Reordering and incomplete preconditioning in serial and parallel adaptive mesh refinement and coarsening flow solutions

The effects of reordering the unknowns on the convergence of incomplete factorization preconditioned Krylov subspace methods are investigated. Of particular interest is the resulting preconditioned iterative solver behavior when adaptive mesh refinement and coarsening (AMR/C) are utilized for serial or distributed parallel simulations. As representative schemes, we consider the familiar reverse Cuthill–McKee and quotient minimum degree algorithms applied with incomplete factorization preconditioners to CG and GMRES solvers. In the parallel distributed case, reordering is applied to local subdomains for block ILU preconditioning, and subdomains are repartitioned dynamically as mesh adaptation proceeds. Numerical studies for representative applications are conducted using the object‐oriented AMR/C software system libMesh linked to the PETSc solver library. Serial tests demonstrate that global unknown reordering and incomplete factorization preconditioning can reduce the number of iterations and improve serial CPU time in AMR/C computations. Parallel experiments indicate that local reordering for subdomain block preconditioning associated with dynamic repartitioning because of AMR/C leads to an overall reduction in processing time. Copyright © 2011 John Wiley & Sons, Ltd.     

STREAMLINE-BASED MATHEMATICAL MODEL FOR CO2 MISCIBLE FLOODING

According to the research theory of improved black oil simulator, a practical mathematical model for C02 miscible flooding was presented. In the model, the miscible process simulation was realized by adjusting oil/gas relative permeability and effective viscosity under the condition of miscible flow. In order to predict the production performance fast, streamline method is employed to solve this model as an alternative to traditional finite difference methods. Based on streamline distribution of steady-state flow through porous media with complex boundary confirmed with the boundary element method (BEM), an explicit total variation diminishing (TVD) method is used to solve the one-dimensional flow problem. At the same time, influences of development scheme, solvent slug size, and injection periods on CO2 drive recovery are discussed. The model has the advantages of less information need, fast calculation, and adaptation to calculate CO2 drive performance of all kinds of patterns in a random shaped porous media with assembly boundary. It can be an effective tool for early stage screening andmiscible oil field.reservoir dynamic management of the CO2 miscible oil field.     

Comparison of the GMRES and ORTHOMIN for the black oil model in porous media

This paper deals with the application of the GMRES algorithm to a three‐dimensional, three‐phase black oil model used in petroleum reservoir simulation. Comparisons between the GMRES and ORTHOMIN algorithms in terms of storage and total flops per restart step are given. Numerical results show that the GMRES is faster than the ORTHOMIN for large‐scale simulation problems. The GMRES uses only as much as 63% of the CPU time of the ORTHOMIN for some of the problems tested. Copyright © 2005 John Wiley & Sons, Ltd.     

Comparison of GMRES and ORTHOMIN for the black oil model on unstructured grids

This paper addresses an application of ORTHOMIN and GMRES to petroleum reservoir simulation using the black oil model on unstructured grids. Comparisons between these two algorithms are presented in terms of storage and total flops per restart step. Numerical results indicate that GMRES is faster than ORTHOMIN for all tested petroleum reservoir problems, particularly for large scale problems. The control volume function approximation method is utilized in the discretization of the governing equations of the black oil model. This method can accurately approximate both the pressure and velocity in the simulation of multiphase flow in porous media, effectively reduce grid orientation effects, and be easily applied to arbitrarily shaped control volumes. It is particularly suitable for hybrid grid reservoir simulation. Copyright © 2006 John Wiley & Sons, Ltd.     

基于Nataf变换的层递响应面法分析结构可靠度

针对现有的随机响应面法(SRSM)和层递响应面法(CRSM)存在的局限性,本文结合预处理随机Krylov子空间法,建立了基于Nataf变换的向量型层递响应面法,并应用于含非高斯型互相关随机变量的结构可靠度分析。首先,利用预处理随机Krylov子空间的层递基向量近似展开结构的总体节点位移向量,建立向量型层递响应面;然后,根据Nataf变换建立非高斯型互相关随机变量与独立标准正态随机变量之间的关系式,将独立标准正态空间内由Hermite多项式的根组合形成的概率配点变换成非高斯空间内的概率配点,并通过回归分析确定层递响应面的待定系数。计算结果表明,本文建立的CRSM属于向量型响应面法,能较好地处理含非高斯型互相关随机变量的结构可靠度分析问题,计算精度和效率均较高,且具有良好的全域性。     

