An overview and generalization of implicit Navier–Stokes algorithms and approximate factorization

A theme of linearization and approximate factorization provides the context for a retrospective overview of the development and evolution of implicit numerical methods for the compressible and incompressible Euler and Navier–Stokes algorithms. This topic was chosen for this special volume commemorating the recent retirements of R.M. Beam and R.F. Warming. A generalized treatment of approximate factorization schemes is given, based on an operator notation for the spatial approximation. The generalization focuses on the implicit structure of Euler and Navier–Stokes algorithms as nonlinear systems of partial differential equations, with details of the spatial approximation left to operator definitions. This provides a unified context for discussing noniterative and iterative time-linearized schemes, and Newton iteration for unsteady nonlinear schemes. The factorizations include alternating direction implicit, LU and line relaxation schemes with either upwind or centered spatial approximations for both compressible and incompressible flows. The noniterative schemes are best suited for steady flows, while the iterative schemes are well suited for either steady or unsteady flows. This generalization serves to unify a large number of schemes developed over the past 30 years.     

Development of implicit methods in CFD NASA Ames Research Center 1970s–1980s

The focus here is on the early development (mid 1970s–1980s) at NASA Ames Research Center of implicit methods in Computational Fluid Dynamics (CFD). A class of implicit finite difference schemes of the Beam and Warming approximate factorization type will be addressed. The emphasis will be on the Euler equations. A review of material pertinent to the solution of the Euler equations within the framework of implicit methods will be presented. The eigensystem of the equations will be used extensively in developing a framework for various methods applied to the Euler equations. The development and analysis of various aspects of this class of schemes will be given along with the motivations behind many of the choices. Various acceleration and efficiency modifications such as matrix reduction, diagonalization and flux split schemes will be presented.     

Uzawa algorithms for coupled Stokes equations from the optimal control problem

Coupled Stokes equations obtained from the optimal control problem subject to Stokes equations can be approximated by Uzawa finite element methods. The linear convergence is provided with an optimal relaxation parameter for the Richardson pressure update. This work was supported by KRF-2005-070-C00017.     

Internal stabilizability of the Navier–Stokes equations

One shows that the steady-state solutions to Navier–Stokes equations in d-dimensional domains Ω, d=2,3 with Dirichlet and slip curl boundary conditions are exponentially stabilizable by proportional controllers with the support in open subsets ωΩ such that Ωω is sufficiently “small”.     

Numerical solution of incompressible Navier–Stokes equations using a fractional-step approach

A fractional step method for the solution of steady and unsteady incompressible Navier–Stokes equations is outlined. The method is based on a finite-volume formulation and uses the pressure in the cell center and the mass fluxes across the faces of each cell as dependent variables. Implicit treatment of convective and viscous terms in the momentum equations enables the numerical stability restrictions to be relaxed. The linearization error in the implicit solution of momentum equations is reduced by using three subiterations in order to achieve second order temporal accuracy for time-accurate calculations. In spatial discretizations of the momentum equations, a high-order (third and fifth) flux-difference splitting for the convective terms and a second-order central difference for the viscous terms are used. The resulting algebraic equations are solved with a line-relaxation scheme which allows the use of large time step. A four color ZEBRA scheme is employed after the line-relaxation procedure in the solution of the Poisson equation for pressure. This procedure is applied to a Couette flow problem using a distorted computational grid to show that the method minimizes grid effects. Additional benchmark cases include the unsteady laminar flow over a circular cylinder for Reynolds numbers of 200, and a 3-D, steady, turbulent wingtip vortex wake propagation study. The solution algorithm does a very good job in resolving the vortex core when fifth-order upwind differencing and a modified production term in the Baldwin–Barth one-equation turbulence model are used with adequate grid resolution.     

