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Author: Matthew Dean Winger (Graduate student) Publisher: ISBN: Category : Computational fluid dynamics Languages : en Pages : 37
Book Description
Abstract: For this work, two-dimensional fluid dynamics within an enclosed square domain are simulated using the incompressible Navier-Stokes equations. The computational domain is discretized using the Marker-And-Cell (MAC) method which places velocity and pressure components in a mesh of staggered grids, offering improved solution stability as opposed to placing these components on the same grid. A modified form of the projection method is used to linearize the incompressible equations which produces a series of systems which are solved for velocity and pressure. With an immersed boundary method we can simulate the effect various surfaces with no-slip boundary conditions have on the fluid flow when placed inside our domain via a forcing term which is added to our systems. Solution of these linear systems is achieved by using the multigrid method, which uses restriction and interpolation operators that account for the staggered grids produced by the MAC method. With these tools we are able to describe a variety of types of fluid simulations, illustrated in this work with a series of numerical examples.
Author: Matthew Dean Winger (Graduate student) Publisher: ISBN: Category : Computational fluid dynamics Languages : en Pages : 37
Book Description
Abstract: For this work, two-dimensional fluid dynamics within an enclosed square domain are simulated using the incompressible Navier-Stokes equations. The computational domain is discretized using the Marker-And-Cell (MAC) method which places velocity and pressure components in a mesh of staggered grids, offering improved solution stability as opposed to placing these components on the same grid. A modified form of the projection method is used to linearize the incompressible equations which produces a series of systems which are solved for velocity and pressure. With an immersed boundary method we can simulate the effect various surfaces with no-slip boundary conditions have on the fluid flow when placed inside our domain via a forcing term which is added to our systems. Solution of these linear systems is achieved by using the multigrid method, which uses restriction and interpolation operators that account for the staggered grids produced by the MAC method. With these tools we are able to describe a variety of types of fluid simulations, illustrated in this work with a series of numerical examples.
Author: R. C. Swanson Publisher: BiblioGov ISBN: 9781289262938 Category : Languages : en Pages : 50
Book Description
A fast multigrid solver for the steady, incompressible Navier-Stokes equations is presented. The multigrid solver is based upon a factorizable discrete scheme for the velocity-pressure form of the Navier-Stokes equations. This scheme correctly distinguishes between the advection-diffusion and elliptic parts of the operator, allowing efficient smoothers to be constructed. To evaluate the multigrid algorithm, solutions are computed for flow over a flat plate, parabola, and a Karman-Trefftz airfoil. Both nonlifting and lifting airfoil flows are considered, with a Reynolds number range of 200 to 800. Convergence and accuracy of the algorithm are discussed. Using Gauss-Seidel line relaxation in alternating directions, multigrid convergence behavior approaching that of O(N) methods is achieved. The computational efficiency of the numerical scheme is compared with that of Runge-Kutta and implicit upwind based multigrid methods.
Author: Ruben S. Montero Publisher: ISBN: Category : Anisotropy Languages : en Pages : 26
Book Description
Anisotropies occur naturally in CFD where the simulation of small scale physical phenomena, such as boundary layers at high Reynolds numbers, causes the grid to be highly stretched leading to a slow down in convergence of multigrid methods. Several approaches aimed at making multigrid a robust solver have been proposed and analyzed in literature using the scalar diffusion equation. However, they have been rarely applied to solving more complicated models, like the incompressible Navier-Stokes equations. This paper contains the first published numerical results of the behavior of two popular robust multigrid approaches (alternating-plane smoothers combined with standard coarsening and plane implicit smoothers combined with semi-coarsening) for solving the 3-D incompressible Navier-Stokes equations in the simulation of the driven cavity and a boundary layer over a flat plate on a stretched grid. The discrete operator is obtained using a staggered-grid arrangement of variables with a finite volume technique and second-order accuracy is achieved using defect correction within the multigrid cycle. Grid size, grid stretching and Reynolds number are the factors considered in evaluating the robustness of the multigrid methods. Both approaches yield large increases in convergence rates over cell-implicit smoothers on stretched grids. The combination of plane implicit smoothers and semi-coarsening was found to be fully robust in the fiat plate simulation up to Reynolds numbers 10(exp 6) and the best alternative in the driven cavity simulation for Reynolds numbers above 10(exp 3). The alternating-plane approach exhibits a better behavior for lower Reynolds numbers (below to 10(exp 3) in the driven cavity simulation. A parallel variant of the smoother, tri-plane ordering, presents a good trade-off between convergence and parallel properties.
