University:
Boston University
Course:
MET CS 201 | Introduction to Programming
Academic year:

2024
Views:

367

Pages:

25
Author:

Kasazami2c24

GPU Computing with CUDA Lecture 9 Outline of lecture ‣ Overview of CFD - Navier Stokes equations - Types of problems - Discretization methods ‣ “Conventional” CFD ‣ Port CFD codes to CUDA ‣ Efforts ‣ Example problem: implicit heat transfer CFD - Introduction ‣ Numerical modeling of fluid systems ‣ Navier-Stokes equation: momentum conservation ∇p ∂u 2 + (u · ∇)u = − + ν∇ u ∂t ρ ‣ Type of problems: - Incompressible - Compressible (non-viscous approximation) - Shallow water - Biphasic flows.... CFD - Introduction ‣ Earliest: Richardson (1910) - Human computers - Quickest averaged 2000 operations a week ‣ CFD development tied with computers! - 50s-60s: use of digital computers, finite difference methods - 70s: finite element methods, spectral methods - 80s: finite volume methods - 90s: application to diverse industries CFD - Main discretization methods ‣ Finite difference ui+1,j − ui,j ∂u = ∂x ∆x ‣ Finite volume " ∂U ∂ F" + =0 ∂t ∂x ∂ ∂t ! Ω " dΩ + U " ∂Ω F" n̂ds = 0 CFD - Main discretization methods ‣ Finite element method ! Ω ∇u · ∇v dx = ‣ Spectral methods ! ∂u = ikû ∂x ! Ω f v dx. CFD - Main discretization methods ‣ Mesh free methods - Smoothed Particle Hydrodynamics - Vortex methods - Radial Basis Functions - ... ∂u = ∂x N ! ∂φi αi ∂x i=0 CFD - Fluid Modeling ‣ Fluid flow is a multi-scale phenomena - We need Re3 mesh points to reproduce all scales! - Turbulence modeling - Approximate turbulence effects VD Re = ν Conventional CFD ‣ Unstructured grids - Unstructured sparse matrices ‣ Incompressible ∇ · u = 0 - Projection methods ‣ Implicit - Linear solvers ‣ Modeled turbulence - Reduced number of points Conventional CFD ‣ CFD is a tough problem for the GPU: - Memory bound problems ‣ Also, needs to convince people - Old legacy codes - How to port old codes to the GPU? ‣ On the other hand, CFD codes are - SIMD - Single precision - Large data sets ‣ Option 1: accelerate the existing code ‣ Option 2: Rewrite code from scratch ‣ Option 3: Rethink algorithms Potential acceleration Porting a code to GPU Option 1: Accelerate existing code ‣ Easiest way ‣ Probably not huge speedup ‣ Libraries like Cusp or CUFFT may be useful Option 1: Accelerate existing code - SpeedIT ‣ SpeedIT (OpenFOAM) - Ported linear solvers to GPU - Supports multi-GPU Mesh Speedup 20x20 -100x 96x96x96 2.4x 128x128x32 2.0x Xeon X5650 CPU M2050 GPU Ville Tossavainen (Seeinside Ltd.) Option 1: Accelerate existing code - FEAST ‣ FEAST (Finite Element Analysis and Solution Tools) - High level abstraction approach - Isolate “accelerable” parts of code - Ports solver to GPU: Multigrid Acceleration fraction: 75% Local speedup: 11.5x Global speedup: 3.8x Opteron 2214 4 nodes CPU GTX 8800 GPU Option 2: Rewrite whole code ‣ First need to think about - What is the total application speedup that you can get - How does rewrite compare to accelerator approach - Good design - What global optimizations are possible Option 2: Rewrite whole code - cuIBM ‣ Immersed Boundary Method on GPU (cuIBM) - Finite difference code with immersed boundary no slip condition - 2 linear systems: implicit diffusion and projection 2.5 AXPY Apply BCs Conversion Force Calculation Force Output Generate bc1 Generate r2 Generate rN MMM Mat−vec Mem Transfer Output Preconditioner Solve 1 Solve 2 Transfer q Transpose Update B Update QT - Reported speedup: 7x 2 Time [s] 1.5 1 0.5 0 Average over 16000 timesteps Option 2: Rewrite whole code - cuIBM 50 CPU 45 40 35 Time [s] 30 25 20 15 GPU 10 5 0 pyAMG blackbox pyAMG Cusp Cusp Smoothed Non−PC Diagonal PC Aggregation Cusp Scaled Bridson Cusp Smoothed Aggregation With good pre-conditioner, GPU is 9x faster, not much difference in other cases (best is 1.6x faster) Option 2: Rewrite whole code - Open Current ‣ Developed by Jonathan Cohen in NVIDIA ‣ Compared a highly optimized CPU code and GPU code - CPU: Fortran, 8-core 2.5 GHz Xeon (8 thredas with MPI and OpenMP) - GPU: CUDA, Tesla C1060 ‣ Solved the Rayleigh-Bernard with a finite difference code Option 2: Rewrite whole code - Open Current Resolution CUDA time/step ms Fortran time/step ms Speedup 64x64x32 24 47 2.0x 128x128x64 79 327 4.1x 256x256x128 498 4070 8.2x 384x384x192 1616 13670 8.5x Option 3: Rethink numerical algorithms ‣ Most time consuming alternative! ‣ Maybe new architectures require new numerics ‣ Find methods that map well to the hardware - Maybe we overlooked something in the past because it was impractical Option 3: Rethink numerical algorithms - DG ‣ Discontinuous Galerkin Methods - Arithmetically intensive - Mainly local ‣ Klockner et al. used DG to solve conservation laws GPU T1060 CPU: Xeon E5472 Implicit heat equation solver ‣ Conventional CFD usually is dominated by Poisson type solvers - Projection methods - Implicit solvers to avoid stability constraints ∂u 2 ‣ Heat equation with Crank-Nicolson = α∇ u ∂t - No stability constraint! αk < 0.5 h2 Implicit heat equation solver T=0 α = 0.645 k = 1e-5 N = 128 T = 200 T=0 3.5 ∂u 2 = α∇ u ∂t T = 200 n+1 ui,j − uni,j = k α 2 + ! 3.5 n n n n un +u +u +u −4u i,j+1 i,j−1 i+1,j i−1,j i,j h2 n+1 n+1 n+1 n+1 un+1 +u +u +u −4u i,j+1 i,j−1 i+1,j i−1,j i,j h2 " Implicit heat equation solver n+1 aui,j−1 + n+1 aui,j+1 αk a=− 2 2h RHSi,j = n ui,j + n+1 aui−1,j + n+1 aui+1,j + n+1 bui,j = RHSi,j 2αk b = 1 − 4a = 1 + 2 h " αk ! n n n n n n+1 + 2 ui,j−1 + ui,j+1 + ui−1,j + ui+1,j − 4ui,j − BC 2h Implicit heat equation solver [A]u n+1 = RHS αk [A] = I + 2 · [Poisson] 2h [A] size (N-2) x (N-2) 2 RHS size (N-2) u n+1 2 size (N-2) 2 2