A comparison of three kinds of local and parallel finite element algorithms based on two-grid discretizations for the stationary Navier–Stokes equations

Based on two-grid discretizations, three kinds of local and parallel finite element algorithms for the stationary Navier–Stokes equations are introduced and discussed. The main technique is first to use a standard finite element discretization on a coarse grid to approximate low frequencies of the solution, then to apply some linearized discretizations on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local and parallel procedures. Three approaches to linearization are discussed. Under the uniqueness condition, error estimates of the finite element solution are derived. Numerical results show that among the three kinds of parallel algorithms, the Oseen-linearized algorithm is preferable if we both consider the computational time and the accuracy of the approximate solution.     

交替方向隐式CFD解法器的GPU并行计算及其优化

交替方向隐格式(ADI)是常见的偏微分方程离散格式之一,目前对ADI格式在计算流体力学（CFD）实际应用中的GPU并行工作开展较少。从一个有限体积CFD应用出发,通过分析ADI解法器的特点和计算流程,基于统一计算架构(CUDA)编程模型设计了基于网格点与网格线的两类细粒度GPU并行算法,讨论了若干性能优化方法。在天河-1A系统上,采用128×128×128网格规模的单区结构网格算例,无粘项、粘性项及ADI迭代计算的GPU并行性能相对于单CPU核,分别取得了100.1、40.1和10.3倍的加速比,整体ADI CFD解法器的GPU并行加速比为17.3     

Exponential stabilization of the linearized Navier–Stokes equation by pointwise feedback noise controllers

We address the problem of exponential stabilization in probability of the linearized Navier–Stokes equations in an equilibrium point. This is done by designing a linear stochastic feedback controller with support in a point or on a discrete set of points of the domain. This controller consists of a steady-state impulse component with support in a finite set of points modulated by an unsteady feedback noise controller.     

Development of a parallelized 2D/2D-axisymmetric Navier–Stokes equation solver for all-speed gas flows

A parallelized 2D/2D-axisymmetric pressure-based, extended SIMPLE finite-volume Navier–Stokes equation solver using Cartesians grids has been developed for simulating compressible, viscous, heat conductive and rarefied gas flows at all speeds with conjugate heat transfer. The discretized equations are solved by the parallel Krylov–Schwarz (KS) algorithm, in which the ILU and BiCGStab or GMRES scheme are used as the preconditioner and linear matrix equation solver, respectively. Developed code was validated by comparing previous published simulations wherever available for both low- and high-speed gas flows. Parallel performance for a typical 2D driven cavity problem is tested on the IBM-1350 at NCHC of Taiwan up to 32 processors. Future applications of this code are discussed briefly at the end.     

Constrained optimal control of Navier–Stokes flow by semismooth Newton methods

We propose and analyze a semismooth Newton-type method for the solution of a pointwise constrained optimal control problem governed by the time-dependent incompressible Navier–Stokes equations. The method is based on a reformulation of the optimality system as an equivalent nonsmooth operator equation. We analyze the flow control problem and prove q-superlinear convergence of the method. In the numerical implementation, adjoint techniques are combined with a truncated conjugate gradient method. Numerical results are presented that support our theoretical results and confirm the viability of the approach.     

A high order Discontinuous Galerkin Finite Element solver for the incompressible Navier–Stokes equations

The paper presents an unsteady high order Discontinuous Galerkin (DG) solver that has been developed, verified and validated for the solution of the two-dimensional incompressible Navier–Stokes equations. A second order stiffly stable method is used to discretise the equations in time. Spatial discretisation is accomplished using a modal DG approach, in which the inter-element fluxes are approximated using the Symmetric Interior Penalty Galerkin formulation. The non-linear terms in the Navier–Stokes equations are expressed in the convective form and approximated through the Lesaint–Raviart fluxes modified for DG methods.Verification of the solver is performed for a series of test problems; purely elliptic, unsteady Stokes and full Navier–Stokes. The resulting method leads to a stable scheme for the unsteady Stokes and Navier–Stokes equations when equal order approximation is used for velocity and pressure. For the validation of the full Navier–Stokes solver, we consider unsteady laminar flow past a square cylinder at a Reynolds number of 100 (unsteady wake). The DG solver shows favourably comparisons to experimental data and a continuous Spectral code.     

