• 한국연소학회:학술대회논문집
• /
• 1999.10a
• /
• pp.219-230
• /
• 1999

The Equations of Chemical kinetics are very stiff, which forces the use of an implicit scheme. The problem of implicit scheme, however, is that the jacobian must be solved at each time step. In this paper, we examined the methodology that can be stable without full chemical jacobian, This method is derived by applying the different time steps to the chemical source term. And the lower triangular chemical jacobian is derived. This is called the preconditioned time differencing method and represents partial implicit method. We show that this method is more stable in chemical kinetics than the full implicit method and that this is more efficient in supersonic combustion problem than the full jacobian method with same accuracy.

• 한국전산유체공학회:학술대회논문집
• /
• 2001.05a
• /
• pp.189-194
• /
• 2001

In this paper, non-reacting and reacting flowfields were computed using a preconditioned Navier-Stokes solver. The preconditioning technique of Merkle et al. and TVD scheme or Chakravarthy and Osher was employed and the results obtained using developed code have a good agreement with the previous results and experimental data. The preconditioned Wavier-Stokes equation set with low Reynolds number $\kappa-\epsilon$ equation and species continuity equations, are discretized with strongly implicit manner and time integrated with LU-SSOR scheme. For the purpose of treating unsteady problem the duel-time stepping scheme was employed. For the validation of the code in incompressible flow regime, steady driven square cavity flow was considered and calculation result shows reasonably good agreement with the result of incompressible code. Shock wave/boundary layer interaction problem was considered to show the shock capturing performance of preconditioned-TVD scheme. To validate unsteady flow, acoustic oscillation problem was calculated, and supersonic premix flame of $H_2$-air reaction problem which is calculated with turbulence model, 9-species/18-reaction step reaction model, shows reasonable agreement with the previous results. As a result, the preconditioning method has an advantage to calculate incompressible and compressible flow through one code and preconditioned solver easily developed from standard compressible code with minor efforts. But additional computational time and computer memory is required due to preconditioning matrix.

• 유체기계공업학회:학술대회논문집
• /
• 2006.08a
• /
• pp.349-352
• /
• 2006

The conventional segregated finite element formulation produces a small and simple matrix at each step than in an integrated formulation. And the memory and cost requirements of computations are significantly reduced because the pressure equation for the mass conservation of the Navier-Stokes equations is constructed only once if the mesh is fixed. However, segregated finite element formulation solves Poisson equation of elliptic type so that it always needs a pressure boundary condition along a boundary even when physical information on pressure is not provided. On the other hand, the conventional integrated finite element formulation in which the governing equations are simultaneously treated has an advantage over a segregated formulation in the sense that it can give a more robust convergence behavior because all variables are implicitly combined. Further it needs a very small number of iterations to achieve convergence. However, the saddle-paint-type matrix (SPTM) in the integrated formulation is assembled and preconditioned every time step, so that it needs a large memory and computing time. Therefore, we newly proposed the P2PI semi-segregation formulation. In order to utilize the fact that the pressure equation is assembled and preconditioned only once in the segregated finite element formulation, a fixed symmetric SPTM has been obtained for the continuity constraint of the present semi-segregation finite element formulation. The momentum equation in the semi-segregation finite element formulation will be separated from the continuity equation so that the saddle-point-type matrix is assembled and preconditioned only once during the whole computation as long as the mesh does not change. For a comparison of the CPU time, accuracy and condition number between the two methods, they have been applied to the well-known benchmark problem. It is shown that the newly proposed semi-segregation finite element formulation performs better than the conventional integrated finite element formulation in terms of the computation time.

• 한국전산유체공학회:학술대회논문집
• /
• 1998.05a
• /
• pp.35-42
• /
• 1998

Time marching algorithms applied to compressible Navier-Stokes equation have a convergence problem at low Mach number. It is mainly due to the eigenvalue stiffness and pressure singularity as Mach number approaches to zero. Among the several methods to overcome the shortcomings of time marching scheme, time derivative preconditioning method have been used successfully. In this numerical analysis, we adopted a preconditioner of K.H. Chen and developed a two-dimensional, axisymmetric Navier-Stokes program. The steady state driven cavity flow and backward facing step flow problems were computed to confirm the accuracy and the robustness of preconditioned algorithm for low Mach number flows. And the transonic and supersonic flows insice the JPL axisymmetric nozzle internal flow is exampled to investigate the effects of preconditioning at high Mach number flow regime. Test results showed excellent agreement with the experimental data.

