Lower-upper Symmetric Gauss-Seidel Scheme Research Articles

Implicit block data-parallel relaxation scheme of Navier–Stokes equations using graphics processing units

The granularity of computational fluid dynamics (CFD) generally refers to the point granularity parallelization as a unit of the grid when graphics processing units (GPUs) are utilized as the computing carrier. In commonly deployed implicit time advancement schemes, the parallel dimensionality must be reduced, resulting in the time advancement procedure becoming the only highly time-consuming step in the whole CFD computing procedures. In this paper, a block data-parallel lower-upper relaxation (BDPLUR) scheme based on Jacobi iteration and Roe's flux scheme is proposed and then implemented on a GPU. Numerical experiments are carried out and show that the convergence speed of the BDPLUR scheme, especially when implemented on a GPU, is approximately ten times higher than that of the original data-parallel lower-upper relaxation scheme and more than 100 times higher than that of the lower-upper symmetric Gauss–Seidel scheme. Moreover, the influence of different Courant–Friedrichs–Lewy numbers on the convergence time is discussed, and different viscous matrices are compared. Standard cases are adopted to verify the effectiveness of the BDPLUR scheme.

Efficient same-dimensional implicit time advancement parallel scheme and optimization methods for the iteration parameters using a graphics-processing unit

Many studies have focused on the acceleration of computational fluid dynamics (CFD) using multicore hardware such as graphics-processing units (GPUs) in the field of parallel computing. In GPU acceleration, CFD parallel granularity generally refers to the point granularity parallelization as a unit of the grid. An implicit time advancement scheme is more efficient and faster than an explicit time advancement scheme for CFD. However, for commonly used implicit schemes, such as the lower-upper symmetric Gauss–Seidel (LUSGS) scheme, the parallel dimensionality is reduced, resulting in a highly time-consuming procedure. In this paper, the data-parallel upper-lower relaxation (DPLUR) scheme based on Jacobi iteration is used and then implemented on a GPU. Numerical experiments are carried out and show that the computing speed of point granularity parallelization using the DPLUR scheme, especially implemented on a GPU, is much higher than that of dimensionality reduction using the LUSGS scheme. Moreover, the influence of different Jacobi inner iteration steps (JIIS) on the convergence time is discussed, and two JIIS optimization algorithms are proposed according to the characteristics of convergence. On the basis of the memory access form, a DPLUR red–black (DPRB) scheme with more stable and faster convergence than the conventional DPLUR scheme is developed. Finally, some standard cases are adopted to verify the effectiveness of DPRB schemes with the JIIS optimization algorithm.

Gas-Kinetic Harmonic Balance Method for Unsteady Supersonic/Hypersonic Flows with Oscillating Motions

In this paper, a novel gas-kinetic harmonic balance method (HBM) is developed for unsteady flows with oscillating motions. Particularly, the emphasis here is on supersonic and hypersonic flows. Because of the presence of an additional source term in the harmonic balance equations, usually very small time steps are needed for explicit algorithms, and some factorization-based implicit algorithms may fail to converge in the case of higher frequencies. To achieve better convergence performance and robustness, a Jacobian-free Newton–Krylov method is adopted. By constructing a simplified preconditioning matrix, the lower–upper symmetric Gauss–Seidel scheme is employed as a preconditioner, so that the computational cost and memory requirement can be reduced. Considering that conventional spatial discretization schemes may encounter numerical difficulties and low robustness in high-Mach-number flows, a more general gas-kinetic scheme (GKS) is extended for the harmonic balance equations. As the HBM solves a mathematically steady problem, the present GKS focuses on evaluation of the time-independent fluxes at each cell interface and each subtime level. The gas-kinetic HBM is applied to a supersonic oscillating flat plate cascade and an oscillating panel in hypersonic flow. By comparison with other methods, the advantages of the developed method are demonstrated.

A conservation-moment-based implicit finite volume lattice Boltzmann method for steady nearly incompressible flows

This paper presents an efficient, low memory cost, implicit finite volume lattice Boltzmann method (FVLBM) based on conservation moments acceleration for steady nearly incompressible flows. In the proposed scheme, not as the conventional implicit schemes, both the micro lattice Boltzmann equations (LBE) and the associated conservation moment equations are solved by the matrix-free, lower-upper symmetric Gauss-Seidel scheme (LUSGS) and the conservation moment equations are used to predict equilibrium distribution functions at the new time, which eliminates the storage of the Jacobian matrix of the collision term in the implicit LBE system and provides a driving force for the fast convergence of the LBE. Moreover, by utilizing the projection matrix and the collision invariant, we can construct the fluxes of the moment equations efficiently from the fluxes of the LBE and avoid the time-consuming reconstruction procedure for obtaining the fluxes of the moment equations. To demonstrate the accuracy and high efficiency of the proposed scheme, comparison studies of simulation results of several two-dimensional testing cases by the present scheme and an explicit FVLBM are conducted and numerical results show that the proposed implicit scheme can be as accurate as its explicit counterpart with 1 or 2 orders times speedup.

