Lower-upper Symmetric Gauss-Seidel Method Research Articles

Acceleration of the data-parallel lower-upper relaxation time-integration method on GPU for an unstructured CFD solver

The acceleration of CFD solvers on GPUs allows for large speedups, which enable more complex and detailed simulations. However, not all numerical methods can be efficiently executed in a massively parallel fashion. The Runge–Kutta (RK) and Lower-Upper Symmetric Gauss–Seidel (LU-SGS) time-integration methods are widely used on GPUs and multicore CPUs, respectively; however, the RK method suffers from low convergence speed, while the LU-SGS method is not adapted to a many-core environment. In this paper, we propose to accelerate the matrix-free version of the Data-Parallel Lower-Upper Relaxation (DP-LUR) time-integration method on the unstructured solver FaSTAR, developed by JAXA. The DP-LUR method outperformed the RK and LU-SGS methods on an NVIDIA Tesla V100 GPU when tested on the ONERA M6 and NASA CRM geometries, and reached up to 3.83% of the performance peak of one device.

High-order gas-kinetic scheme on three-dimensional unstructured meshes for compressible flows

In this paper, a high-order gas-kinetic scheme is developed on three-dimensional unstructured meshes for compressible Euler and Navier–Stokes equations. To achieve the high-order spatial accuracy, the three-dimensional weighted essentially non-oscillatory (WENO) reconstruction is extended to the unstructured tetrahedral and hexahedral meshes. A simple strategy is adopted for the selection of candidate stencils, and the topologically independent linear weights are used for the spatial reconstruction. The efficiency and robustness of the classical WENO reconstruction are improved. In addition to the two-stage fourth-order temporal discretization and lower–upper symmetric Gauss–Seidel method, the explicit and implicit high-order gas-kinetic schemes are developed for unsteady and steady problems. Accuracy tests on hexahedral and tetrahedral grids validate the third-order of accuracy, and various three-dimensional incompressible and compressible numerical experiments are also presented. The results validate the accuracy and robustness of the proposed scheme for both inviscid and viscous flows. In the future, the current scheme will be extended to the hybrid unstructured meshes and Reynolds-averaged Navier–Stokes simulation with high Reynolds numbers.

An implicit simplified sphere function-based gas kinetic scheme for simulation of 3D incompressible isothermal flows

In this work, an implicit simplified sphere function-based gas kinetic scheme (SGKS) is presented for simulation of 3D incompressible isothermal flows. At first, the numerical fluxes of governing equations are reconstructed by the local solution of Boltzmann equation with sphere function distribution. Due to incompressible limit, the sphere at cell interface can be approximately considered to be symmetric as shown in the work. Besides that, the energy equation is usually not needed for simulation of incompressible isothermal flows. With all these simplifications, the formulations of the simplified SGKS can be expressed concisely and explicitly. Secondly, the commonly-used implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) method is adopted to further improve the computational efficiency and numerical stability of present scheme. In LU-SGS method, only a forward and a backward sweep are needed for marching the conservative variables in time. As a result, the simplified SGKS with the LU-SGS method can be implemented easily. Numerical experiments, including the 3D lid-driven cavity flow and flow over a backward-facing step, showed that the incompressible isothermal flows can be well simulated by the developed scheme and its computational efficiency is significantly higher than that of the original SGKS and the lattice Boltzmann flux solver (LBFS). In addition, it was found that the present scheme with the LU-SGS method is more efficient than that with the explicit Euler method, and the speedup ratio is about 2 to 5.

Numerical investigation of heat flux distribution in a deep gap based on chemical equilibrium

PurposeThe purpose of this paper is to perform the simulation to explore the gap flow field under a hypersonic air flow. Thermal protection systems of hypersonic vehicles generally consist of thermal insulation tiles, and gaps between these tiles probably cause a severe local aerodynamic thermal effect.Design/methodology/approachThe discretizations of convection flux term and temporal term in the governing equation with chemical equilibrium, respectively, take AUSM+-up flux-vector splitting scheme and the implicit lower-upper symmetric Gauss–Seidel method. Based on these, the flow field in a deep gap is simulated by means of the computer codes that the authors have written.FindingsThe numerical results show that the heat flux distribution in a gap has a good agreement with experimental results. Importantly, the distribution of heat flux is “U” shaped and the maximum of the heat flux occurs at the windward corner of a gap.Originality/valueTo explore the gap flow field under a hypersonic air flow, which is a chemically reacting, all speed and viscous flow, a novel model with an equivalent ratio of specific heats is presented. The investigation in this paper has a guide for the design of the thermal protection system in hypersonic vehicles.

