A method for finite element parallel viscous compressible flow calculations

This paper presents the parallelization aspects of a solution method for the fully coupled 3D compressible Navier-Stokes equations. The algorithmic thrust of the approach, embedded in a finite element code NS3D, is the linearization of the governing equations through Newton methods, followed by a fully coupled solution of velocities and pressure at each non-linear iteration by preconditioned conjugate gradient-like iterative algorithms. For the matrix assembly, as well as for the linear equation solver, efficient coarse-grain parallel schemes have been developed for shared memory machines, as well as for networks of workstations, with a moderate number of processors. The parallel iterative schemes, in particular, circumvent some of the difficulties associated with domain decomposition methods, such as geometry bookkeeping and the sometimes drastic convergence slow-down of partitioned non-linear problems.     

建筑风压数值模拟的几种并行化策略

针对四个处理机的Ｔｒａｎｓｐｕｔｅｒ并行计算机系统，建立了建筑风压数值模拟问题基于ＳＩＭ－ＰＬＥＣ算法的几种并行化策略：分区并行策略、方程并行策略和双重并行策略。对各种策略的计算流程、数据通讯及并行效率等进行了分析和比较，并通过实例计算作了验证。     

The implementation of an aeronautical CFD flow code onto distributed memory parallel systems

The parallelization of an industrially important in‐house computational fluid dynamics (CFD) code for calculating the airflow over complex aircraft configurations using the Euler or Navier–Stokes equations is presented. The code discussed is the flow solver module of the SAUNA CFD suite. This suite uses a novel grid system that may include block‐structured hexahedral or pyramidal grids, unstructured tetrahedral grids or a hybrid combination of both. To assist in the rapid convergence to a solution, a number of convergence acceleration techniques are employed including implicit residual smoothing and a multigrid full approximation storage scheme (FAS). Key features of the parallelization approach are the use of domain decomposition and encapsulated message passing to enable the execution in parallel using a single programme multiple data (SPMD) paradigm. In the case where a hybrid grid is used, a unified grid partitioning scheme is employed to define the decomposition of the mesh. The parallel code has been tested using both structured and hybrid grids on a number of different distributed memory parallel systems and is now routinely used to perform industrial scale aeronautical simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

An Efficient Parallel ADI Algorithm for Turbomachinery Flows

The paper describes a domain decomposition strategy which allows high efficiency parallel simulations of turbomachinery flows. The implicit ADI parallel solver is based on the scalar approximate factorisation. The Navier-Stokes (NS) and turbulence model equations are discretised by centred finite differences. The results prove that the parallel calculations with domain decomposition, in which each sub-domain explicitly communicates with the adjacent ones at the end of each implicit iteration, may suffer from a considerable deterioration of the convergence rate. A simple sub-iterative domain recoupling strategy allows recovering the convergence rate of a single processor computation. The strategy is carefully analysed and optimised in terms of inter-processor data communication rate and algorithm memory requirements. The span-wise domain decomposition is particularly suited for turbomachinery flows and is applied to a radial impeller and to an axial turbine stator and stage to prove the validity and the accuracy of the proposed approach. The results indicate that the parallel recoupled algorithm usually reach efficiencies of 0.8, with peaks over 0.9 with up to 16 processors, thereby allowing a considerable speed-up of design and verification calculations.     

Large Eddy Simulation of an Internal Combustion Engine Using an Efficient Immersed Boundary Technique

This paper presents highly resolved large eddy simulations (LES) of an internal combustion engine (ICE) using an immersed boundary method (IBM), which can describe moving and stationary boundaries in a simple and efficient manner. In this novel approach, the motion of the valves and the piston is modeled by Lagrangian particles, whilst the stationary parts of the engine are described by a computationally efficient IBM. The proposed mesh-free technique of boundary representation is simple for parallelization and suitable for high performance computing (HPC). To demonstrate the method, LES results are presented for the flow and the combustion in an internal combustion engine. The Favre-filtered Navier-Stokes equations are solved for a compressible flow employing a finite volume method on Cartesian grids. Non-reflecting boundary conditions are applied at the intake and the exhaust ports. Combustion is described using a flame surface density (FSD) model with an algebraic reaction rate closure. A simplified engine with a fixed axisymmetric valve (see Appendix A) is employed to show the correctness of the method while avoiding the uncertainties which may be induced by the complex engine geometry. Three test-cases using a real engine geometry are investigated on different grids to evaluate the impact of the cell size and the filter width. The simulation results are compared against the experimental data. A good overall agreement was found between the measurements and the simulation data. The presented method has particular advantages in the efficient generation of the grid, high resolution and low numerical dissipation throughout the domain and an excellent suitability for massively parallel simulations.     

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.     