Dynamic stall analysis using harmonic balance and correlation-basedγ- Reθt¯ transition models for wind turbine applications

In this paper, we use the harmonic balance method to study an oscillating S809 airfoil in dynamic stall. The periodic behavior of this problem makes it well suited for the harmonic balance method, which is able to model unsteady aerodynamics at greatly reduced computational costs when compared with time-accurate unsteady-flow solvers. A finite-volume technique based on the lower–upper symmetric Gauss–Seidel scheme with Roe fluxes is used to solve the Reynolds-averaged Navier–Stokes equations. The turbulent viscosity is computed with the one-equation Spalart–Allmaras turbulence model. In addition, the laminar–turbulent transition is modeled using a correlation-based approach originally developed by Langtry and Menter. Comparisons with experimental data for steady flows with the S809 airfoil highlight the necessity of the transition model to accurately predict the onset of static stall. For unsteady cases, the transition model provides improved agreement with experimental data, predicting dynamic stall when the fully turbulent model cannot. Copyright © 2014 John Wiley & Sons, Ltd.

Application of weighted non-oscillatory and non-free-parameter dissipation difference scheme in calculating the flow of vibrating flat cascade

A dual-time method is introduced to calculate the unsteady flow in a certain vibrating fiat cascade.An implicit lower-upper symmetric-gauss-seidel scheme(LU-SGS) is applied for time stepping in pseudo time domains,and the convection items are discretized with the spatial three-order weighted non-oscillatory and non-free-parameter dissipation difference (WNND) scheme.The turbulence model adopts q-ωlow-Reynolds-number model.The frequency spectrums of lift coefficients and the unsteady pressure-difference coefficients at different spanwise heights as well as the entropy contours at blade tips on different vibrating instants,are obtained.By the analysis of frequency spectrums of lift coefficients at three spanwise heights,it is considered that there exist obvious non-linear perturbations in the flow induced by the vibrating,and the perturbation frequencies are higher than the basic frequency.The entropy contours at blade tips at different times display an intensively unsteady attribute of the flow under large amplitudes.

Fast, Block Lower-Upper Symmetric Gauss-Seidel Scheme for Arbitrary Grids

Ablock implicit lower-uppersymmetric Gauss ‐Seidel(LU-SGS)approximatefactorization schemeisdeveloped and implemented for unstructured grids of arbitrary topology including viscous adaptive Cartesian grids. CPU times and memory requirements for the block LU-SGS (BLU-SGS), the original LU-SGS, and a fully implicit scheme (FIS) with a preconditioned conjugate gradient squared solver for several representative test examples are compared. Computational results showed that the block LU-SGS scheme requires about 20 ‐30% more memory than the original LU-SGS but converges many times faster. The BLU-SGS scheme has a convergence rate competitive to and in many cases faster than the FIS while requiring much less memory.

Fast, block lower-upper symmetric Gauss-Seidel scheme for arbitrary grids

Fast, block lower-upper symmetric Gauss-Seidel scheme for arbitrary grids

Numerical Simulation of Hypersonic Shock-Induced Combustion Ramjets

Hypersonic air-breathing propulsion utilizing shock-induced combustion ramjets is investigated. Twodimensional geometries are simulated with planar and axisymmetric configurations, as well as external and mixed-compression configurations. The lower-upper Symmetric Gauss-Seidel scheme combined with a symmetric shock-capturing total variation diminishing scheme are used to solve the Euler equations, with nonequilibrium chemical reactions. The finite rate chemistry model includes 13 species (H2, O2, H, O, OH, H2O, HO2, H2O2, N, NO, HNO, N2, and NO2) and 33 reactions. The numerical method has been verified by comparison with H2/air induction delay times, analytical solutions to wedge problems, and exothermic blunt body flows. Results obtained with an inviscid, chemically nonequilibrium numerical approach and with realistic geometries demonstrate that shock-induced combustion can be used as a viable means of hypersonic propulsion. As part of the combustor design, it has also been numerically demonstrated that a minimum-entrop y, or Chapman-Jouguet condition exists for oblique-detonation waves generated by wedges in nonequilibrium chemically reacting H2/air flowfields.

Lower-upper symmetric Gauss-Seidel scheme exhibiting asymmetric vortices over slender bodies

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