非结构粘性直角网格并行算法建立与应用

开展了基于粘性直角非结构网格的并行CFD解算软件的开发研究,工作分两部分：网格分区实现和解算器并行实施,文章介绍了关于CFD解算器并行的实施情况,给出了并行过程中的操作流程,并对一些关键问题进行了讨论。并行计算结果表明项目所采用的并行途径和方法有效,计算结果可靠。     

基于光滑聚集代数多重网格的有限元 并行计算实现方法

基于光滑聚集代数多重网格法实现一种用于结构有限元并行计算的预条件共轭梯度求解方法。对计算区域进行均匀划分,将这些子区域分配给各个进程同时进行单元刚度矩阵的计算,并组合形成分布式存储的整体平衡方程。采用光滑聚集代数多重网格预条件共轭梯度法对整体平衡方程进行并行求解,在天河二号超级计算机上进行数值试验,分析代数多重网格的主要参数对算法性能的影响,测试程序的并行计算性能。试验结果表明该方法具有较好的并行性能和可扩展性,适合于大规模实际应用。     

基于GPU的多重网格Navier-Stokes解算器并行优化方法研究

随着工业计算需求的激增,计算流体力学 (Computational Fluid Dynamics, CFD) 学科对计算效率问题越来越重视。作者基于自行开发的 Navier-Stokes 解算器,引入多重网格加速收敛算法,并结合NVIDIA GPU 计算平台,从数值方法和高性能计算两个方面为 CFD 实现加速。数值加速算例测试结果表明,基于多重网格算法的 GPU 解算器相对 CPU 版本代码双精度可获得 45 倍以上的加速。     

基于方程求解的非结构网格并行处理方案与实现

非结构网格在微分方程数值求解领域有着重要的应用前景,但若干关键的具体应用技术仍缺乏足够的研究。本文对非结构网格的并行处理问题做了深入探讨,特别的,从应用流行的分布式消息传递并行编程模型出发,提出了一套非结构网格划分和管理的综合应用方案,并给出了相应的实现与应用步骤。以本文方案为基础,能方便地实现微分方程的大规模并行求解。     

用于图像重构的代数多重网格算法

通过分析代数多重网格(algebraic multi-grid,AMG)算法中粗网格提取过程,提出了一种基于代数多重网格算法的图像重构算法.在代数多重网格算法的粗网格序列中,下一层粗网格保留上一层网格的强连接部分.将这种机制运用到图像,提取的粗网格可以较好的保留图像的有效信息部分,在图像变化剧烈的细节区域网格点分布不均匀,平滑模糊部分网格点分布均匀一致.以粗网格像素点进行插值,可以得到较好的重建结果.以均方误差为评价参数,与小波算法进行了比较,比较结果表明该算法在一定程度上优于传统的小波算法,且有一个图像融合应用实例,优于小波融合方法.     

结构化多重网格粘性流场数值模拟

流体力学控制方程的数值求解过程中,当网格加密或者粘性效应强的时候,流场收敛非常缓慢.为了解决计算的效率问题,在结构网格的基础上采用多重网格技术,模拟了二维RAE2822超临界翼型的亚音速绕流及三维M6机翼跨音速流场,仿真结果表明,采用多重网格方法在二维,三维粘性流场的计算结果都与实验结果吻合良好,与不采用多重网格方法比较,在求解中获得了相当满意的加速收敛效果.还比较了两种不同循环方式:V循环,W循环的加速效率,为多重网格的工程应用奠下基础.     

智能网格系统组建研究

网格是一个计算资源池,网格技术适于大型的科学计算和项目研究,它是科技发展的必然产物.通过对网格技术的国内外研究现状分析,提出了智能网格的环境和AGS结构体系模型、并对此AGS结构体系模型进行了详细的研究设计.同时从网格的应用背景出发、选择合适的网格平台、从局部到整体规划的原则进行了智能网格系统的组建研究;指出了网格技术的应用,给网格内的用户提供一体化的高性能计算环境和信息服务;同时由于网格的实现,也将为解决复杂的系统问题的综合集成技术打下坚实基础.     

基于非等距网格高阶紧致差分格式的多重网格算法研究

本文结合非等距网格高精度紧致差分格式的优越性与多重网格方法的快速收敛性,求解二维对流扩散方程。研究结果表明,对于处理物理量在不同的空间方向呈现不同的性态特征或不同变化规律的物理问题时,用非等距网格离散的四阶紧致格式的多重网格算法和二阶中心差分格式的多重网格算法都比等距网格离散得高效。同时,在非等距网格下下,部分半粗化多重网格算法比完全粗化多重网格算法具有更高的计算效率。针对不同的松弛算子对误差残量的磨光效果比较研究表明,线松弛算子是最高效的。而且,非等距网格离散的高精度紧致格式的多重网格算法对于对流扩散问题中大网格雷诺数情形也是收敛的。     