Quasi-real-time simulation of rotating drum using discrete element method with parallel GPU computing

Real-time simulation of industrial equipment is a huge challenge nowadays. The high performance and fine-grained parallel computing provided by graphics processing units (GPUs) bring us closer to our goals. In this article, an industrial-scale rotating drum is simulated using simplified discrete element method (DEM) without consideration of the tangential components of contact force and particle rotation. A single GPU is used first to simulate a small model system with about 8000 particles in real-time, and the simulation is then scaled up to industrial scale using more than 200 GPUs in a 1D domain-decomposition parallelization mode. The overall speed is about 1/11 of the real-time. Optimization of the communication part of the parallel GPU codes can speed up the simulation further, indicating that such real-time simulations have not only methodological but also industrial implications in the near future.     

A comparison of coarse and fine grain parallelization strategies for the simple pressure correction algorithm

The primary aim of this work was to determine the simplest and most effective parallelization strategy for control-volume-based codes solving industrial problems. It has been found that for certain classes of problems, the coarse-grain functional decomposition strategy, largely ignored due to its limited scaling capability, offers the potential for significant execution speed-ups while maintaining the inherent structure of traditional serial algorithms. Functional decomposition requires only minor modification of the existing serial code to implement and, hence, code portability across both concurrent and serial computers is maintained. Fine-grain parallelization strategies at the ‘DO loop’ level are also easy to implement and largely preserve code portability. Both coarse-grain functional decomposition and fine-grain loop-level parallelization strategies for the SIMPLE pressure correction algorithm are demonstrated on a Silicon Graphics 4D280S eight CPU shared memory computer system for a highly coupled, transient two-dimensional simulation involving melting of a metal in the presence of thermal-buoyancy-driven laminar convection. Problems requiring the solution of a larger number of transport equations were simulated by including further scalar variables in the calculation. While resulting in slight degradation of the convergence rate, the functional decomposition strategy exhibited higher parallel efficiencies and yielded greater speed-ups relative to the original serial code. Initially, this strategy showed a significant degradation in convergence rate due to an inconsistency in the parallel solution of the pressure correction equation. After correcting for this inconsistency, the maximum speed-up for 16 dependent variables was a factor of 5·28 with eight processors, representing a parallel efficiency of 67%. Peak efficiency of 76% was achieved using five processors to solve for 10 dependent variables.     

Development and assessment of algorithms for DEM-LES simulations of fluidized bed

The use of high-fidelity Discrete Element Method (DEM) coupled with Computational Fluid Dynamics (CFD) for particle-scale simulations demands extensive simulation times and restricts application to small particulate systems. DEM-CFD simulations require good performance and satisfactory scalability on high-performance computing platforms. A reliable parallel computing strategy must be developed to calculate the collision forces, since collisions can occur between particles that are not on the same processor, or even across processors whose domains are disjoint. The present paper describes a parallelization technique and a numerical verification study based on a number of tests that allow for the assessment of the numerical performance of DEM used in conjunction with Large-Eddy Simulation (LES) to model dense flows in fluidized beds. The fluid phase is computed through solving the volume-averaged four-way coupling Navier-Stokes equations, in which the Smagorinsky sub-grid scale tensor model is used. Furthermore, the performance of Sub-Grid Scale (SGS) turbulence models applied to Fluidized Bed Reactor (FBR) configurations has been assessed and compared. The developed numerical solver represents an interesting combination of techniques that work well for the present purpose of studying particle formation in fluidized beds.     

Schwarz domain decomposition for the incompressible Navier–Stokes equations in general co‐ordinates

This paper describes a domain decomposition method for the incompressible Navier–Stokes equations in general co‐ordinates. Domain decomposition techniques are needed for solving flow problems in complicated geometries while retaining structured grids on each of the subdomains. This is the so‐called block‐structured approach. It enables the use of fast vectorized iterative methods on the subdomains. The Navier–Stokes equations are discretized on a staggered grid using finite volumes. The pressure‐correction technique is used to solve the momentum equations together with incompressibility conditions. Schwarz domain decomposition is used to solve the momentum and pressure equations on the composite domain. Convergence of domain decomposition is accelerated by a GMRES Krylov subspace method. Computations are presented for a variety of flows. Copyright © 2000 John Wiley & Sons, Ltd.     

A PARALLEL BLOCK-STRUCTURED MULTIGRID METHOD FOR THE PREDICTION OF INCOMPRESSIBLE FLOWS

In this paper a parallel multigrid finite volume solver for the prediction of steady and unsteady flows in complex geometries is presented. For the handling of the complexity of the geometry and for the parallelization a unified approach connected with the concept of block-structured grids is employed. The parallel implementation is based on grid partitioning with automatic load balancing and follows the message-passing concept, ensuring a high degree of portability. A high numerical efficiency is obtained by a non-linear multigrid method with a pressure correction scheme as smoother. By a number of numerical experiments on various parallel computers the method is investigated with respect to its numerical and parallel efficiency. The results illustrate that the high performance of the underlying sequential multigrid algorithm can largely be retained in the parallel implementation and that the proposed method is well suited for solving complex flow problems on parallel computers with high efficiency.     

