Incomplete Lower-upper Factorization Research Articles

Numerical Simulation of Multiphase Multicomponent Flow in Porous Media: Efficiency Analysis of Newton-Based Method

Newton’s method has been widely used in simulation multiphase, multicomponent flow in porous media. In addition, to solve systems of linear equations in such problems, the generalized minimal residual method (GMRES) is often used. This paper analyzed the one-dimensional problem of multicomponent fluid flow in a porous medium and solved the system of the algebraic equation with the Newton-GMRES method. We calculated the linear equations with the GMRES, the GMRES with restarts after every m steps—GMRES (m) and preconditioned with Incomplete Lower-Upper factorization, where the factors L and U have the same sparsity pattern as the original matrix—the ILU(0)-GMRES algorithms, respectively, and compared the computation time and convergence. In the course of the research, the influence of the preconditioner and restarts of the GMRES (m) algorithm on the computation time was revealed; in particular, they were able to speed up the program.

Performance Analysis of the χMD Matrix Solver Package for MODFLOW-USG.

The χMD matrix solver package is incorporated into USGS groundwater modeling software, such as MODFLOW-NWT, MODFLOW-USG, and MT3D. The solver is used to solve matrices assembled through numerical discretization of the groundwater flow equation, and solute transport equations. χMD has demonstrated its higher robustness, faster execution speed, and more efficient memory usage compared to the existing solvers for many types of groundwater flow problems. χMD uses preconditioned iterative Krylov-subspace methods and consists of preconditioning and acceleration modules. Because the solver package uses a variety of preconditioning features including level-based incomplete lower-upper (ILU) factorization method with a drop tolerance scheme, users must choose optimal preconditioning parameters to improve execution speed and robustness. In order to examine how the preconditioning parameters, ILU factorization level, and drop tolerance values affect the overall performance of the matrix solver, we evaluated five different groundwater model applications using MODFLOW-USG that include different numerical complexities. For those five cases, the number of discretization nodes varied from 10,000 cells to 730,300 cells. From the analysis, we found that the preconditioning parameters greatly affect execution times and memory usage of the preconditioning and acceleration procedures. In addition, a combination of the ILU level between five to seven and the drop tolerance value between 10-2 and 10-3 usually resulted in shorter overall execution time. Our study suggests that the users can elicit higher performance and robustness of the χMD matrix solver using this combination of the parameters and enhance computational efficiency of solving groundwater and solute transport problems.

Improving Performance of Simplified Computational Fluid Dynamics Models via Symmetric Successive Overrelaxation

The ability to model fluid flow and heat transfer in process equipment (e.g., shell-and-tube heat exchangers) is often critical. What is more, many different geometric variants may need to be evaluated during the design process. Although this can be done using detailed computational fluid dynamics (CFD) models, the time needed to evaluate a single variant can easily reach tens of hours on powerful computing hardware. Simplified CFD models providing solutions in much shorter time frames may, therefore, be employed instead. Still, even these models can prove to be too slow or not robust enough when used in optimization algorithms. Effort is thus devoted to further improving their performance by applying the symmetric successive overrelaxation (SSOR) preconditioning technique in which, in contrast to, e.g., incomplete lower–upper factorization (ILU), the respective preconditioning matrix can always be constructed. Because the efficacy of SSOR is influenced by the selection of forward and backward relaxation factors, whose direct calculation is prohibitively expensive, their combinations are experimentally investigated using several representative meshes. Performance is then compared in terms of the single-core computational time needed to reach a converged steady-state solution, and recommendations are made regarding relaxation factor combinations generally suitable for the discussed purpose. It is shown that SSOR can be used as a suitable fallback preconditioner for the fast-performing, but numerically sensitive, incomplete lower–upper factorization.

Edge-Based Finite Element Modeling of Magnetogasdynamic-Based Propulsion Systems

The paper presents an edge-based finite element method formulation for the coupled magnetohydrodynamic equations used for the simulation of magnetogasdynamic propulsion systems. The flow is modeled by the Reynolds-averaged Navier–Stokes equations along with the electromagnetic Lorentz force and joule heating resulting from the magnetic/electric interactions represented by source terms, whereas the electromagnetic phenomena are modeled through the magnetic induction equation. Turbulence is treated through the one-equation Spalart–Allmaras model. Both the flow and the electromagnetic problems are transformed into their pseudo-unsteady forms, and a segregated solution strategy is employed for the coupled system. The algebraic systems are solved by applying the generalized minimum residual with an incomplete lower upper factorization as a right preconditioner. After using the Hartmann flow as a validation test, results have been obtained for an magnetohydrodynamic converging channel accelerator, a two-dimensional inviscid flow in a scramjet, and a three-dimensional viscous flow in the same scramjet. The numerical results are compared against published numerical ones and show good agreement.

