title: Παραλληλοποιήσιμες πολυπλεγματικές τεχνικές σε επιλυτές Navier − Stokes για την μοντελοποίηση ασυμπίεστων ροών
title: Parallel multigrid techniques for Navier - Stokes solvers simulating incompressive flows
Creators: name: Μανδικας Βασιλειος
Publisher: name: Technical University of Crete
Year: 2017

description: The Navier-Stokes equations, that govern the motion of an incompressible or compressible fluid, were introduced more than a hundred years ago. Their nonlinearity leads to signifi- cant difficulties, sometimes insurmountable, when trying to find an analytical solution. In order to obtain an exact solution, specific geometries and initial/boundary conditions must be considered. In more complex situations, such as applications in fluid structure inter- action, low speed aerodynamics, biomechanics etc., the application of numerical methods may provide reliable solutions for the Navier-Stokes equations. Furthermore, the necessity for a fast high-resolution numerical solver for the incompressible Navier-Stokes equations emerges from real-life simulations, such as flows over hydrofoils, wind-turbine blades, and aircraft wings during takeoff and landing. Thus, the objective of this thesis is to develop, explicate and demonstrate the performance of a highly efficient flow-solver, which exploits the computational power of architectures with computing accelerators.
In the first part of this thesis, and after the preliminaries, an elliptic PDE multigrid solver is developed and demonstrated. The solver is capable of handling highly anisotropic 2D Boundary Value Problems (BVPs) and is based on high-order cell-centered Finite Difference Compact schemes and Multigrid techniques. Compact schemes provide a representation of the shorter length scales, when applied to problems with a range of spatial scales, compared to traditional finite difference approximations. An improvement of the proposed method is the treatment of PDE boundary conditions. Boundary closure formulas for Dirichlet, Neumann, Robin or mixed-type boundary conditions applied to the physical boundary are derived and tested herein for several simulation problems. Moreover, novel multigrid components for cell-centered discretization are being constructed, which could also be generalized to three dimensional problems in a straightforward manner. In particular, the new intergrid operators involve less non-zero entries than common operators in cell-centered grids, preserving at the same time the accuracy of high-order operators. Next, some theoretical convergence results for the multigrid solver using Local Fourier Analysis (LFA) are given in order to improve its multigrid convergence factors. The analysis focuses on both the relaxation method used within the multigrid, as well as in the remaining components of the multigrid method. The proposed multigrid elliptic solver is evaluated on two classical BVPs, so that the fourth-order accuracy of the solver, as well as the boundary treatments, could be validated. The multigrid convergence rates for every acknowledged transfer operator, along with every novel one, are evaluated as to determine the convergence behavior of the corresponding multigrid solver. These results are also compared to the theoretical convergence factors based on the LFA. The concordance of the analytical and numerical results is acceptable. A comparison between the calculated convergence rates to the corresponding values, obtained from the second-order compact scheme, indicates the superiority of the high-order compact scheme. In addition, these convergence rates are as good as the factors one obtains from the vertex-centered case. It is noted, that the cell-centered multigrid numerical analysis is presented for a high-order scheme, opposed to earlier studies. Along with the investigation on the improvement of multigrid techniques for the high-order scheme, iterative methods are also tested for the resulting sparse linear system and compared with the multigrid numerical solver. The comparison results indicate the necessity of a multigrid solver for this kind of problems, as it is proved to be hundreds of times faster than the optimal iterative method when fine discretizations are used.
The proposed spatial discretization solver is incorporated in an effective Navier-Stokes solver capable of handling highly anisotropic flow problems. The solver is based on the pressure-velocity coupling and uses fourth-order compact schemes for discretizing each spatial dimension, formulated on a staggered grid arrangement. The temporal discretization is carried out by a fourth-order Runge-Kutta (RK4) method. Incompressibility is enforced at each time step using a global pressure correction method solving a Poisson-type PDE. In this method, the multigrid solver is being used within each stage of the RK4 method to compute the pressure correction. The spatial and temporal fourth-order accuracy of this Navier-Stokes solver are validated for a set of steady and unsteady classical test problems. Further, the performance results indicate that multigrid accelerates the solution procedure more than 10 times, comparing with other solvers in the literature.
The numerical study of the sequential Navier-Stokes solver evince that, in cases of grid sizes up to 256 × 256, the incorporation of the multigrid scheme handles the increasing execution time moderately. However, in case of finer grid sizes, the computational cost becomes intolerable despite the high convergence rates of the Multigrid method. It is noted that the most time-consuming part of the solver is the pressure correction procedure. This time restriction gives motive to redesign and develop an efficient parallel multigrid based Navier-Stokes algorithm, to exploit the benefits of modern parallel computer architectures with accelerators. In order to increase parallelism at each computing phase of the algo- rithm, the horizontal red-black coloring scheme for grid nodes is chosen. The Block Cyclic Reduction method is also applied for the solution of the arising linear sub-systems, without modifying the multigrid cycling nature in the algorithm. This enables the execution of the entire simulation in the acceleration device, minimizing the communication cost between memory units. In addition, the proposed parallelization exploits the block structure of the coefficient matrix, minimizing data storage and increasing again the parallelism. The solver is implemented and examined on three parallel machines with different type of accelerator devices. The realization is developed using the OpenACC and OpenMP APIs. The effect of several multigrid components on modern and legacy acceleration architectures is investigated
Application’s performance investigation demonstrates that the proposed parallel multi- grid solver accomplishes an acceleration of more than ten times over the sequential solver and more than four times over multi-core CPU-only realizations. In case of highly anisotropic problems, the parallel semi-coarsening multigrid solver is preferred to the full-coarsening one, as the division of labor by the accelerator device provides faster computational rates, in case of non-uniform discretizations. 
The proposed numerical algorithm can be easily extended for the case of three dimensional flow problems, on curvilinear coordinates, expanding the applicability of the current methodology. Furthermore, the proposed numerical methodology can be applied for solving comparable problems, e.g. Maxwells equations. The design of a parallel algorithm for the utilization of a Heterogeneous Multi-Accelerator architecture using the MPI, OpenMP and OpenACC APIs, is considered to be a promising improvement.