Author: Stephen F. McCormick Publisher: SIAM ISBN: 1611971055 Category : Mathematics Languages : en Pages : 292
Book Description
A thoughtful consideration of the current level of development of multigrid methods, this volume is a carefully edited collection of papers that addresses its topic on several levels. The first three chapters orient the reader who is familiar with standard numerical techniques to multigrid methods, first by discussing multigrid in the context of standard techniques, second by detailing the mechanics of use of the method, and third by applying the basic method to some current problems in fluid dynamics. The fourth chapter provides a unified development, complete with theory, of algebraic multigrid (AMG), which is a linear equation solver based on multigrid principles. The last chapter is an ambitious development of a very general theory of multigrid methods for variationally posed problems. Included as an appendix is the latest edition of the Multigrid Bibliography, an attempted compilation of all existing research publications on multigrid.
Author: Sergey I. Martynenko Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110539268 Category : Mathematics Languages : en Pages : 212
Book Description
This book presents a detailed description of a robust pseudomultigrid algorithm for solving (initial-)boundary value problems on structured grids in a black-box manner. To overcome the problem of robustness, the presented Robust Multigrid Technique (RMT) is based on the application of the essential multigrid principle in a single grid algorithm. It results in an extremely simple, very robust and highly parallel solver with close-to-optimal algorithmic complexity and the least number of problem-dependent components. Topics covered include an introduction to the mathematical principles of multigrid methods, a detailed description of RMT, results of convergence analysis and complexity, possible expansion on unstructured grids, numerical experiments and a brief description of multigrid software, parallel RMT and estimations of speed-up and efficiency of the parallel multigrid algorithms, and finally applications of RMT for the numerical solution of the incompressible Navier Stokes equations. Potential readers are graduate students and researchers working in applied and numerical mathematics as well as multigrid practitioners and software programmers. Contents Introduction to multigrid Robust multigrid technique Parallel multigrid methods Applications of multigrid methods in computational fluid dynamics
Author: Anton Schüller Publisher: ISBN: Category : Navier-Stokes equations Languages : en Pages : 13
Book Description
Abstract: "In the applications software development of the SUPRENUM project many parallel, grid-oriented algorithms are being programmed, especially multigrid and computational fluid dynamics codes. Special emphasis is placed on the solution of the Navier-Stokes equations. In this paper a reformulation of the incompressible Navier-Stokes problem is derived, in which the continuity equation is replaced by a Poisson-like equation for the pressure. The corresponding discrete nonlinear system of equations has some advantageous features for the numerical solution. Results computed by a suitable multigrid algorithm are shown and discussed."
Author: Achi Brandt Publisher: SIAM ISBN: 1611970741 Category : Mathematics Languages : en Pages : 217
Book Description
This revised edition of a classic text presents the best practices of developing multigrid solvers for large-scale computational problems. This book will be useful to practitioners and researchers, as well as students and instructors, in many areas of computational science and engineering, applied mathematics and numerical analysis.
Author: Matthew Dean Winger (Graduate student)
Author: Matthew Dean Winger (Graduate student)
Author: R. C. Swanson
Author: Ruben S. Montero
Author: Institute for Computer Applications in Science and Engineering
Author: Institute for Computer Applications in Science and Engineering
Author: Stephen F. McCormick
Author: Sergey I. Martynenko
Author: 
Author: Anton Schüller
Author: Achi Brandt