Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.     

Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

Based on local algorithms, some parallel finite element(FE) iterative methods for stationary incompressible magnetohydrodynamics(MHD) are presented. These approaches are on account of two-grid skill include two major phases: find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution, and then locally solve linearized residual subproblems by one of three iterations(Stokes-type, Newton, and Oseen-type) on subdomains with fine...     

Adaptive variational multiscale method for the Stokes equations

An adaptive variational multiscale method for the Stokes equations is presented in this paper. We solve the coarse scale problem on the coarse mesh and approximate the fine scale solution by solving a series of local residual equations defined on some local fine grids, which can be implemented in parallel. In addition, we also propose a reliable local a posteriori error estimator and construct an adaptive algorithm based on the corresponding a posterior error estimate. Finally, numerical examples are presented to verify the algorithm.Copyright © 2012 John Wiley & Sons, Ltd.     

A parallel two-level finite element method for the Navier-Stokes equations

Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.     

NESTED GRID METHODS FOR AN OCEAN MODEL: A COMPARATIVE STUDY

In this paper a comparison is carried out between three correction methods for multigrid local mesh refinement in oceanic applications: FIC, LDC and the direct method (DM) proposed by Spall and Holland. This study is based on a nested primitive equation model developed by Laugier on the basis of the code OPA (LODYC). The external barotropic problem is solved using any of the three local grid correction algorithms yielding an interactive nested grid model. The non-linear elliptic equation for the barotropic streamfunction tendency is solved on two nested grids, called the global and the zoom grid, that interact between themselves. The zoom grid is entirely embedded within the global domain with a horizontal grid step ratio of 3:1. The computation on the global grid supplies the boundary conditions for the zoom grid region and the fine grid fields are used to correct the global coarse solution. The three local correction methods are tested on two problems relevant to oceanic circulation phenomena proposed by Spall and Holland: a barotropic modon and an anticyclonic vortex. The results show that the nesting technique is a very efficient way to solve these problems in terms of a gain in precision compared with the required CPU time. The two-domain model with local mesh refinement allows one both to manage effectively the open boundary conditions for the local grid and to correct the global solution thanks to the zoom solution. In the case of the modon propagation the three local correction methods provide approximately the same results. For the baroclinic vortex it appears that the two iterative methods are more efficient than the direct one.     

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.     

Variational multiscale analysis of elastoplastic deformation using meshfree approximation

A variational multiscale method has been presented for efficient analysis of elastoplastic deformation problems. Severe deformation occurs in plastic region and leads to high gradient displacement. Therefore, solution needs to be refined to properly capture local deformation in plastic region. In this work, scale decomposition based on variational formulation is presented. A coarse scale and a fine scale are introduced to represent global and local behavior, respectively. The displacement is decomposed into a coarse and a fine scale. Subsequently the problem is also decomposed into a coarse and a fine scale from the variational formulation. Each scale variable is approximated using meshfree method. Adaptivity can easily and nicely be implemented in meshfree method. As a method of increasing resolution, extrinsic enrichment of partition of unity is used. Each scale problem is solved iteratively and conversed results are obtained consequently. Iteration procedure is indispensable for the elastoplastic deformation analysis. Therefore iterative solution procedure of each scale problem is naturally adequate. The proposed method is applied to the Prandtl’s punch test and shear band problem. The results are compared with those of other methods and the validity of the proposed method is demonstrated.     

NORMAL GRID RULE FOR HYPERSONIC HEAT FLUX NUMERICAL SIMULATION

采用数值求解雷诺平均Navier-Stokes 方程方法对高超声速钝双锥模型进行了计算,比较了不同法向网格尺度对热流计算的影响,并在此基础上提出了一种适于热流计算的网格准则. 研究表明:在实际外形的热流计算中,使用过密或过粗的法向网格均不利于得到准确的热流数据;采用该文提出的基于当地温度梯度的法向网格尺度准则可以在保证热流计算精度的同时实现较高的收敛速度.     

Generation of Voronoi Grid Based on Vorticity for Coarse-Scale Modeling of Flow in Heterogeneous Formations

We present a novel unstructured coarse grid generation technique based on vorticity for upscaling two-phase flow in permeable media. In the technique, the fineness of the gridblocks throughout the domain is determined by vorticity distribution such that where the larger is the vorticity at a region, the finer are the gridblocks at that region. Vorticity is obtained from single-phase flow on original fine grid, and is utilized to generate a background grid which stores spacing parameter, and is used to steer generation of triangular and finally Voronoi grids. This technique is applied to two channelized and heterogeneous models and two-phase flow simulations are performed on the generated coarse grids and, the results are compared with the ones of fine scale grid and uniformly gridded coarse models. The results show a close match of unstructured coarse grid flow results with those of fine grid, and substantial accuracy compared to uniformly gridded coarse grid model.     