An Efficient Discretization of the Navier–Stokes Equations in an Axisymmetric Domain. Part 1: The Discrete Problem and its Numerical Analysis

Any solution of the Navier–Stokes equations in a three-dimensional axisymmetric domain admits a Fourier expansion with respect to the angular variable, and it can be noted that each Fourier coefficient satisfies a variational problem on the meridian domain, all problems being coupled due to the nonlinear convection term. We propose a discretization of these equations which combines Fourier truncation and finite element methods applied to each two-dimensional system. We perform the a priori and a posteriori analysis of this discretization.     

Computational simulation of model and full scale Class 8 trucks with drag reduction devices

Numerical solutions of the unsteady Reynolds-averaged Navier–Stokes equations using a parallel implicit flow solver are given to investigate unsteady aerodynamic flows affecting the fuel economy of Class 8 trucks. Both compressible and incompressible forms of the equations are solved using a finite-volume discretization for unstructured grids and using Riemann-based interfacial fluxes and characteristic-variable numerical boundary conditions. A preconditioned primitive-variable formulation is used for compressible solutions, and the incompressible solutions employ artificial compressibility. Detached eddy simulation (DES) versions of the one-equation Menter SAS and the two-equation k − ?/k − ω hybrid turbulence models are used. A fully nonlinear implicit backward-time approximation is solved using a parallel Newton-iterative algorithm with numerically computed flux Jacobians. Unsteady three-dimensional aerodynamic simulations with grids of 18–20 million points and 50,000 time steps are given for the Generic Conventional Model (GCM), a 1:8 scale tractor–trailer model that was tested in the NASA Ames 7 × 10 tunnel. Computed pressure coefficients and drag force are in good agreement with measurements for a zero-incidence case. Similar computations for a case with 10° yaw gave reasonable agreement for drag force, while the pressure distributions suggested the need for tighter grid resolution or possibly improved turbulence models. Unsteady incompressible flow simulations were performed for a modified full scale version of the GCM geometry to evaluate drag reduction devices. All of these simulations were performed with a moving ground plane and rotating rear wheels. A simulation with trailer base flaps is compared with drag reduction data from wind tunnels and track and road tests. A front spoiler and three mud-flap designs with modest drag reduction potential are also evaluated.     

Fast algorithms for computing the diameter of a finite planar set

Three algorithms for computing the diameter of a finite planar set are proposed. Although all three algorithms have (O(n ²) worst-case running time, an expected-complexity analysis shows that, under reasonable probabilistic assumptions, all three algorithms have linear expected running time. Experimental results indicate that two of these algorithms perform very well for some distributions, and are competitive with an existing method. Finally, we exhibit situations where these exact algorithms out-perform a published approximate algorithm.Research of the first author was supported by grant NSERC A 2422. Research of the second author was supported by grants NSERC A 9293, FCAC EQ-1678 and a Killam Senior Research Fellowship awarded by the Canada Council     

Continuity of the spectral factorization on a vertical strip

The continuity of the mapping which associates a spectral factor to a spectral density is investigated. This mapping can be defined on several classes of spectral densities and spectral factors. For the usual largest class of spectral densities, i.e., essential bounded functions on the imaginary axis that are bounded away from zero, it is known that this mapping is not continuous. It is shown here that for slightly smaller, but still generic class the mapping becomes continuous.     