Analysis of preconditioned iterative solvers for incompressible flow problems

Solving efficiently the incompressible Navier–Stokes equations is a major challenge, especially in the three‐dimensional case. The approach investigated by Elman et al. (Finite Elements and Fast Iterative Solvers. Oxford University Press: Oxford, 2005) consists in applying a preconditioned GMRES method to the linearized problem at each iteration of a nonlinear scheme. The preconditioner is built as an approximation of an ideal block‐preconditioner that guarantees convergence in 2 or 3 iterations. In this paper, we investigate the numerical behavior for the three‐dimensional lid‐driven cavity problem with wedge elements; the ultimate motivation of this analysis is indeed the development of a preconditioned Krylov solver for stratified oceanic flows which can be efficiently tackled using such meshes. Numerical results for steady‐state solutions of both the Stokes and the Navier–Stokes problems are presented. Theoretical bounds on the spectrum and the rate of convergence appear to be in agreement with the numerical experiments. Sensitivity analysis on different aspects of the structure of the preconditioner and the block decomposition strategies are also discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Jacobian-free Newton-Krylov subspace method with wavelet-based preconditioner for analysis of transient elastohydrodynamic lubrication problems with surface asperities

This paper presents an investigation into the effect of surface asperities on the over-rolling of bearing surfaces in transient elastohydrodynamic lubrication(EHL) line contact. The governing equations are discretized by the finite difference method. The resulting nonlinear system of algebraic equations is solved by the Jacobian-free Newtongeneralized minimal residual(GMRES) from the Krylov subspace method(KSM). The acceleration of the GMRES iteration is accomplished by a wavelet-based preconditioner.The profiles of the lubricant pressure and film thickness are obtained at each time step when the indented surface moves through the contact region. The prediction of pressure as a function of time provides an insight into the understanding of fatigue life of bearings.The analysis confirms the need for the time-dependent approach of EHL problems with surface asperities. This method requires less storage and yields an accurate solution with much coarser grids. It is stable, efficient, allows a larger time step, and covers a wide range of parameters of interest.     

Assessment of some iterative methods for non-symmetric linear systems arising in computational fluid dynamics

Various tests have been carried out in order to compare the performances of several methods used to solve the non-symmetric linear systems of equations arising from implicit discretizations of CFD problems, namely the scalar advection-diffusion equation and the compressible Euler equations. The iterative schemes under consideration belong to three families of algorithms: relaxation (Jacobi and Gauss-Seidel), gradient and Newton methods. Two gradient methods have been selected: a Krylov subspace iteration method (GMRES) and a non-symmetric extension of the conjugate gradient method (CGS). Finally, a quasi-Newton method has also been considered (Broyden). The aim of this paper is to provide indications of which appears to be the most adequate method according to the particular circumstances as well as to discuss the implementation aspects of each scheme.     

Nonlinear Multigrid Methods for Numerical Solution of the Variably Saturated Flow Equation in Two Space Dimensions

The need of accurate and efficient numerical schemes to solve Richards’ equation is well recognized. This study is carried out to examine the numerical performances of the nonlinear multigrid method for numerical solving of the two-dimensional Richards’ equation modeling water flow in variably saturated porous media. The numerical approach is based on an implicit, second-order accurate time discretization combined with a vertex centered finite volume method for spatial discretization. The test problems simulate infiltration of water in 2D saturated–unsaturated soils with hydraulic properties described by van Genuchten–Mualem models. The numerical results obtained are compared with those provided by the modified Picard–preconditioned conjugated gradient (Krylov subspace) approach.     

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.     

CONTROLLABILITY OF A CLASS OF HYBRID DYNAMIC SYSTEMS (Ⅰ)—BASIC PROPERTIES AND PRELIMINARY RESULTS

The controllability for switched linear system with time-delay in controls was first investigated. The whole work contains three parts. This is the first part, including problem formulation and some preliminaries. Firstly, the mathematical model of switched linear systems with time-delay in control functions was presented. Secondly, the concept of column space, cyclic invariant subspace and generalized cyclic invariant subspace were introduced. And some basic properties, such as separation lemma, were presented. Finally, a basic lemma was given to reveal the relation between the solution set of a centain integral equations and the generalized cyclic invariant subspace. This lemma will play an important role in the determination of controllability. All these definitions and lemmas are necessary research tools for controllability analysis. Contributed by YE Qing-kai Foundation items: the National Natural Science Foundation of China (69925307, 60274001); the National Key Basic Reasearch and Development Program (2002CB312200); the Postdoctoral Program Foundation of China Biography: XIE Guang-ming (1972∼), Doctor (E-mail: xiegming@mech.pku.edu.cn)     

Numerical flux functions for Reynolds‐averaged Navier–Stokes and kω turbulence model computations with a line‐preconditioned p‐multigrid discontinuous Galerkin solver

We present an eigen‐decomposition of the quasi‐linear convective flux formulation of the completely coupled Reynolds‐averaged Navier–Stokes and kω turbulence model equations. Based on these results, we formulate different approximate Riemann solvers that can be used as numerical flux functions in a DG discretization. The effect of the different strategies on the solution accuracy is investigated with numerical examples. The actual computations are performed using a p‐multigrid algorithm. To this end, we formulate a framework with a backward‐Euler smoother in which the linear systems are solved with a general preconditioned Krylov method. We present matrix‐free implementations and memory‐lean line‐Jacobi preconditioners and compare the effects of some parameter choices. In particular, p‐multigrid is found to be less efficient than might be expected from recent findings by other authors. This might be due to the consideration of turbulent flow. Copyright © 2012 John Wiley & Sons, Ltd.     