Three-Dimensional Wide-Band Electromagnetic Forward Modelling Using Potential Technique

The efficacy of Krylov subspace solvers is strongly dependent on the preconditioner applied to solve the large sparse linear systems of equation for electromagnetic problems. In this study, we present a three-dimensional (3-D) plane wave electromagnetic forward simulation over a broadband frequency range. The Maxwell’s equation is solved in a secondary formulation of the Lorentz gauge coupled-potential technique. A finite-volume scheme is employed for discretizing the system of equations on a structured rectilinear mesh. We employed a block incomplete lower-upper factorization (ILU) preconditioner that is suitable for our potential formulation to enhance the computing time and convergence of the systems of equation by comparing with other preconditioners. Furthermore, we observe their effect on the iterative solvers such as the quasi-minimum residual and bi-conjugate gradient stabilizer. Several applications were used to validate and test the effectiveness of our method. Our scheme shows good agreement with the analytical solution. Notably, from the marine hydrocarbon and the crustal model, the utilisation of the bi-conjugate gradient stabilizer with block ILU preconditioner is the most appropriate. Thus, our approach can be incorporated to optimize the inversion process.

2D Zero-Inertia Model for Solution of Overland Flow Problems in Flexible Meshes

AbstractA study of the efficiency of a zero-inertia model (ZI) for two-dimensional (2D) overland flow simulation is presented in this work. An upwind numerical scheme is used for the spatial discretization in the frame of finite-volume methods and an implicit formulation is chosen to avoid numerical instability. The scheme is applied in both structured and unstructured meshes, focusing in the latter ones due to their good adaptability. The ZI equation has a nonlinear character; hence, a linearization is required in the implicit procedure. This is carried out by means of Picard iterations method as a previous step to the system matrix resolution, characteristic of implicit techniques. The BiConjugate Gradient Stabilized (BiCGStab) method combined with sparse storage strategies is selected for the system resolution. A dual-threshold incomplete lower upper factorization (ILUT) is chosen as matrix preconditioner. Computational efficiency of the implicit temporal discretization for ZI model is explored under b...

Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations

This article studies the effect of discretization order on preconditioning and convergence of a high-order Newton–Krylov unstructured flow solver. The generalized minimal residual (GMRES) algorithm is used for inexactly solving the linear system arising from implicit time discretization of the governing equations. A first-order Jacobian is used as the preconditioning matrix. The complete lower–upper factorization (LU) and an incomplete lower–upper factorization (ILU(4)) techniques are employed for preconditioning of the resultant linear system. The solver performance and the conditioning of the preconditioned linear system have been compared in detail for second, third, and fourth-order accuracy. The conditioning and eigenvalue spectrum of the preconditioned system are examined to investigate the quality of preconditioning.

High‐order ILU preconditioners for CFD problems

This paper tests a number of incomplete lower–upper (ILU)-type preconditioners for solving indefinite linear systems, which arise from complex applications such as computational fluid dynamics (CFD). Both point and block preconditioners are considered. The paper focuses on ILU factorization that can be computed with high accuracy by allowing liberal amounts of fill-in. A number of strategies for enhancing the stability of the factorizations are examined. Copyright © 2000 John Wiley & Sons, Ltd.

High-order ILU preconditioners for CFD problems

This paper tests a number of incomplete lower–upper (ILU)-type preconditioners for solving indefinite linear systems, which arise from complex applications such as computational fluid dynamics (CFD). Both point and block preconditioners are considered. The paper focuses on ILU factorization that can be computed with high accuracy by allowing liberal amounts of fill-in. A number of strategies for enhancing the stability of the factorizations are examined. Copyright © 2000 John Wiley & Sons, Ltd.

Efficient Newton-Krylov Solver for Aerodynamic Computations

An efficient inexact Newton-Krylov algorithm is presented for the computation of steady two-dimensional aerodynamic flows. The algorithm uses the preconditioned, restarted generalized minimal residual method in matrix-free form to solve the linear system arising at each Newton iteration. The preconditioner is formed using a block-fill incomplete lower-upper factorization of an approximate Jacobian matrix with two levels of fill after applying the reverse Cuthill-McKee reordering. The algorithm has been successfully applied to a wide range of test cases, which include inviscid, laminar, and turbulent aerodynamic flows. In all cases, convergence of the residual to 10 -12 is achieved with a computing cost equivalent to fewer than 1000 function evaluations. The matrix-free inexact Newton-Krylov algorithm is shown to converge faster and more reliably than an approximate Newton algorithm and an approximately factored multigrid algorithm

Solution of the Unsteady Euler Equations Using an Implicit Dual-Time Method

An unfactored implicit time-marching method for the solution of the unsteady two-dimensional Euler equations on deforming grids is described. The present work is placed into a multiblock framework and e ts into the development of a generally applicable parallel multiblock e ow solver. The convective terms are discretized using an upwind total variation diminishing scheme, whereas the unsteady governing equations are discretized using an implicit dual-time approach. The large sparse linear system arising from the implicit time discretization at each pseudotime step is solved efe ciently by using a conjugate-gradient-type method with a preconditioning based on a block incomplete lower-upper factorization. Results are shown for a series of pitching airfoil test cases selected from the AGARD aeroelastic cone gurations for the NACA 0012 airfoil. Comparisons with experimental data and previous published results are presented. The efe ciency of the method is demonstrated by looking at the effect of a number of numerical parameters, such as the conjugate gradient tolerance and the size of the global time step and by carrying out a grid ree nement study. Finally, a demonstration test case forthe Williamsairfoil (Williams, B. R., “ An Exact Test Case for the Plane Potential Flow About Two Adjacent Lifting Aerofoils,” National Physical Lab., Aeronautical Research Council, Research Memorandum 3717, London, 1973 )with an oscillating e ap is presented, highlighting the capability of the grid deformation technique.