基于结构网格的大规模并行计算研究

通过求解RANS方程和Menter's k-Omega SST两方程湍流模型,以及采用多重网格加速收敛技术、基于多块结构网格的通用数据传输方法和区域分解负载平衡技术,实现CFD软件的并行计算。在国家超算长沙中心的"天河"系统上完成了软件的移植、测试,并实现翼身组合体外形的2048处理器核数、网格规模上亿单元的大规模并行计算,并行效率达到48%,较大幅度地缩短了计算周期,提高了工作效率。通过对DLR-F6的模拟,在气动力系数精确求解、超大规模网格模拟的快速收敛和网格收敛性研究等方面取得了初步结果,为下一步大规模工程实际应用打下了坚实基础。     

基于非结构网格隐式算法的GPU加速研究

针对非结构网格隐式算法在GPU上的加速效果不佳的问题,通过分析GPU的架构及并行模式,研究并实现了基于非结构网格格点格式的隐式LU-SGS算法的GPU并行加速.通过采用RCM和Metis网格重排序（重组）方法,优化非结构网格的数据局部性,改善非结构网格的隐式算法在GPU上的并行加速效果.通过三维机翼算例验证了本文实现的正确性及效率.结果表明两种网格重排序（重组）方法分别得到了63%和69%的加速效果提高.优化后的LU-SGS隐式GPU并行算法获得了相较于CPU串行算法27倍的加速比,充分说明了本文方法的高效性.     

A balanced accumulation scheme for parallel PDE solvers

We present a tailored load balancing technique that addresses specific performance issues in the boundary data accumulation algorithm for non-overlapping domain decompositions. The technique is used to speed up a parallel conjugate gradient algorithm with an algebraic multigrid preconditioner to solve a potential problem on an unstructured tetrahedral finite element mesh. The optimized accumulation algorithm significantly improves the performance of the parallel solver and we show up to 50 % runtime improvements over the standard approach in benchmark runs with up to 48 MPI processes. The load balancing problem itself is a global optimization problem that is solved approximately by local optimization algorithms in parallel that require no communication during the optimization process.     

An Algebraic Multigrid Method for Higher-order Finite Element Discretizations

In this paper, we will design and analyze a class of new algebraic multigrid methods for algebraic systems arising from the discretization of second order elliptic boundary value problems by high-order finite element methods. For a given sparse stiffness matrix from a quadratic or cubic Lagrangian finite element discretization, an algebraic approach is carefully designed to recover the stiffness matrix associated with the linear finite element disretization on the same underlying (but nevertheless unknown to the user) finite element grid. With any given classical algebraic multigrid solver for linear finite element stiffness matrix, a corresponding algebraic multigrid method can then be designed for the quadratic or higher order finite element stiffness matrix by combining with a standard smoother for the original system. This method is designed under the assumption that the sparse matrix to be solved is associated with a specific higher order, quadratic for example, finite element discretization on a finite element grid but the geometric data for the underlying grid is unknown. The resulting new algebraic multigrid method is shown, by numerical experiments, to be much more efficient than the classical algebraic multigrid method which is directly applied to the high-order finite element matrix. Some theoretical analysis is also provided for the convergence of the new method.     

Parallel multigrid on hierarchical hybrid grids: a performance study on current high performance computing clusters

This article studies the performance and scalability of a geometric multigrid solver implemented within the hierarchical hybrid grids (HHG) software package on current high performance computing clusters up to nearly 300,000 cores. HHG is based on unstructured tetrahedral finite elements that are regularly refined to obtain a block‐structured computational grid. One challenge is the parallel mesh generation from an unstructured input grid that roughly approximates a human head within a 3D magnetic resonance imaging data set. This grid is then regularly refined to create the HHG grid hierarchy. As test platforms, a BlueGene/P cluster located at Jülich supercomputing center and an Intel Xeon 5650 cluster located at the local computing center in Erlangen are chosen. To estimate the quality of our implementation and to predict runtime for the multigrid solver, a detailed performance and communication model is developed and used to evaluate the measured single node performance, as well as weak and strong scaling experiments on both clusters. Thus, for a given problem size, one can predict the number of compute nodes that minimize the overall runtime of the multigrid solver. Overall, HHG scales up to the full machines, where the biggest linear system solved on Jugene had more than one trillion unknowns. Copyright © 2012 John Wiley & Sons, Ltd.     

A Robust Multigrid Algorithm for the Simulation of a Yawed Flat Plate

This paper presents a full multigrid solver for the simulation of a flow over a yawed flat plate. The two problems associated with this simulation; boundary layers and entering flows with non-aligned characteristics, have been successfully overcome through the combination of a plane-implicit solver and semicoarsening. In fact, this multigrid algorithm exhibits a textbook multigrid convergence rate, i.e., the solution of the discrete system of equations is obtained in a fixed amount of computational work, independently of the grid size, grid stretching factor and non-alignment parameter. Also, a parallel variant of the smoother based on a four-color ordering of planes is investigated.     