Squeeze film pressures

Summary As parallel faces separated by a fluid film approach one another, a pressure is developed in the film. The reduced Navier-Stokes equations are solved by transform methods for both rectangular and cylindrical geometries. Relatively high pressures can exist at small asperities on the surfaces as they approach one another.     

Development of a parallel computational fluid dynamics algorithm on a hypercube computer

One of the main factors limiting the widespread use of computational fluid dynamics codes for engineering design is their very large requirements both in terms of computer memory and CPU time. Distributed memory parallel computers offer both the potential for a dramatic improvement in cost/performance over conventional supercomputers and the scalability to large numbers of processors that is required if performance beyond that of current supercomputers is to be achieved. As part of an evaluation to explore the potential of such machines for computational fluid mechanics applications, a concurrent algorithm for the solution of the Navier-Stokes equations has been developed and demonstrated on a hypercube parallel computer. The algorithm is based on a domain decomposition of a well-established serial pressure correction algorithm. The algorithm is demonstrated on both a 32-node scalar and eight-node vector Intel iPSC/2 for complicated two-dimensional laminar and turbulent flow problems with different grid sizes and numbers of processors. Speed-ups relative to a single processor of 12.9 with 16 processors and 20.2 with 32 processors are achieved on a scalar iPSC/2, demonstrating the parallel efficiency of the algorithm. Measured performance on a 32-node scalar iPSC/2 exceeds one-sixth that of a Cray X-MP running the original serial algorithm. The performance of the algorithm on an eight-node vector iPSC/2 exceeds that of the larger scalar hypercube and is about one-fifth that of the Cray X-MP. With cost/performance more than 10 times better than the Cray, these results dramatically show the cost effectiveness of vector hypercubes for this class of fluid mechanics algorithm.     

A HIERARCHICAL PARALLEL COMPUTING APPROACH FOR STRUCTURAL STATIC FINITE ELEMENT ANALYSIS

根据分布式存储并行计算机的体系结构特点,提出了一种结构静力有限元分层并行计算方法. 该方法在两级分区两次缩聚策略的基础上不仅实现了大量数据的分布式存储,提高了数据的内存访问速率;而且实现了计算过程的三层并行,有效提高了通信效率;此外,它还进一步降低了界面方程的规模,大幅度减少了界面方程的求解时间. 因此,它能够充分利用分布式存储并行计算机的体系结构特点提升大规模并行计算效率. 最后通过典型数值算例验证了该方法的正确性和有效性.     

结构静力有限元分层并行计算方法

根据分布式存储并行计算机的体系结构特点,提出了一种结构静力有限元分层并行计算方法. 该方法在两级分区两次缩聚策略的基础上不仅实现了大量数据的分布式存储,提高了数据的内存访问速率;而且实现了计算过程的三层并行,有效提高了通信效率;此外,它还进一步降低了界面方程的规模,大幅度减少了界面方程的求解时间. 因此,它能够充分利用分布式存储并行计算机的体系结构特点提升大规模并行计算效率. 最后通过典型数值算例验证了该方法的正确性和有效性.      

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.     

Numerical solutions of high-Re recirculating flows in vorticity-velocity form

A numerical method for computing high-Re laminar steady flows is presented. The incompressible Navier-Stokes equations are expressed in terms of vorticity-velocity variables, discretized in space by finite differences on a staggered grid and advanced in time by a scalar alternating direction implicit (ADI) procedure, which allows a fully vectorized computer code. The accuracy and efficiency of the present formulation are discussed in comparison with the standard ω-ψ and u, v, P forms. Numerical results are presented for two test cases: the driven cavity at Re up to 5000 and the backward-facing step at Re up to 800.     

三维扰动波的非平行边界层稳定性研究

导出了三维扰动波的原始变量形式的抛物化稳定性方程（PSE）,研究了三维空间模态TS波的非平行边界层稳定性问题.采用了法向四阶紧致格式,以提高计算精度.通过给出不会导致奇性的坐标变换、修改外边界条件以及克服平行流初始值的瞬态影响和推进步长的限制,保证了计算的数值稳定.用补全元素带状矩阵法求解块三对角矩阵,大大提高了速度.计算结果清楚地显示了三维扰动波的演化过程和非平行性对边界层稳定性的影响,特别是,观察到非平行性对三维扰动波的影响,有时会使其稳定性出现逆转的现象.还研究了逆压梯度的作用.算例的结果与其他结果符合良好.