三维热传导方程的Krylov子空间方法并行分析

热传导方程在地下水流动数值模拟、油藏数值模拟等工程计算中有着广泛应用,其并行实现是加速问题求解速度、提高问题求解规模的重要手段,因此热传导方程的并行求解具有重要意义。对Krylov子空间方法中的CG和GMRES算法进行并行分析,并对不同的预处理CG算法作了比较。在Linux集群系统上,以三维热传导模型为例进行了数值实验。实验结果表明,CG算法比GMRES算法更适合建立三维热传导模型的并行求解。此外,CG算法与BJACOBI预条件子的整合在求解该热传导模型时,其并行程序具有良好的加速比和效率。因此,采用BJACOBI预处理技术的CG算法是一种较好的求解三维热传导模型的并行方案。     

有限差分法的并行化计算实现

有限差分法是求解偏微分方程近似解的一种重要的数值方法。串行算法并不能高效的解决大规模复杂计算问题,并行化计算方法可提高复杂计算问题的效率,从而使并行机上计算有限差分问题成为可能。二维场中拉普拉斯方程的差分格式非常适合并行化方法的计算,将串行部分并行化以提高大规模计算的效率具有重要的现实意义。MPI(消息传递接口)是实现并行程序设计的标准之一。虚拟进程(MPI_PROC_NULL)的引用简化了MPI编程中的通信部分,串行算法可更改为并行化计算方法,最终实现有限差分方法的并行化计算。     

有限差分法的并行化计算实现

有限差分法是求解偏微分方程近似解的一种重要的数值方法。串行算法并不能高效的解决大规模复杂计算问题,并行化计算方法可提高复杂计算问题的效率．从而使并行机上计算有限差分问题成为可能。二维场中拉普拉斯方的差分程格式非常适合并行化方法的计算,将串行部分并行化以提高大规模计算的效率具有重要的现实意义。MPI（消息传递接口）是实现并行程序设计的标准之一。虚拟进程（MPI_PROC_NULL）的引用简化了MPI编程中的通信部分,串行算法可更改为并行化计算方法,最终实现有限差分方法的并行化计算。     

并行代数多重网格算法:大规模计算应用现状与挑战

代数多重网格(AMG)是求解偏微分方程离散线性代数方程组最有效的算法之一,广泛应用于科学与工程计算领域实际问题的大规模数值模拟.随着超级计算机性能不断提升,实际数值模拟的计算规模和并行规模越来越大,同时,实际问题应用特征和计算机体系结构特征越来越复杂,AMG面临并行可扩展、算法可扩展和浮点性能优化的严峻挑战.本文结合大规模计算的发展趋势,特别是面向即将到来的百亿亿次(E级)计算,分析AMG算法在这三个方面的挑战,总结研究现状与进展,展望未来研究重点.     

一种孔隙介质中地下水流并行计算方法

针对孔隙介质中地下水流动问题提出了一种并行数值计算方法,并基于此设计了一套专用于求解大规模三维地下水流动方程的并行计算模块。计算模块基于区域分解的方法实现对模型区域的并行求解,采用了分布式内存和压缩矩阵技术解决大规模稀疏矩阵的存储及其计算,整合多种并行Krylov子空间方法和预条件子技术迭代求解大规模线性方程组。在Linux集群系统上进行了数值模拟实验,性能测试结果表明,程序具有良好的加速比和可扩展性。     

边缘海静力数值预报模式并行算法研究

边缘海静力数值模式是国内针对边缘海特点自主开发的数值预报模式,但该模式因物理求解方程较多且采用不宜并行化的SOR求解算法而程序计算时间过长。针对上述问题,提出基于三维网格和海洋模式特点的SOR并行求解算法,该算法在保留三维网格数据间依赖关系的同时,有效解决了SOR迭代算法难以并行化的问题。同时,引入通信避免算法,采用MPI非阻塞通信方式,细分计算和通信过程,利用计算有效隐藏通信开销,提高了并行程序效率。实验结果表明,并行后的边缘海静力数值模式程序的性能相对串行程序提升了60.71倍,3天(25920计算时间步)预报结果的均方根误差低于0.001,满足海洋数值预报的时效性和精度要求。     

