Stabilizing and Accelerating Solution of Harmonic Balance Equation System Using the LU-SGS and Block Jacobi Methods

2016 ◽  
Author(s):  
Xiuquan Huang ◽  
Ding Xi Wang
Keyword(s):  
Balance Equation ◽  
Harmonic Balance ◽  
Equation System ◽  
Jacobi Method ◽  
Implicit Time ◽  
Solution Stability ◽  
Reduced Frequency ◽  
Solution Convergence ◽  
Time Marching ◽  
Jacobi Iterations

The paper presents the combination of the Lower Upper Symmetric Gauss Seidel (LU-SGS) method and the block Jacobi method for stabilizing and accelerating the solution of a harmonic balance equation system. First the baseline LU-SGS procedure for a harmonic balance equation system with explicit discretization of the time spectral source term is derived. The LU-SGS method, different from its original application as an implicit time marching scheme, is used as an implicit residual smoother, allowing big CFL numbers (in order of 1000s), within the Runge-Kutta explicit time marching loops. Then the block Jacobi method is introduced to augment solution stability for the solution of a harmonic balance equation system under situation where grid reduced frequency is in the order of 10. Results are presented to show the effectiveness of the combination of LU-SGS and the block Jacobi method on the solution stabilization and acceleration. The influence of the number of Jacobi iterations on solution convergence is also investigated.

Solution Stabilization and Convergence Acceleration for the Harmonic Balance Equation System

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Ding Xi Wang ◽  
Xiuquan Huang
Keyword(s):  
Balance Equation ◽  
Harmonic Balance ◽  
Source Term ◽  
Equation System ◽  
Jacobi Method ◽  
Implicit Time ◽  
Transonic Compressor ◽  
Blade Row ◽  
Jacobi Iterations ◽  
Blade Row Interaction

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.

A Time-Space Multigrid Method for Efficient Solution of the Harmonic Balance Equation System

2021 ◽  
Author(s):  
Hangkong Wu ◽  
Dingxi Wang ◽  
Xiuquan Huang ◽  
Shenren Xu
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Harmonic Balance ◽  
Multigrid Method ◽  
Equation System ◽  
Solution Stability ◽  
Inverse Discrete Fourier Transform ◽  
Time Space ◽  
Solution Accuracy ◽  
Coarse Grids

Abstract In this paper, an efficient time-space multigrid (TS-MG) method for solving a harmonic balance (HB) equation system is proposed. The principle of the time-space multigrid method is to coarsen grids in both space and time dimensions simultaneously when coarse grids are formed. The inclusion of time in the time-space multigrid is to address the instability issue or diminished convergence speedup of the spatial multigrid (S-MG) due to larger grid reduced frequencies on coarse grids. With the proposed method, the unsteady governing equation will be solved on all grid levels. Comparing to the finest grid, fewer harmonics and thus fewer equations will be solved consequently on coarse grids. Discrete Fourier transform (DFT) and inverse discrete Fourier transform (IDFT) are used to achieve solution prolongation and restriction between different time grid levels. Results from the proposed method are compared with those obtained from the traditional spatial multigrid and time domain methods. It is found that the TS-MG method can increase solution stability, reduce analysis time cost required for convergence, save memory usage and has no adverse effect on solution accuracy.

scholarly journals ON THE SOLUTION OF ENERGY BALANCE EQUATION SYSTEM TO PREDICT TEMPERATURE DISTRIBUTION IN THE BUILDING ELEMENTS WITH VENTILATED AIR GAP

2014 ◽  
Vol 21 (1) ◽  
pp. 21-30 ◽  
Author(s):  
Edmundas Monstvilas ◽  
Karolis Banionis ◽  
Jurga Poderytė ◽  
Raimondas Bliūdžius ◽  
Arūnas Burlingis
Keyword(s):  
Heat Exchange ◽  
Balance Equation ◽  
Heat Balance ◽  
Internal Surface ◽  
Equation System ◽  
Energy Balance Equation ◽  
Air Gap ◽  
Exchange Processes ◽  
Calculation Results ◽  
Thermal Indicators

The article presents the solution of heat balance equation system, describing heat exchange processes in ventilated envelopes, which was applied to derive formulas for the calculation of temperatures in the ventilated layers of the envelopes. The accurateness of the formulas was assessed by experimental research and analysis of the calculation results. During the process of heat exchange balance equation solution, the equations were simplified by introducing the following restriction into the derived formulas: they may only be applied for the ventilated envelopes with steel or similar coatings as their external layers, i.e. coatings having small heat capacity and minor difference between the external and internal surface temperatures. The derived formulas enable the calculation of the temperatures of the ventilated envelopes in the distance which does not exceed a half of the ventilated air gap length measuring from the air entrance into the gap. However, this restriction does not impede the estimation of the average thermal indicators of the ventilated envelopes.

