation, which provides good convergence properties with iterative solvers. The result is a method that can track large deformations and rotations of free surface boundaries in three dimensions. The method is applied to two sample problems: solid body rotation of a fluid and extrusion from a nozzle with a rectangular cross‐section. The extrusion example exhibits a variety of free surface shapes that arise from changing processing conditions. Copyright © 2000 John Wiley & Sons, Ltd.