SMF中的Stokes光频移的温度传感特性研究

本文首先从单模光纤中声学声子、泵浦光子和stokes光子的频率之间的波矢关系出发,理论推导了单模光纤中的stokes光频移与工作环境温度之间的变化关系式。接着设计了一个基于受激布里渊散射效应的多波长掺铒光纤激光器的结构装置,并实验测试了该多波长光纤激光器的Stokes光频移随外界温度的变化曲线,实验结果表明：Stokes光频率漂移量和温度之间存在很好的线性关系,且随着温度的升高,高阶Stokes光频移对温度的频移系数高于低阶Stokes光。     

On the design of high-performance algorithms for aligning multiple protein sequences on mesh-based multiprocessor architectures

In this paper, we address the problem of multiple sequence alignment (MSA) for handling very large number of proteins sequences on mesh-based multiprocessor architectures. As the problem has been conclusively shown to be computationally complex, we employ divisible load paradigm (also, referred to as divisible load theory, DLT) to handle such large number of sequences. We design an efficient computational engine that is capable of conducting MSAs by exploiting the underlying parallelism embedded in the computational steps of multiple sequence algorithms. Specifically, we consider the standard Smith–Waterman (SW) algorithm in our implementation, however, our approach is by no means restrictive to SW class of algorithms alone. The treatment used in this paper is generic to a class of similar dynamic programming problems. Our approach is recursive in the sense that the quality of solutions can be refined continuously till an acceptable level of quality is achieved. After first phase of computation, we design a heuristic scheme that renders the final solution for MSA. We conduct rigorous simulation experiments using several hundreds of homologous protein sequences derived from the Rattus Norvegicus and Mus Musculus databases of olfactory receptors. We quantify the performance based on speed-up metric. We compare our algorithms to serial or single machine processing approaches. We testify our findings by comparing with conventional equal load partitioning (ELP) strategy that is commonly used in the parallel processing literature. Based on our extensive simulation study, we observe that DLT paradigm offers an excellent speed-up characteristics and provides avenues for its use in several other biological sequence processing related problem. This study is a first time attempt in using the DLT paradigm to devise efficient strategies to handle large scale multiple protein sequence alignment problem on mesh-based multiprocessor systems.     

一种基于同层网页相似性去除网页噪音的方法

一个普通的Web页面可以被分成信息块和噪音块两部分。基于web信息检索的第1步就是过滤掉网页中的噪音块。通过网页的特性可以看出，同层网页大多具有相似的显示风格和噪音块。在VIPS算法的基础上，该文提出一种基于同层网页相似性的匹配算法，这个算法可以被用来过滤网页中的噪音块。通过实验检测，算法可以达到95%以上的准确率。     

Numerical simulation of low-Reynolds number flows past rectangular cylinders based on adaptive finite element and finite volume methods

The flow past rectangular cylinders has been investigated by two different numerical techniques, an adaptive finite-element (AFEM) and a finite-volume method (FVM). A square and a rectangular cylinder with width-to-height equal to 5 are taken into account. 2D computations have been performed for different Reynolds numbers in order to consider different flow regimes, i.e. the stationary, the periodic and the turbulent flow. The comparison between the two methods regarded both the reliability of the computed solutions and the overall resulting efficiency of the methods. Velocity profiles and integral parameters such as Strouhal number, drag coefficient and recirculation length have been compared. A good agreement between the adaptive FEM and the FVM computations, as well as with the available literature results, has been found. The computational effort has been evaluated in terms of used degrees of freedom in space and time and human resources employed to reach the mesh and timestep-length independence of the solutions. Relevant outcomes of this work are the cross validation of an adaptive FE method and a popular open source FV code.     

On the impact of objective function transformations on evolutionary and black-box algorithms

Different objective functions characterize different problems. However, certain fitness transformations can lead to easier problems although they are still a model of the considered problem. In this article, the class of not worsening transformations for a simple population-based evolutionary algorithm (EA) is described completely. That is the class of functions that transfers easy problems in easy ones and difficult problems in difficult ones. Surprisingly, this class for the rank-based EA equals that for all black-box algorithms. The importance of the black-box algorithms' knowledge of the transformation is also pointed out. Hence, a comparison with the class of not worsening transformations for a similar EA which applies fitness-proportional selection, shows that is a proper superset of . Moreover, is a proper subset of the corresponding class for random search. Finally, the minimal and maximal classes of not worsening transformations are described completely, too.