Lower-upper Symmetric Gauss-Seidel Research Articles

Reconstructed discontinuous Galerkin methods for compressible flows based on a new hyperbolic Navier-Stokes system

A new first-order hyperbolic system (FOHS) is formulated for the compressible Navier-Stokes equations. The resulting hyperbolic Navier-Stokes system (HNS), termed HNS20G in this paper, introduces the gradients of density, velocity, and temperature as auxiliary variables. Efficient, accurate, compact and robust reconstructed discontinuous Galerkin (rDG) methods are developed for solving this new HNS system. The newly introduced variables are recycled to obtain the gradients of the primary variables. The gradients of these gradient variables are reconstructed based on a newly developed variational formulation in order to obtain a higher order polynomial solution for these primary variables without increasing the number of degrees of freedom. The implicit backward Euler method is used to integrate solution in time for steady flow problems, while the third-order explicit first stage singly diagonally Runge-Kutta (ESDIRK) time marching method is implemented for advancing solutions in time for unsteady flows. The flux Jacobian matrices are obtained with an automatic differentiation toolkit TAPENADE. The approximate system of linear equations is solved with either symmetric Gauss-Seidel (SGS) method or general minimum residual (GMRES) algorithm with a lower-upper symmetric Gauss-Seidel (LU-SGS) preconditioner. A number of test cases are presented to assess accuracy and performance of the newly developed HNS+rDG methods for both steady and unsteady compressible viscous flows. Numerical experiments demonstrate that the developed HNS+rDG methods are able to achieve the designed order of accuracy for both primary variables and the their gradients, and provide an attractive and viable alternative for solving the compressible Navier-Stokes equations.

Fully implicit multiple graphics processing units’ schemes for hypersonic flows with lower upper symmetric Gauss-Seidel preconditioner on unstructured non-orthogonal grids

The paper discusses the simulation of the hypersonic flow problems on unstructured grids using fully implicit scheme and the analysis of the orthogonal corrections on the quality of heat fluxes on the surfaces of the bodies subject to hypersonic flow. The governing equations are the Navier-Stokes equations for viscous calorically perfect gas. Second order finite volume discretization is applied to the problem on arbitrary unstructured grids with implicit temporal treatment. The problem is solved using the Newton-Raphson method. Such methods require the solution of large linear systems with iterative solvers and require a preconditioning operator to converge. In this paper a well known Lower-Upper Symmetric Gauss Seidel (LU-SGS) preconditioner is applied. The new method is based on the tricky reordering for the factored Jacobi matrix that allows one to execute block triangular matrix solvers on GPUs in parallel without the loss of algebraic properties of the original non-factored operator. Good convergence properties are demonstrated for large scale problems of external aerodynamics with implicit Courant number around 1000 - 10000 for flows with Mach number around 13-25. The influence of the non-orthogonal corretions on the fluxes for highly scewed anisotropic grids is demonstrated at Mach numbers around 20. It is shown that the non-orthogonality corrections are of the second infinitesimal order compared to the effects of the shock fix influence on the heat fluxes on the body.

Preconditioned gridless methods for solving three-dimensional Euler equations at low Mach numbers

In this paper, preconditioned gridless methods are developed for solving the three-dimensional (3D) Euler equations at low Mach numbers. The preconditioned system is obtained by multiplying a preconditioning matrix of the type of Weiss and Smith to the time derivative of the 3D Euler equations, which are discretized under the clouds of points distributed in the computational domain by using a gridless technique. The implementations of the preconditioned gridless methods are mainly based on the frame of the traditional gridless method without preconditioning, which may fail to have convergence for flow simulations at low Mach numbers, therefore the modifications corresponding to the affected terms of preconditioning are mainly addressed in the paper. An explicit four-stage Runge–Kutta scheme is first applied for time integration, and the lower-upper symmetric Gauss-Seidel (LU-SGS) algorithm is then introduced to form the implicit counterpart to have the further speed up of the convergence. Both the resulting explicit and implicit preconditioned gridless methods are validated by simulating flows over two academic bodies like sphere or hemispherical headform, and transonic and nearly incompressible flows over one aerodynamic ONERA M6 wing. The gridless clouds of both regular and irregular points are used in the simulations, which demonstrates the ability of the method presented for coping with flows over complicated aerodynamic geometries. Numerical results of surface pressure distributions agree well with available experimental data or simulated solutions in the literature. The numerical results also show that the preconditioned gridless methods presented still functions for compressible transonic flow simulations and additionally, for nearly incompressible flow simulations at low Mach numbers as well. The convergence of the implicit preconditioned gridless method, as expected, is much faster than its explicit counterpart.