A finite-volume ELLAM for non-linear flux convection-diffusion problems

In this paper, a modified finite-volume Eulerian-Lagrangian localized adjoint method (FVELLAM) extended for convection-diffusion problems with a non-linear flux function is introduced. Tracking schemes are discussed using viscous Burgers’ equation. It is shown that in order to have smooth results, only the new time level values should be used in tracking process. Then, the proposed method is employed to study immiscible incompressible two-phase flows in porous media. Various one- and two-dimensional test cases involving internal sources and sinks are solved and accuracy of solution and performance of the method are investigated by comparing the results obtained using FVELLAM with those of fine grid solutions. Finally, it is concluded that although proposed FVELLAM produces satisfactory results even on coarse grids and allows fairly large time-steps, its major advantage is in solving convection-dominated problems in heterogeneous media.     

Convergence acceleration of segregated algorithms using dynamic tuning additive correction multigrid strategy

A convergence acceleration method based on an additive correction multigrid–SIMPLEC (ACM‐S) algorithm with dynamic tuning of the relaxation factors is presented. In the ACM‐S method, the coarse grid velocity correction components obtained from the mass conservation (velocity potential) correction equation are included into the fine grid momentum equations before the coarse grid momentum correction equations are formed using the additive correction methodology. Therefore, the coupling between the momentum and mass conservation equations is obtained on the coarse grid, while maintaining the segregated structure of the single grid algorithm. This allows the use of the same solver (smoother) on the coarse grid. For turbulent flows with heat transfer, additional scalar equations are solved outside of the momentum–mass conservation equations loop. The convergence of the additional scalar equations is accelerated using a dynamic tuning of the relaxation factors. Both a relative error (RE) scheme and a local Reynolds/Peclet (ER/P) relaxation scheme methods are used. These methodologies are tested for laminar isothermal flows and turbulent flows with heat transfer over geometrically complex two‐ and three‐dimensional configurations. Savings up to 57% in CPU time are obtained for complex geometric domains representative of practical engineering problems. Copyright © 1999 John Wiley & Sons, Ltd.     

应变局部化分析的嵌入强间断多尺度有限元法

固体材料的应变局部化行为是导致结构破坏失效的重要因素之一,开展相关数值模拟分析对于结构安全性评估具有重要意义.然而由于材料的非均质和多尺度特性,采用传统数值方法进行求解时通常需要从最小特征尺度离散求解的结构,这将大幅度增加计算规模和成本.针对这一问题,本文提出了一种基于嵌入强间断模型的多尺度有限元方法.该方法从粗细两个尺度离散求解模型,首先在细尺度单元上引入嵌入强间断模型来描述单元间断特性,所附加的跳跃位移自由度则通过凝聚技术进行消除,从而保持细尺度单元刚度阵维度不变.其次,提出了一种增强多节点粗单元技术,其可根据局部化带与粗单元边界相交情况自适应动态地增加粗节点,新构造的增强数值基函数可以捕捉细尺度间断特性,完成物理信息从细单元到粗单元的准确传递以及宏观响应的快速分析;再次,在细尺度解的计算中,将细尺度解分解为降尺度解与单胞局部摄动解,从而消除弹塑性分析时单胞内部的不平衡力.最后,通过两个典型算例分析,并与完全采用细单元的嵌入有限元结果进行对比,验证了所提出算法的正确性与有效性.     

Meshfree analysis of softening elastoplastic solids using variational multiscale method

A meshfree multiscale method is presented for efficient analysis of elastoplastic solids. In the analysis of softening elastoplastic solids, standard finite element methods or meshfree methods typically yield mesh-dependent results. The reason for this well-known effect is the loss of ellipticity of the boundary value problem. In this work, the scale decomposition is carried out based on a variational form of the problem. A coarse scale is designed to represent global behavior and a fine scale to represent local behavior. A fine scale region is detected from the local failure analysis of an acoustic tensor to indicate a region where deformation changes abruptly. Each scale variable is approximated using a meshfree method. Meshfree approximation is well-suited for adaptivity. As a method of increasing the resolution, a partition of unity based extrinsic enrichment is used. In particular, fine scale approximations are designed to appropriately represent local behavior by using a localization angle. Moreover, the regularization effect through the convexification of non-convex potential is embedded to represent fine scale behavior. Each scale problem is solved iteratively. The proposed method is applied to shear band problems. In the results of analysis about pure shear and compression problems, straight shear bands can be captured and mesh-insensitive results are obtained. Curved shear bands can also be captured without mesh dependency in the analysis of indentation problem.     

含泥质夹层油藏蒸汽注采过程的自适应网格算法研究

根据泥质夹层的低渗特性及空间分布，本文提出了一种含泥质夹层油藏网格渗透率的粗化计算方法，并在此基础上，将自适应网格算法应用于含泥质夹层油藏的数值模拟，提升其计算效率．在计算过程中，网格的动态划分仅依据流体物理量的变化，泥质夹层区域不全部采用细网格，仅针对流动锋面处的泥质夹层采用细网格，其余泥质夹层处采用不同程度的粗网格．相较于传统算法，网格数大幅下降．数值算例表明，自适应网格算法的计算结果精度与全精细网格一致，能够准确模拟出泥质夹层对于流体的阻碍作用，同时计算效率得到大幅提升，约为全精细网格算法的3~7 倍．     

Hybrid Parallel Linear System Solvers

This paper presents a new approach to the solution of nonsymmetric linear systems that uses hybrid techniques based on both direct and iterative methods. An implicitly preconditioned modified system is obtained by applying projections onto block rows of the original system. Our technique provides the flexibility of using either direct or iterative methods for the solution of the preconditioned system. The resulting algorithms are robust, and can be implemented with high efficiency on a variety of parallel architectures. The algorithms are used to solve linear systems arising from the discretization of convection-diffusion equations as well as those systems that arise from the simulation of particulate flows. Experiments are presented to illustrate the robustness and parallel efficiency of these methods.     