Numerical Modeling of Radiative and Reactive Flow Field Around a Blunt Body at Hypersonic Regimes

2010 ◽  
Author(s):  
Morteza Rahmanpour ◽  
Reza Ebrahimi ◽  
Mehrzad Shams
Keyword(s):  
Flow Field ◽  
Differential Equations ◽  
Heat Fluxes ◽  
Transitional Flow ◽  
Navier Stokes ◽  
Implicit Time ◽  
Discrete Ordinates ◽  
Two Dimensional ◽  
Fully Implicit ◽  
Time Marching

A numerical method for calculation of strong radiation for two-dimensional reactive air flow field is developed. The governing equations are taken to be two dimensional, compressible Reynolds-average Navier-Stokes and species transport equations. Also, radiation heat flux in energy equation is evaluated using a model of discrete ordinate method. The model used S4 approximation to reduce the governing system of integro-differential equations to coupled set of partial differential equations. A multiband model is used to construct absorption coefficients. Tangent slab approximation is assumed to determine the characteristic parameters needed in the Discrete Ordinates Method. The turbulent diffusion and heat fluxes are modeled by Baldwin and Lomax method. The flow solution is obtained with a fully implicit time marching method. A thermochemical nonequilibrium formulation appropriate to hypersonic transitional flow of air is presented. The method is compared with existing experimental results and good agreement is observed.

scholarly journals Navier-Stokes Analysis of Unsteady Transonic Flows Through Gas Turbine Cascades With and Without Coolant Ejection

1997 ◽  
Author(s):  
T. Tanuma ◽  
N. Shibukawa ◽  
S. Yamamoto
Keyword(s):  
Gas Turbine ◽  
Stokes Equations ◽  
Viscous Flows ◽  
Navier Stokes ◽  
Implicit Time ◽  
Trailing Edge ◽  
Navier Stokes Equations ◽  
Time Marching ◽  
Turbine Cascades ◽  
Annular Cascade

An implicit time-marching higher-order accurate finite-difference method for solving the two-dimensional compressible Navier-Stokes equations was applied to the numerical analyses of steady and unsteady, subsonic and transonic viscous flows through gas turbine cascades with trailing edge coolant ejection. Annular cascade tests were carried out to verify the accuracy of the present analysis. The unsteady aerodynamic mechanisms associated with the interaction between the trailing edge vortices and shock waves and the effect of coolant ejection were evaluated with the present analysis.

scholarly journals A Parallel Adaptive Newton-Krylov-Schwarz Method for 3D Compressible Inviscid Flow Simulations

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Marzio Sala ◽  
Pénélope Leyland ◽  
Angelo Casagrande
Keyword(s):  
Compressible Flows ◽  
Time Derivative ◽  
Inviscid Flow ◽  
Implicit Time ◽  
Residual Distribution ◽  
Preconditioning Techniques ◽  
Flow Simulations ◽  
Efficient Data ◽  
Time Marching ◽  
2D And 3D

A parallel adaptive pseudo transient Newton-Krylov-Schwarz (αΨNKS) method for the solution of compressible flows is presented. Multidimensional upwind residual distribution schemes are used for space discretisation, while an implicit time-marching scheme is employed for the discretisation of the (pseudo)time derivative. The linear system arising from the Newton method applied to the resulting nonlinear system is solved by the means of Krylov iterations with Schwarz-type preconditioners. A scalable and efficient data structure for theαΨNKS procedure is presented. The main computational kernels are considered, and an extensive analysis is reported to compare the Krylov accelerators, the preconditioning techniques. Results, obtained on a distributed memory computer, are presented for 2D and 3D problems of aeronautical interest on unstructured grids.

High Dimensional Harmonic Balance Analysis for Dynamic Piecewise Aeroelastic Systems

2008 ◽  
Author(s):  
Liping Liu ◽  
Earl H. Dowell
Keyword(s):  
Frequency Domain ◽  
Limit Cycle ◽  
Time Domain ◽  
Harmonic Balance ◽  
Solution Method ◽  
Computational Time ◽  
High Dimensional ◽  
Aeroelastic System ◽  
Time Marching ◽  
The Time Domain

This paper describes the extension and application of a novel solution method for the periodic nonlinear oscillations of an aeroelastic system. This solution method is a very attractive alternative to time marching algorithms in that it is much faster and may track unstable as well as stable limit cycles. The method is employed to analyze the nonlinear aeroelastic response of a two dimensional airfoil including a control surface with freeplay placed in an incompressible flow. The mathematical model for this piecewise aeroelastic system is initially formulated as a set of first order ordinary differential equations. A frequency domain solution for the limit cycle oscillations is derived by a novel high dimensional harmonic balance (HDHB) method. By an inverse Fourier transformation, the system in the frequency domain is then converted into the time domain. Finally, the airfoil motions are obtained by solving the system in the time domain for only one period of limit cycle oscillation. This process can be easily implemented into computer programs without going through the complex algebraic manipulations for the nonlinearities typical of a more conventional harmonic balance solution method. The solutions found using this new HDHB method have been shown to be the same as those found using a more traditional time marching (e.g. Runge-Kutta) approach and also a conventional harmonic balance approach in the frequency domain with a considerable computational time saving.

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