Implicit method for the solution of supersonic and hypersonic 3D flow problems with Lower-Upper Symmetric-Gauss-Seidel preconditioner on multiple graphics processing units

The paper describes a numerical method for the solution of stationary gas dynamics 3D spatial equations on unstructured grids that is designed for multiple graphics processing unit (GPU) computational architecture. Discretization of governing equations is done using first and second order TVD schemes. The Newton's method with simple pseudo time-step homotopy is used to solve the problem. Each iteration step involves solution of the linear system originated from the linearization of gas dynamics equations. Krylov subspace iterative methods are used to solve the linear system. The main aim of the paper is to describe a preconditioning Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) method and its adaptation on multiple GPU computational systems. It is shown that deliberately reordered matrices with rearranged solution process of arising lower and upper triangular linear systems allow one to obtain close algebraic properties to the original single threaded LU-SGS. The method is benchmarked against published results. The analysis of computational efficiency and acceleration is presented for different flows with Mach number ranging from 1.2 up to 25.

An implicit free element method for simulation of compressible flow

A novel implicit scheme based on free element method (FECM) is proposed to solve Navier–Stokes equations. The proposed method has the mesh-free features but elements are essential instead of point clouds. The isoparametric transformations, which are usually used in finite element method, are employed to obtain the coefficients in the spatial derivative approximation. The method satisfies the geometric conservation law and first-order consistency. An approximate Riemann solver is employed to compute convective fluxes at the midpoints between collocated points and field points. Meanwhile, a reconstruction approach is used to reach higher order discretization. Viscous terms are approximated with a corrected average progress of gradients. Furthermore, a matrix-free Lower-Upper Symmetric Gauss–Seidel (LU-SGS) scheme is introduced to speed up the convergence. In order to validate the proposed method, three test cases varying from subsonic to supersonic flows are simulated and discussed in detail. The test cases validate that the proposed method is as accurate as second order finite volume solver and is robust with different node sets. What's more, LU-SGS algorithm can significantly improve convergence speed compared with the explicit scheme.

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.

Implicit high-order gas kinetic scheme for turbulence simulation

In recent years, coupled with traditional turbulence models, the second-order gas-kinetic scheme (GKS) has been used in the turbulent flow simulations. At the same time, high-order GKS has been developed, such as the two-stage fourth-order scheme (S2O4) GKS, and used for laminar flow calculations. In this paper, targeting on the high-Reynolds number engineering turbulent flows, an implicit high-order GKS with Lower-Upper Symmetric Gauss-Seidel (LU-SGS) technique is developed under the S2O4 framework. Based on Vreman-type LES model and k−ω SST model, a turbulent relaxation time is obtained and used for the determination of an enlarged particle collision time for the high-Reynolds number turbulent flow simulation. Numerical experiments include incompressible decaying homogeneous isotropic turbulence, incompressible high-Reynolds number flat plate turbulent flow, incompressible turbulence around NACA0012 airfoil, transonic turbulence around RAE2822 airfoil, and transonic high-Reynolds number ARA M100 wing-body simulation. Comparisons among the numerical solutions from current implicit high-order GKS, the explicit high-order GKS, the implicit second-order GKS, and experimental measurements have been conducted. Through these examples, it is concluded that the high-order GKS has high accuracy in space and time, especially for smooth flows, and obtains more accurate turbulent flow fields on coarse grids than the second-order GKS. In addition, significant acceleration on computational efficiency, as well as super robustness in simulating complex flows are confirmed from the current implicit high-order GKS. This study also indicates that turbulence modeling plays a dominant role in capturing physical solution, such as in the transonic three-dimensional complex RANS simulation, which is beyond the numerical discretization error, such as the differences between the second and fourth-order GKS.

Fully Implicit Chebyshev Time-Spectral Method for General Unsteady Flows

An efficient fully implicit Chebyshev time-spectral method (TSM) is developed to solve general unsteady flows with or without periodicity. High accuracy of the Chebyshev time-spectral operator is demonstrated by simulating the one-dimensional shock motion at constant speed. A space-time lower–upper symmetric Gauss–Seidel (LU-SGS) fully implicit scheme with multigrid acceleration is used to efficiently solve the equations resulting from the Chebyshev TSM. The implicit temporal coupling term is properly split so that the LU-SGS sweeps are extended to the time domain. For cases in which the physical time step is small, a modification to the implicit scheme is suggested to enhance numerical stability. Computational results of nonperiodic flows over a pitching airfoil are presented for different physical time steps and various numbers of Chebyshev collocation points. The fully implicit Chebyshev TSM is also tested to predict the natural vortex shedding frequency for a stationary circular cylinder. This efficient approach can be easily applied to the time-accurate computation of general nonperiodic unsteady flows with a given initial condition.