• Journal of applied mathematics & informatics
• /
• v.16 no.1_2
• /
• pp.307-321
• /
• 2004

In this paper, the unsteady Stokes problem is considered and also the pressure-correction method for the problem is described. At a fixed time level, we reduce the problem to two symmetric positive definite problems which depend on a time step parameter. Linear systems that arise from the problems are large, sparse, symmetric, positive definite and ill-conditioned as the time step tends to zero. Preconditioned problems based on an additive Schwarz method for solving the symmetric positive definite problems are derived and preconditioners are defined implicitly. It will be shown that the rate of convergence is independent of the mesh parameters as well as the time step size.

• 한국전산유체공학회:학술대회논문집
• /
• 2002.10a
• /
• pp.95-102
• /
• 2002

Studies of the numerical characteristics of implicit upwind schemes, such as upwind ADI, Line Gauss-Seidel(LGS) and Point Gauss-Seidel(LU) algorithms, for preconditioned Navier-Stokes equations ate performed. All the algorithms are expressed in approximate factorization form and Von Neumann stability analysis and convergence studies are made. Preconditioning is applied for efficient convergence at low Mach numbers and low Reynolds numbers. For high aspect ratio computations, the ADI and LGS algorithms show efficient and uniform convergence up to moderate aspect ratio if we adopt viscous preconditioning based on min- CFL/max- VNN time-step definition. The LU algorithm, on the other hand, shows serious deterioration in convergence rate as the grid aspect ratio increases. Computations for practical applications also verify these results.

• 한국전산유체공학회:학술대회논문집
• /
• 2003.10a
• /
• pp.181-182
• /
• 2003

A preconditioned numerical method for gas-liquid to-phase flow is applied to solve cavitating flow. The present method employs a density based finite-difference method of dual time-stepping integration procedure and Roe's flux difference splitting approximation with MUSCL-TVD scheme. A homogeneous equilibrium cavitation model is used. The method permits simple treatment of the whole gas-liquid two-phase flow field including wave propagation, large density changes and incompressible flow characteristics at low Mach number. By this method, two-dimensional internal flows through a venturi tuve and decelerating cascades are computed and discussed.

• Journal of computational fluids engineering
• /
• v.19 no.4
• /
• pp.20-28
• /
• 2014

An efficient solution algorithm for simulating free surface problem is presented. Navier-Stokes equations for variable density incompressible flow are employed as the governing equation on Cartesian meshes. In order to describe the free surface motion efficiently, VOF(Volume Of Fluid) method utilizing THINC(Tangent of Hyperbola for Interface Capturing) scheme is employed. The most time-consuming part of the current free surface flow simulations is the solution step of the linear system, derived by the pressure Poisson equation. To solve a pressure Poisson equation efficiently, the PCG(Preconditioned Conjugate Gradient) method is utilized. This study showed that the proper application of the preconditioner is the key for the efficient solution of the free surface flow when its pressure Poisson equation is solved by the CG method. To demonstrate the efficiency of the current approach, we compared the convergence histories of different algorithms for solving the pressure Poisson equation.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.34 no.4
• /
• pp.191-197
• /
• 2021

To calculate the induced charge of atoms in molecular dynamics, linear equations for the induced charges need to be solved. As induced charges are determined at each time step, the process involves considerable computational costs. Hence, an efficient method for calculating the induced charge distribution is required when analyzing large systems. This paper introduces the Uzawa method for solving saddle point problems, which occur in linear systems, for the solution of the Lagrange equation with constraints. We apply the accelerated Uzawa algorithm, which reduces computational costs noticeably using the Schur complement and preconditioned conjugate gradient methods, in order to overcome the drawback of the Uzawa parameter, which affects the convergence speed, and increase the efficiency of the matrix operation. Numerical models of molecular dynamics in which two gold nanoparticles are placed under external electric fields reveal that the proposed method provides improved results in terms of both convergence and efficiency. The computational cost was reduced by approximately 1/10 compared to that for the Gaussian elimination method, and fast convergence of the conjugate gradient, as compared to the basic Uzawa method, was verified.

• Computers and Concrete
• /
• v.18 no.5
• /
• pp.1019-1039
• /
• 2016

In this paper, a parallel algorithm of nonlinear dynamic analysis of three-dimensional (3D) reinforced concrete (RC) frame structures based on the platform of graphics processing unit (GPU) is proposed. Time integration is performed using Newmark method for nonlinear implicit dynamic analysis and parallelization strategies are presented. Correspondingly, a parallel Preconditioned Conjugate Gradients (PCG) solver on GPU is introduced for repeating solution of the equilibrium equations for each time step. The RC frames were simulated using fiber beam model to capture nonlinear behaviors of concrete and reinforcing bars. The parallel finite element program is developed utilizing Compute Unified Device Architecture (CUDA). The accuracy of the GPU-based parallel program including single precision and double precision was verified in comparison with ABAQUS. The numerical results demonstrated that the proposed algorithm can take full advantage of the parallel architecture of the GPU, and achieve the goal of speeding up the computation compared with CPU.