Investigation on Propeller Slipstream by Using an Unstructured Rans Solver Based on Overlapping Grids

AbstractBased on unstructured hybrid grid and dynamic overlapping grid technique, numerical simulations of Unsteady Reynolds Averaged Navier-Stokes equations were performed and investigation on isolated propeller aerodynamic characteristics and effects of propeller slipstream on turboprops were undertaken. The computational grid consisted of rotational subzone of propeller and stationary major-zone of aircraft, and walls criterion was used in the automatic hole-cutting procedure. Distance weight interpolation and tri-linear interpolation were developed to transfer information between the rotational and stationary subzones. The boundaries of overlapping grids were optimized for fixed axis rotation. The governing equations were solved by dual-time method and Lower Upper-Symmetric Gauss-Seidel method. The method and grid technique were verified by isolated propeller configuration and the computational results were in well agreement with the experimental data. The grid independence was studied to establish the numerical results. Finally, the flow around a turboprop case was simulated and the influence of propeller slipstream was presented by analyzing the surface pressure contours, profile pressure distribution, vorticity contours and profile streamline. It's indicated that the slipstream accelerates and rotates the free stream flow, changing the local angle of attack, enhancing the downwash effects, affecting the pressure distribution on wing and horizontal tail, as well as increasing the drag coefficient, pitching moment coefficient and the slope of lift coefficient.

Solution Stabilization and Convergence Acceleration for the Harmonic Balance Equation System

This paper presents an efficient approach for stabilizing solution and accelerating convergence of a harmonic balance equation system for an efficient analysis of turbomachinery unsteady flows due to flutter and blade row interaction. The proposed approach combines the Runge–Kutta method with the lower upper symmetric Gauss Seidel (LU-SGS) method and the block Jacobi method. The LU-SGS method, different from its original application as an implicit time marching scheme, is used as an implicit residual smoother with under-relaxation, allowing big Courant–Friedrichs–Lewy (CFL) numbers (in the order of hundreds), leading to significant convergence speedup. The block Jacobi method is introduced to implicitly integrate the time spectral source term of a harmonic balance equation system, in order to reduce the complexity of the direct implicit time integration by the LU-SGS method. The implicit treatment of the time spectral source term thus greatly augments the stability region of a harmonic balance equation system in the case of grid-reduced frequency well above ten. Validation of the harmonic balance flow solver was carried out using linear cascade test data. Flutter analysis of a transonic rotor and blade row interaction analyses for a transonic compressor stage were presented to demonstrate the stabilization and acceleration effect by the combination of the LU-SGS and the block Jacobi methods. The influence of the number of Jacobi iterations on solution stabilization is also investigated, showing that two Jacobi iterations are sufficient for stability purpose, which is much more efficient than existing methods of its kind in the open literature.

On the implicit density based OpenFOAM solver for turbulent compressible flows

The contribution deals with the development of coupled implicit density based solver for compressible flows in the framework of open source package OpenFOAM. However the standard distribution of OpenFOAM contains several ready-made segregated solvers for compressible flows, the performance of those solvers is rather week in the case of transonic flows. Therefore we extend the work of Shen [15] and we develop an implicit semi-coupled solver. The main flow field variables are updated using lower-upper symmetric Gauss-Seidel method (LU-SGS) whereas the turbulence model variables are updated using implicit Euler method.

An efficient wavefront parallel algorithm for structured three dimensional LU-SGS

Parallel computing is a useful technology for scientific and engineering algorithms/applications. LU-SGS (lower-upper Symmetric-Gauss–Seidel method) is an efficient and robust scheme for CFD (Computational fluid dynamics) and has strong data dependence in its computation. In this paper, we present an efficient wavefront parallel algorithm for 3D (three dimensional) LU-SGS with structured meshes. The corresponding data structure and memory access method with better data locality and communication optimization is designed. The performances of the presented parallel algorithm are reported with different problem sizes. Some discussion and performance issues are also reported. The results show that the overall performance speedup of one Intel E5540 CPU (4 CPU cores) ranges from 2.23 to 2.95 compared with one E5540 core. The parallel efficiency of 1024, 128 processes are up to 35.68%, 72.69% compared with 32 processes on a distributed memory cluster system. The CFD simulation of M6 wing model shows the effect of the presented parallel algorithm.