Developing a parallel density-based implicit solver with mesh deformation in OpenFOAM

Numerical simulations like the aeroelastic computation and aerodynamic shape optimization usually involve moving boundaries, and are always carried out with supersonic compressible flows. In this paper, based on the OpenFOAM platform, we present a parallel density-based implicit solver with mesh deformation to address this kind of problems, and fill the gap for the deficiencies of implicit solvers. The core implementation details of the solver based on OpenFOAM are introduced. The Godunov type schemes in ALE (Arbitrary Lagrangian Eulerian) formulation and the dual-time stepping LU-SGS (Lower-Upper Symmetric Gauss-Seidel) algorithm are applied to solve the Navier-Stokes equations with turbulence model. For dynamic meshes, the novel parallel mesh deformation approach based on the Support Vector Machine is implemented. Four typical cases are tested to demonstrate the validity and parallel efficiency of the proposed solver. The result shows that the scalability of the solver reaches its limit at around 128 MPI (Message Passing Interface) tasks, and the cost of MPI communication is the bottleneck for large-scale simulations. Overall, our solver is applicable for various dynamic mesh compressible problems and all types of grids.

Implicit solution of harmonic balance equation system using the LU-SGS method and one-step Jacobi/Gauss-Seidel iteration

ABSTRACTThis paper presents efficient alternative numerical methods for an implicit solution of the harmonic balance equation system for analysing temporal periodic unsteady flows. The proposed method employs approximate factorisation to decouple the common residual term and the time spectral source term of a harmonic balance equation system when it is discretised implicitly. With this approximate factorisation, the complexity of implicit solution of the discrete system is greatly reduced. The common residual term can be dealt with using a lower-upper symmetric-Gauss-Seidel (LU-SGS) method and the time spectral source term is integrated using a Jacobi iteration (JI) or one step Gauss-Seidel (GS) iteration, leading to the LU-SGS/JI method or LU-SGS/GS method. The NASA stage 35 compressor and the 1.5 stage Aachen turbine were used to demonstrate the effectiveness of the proposed methods in stabilising solution and its advantages in comparison with the existing lower-upper symmetric-Gauss-Seidel/block Jacobi (LU-SGS/BJ) method. The LU-SGS/GS method and the LU-SGS/JI method are more robust than the LU-SGS/BJ method in stabilising solution. The LU-SGS/GS method also has faster and tighter convergence and lower memory consumption in comparison with the LU-SGS/BJ method.

Parallel Versions of Implicit LU-SGS Method

Different parallel distributed-memory versions of Lower-Upper Symmetric Gauss–Seidel (LU-SGS) method for solution of discrete equations in finite-volume framework are compared in terms of parallel structure of algorithms. New version based on multilevel recursive decomposition is proposed.

Development of a coupled matrix-free LU-SGS solver for turbulent compressible flows

The paper deals with the development of the lower-upper symmetric Gauss–Seidel (LU-SGS) matrix-free finite volume solver for the simulation of compressible flows within the framework of the OpenFOAM package. The solver evaluates the convective fluxes using approximate Riemann solvers with limited piece-wise linear reconstructions whereas the viscous fluxes are approximated using a central scheme. The time evolution is realised through the backward differentiation formula of first or second order. The system of non-linear equations is then solved with the help of the matrix-free LU-SGS method. The developed solver is used to solve several flow problems and compared to a pressure-based segregated solver. Our numerical experiments indicate that the LU-SGS solver is more efficient for flows with higher Mach numbers and provides better resolution of shock waves. Moreover the LU-SGS solver benefits from the low memory footprint and does not use any problem specific setup.

An implicit gas-kinetic scheme for turbulent flow on unstructured hybrid mesh

In this study, an implicit scheme for the gas-kinetic scheme (GKS) on the unstructured hybrid mesh is proposed. The Spalart–Allmaras (SA) one equation turbulence model is incorporated into the implicit gas-kinetic scheme (IGKS) to predict the effects of turbulence. The implicit macroscopic governing equations are constructed and solved by the matrix-free lower-upper symmetric-Gauss–Seidel (LU-SGS) method. To reduce the number of cells and computational cost, the hybrid mesh is applied. A modified non-manifold hybrid mesh data(NHMD) is used for both unstructured hybrid mesh and uniform grid. Numerical investigations are performed on different 2D laminar and turbulent flows. The convergence property and the computational efficiency of the present IGKS method are investigated. Much better performance is obtained compared with the standard explicit gas-kinetic scheme. Also, our numerical results are found to be in good agreement with experiment data and other numerical solutions, demonstrating the good applicability and high efficiency of the present IGKS for the simulations of laminar and turbulent flows.