A structured but non‐uniform Cartesian grid‐based model for the shallow water equations

In large‐scale shallow flow simulations, local high‐resolution predictions are often required in order to reduce the computational cost without losing the accuracy of the solution. This is normally achieved by solving the governing equations on grids refined only to those areas of interest. Grids with varying resolution can be generated by different approaches, e.g. nesting methods, patching algorithms and adaptive unstructured or quadtree gridding techniques. This work presents a new structured but non‐uniform Cartesian grid system as an alternative to the existing approaches to provide local high‐resolution mesh. On generating a structured but non‐uniform Cartesian grid, the whole computational domain is first discretized using a coarse background grid. Local refinement is then achieved by directly allocating a specific subdivision level to each background grid cell. The neighbour information is specified by simple mathematical relationships and no explicit storage is needed. Hence, the structured property of the uniform grid is maintained. After employing some simple interpolation formulae, the governing shallow water equations are solved using a second‐order finite volume Godunov‐type scheme in a similar way as that on a uniform grid. Copyright © 2010 John Wiley & Sons, Ltd.     

Using Vorticity as an Indicator for the Generation of Optimal Coarse Grid Distribution

An improved vorticity-based gridding technique is presented and applied to create optimal non-uniform Cartesian coarse grid for numerical simulation of two-phase flow. The optimal coarse grid distribution (OCGD) is obtained in a manner to capture variations in both permeability and fluid velocity of the fine grid using a single physical quantity called “vorticity”. Only single-phase flow simulation on the fine grid is required to extract the vorticity. Based on the fine-scale vorticity information, several coarse grid models are generated for a given fine grid model. Then the vorticity map preservation error is used to predict how well each coarse grid model reproduces the fine-scale simulation results. The coarse grid model which best preserves the fine-scale vorticity, i.e. has the minimum vorticity map preservation error is recognized as an OCGD. The performance of vorticity-based optimal coarse grid is evaluated for two highly heterogeneous 2D formations. It is also shown that two-phase flow parameters such as mobility ratio have only minor impact on the performance of the predicted OCGD.     

Dual Mesh Method for Upscaling in Waterflood Simulation

Detailed geological models typically contain many more cells than can be accommodated by reservoir simulation due to computer time and memory constraints. However, recovery predictions performed on a coarser upscaled mesh are inevitably less accurate than those performed on the initial fine mesh. Recent studies have shown how to use both coarse and fine mesh information during waterflooding simulations. In this paper, we present an extension of the dual mesh method (Verdière and Guérillot, 1996) which simulates water flooding injection using both the coarse and the original fine mesh information. The pressure field is first calculated on the coarse mesh. This information is used to estimate the pressure field within each coarse cell and then phase saturations are updated on the fine mesh. This method avoids the most time consuming step of reservoir simulation, namely solving for the pressure field on the fine grid. A conventional finite difference IMPES scheme is used considering a two phase fluid with gravity and vertical wells. Two upscaling methodologies are used and compared for averaging the coarse grid properties: geometric average and the pressure solve method. A series of test cases show that the method provides predictions similar to those of full fine grid simulations but using less computer time.     

Three-dimensional analysis of the flow in a curved hydraulic turbine draft tube

The three-dimensional turbulent flow in a curved hydraulic turbine draft tube is studied numerically. The analysis is based on the steady Reynolds-averaged Navier–Stokes equations closed with the κ-ε model. The governing equations are discretized by a conservative finite volume formulation on a non-orthogonal body-fitted co-ordinate system. Two grid systems, one with 34 × 16 × 12 nodes and another with 50 × 30 × 22 nodes, have been used and the results from them are compared. In terms of computing effort, the number of iterations needed to yield the same degree of convergence is found to be proportional to the square root of the total number of nodes employed, which is consistent with an earlier study made for two-dimensional flows using the same algorithm. Calculations have been performed over a wide range of inlet swirl, using both the hybrid and second-order upwind schemes on coarse and fine grids. The addition of inlet swirl is found to eliminate the stalling characteristics in the downstream region and modify the behaviour of the flow markedly in the elbow region, thereby affecting the overall pressure recovery noticeably. The recovery factor increases up to a swirl ratio of about 0˙75, and then drops off. Although the general trends obtained with both finite difference operators are in agreement, the quantitative values as well as some of the fine flow structures can differ. Many of the detailed features observed on the fine grid system are smeared out on the coarse grid system, pointing out the necessity of both a good finite difference operator and a good grid distribution for an accurate result.