基于进化策略的线性变分不等式求解算法

基于在求解变分不等式过程中存在着传统数字计算机的迭代算法很难满足并行性要求的问题,提出了求解一类线性变分不等式问题的进化策略算法.将进化策略算法用于求解线性变分不等式的数值方法,充分发挥了进化策略算法的全局收敛和并行搜索的特性,满足了工程技术中并行求解变分不等式问题的要求.数值计算结果表明,该算法收敛速度快、精度高,稳定性好,是一种解决线性变分不等式问题的有效方法.     

非均匀区域油藏模拟负载平衡的分区并行算法*

基于分布式并行计算机系统,对一类非均匀区域的油藏数值模拟问题,采用了区域分解方法并行求解,给出了并行求解的负载平衡模型及区域负载平衡的一种有效分区算法,从而将这类油藏数值模拟问题均衡映射到并行环境中,高效地并行求解.在黑油油藏数值模拟并行软件的研究中,实验结果表明,该算法有利于提高加速比.     

一种基于GPU加速的细粒度并行蚁群算法

为改善蚁群算法对大规模旅行商问题的求解性能,提出一种基于图形处理器(GPU)加速的细粒度并行蚁群算法.将并行蚁群算法求解过程转化为统一计算设备架构的线程块并行执行过程,使得蚁群算法在GPU中加速执行.实验结果表明,该算法能提高全局搜索能力,增大细粒度并行蚁群算法的蚂蚁规模,从而提高了算法的运算速度.     

基于非结构滑移网格技术的旋翼型无人机空气动力学并行数值模拟方法

在现代飞行器设计中,数值模拟方法以低成本、高效率和高灵活性等优点成为研究飞行器空气动力学的重要方法.在旋翼型无人机流场模拟中,由于旋翼与机身存在相互作用,为获得精确模拟结果需要对整个无人机的流场进行模拟,因此,有效地模拟旋翼与机身的相对运动是实现成功模拟的关键步骤,这使得此类模拟问题极具挑战性.文章设计了一套求解旋翼型无人机空气动力学数值模拟问题的基于非结构滑移网格技术的高可扩展并行计算方法.该方法对控制方程的离散,在空间方向采用非结构移动网格有限元方法,时间推进采用全隐式二阶向后差分格式,最后采用一种并行Newton-Krylov-Schwarz方法求解离散后的非线性方程组.作为应用,文章对一个真实旋翼型无人机模型在悬停状态下的外流场进行了数值模拟,获得了一些非常详细的流场信息.数值结果显示,算法在天河2号上使用4 096个处理器核时仍具有接近线性的并行加速比,这为下一步开展旋翼型无人机的高保真度快速模拟奠定了良好的基础.     

A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Numerical simulation of three-dimensional incompressible flows at high Reynolds number using the unsteady Navier–Stokes equations is challenging. In order to obtain accurate simulations, very fine meshes are necessary, and such simulations are increasingly important for modern engineering practices, such as understanding the flow behavior around high speed trains, which is the target application of this research. To avoid the time step size constraint imposed by the CFL number and the fine spacial mesh size, we investigate some fully implicit methods, and focus on how to solve the large nonlinear system of equations at each time step on large scale parallel computers. In most of the existing implicit Navier–Stokes solvers, segregated velocity and pressure treatment is employed. In this paper, we focus on the Newton–Krylov–Schwarz method for solving the monolithic nonlinear system arising from the fully coupled finite element discretization of the Navier–Stokes equations on unstructured meshes. In the subdomain, LU or point-block ILU is used as the local solver. We test the algorithm for some three-dimensional complex unsteady flows, including flows passing a high speed train, on a supercomputer with thousands of processors. Numerical experiments show that the algorithm has superlinear scalability with over three thousand processors for problems with tens of millions of unknowns.     