Investigation and Comparison of Implicit Smoothers Applied in Agglomeration Multigrid

A straightforward implicit smoothing method implemented in several codes solving the Reynolds-averaged Navier–Stokes equations is the lower–upper symmetric Gauss–Seidel method. It was proposed several years ago and is attractive to implement because of its low memory requirements and low operation count. Since, for many examples, often a severe restriction of the Courant–Friedrichs–Lewy number and loss of robustness are observed, it is the goal of this paper to revisit the lower–upper symmetric Gauss–Seidel implementation and to discuss several alternative implicit smoothing strategies used within an agglomeration multigrid for unstructured meshes. The starting point is a full implicit multistage Runge–Kutta method. Based on this method, several additional features and simplifications are developed and suggested, such that the implicit method is applicable to high-Reynolds-number viscous flows; that is, the required matrices fit into the fast memory of the cluster hardware and the arising linear systems can be approximately solved efficiently. To this end, the focus is on simplifications of the Jacobian, as well as efficient iterative approximate solution methods. To significantly improve the approximate linear solution methods, grid anisotropy is taken care of for approximately solving both the linear systems and the agglomeration strategy. The procedure creating coarse-grid meshes is extended by strategies identifying structured parts of the mesh. This seems to improve the quality of coarse-grid meshes in the way that an overall better reliability of the multigrid can be observed. Furthermore, grid information is exploited within the iterative solution methods for the linear systems. Numerical examples demonstrate the gain with respect to reliability and efficiency.

Overrelaxation Applied to Lower-Upper Symmetric Gauss-Seidel Method for Hypersonic Flow

An implicit time-marching method for acceleration of the convergence of solution for the compressible Navier-Stokes equations is presented. This is based on the lower-upper symmetric-Gauss-Seidel (LU-SGS) method proposed by Yoon and Jameson. A very simple overrelaxation is additionally executed in the LU-SGS method coupled with diagonal flux splitting. Then, an overrelaxation parameter w, a parameter similar to that in the well-known successive-overrelaxation method is introduced. The calculated results for a hypersonic viscous flows around a cylinder and through a compression corner indicate that the convergence is accelerated by a proper value of ω(1 < ω < 2). This method is also applied to a hypersonic thermochemical nonequilibrium flow around a sphere

Data-parallel lower-upper relaxation method for the Navier-Stokes equations

The lower-upper symmetric Gauss-Seidel method is modified for the simulation of viscous flows on massively parallel computers. The resulting diagonal data-parallel lower-upper relaxation (DP-LUR) method is shown to have good convergence properties on many problems. However, the convergence rate decreases on the high cell aspect ratio grids required to simulate high Reynolds number flows. Therefore, the diagonal approximation is relaxed, and a full matrix version of the DP-LUR method is derived. The full matrix method retains the data-parallel properties of the original and reduces the sensitivity of the convergence rate to the aspect ratio of the computational grid. Both methods are implemented on the Thinking Machines CM-5, and a large fraction of the peak theoretical performance of the machine is obtained. The low memory use and high parallel efficiency of the methods make them attractive for large-scale simulation of viscous flows.

Comparison of nonequilibrium solution algorithms applied to chemically stiff hypersonic flows

Three solution algorithms, explicit under-relaxation, point implicit, and lower-upper symmetric Gauss-Seidel, are used to compute nonequilibrium flow around the Apollo 4 return capsule at the 62-km altitude point in its descent trajectory. By varying the Mach number, the efficiency and robustness of the solution algorithms were tested for different levels of chemical stiffness. The performance of the solution algorithms degraded as the Mach number and stiffness of the flow increased. At Mach 15 and 30, the lower-upper symmetric Gauss-Seidel method produces an eight order of magnitude drop in the energy residual in one-third to one-half the Cray C-90 computer time as compared to the point implicit and explicit under-relaxation methods. The explicit under-relaxation algorithm experienced convergence difficulties at Mach 30 and above. At Mach 40 the performance of the lower-upper symmetric Gauss-Seidel algorithm deteriorates to the point that it is out performed by the point implicit method. The effects of the viscous terms are investigated. Grid dependency questions are explored.