IMPLEMENTATION OF A PARALLEL AMG SOLVER

Transcription

1 IMPLEMENTATION OF A PARALLEL AMG SOLVER Tony Saad May tsaad@utsi.edu

2 PLAN INTRODUCTION 2 min. MULTIGRID METHODS.. 3 min. PARALLEL IMPLEMENTATION PARTITIONING. 1 min. RENUMBERING... 1 min. THE SOLVER... 3 min. RESULTS... 3 min. CONCLUSION. 2 min.

3 INTRODUCTION Computational Fluid Dynamics (CFD) is an advanced computing technology developed from traditional fluid mechanics. It uses a set of numerical methods for the solution of PDEs arising from general transport phenomena CFD is being used to model several applications

4 Aerospace INTRODUCTION

5 INTRODUCTION Aerospace Appliances

6 INTRODUCTION Aerospace Appliances Sports

7 INTRODUCTION Aerospace Appliances Sports Environment

8 INTRODUCTION The Finite Volume Method is a very popular method used in CFD It is based on conservation princinples

9 INTRODUCTION ρφ ( ) ( ) t Transient Term + ρuφ = Γ φ + Convection Term Diffusion Term S φ Source Term

10 ρφ ( ) ( ) t Transient Term + ρuφ = Γ φ + Convection Term Diffusion Term INTRODUCTION S φ Source Term a P t CD + a P urf φ + P i= NB( P) a CD φ Ni Ni = Q P V P + a Pt φ P urf urf t CD * ( a P + a P )φ P a p φ P + i= NB( P) a Ni φ Ni = b P a Ni = Γ f E f d PNi &m f,0 b P = Q P V P + a Pt φ 0 P + 1 urf urf t a P = a P a Ni + &m fi i= NB( P) i= nb( P) t CD ( a P + a P )φ * P + Γ φ i= nb( P) ( ) fi T fi

11 INTRODUCTION Discretization yields a linear system of equations a P φ P + a Ni φ Ni = b P i= NB( P) A φ = b The solution of this system requires an iterative procedure as the coefficient matrix is non-linear Famous solvers: SOR, ILU Convergence rate slows down as error becomes smooth

12 SOR Starts with a Gauss-Seidl Iterate k b P a Ni φ Ni ( * φ ) k P = i= NB( P) a P Relax and update the value (k φ ) P = ω( * φ ) k P + ( (k 1) 1 ω)φ P φ P (k ) = ω b P i= NB( P) a P k a Ni φ Ni + ( (k 1) 1 ω)φ P

13 MULTIGRID METHODS Multigrid Methods are designed to overcome this problem by deriving a set of coarse grids from the original grid on which the solution is not smooth Algebraic Multigrid (AMG) is suitable for unstructured grids and FV formulation

14 MULTIGRID METHODS

15 MULTIGRID METHODS At each level, a linear system of equations is solved The coarse grid linear systems are derived from the finer grids using an agglomeration of the coefficients i G I j G J (l A +1) l I,J = A i, j b I (l +1) = i G I r i (l) r k = b A x k

16 PARALLEL IMPLEMENTATION Why think parallel? CFD models are becoming computationally expensive Parallel computing is the future of scientific computing Parallelization requires several steps: Partitioning Partition Renumbering Iterative Solver

17 PARTITIONING Partitioning Divides domain into partitions of equal size Each partition is then assigned to a different processor

18 PARTITIONING

19 PARTITIONING Partitioning Divides domain into partitions of equal size Each partition is then assigned to a different processor

20 PARTITION RENUMBERING Partition Renumbering Each partition is renumbered locally so as to have an independent problem on each process Each processor solves the same set of equations but on its own partition Coupling between processors is done by defining shadow and sender elements at the interface

21 PARTITIONG RENUMBERING

22 SOLVER SYNCHRONIZATION As each processor iterates on its system of equations, updates at the interface need to be made

23 SOLVER SYNCHRONIZATION As each processor iterates on its system of equations, updates at the interface need to be made =

24 AMG SYNCHRONIZATION Linear solver synchronization affects the number of outer iterations and thus scalability degenerates Solution is to parallelize AMG solver Each processor performs agglomeration while enforcing same number of levels across the domain Shadow and sender elements are defined for each level Updates at the interface are made at each level

25 RESULTS Non-linear Diffusion problem is a square domain ( Γ φ)= 0 Γ = φ 0.1

26 RESULTS Speedup Solution Time Shadow-to-core Ratio Efficiency Solution Time Ideal ,454 99, ,222 99, , , , , , , , , Maximum Number shadow/core Processors of Partitions elements ratio

27 CONCLUSION A Parallel AMG solver has been implemented and tested using available resources Good scalability was achieved Improvement by using better network switch Future plans included implementation of the solution of the Navier-Stokes equations

28 THANK YOU FOR ATTENDING