Gas-kinetic numerical study of complex flow problems covering various flow regimes

The Boltzmann simplified velocity distribution function equation, as adapted to various flow regimes, is described on the basis of the Boltzmann–Shakhov model from the kinetic theory of gases in this study. The discrete velocity ordinate method of gas-kinetic theory is studied and applied to simulate complex multi-scale flows. On the basis of using the uncoupling technique on molecular movements and collisions in the DSMC method, the gas-kinetic finite difference scheme is constructed by extending and applying the unsteady time-splitting method from computational fluid dynamics, which directly solves the discrete velocity distribution functions. The Gauss-type discrete velocity numerical quadrature technique for flows with different Mach numbers is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established for studying the three-dimensional complex flows with high Mach numbers from rarefied transition to continuum regimes. On the basis of the parallel characteristics of the respective independent discrete velocity points in the discretized velocity space, a parallel strategy suitable for the gas-kinetic numerical method is investigated and, then, the HPF (High Performance Fortran) parallel programming software is developed for simulating gas dynamical problems covering the full spectrum of flow regimes. To illustrate the feasibility of the present gas-kinetic numerical method and simulate gas transport phenomena covering various flow regimes, the gas flows around three-dimensional spheres and spacecraft-like shapes with different Knudsen numbers and Mach numbers are investigated to validate the accuracy of the numerical methods through HPF parallel computing. The computational results determine the flow fields in high resolution and agree well with the theoretical and experimental data. This computing, in practice, has confirmed that the present gas-kinetic algorithm probably provides a promising approach for resolving hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view for solving the mesoscopic Boltzmann model equation.     

A parallel fully coupled implicit domain decomposition method for numerical simulation of microfluidic mixing in 3D

A parallel fully coupled implicit fluid solver based on a Newton–Krylov–Schwarz algorithm is developed on top of the Portable, Extensible Toolkit for Scientific computation for the simulation of microfluidic mixing described by the three-dimensional unsteady incompressible Navier–Stokes equations. The popularly used fractional step method, originally designed for high Reynolds number flows, requires some modification of the inviscid-type pressure boundary condition in order to reduce the divergence error near the wall. On the other hand, the fully coupled approach works well without any special treatment of the boundary condition for low Reynolds number microchannel flows. A key component of the algorithm is an additive Schwarz preconditioner, which is used to accelerate the convergence of a linear Krylov-type solver for the saddle-point-type Jacobian systems. As a test case, we carefully study a three-dimensional passive serpentine micromixer and report the parallel performance of the algorithm obtained on a parallel machine with more than one hundred processors.     

Interactive time-dependent particle tracing using tetrahedraldecomposition

Streak lines and particle traces are effective visualization techniques for studying unsteady fluid flows. For real time applications, accuracy is often sacrificed to achieve interactive frame rates. Physical space particle tracing algorithms produce the most accurate results although they are usually too expensive for interactive applications. An efficient physical space algorithm is presented which was developed for interactive investigation and visualization of large, unsteady, aeronautical simulations. Performance has been increased by applying tetrahedral decomposition to speed up point location and velocity interpolation in curvilinear grids. Preliminary results from batch computations showed that this approach was up to six times faster than the most common algorithm which uses the Newton-Raphson method and trilinear interpolation. Results presented show that the tetrahedral approach also permits interactive computation and visualization of unsteady particle traces. Statistics are given for frame rates and computation times on single and multiprocessors. The benefits of interactive feature detection in unsteady flows are also demonstrated     

Applications of stencil-adaptive finite difference method to incompressible viscous flows with curved boundary

This paper investigates the applicability of the stencil-adaptive finite difference method for the simulation of two-dimensional unsteady incompressible viscous flows with curved boundary. The adaptive stencil refinement algorithm has been proven to be able to continuously adapt the stencil resolution according to the gradient of flow parameter of interest [Ding H, Shu C. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J Comput Phys 2006;214:397-420], which facilitates the saving of the computational efforts. On the other hand, the capability of the domain-free discretization technique in dealing with the curved boundary provides a great flexibility for the finite difference scheme on the Cartesian grid. Here, we show that their combination makes it possible to simulate the unsteady incompressible flow with curved boundary on a dynamically changed grid. The methods are validated by simulating steady and unsteady incompressible viscous flows over a stationary circular cylinder.     