HIGH-ORDER SCHEME IMPLEMENTATION USING NEWTON-KRYLOV SOLUTION METHODS

Implementation of high-order discretization for the connective transport terms in the inexact Newton method for a benchmark fluid flow and heat transfer problem using various solution configurations at two Reynolds numbers has been investigated. These configurations include fully consistent discretization of the Jacobian, preconditioner and residual of the Newton method, low-order preconditioning using a matrix-free method to approximate the action of the Jacobian, and defect correction or low-order Jacobian and preconditioning. The residual in each case employs high-order discretization to preserve the high-order solution. Two preconditioners, point incomplete lower-upper factorization ILU( k) and block incomplete lower-upper factorization BILUO(K) for k = 0,1,2 were applied. Also, “one-way” muitigrid-ing and capping the inner iterations was applied to determine the behavior of the solution performance. It was determined that, overall, the configuration using low-order preconditioning with ILU(1), BILU(1), or BILU(2), mesh sequencing, and inner linear solve iterations capped at the same value of the dimension n, used with the GMRES(n) iterative solver( i.e., no restarts), performed best for time, memory, and robustness considerations.

Newton-Krylov methods for low-Mach-number compressible combustion

Fully coupled numerical techniques are used to compute steady-state solutions to a combusting, low-Mach-number compressible flow through a channel. The nonlinear governing equations are discretized on a staggered mesh via integration over discrete finite volumes. The resulting nonlinear algebraic equations are linearized with Newton's method and solved with a preconditioned Krylov algorithm. The selected Krylov solver is the generalized minimum residual algorithm. A matrix-free Newton-Krylov method and a modified Newton-Krylov method are employed as a means of reducing the required number of expensive Jacobian evaluations. The matrix-free implementation is shown to be superior to the modified Newton-Krylov method when starting from a poor initial guess. The technique of mesh sequencing is shown to provide significant CPU savings for fine grid calculations. Additionally, the domain-based multiplicative Schwarz preconditioning strategy was found to be more effective than incomplete lower-upper factorization type preconditioning at lower Mach numbers.

Comparison of implicit schemes for the incompressible Navier-Stokes equations

For a computational flow simulation tool to be useful in a design environment, it must be very robust and efficient. To develop such a tool for incompressible flow applications, a number of different implicit schemes are compared for several two-dimensional flow problems in the current study. The schemes include Point-Jacobi relaxation, Gauss-Seidel line relaxation, incomplete lower-upper decomposition, and the generalized minimum residual method preconditioned with each of the three other schemes. The efficiency of the schemes is measured in terms of the computing time required to obtain a steady-state solution for the laminar flow over a backward-facing step, the flow over a NACA 4412 airfoil, and the flow over a three-element airfoil using overset grids. The flow solver used in the study is the INS2D code that solves the incompressible Navier-Stokes equations using the method of artificial compressibility and upwind differencing of the convective terms. The results show that the generalized minimum residual method preconditioned with the incomplete lower-upper factorization outperforms all other methods by at least a factor of 2.

Fully coupled finite volume solutions of the incompressible Navier–Stokes and energy equations using an inexact Newton method

AbstractAn inexact Newton method is used to solve the steady, incompressible Navier–Stokes and energy equation. Finite volume differencing is employed on a staggered grid using the power law scheme of Patankar. Natural convection in an enclosed cavity is studied as the model problem. Two conjugate‐gradient ‐like algorithms based upon the Lanczos biorthogonalization procedure are used to solve the linear systems arising on each Newton iteration. The first conjugate‐gradient‐like algorithm is the transpose‐free quasi‐minimal residual algorithm (TFQMR) and the second is the conjugate gradients squared algorithm (CGS). Incomplete lower‐upper (ILU) factorization of the Jacobian matrix is used as a right preconditioner. The performance of the Newton‐ TFQMR algorithm is studied with regard to different choices for the TFQMR convergence criteria and the amount of fill‐in allowed in the ILU factorization. Performance data are compared with results using the Newton‐CGS algorithm and previous results using LINPACK banded Gaussian elimination (direct‐Newton). The inexact Newton algorithms were found to be CPU competetive with the direct‐Newton algorithm for the model problem considered. Among the inexact Newton algorithms, Newton‐CGS outperformed Newton‐ TFQMR with regard to CPU time but was less robust because of the sometimes erratic CGS convergence behaviour.