Evolution of a 2-D disturbance in a supersonic boundary layer and the generation of shocklets

Through direct numerical simulation, the evolution of a 2-D disturbance in a supersonic boundary layer has been investigated. At a chosen location, a small amplitude T-S wave was fed into the boundary layer to investigate its evolution. Characteristics of nonlinear evolution have been found. Two methods were applied for the detection of shocklets, and it was found that when the amplitude of the disturbance reached a certain value, shocklets would be generated, which should be taken into consideration when nonlinear theory of hydrodynamic stability for compressible flows is to be established.     

Generalized fatigue constant life curve and two-dimensional probability distribution of fatigue limit

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An implicit meshless method for application in computational fluid dynamics

An implicit meshless scheme is developed for solving the Euler equations, as well as the laminar and Reynolds‐averaged Navier–Stokes equations. Spatial derivatives are approximated using a least squares method on clouds of points. The system of equations is linearised, and solved implicitly using approximate, analytical Jacobian matrices and a preconditioned Krylov subspace iterative method. The details of the spatial discretisation, linear solver and construction of the Jacobian matrix are discussed; and results that demonstrate the performance of the scheme are presented for steady and unsteady two dimensional fluid flows. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient stochastic FEM for flow in heterogeneous porous media. Part 1: random Gaussian conductivity coefficients

This paper is concerned with the development of efficient iterative methods for solving the linear system of equations arising from stochastic FEMs for single‐phase fluid flow in porous media. It is assumed that the conductivity coefficient varies randomly in space according to some given correlation function and is approximated using a truncated Karhunen–Loève expansion. Distinct discretizations of the deterministic and stochastic spaces are required for implementations of the stochastic FEM. In this paper, the deterministic space is discretized using classical finite elements and the stochastic space using a polynomial chaos expansion. The highly structured linear systems which result from this discretization mean that Krylov subspace iterative solvers are extremely effective. The performance of a range of preconditioned iterative methods is investigated and evaluated in terms of robustness with respect to mesh size and variability of the conductivity coefficient. An efficient symmetric block Gauss–Seidel preconditioner is proposed for problems in which the conductivity coefficient has a large standard deviation.The companion paper, herein, referred to as Part 2, considers the situation in which Gaussian random fields are transformed into lognormal ones by projecting the truncated Karhunen–Loève expansion onto a polynomial chaos basis. This results in a stochastic nonlinear problem because the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Copyright © 2013 John Wiley & Sons, Ltd.     

A class of nonlinear boundary value problems for the second-order<Emphasis Type="Italic">E</Emphasis><Subscript>2</Subscript> class elliptic systems in general form

A class of nonlinear boundary value problems (BVP) for the second-order E₂ class elliptic systems in general form is discussed. By introducing a kind of transformation, this kind of BVP is reduced to a class of generalized nonlinear Riemann-Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods. Foundation items: the National Natural Science Foundation of China (19671056); Shanghai Municipal Natural Scientific Foundation (99ZA14030, 01ZA14023); Jiangxi Provincial Natural Scientific Foundation (981102, 0211014) Biographies: LI Ming-zhong (1935−); XU Ding-hua (1967−)     

Unstructured finite element method for the solution of the Boussinesq problem in three dimensions

We present a numerical method for the monolithic discretisation of the Boussinesq system in three spatial dimensions. The key ingredients of the proposed methodology are the finite element discretisation of the spatial part of the problem using unstructured tetrahedral meshes, an implicit time integrator, based on adaptive predictor–corrector scheme (the explicit second‐order Adams–Bashforth method with the implicit stabilised trapezoid rule), and a new preconditioned Krylov subspace solver for the resulting linearised discrete problem. We test the proposed methodology on a number of physically relevant cases, including laterally heated cavities and the Rayleigh–Bénard convection. Copyright © 2013 John Wiley & Sons, Ltd.     

The 3-layered explicit difference scheme for 2-D heat equation

A 3-layered explicit difference scheme for the numerical solution of 2-D heat equation is proposed. Firstly, a general symmetric difference scheme is constructed and its optimal error is obtained. Then two kinds of condition for choosing the parameters for optimal error and stable difference scheme are given. Finally, some numerical results are presented to show the advantage of the schemes Foundation items: the Science Foundation of Chinese Postdoctoral (2002031224); the Science Foundations of Southeast University (9209011148, 3007011043) Biography: Liu Ji-jun (1965-)