Local penalty methods for flows interacting with moving solids at high Reynolds numbers

An original numerical modelling of multiphase flows interacting with solids in unsteady regimes is presented. Based on the generalized Navier-Stokes equations for multiphase flows and Volume of Fluid (VOF) formulations, an Uzawa minimization algorithm is implemented for the treatment of incompressibility and solid constraints. Augmented Lagrangian terms are added in the momentum equations to speed the convergence of the iterative solver. Defining a priori the penalty parameters which are dedicated to incompressibility and solid constraints is difficult, or impossible, as soon as the flow involves more than one phase and inertia becomes predominant compared to viscous and gravity forces. The Lagrangian penalty terms are calculated automatically according to an original local estimate of the various physical contributions. Numerical validations have been carried out for single particle settling in confined media and viscous flow through a fixed Cubic Faced Centered array. A very good agreement is obtained between experimental, theoretical and numerical results. Extension to unsteady free surface flow interacting with particles is illustrated with the simulation of a dam break flow over moving obstacles.     

Parallel 3D computation of unsteady flows around circular cylinders

In this article we present parallel 3D finite element computation of unsteady incompressible flows around circular cylinders. We employ stabilized finite element formulations to solve the Navier-Stokes equations on a thinking machine CM-5 supercomputer. The time integration is based on an implicit method, and the coupled, nonlinear equations generated every time step are solved iteratively, with an element-vector based evaluation technique. This strategy enables us to carry out these computations with millions of coupled, nonlinear equations, and thus resolve the flow features in great detail. At Reynolds number 300 and 800, our results indicate strong 3D features arising from the instability of the columnar vortices forming the Karman street. At Re = 10 000 we employ a large eddy simulation (LES) turbulence model.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.     

Towards a three-dimensional moving body incompressible flow solver with a linear deformable model

In this study, a three-dimensional fluid–structured parallelized solver is extended from the previous work (Niu et al., 2009 [1]) for moving body simulations. Based on the unified Eulerian and Lagrangian coordinate transformations, the unsteady three-dimensional incompressible Navier–Stokes equations with artificial compressibility (Chorin, 1967 [2]) in a dual-time stepping approach are first derived. To implement unsteady flow calculations, the dual-time stepping strategy including the LU decomposition method is used in the pseudo-time iteration and the second-order accurate backward difference is adopted to discretize the unsteady flow terms. Also, a third-order Roe type flux limited splitting is derived to evaluate the spatial difference of the convective fluxes. The original FORTRAN code is converted to the MPI code and tested on a 64-CPU IBM SP2. The parallel strategy here is based on the partitions of all do-loops in the original FORTRAN code and transferring the calculations inside the do-loop into different CPUs. The partition of the do-loop can be applied on the innermost loop, only or the last two inner loops depending on two-dimensional or three-dimensional problems. This kind of the parallel data partition of the loops is independent of what kind of the explicit or implicit type numerical algorithm used. Therefore, the current parallel approach can take advantage of the MPI language fully to transfer data efficiently among CPUs even for solving the governing equation implicitly. The test results show that a significant reduction of computing time in running the model and a near-linear speed up rate is achieved up to 32 CPUs at IBM SP2. The speed up rate is as high as 31 for using 64 IBM SP2 processors The test shows efficient parallel processing to provide prompt simulation of 3D cavity, unsteady dropping airfoil and blood flows in an aortic tube with a linear elastic modeling of wall motion is included here.     

MPI并行调试与优化策略在三维绕流气体运动 论数值模拟中的应用

从求解三维绕流问题的Boltzmann模型方程的数值模拟程序出发,通过研究区域分解并行计算策略,引入输入/输出、通信与CACHE等优化策略,对数值模拟程序进行MPI并行化移植与高性能计算调试。以高空稀薄过渡流区飞行器绕流状态为算例,进行了MPI大规模并行计算测试,证实了所发展的MPI并行化区域分解策略及程序优化途径的正确性。研究表明开展的并行化实现能明显地缩短模式计算时间,并取得较好的效果。