Semicoarsening smoothers for the simulation of a flat plate at yaw

This paper presents a full multigrid solver for the simulation of flow over a yawed flat plate. The two problems associated with this simulation; boundary layers and entering flows with non-aligned characteristics, have been successfully overcome through the combination of a plane-implicit solver and semicoarsening. In fact, this multigrid algorithm exhibits a textbook multigrid convergence rate, i.e., the solution of the discrete system of equations is obtained in a fixed amount of computational work, independently of the grid size, grid stretching factor and non-alignment parameter. Also, a parallel variant of the smoother based on a four-color ordering of planes is investigated.     

Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method

The development and validation of a parallel unstructured tetrahedral non-nested multigrid (MG) method for simulation of unsteady 3D incompressible viscous flow is presented. The Navier-Stokes solver is based on the artificial compressibility method (ACM) and a higher-order characteristics-based finite-volume scheme on unstructured MG. Unsteady flow is calculated with an implicit dual time stepping scheme. The parallelization of the solver is achieved by a MG domain decomposition approach (MG-DD), using the Single Program Multiple Data (SPMD) programming paradigm. The Message-Passing Interface (MPI) Library is used for communication of data and loop arrays are decomposed using the OpenMP standard. The parallel codes using single grid and MG are used to simulate steady and unsteady incompressible viscous flows for a 3D lid-driven cavity flow for validation and performance evaluation purposes. The speedups and efficiencies obtained by both the parallel single grid and MG solvers are reasonably good for all test cases, using up to 32 processors on the SGI Origin 3400. The parallel results obtained agree well with those of serial solvers and with numerical solutions obtained by other researchers, as well as experimental measurements.     

Array-based,parallel hierarchical mesh refinement algorithms for unstructured meshes

In this paper, we describe an array-based hierarchical mesh refinement capability through uniform refinement of unstructured meshes for efficient solution of PDE’s using finite element methods and multigrid solvers. A multi-degree, multi-dimensional and multi-level framework is designed to generate the nested hierarchies from an initial coarse mesh that can be used for a variety of purposes such as in multigrid solvers/preconditioners, to do solution convergence and verification studies and to improve overall parallel efficiency by decreasing I/O bandwidth requirements (by loading smaller meshes and in-memory refinement). We also describe a high-order boundary reconstruction capability that can be used to project the new points after refinement using high-order approximations instead of linear projection in order to minimize and provide more control on geometrical errors introduced by curved boundaries.The capability is developed under the parallel unstructured mesh framework “Mesh Oriented dAtaBase” (MOAB Tautges et al. (2004)). We describe the underlying data structures and algorithms to generate such hierarchies in parallel and present numerical results for computational efficiency and effect on mesh quality. We also present results to demonstrate the applicability of the developed capability to study convergence properties of different point projection schemes for various mesh hierarchies and to a multigrid finite-element solver for elliptic problems.     

Parallel multilevel methods with adaptivity on unstructured grids

We present two parallel multilevel methods for solving large-scale discretized partial differential equations on unstructured 2D/3D grids. The presented methods combine three powerful numerical algorithms: overlapping domain decomposition, multigrid method and adaptivity. As the foundation of the methods we propose an algorithm for generating and partitioning a hierarchy of adaptively refined unstructured grids, so that adaptivity can be incorporated up to a certain grid level. We ensure that the resulting subgrid hierarchies are well balanced and no inter-processor communication is needed across different grid levels, thus obtaining high parallel efficiency. Numerical experiments show that the parallel multilevel methods offer almost equally fast convergence as their sequential multigrid counterpart. And the resulting implementation has reasonably good scalability. Received: 4 December 1998 / Accepted: 12 January 2000     

A parallel multigrid solver for incompressible flows on computing architectures with accelerators

An efficient parallel multigrid pressure correction algorithm is proposed for the solution of the incompressible Navier–Stokes equations on computing architectures with acceleration devices. The pressure correction procedure is based on the numerical solution of a Poisson-type problem, which is discretized using a fourth-order finite difference compact scheme. Since this is the most time-consuming part of the solver, we propose a parallel pressure correction algorithm using an iterative method based on a block cyclic reduction solution method combined with a multigrid technique. The grid points are numbered with respect to the red–black ordering scheme for the parallel Gauss–Seidel smoother. These parallelization techniques allow the execution of the entire simulation computations on the acceleration device, minimizing memory communication costs. The realization is developed using the OpenACC API, and the numerical method is demonstrated for the solution of two classical incompressible flow test problems. The first is the two-dimensional lid-driven cavity problem over equal mesh sizes while the other is the Stokes boundary layer, which is a decent benchmark problem for unequal mesh spacing. The effect of several multigrid components on modern and legacy acceleration architectures is examined. Eventually the performance investigation demonstrates that the proposed parallel multigrid solver achieves an acceleration of more than 10\(\times \) over the sequential solver and more than 4\(\times \) over multi-core CPU only realizations for all tested accelerators.     

An algorithm for ideal multigrid convergence for the steady Euler equations

A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Flows in two-dimensional channels and over airfoils have been computed. Using Gauss–Seidel relaxation with the grid vertices ordered in the flow direction, ideal multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier–Stokes equations.