CFD analysis of a twin scroll radial turbine

The contribution deals with the application of coupled implicit solver for compressible flows to CFD analysis of a twin scroll radial turbine. The solver is based on the finite volume method, convective terms are approximated using AUSM+up scheme, viscous terms use central approximation and the time evolution is achieved with lower-upper symmetric Gauss-Seidel (LU-SGS) method. The solver allows steady simulation with the so called frozen rotor approach as well as the fully unsteady solution. Both approaches are at first validated for the case of ERCOFTAC pump [1]. Then the CFD analysis of the flow through a twin scroll radial turbine and the predictions of the efficiency and turbine power is performed and the results are compared to experimental data obtained in the framework of Josef Božek – Competence Centre for Automotive Industry.

Application of nonlinear Krylov acceleration to a reconstructed discontinuous Galerkin method for compressible flows

A variant of Anderson Mixing, namely the Nonlinear Krylov Acceleration (NKA), is presented and implemented in a reconstructed Discontinuous Galerkin (rDG) method to solve the compressible Euler and Navier–Stokes equations on hybrid grids. A nonlinear system of equations as a result of a fully implicit temporal discretization at each time step is solved using the NKA method with a lower-upper symmetric Gauss–Seidel (LU-SGS) preconditioner. The developed NKA method is used to compute a variety of flow problems and compared with a well-known Newton-GMRES method to demonstrate the performance of the NKA method. Our numerical experiments indicate that the NKA method outperforms its Newton-GMRES counterpart for transient flow problems, and is comparable to Newton-GMRES for steady cases, and thus provides an attractive alternative to solve the system of nonlinear equations arising from the rDG approximation.

Improvement of computational efficiency of circular function-based gas kinetic scheme by using Jacobian-free Newton–Krylov method

The gas-kinetic BGK scheme is a promising method for simulation of inviscid and viscous flows. Different from conventional Navier–Stokes (N-S) solvers, it evaluates the inviscid and viscous fluxes simultaneously by reconstructing the local solution of BGK Boltzmann equation at the cell interface. Due to its inherently superior dissipation property, it usually gives accurate and robust numerical results. However, a notable drawback of the BGK-type scheme is the low computational efficiency. Recently, aimed at reducing the computational effort, a circular function-based BGK (CBGK) scheme was developed. Nevertheless, it is still time consuming because the original scheme used an explicit way in the time integration as in most existing BGK schemes and the convergence speed can be slow. To improve the computational efficiency, an implicit CBGK scheme is developed in this paper by incorporating the Jacobian-free Newton–Krylov (JFNK) method into the scheme. Particularly, the generalized minimal residual (GMRES) approach is employed to iteratively solve the large linear equation system. With the help of the Jacobian-free approach, a faster convergence speed can be achieved without explicitly computing and storing the flux Jacobian, which is usually a large and sparse matrix. In order to reduce the number of GMRES iterations, the preconditioning is also adopted and the Lower-Upper symmetric Gauss-Seidel (LUSGS) scheme is employed as a preconditioner. For validation of the present CBGK-JFNK method, several two-dimensional inviscid and viscous test cases are investigated. The numerical results show that the needed computational time is significantly reduced as compared with the original explicit CBGK scheme and the CBGK-LUSGS method.

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.

A multithreaded OpenMP implementation of the LU-SGS method using the multilevel decomposition of the unstructured computational mesh

A new parallel version of the method LU-SGS (Lower-upper symmetric Gauss-Seidel) based on a multilevel decomposition of the unstructured computational mesh is proposed. The advantages of the proposed approach are demonstrated by computing the supersonic flow around the RAM-C geometry. The method is well scalable when a large number of threads are used on the processor Intel Xeon Phi.

Numerical Simulation of the Supersonic Disk-Gap-Band Parachute by Using Implicit Coupling Method

Abstract The implicit coupling method is applied to model the 0.8 m disk-band-gap parachute at Mach 2.0. The fluid and structure governing equations are solved by the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) algorithm and Newmark scheme, respectively. By exchanging the numerical results of the coupling surface with Gauss-Seidel algorithm, high accuracy solutions at every physical time step are obtained. The numerical results of the canopy drag coefficient and projected area fit well with the wind tunnel test results. The simulation reproduces the shock oscillation and breathing phenomenon of the canopy that are usually observed in these systems at Mach 2.0. Furthermore, it is found that the unstable saddle point is the main reason for the shock oscillation of the canopy. And the unsynchronized phases of the canopy area and shock oscillation curves lead to the drag of the canopy oscillate in irregular state.

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.