Mesh-free numerical methods for large-scale engineering problems.  

Basics of GPU-based computing. D.Stevens, M.Lees, P. Orsini

Supervisors: Prof. Henry Power, Herve Morvan

January 2008

Outline:

Meshless numerical methods using Radial Basis Functions (RBFs) Basic RBF interpolation Brief overview of the work done in our research group, under the direction of Prof. Henry Power Example results Large scale future problems

GPU computing Introduction to the principle of using graphics processors (GPUs), as floating point co-processors Current state of GPU hardware and software Basic implementation strategy for numerical simulations

GABARDINE - EU project GABARDINE:

Groundwater Artificial recharge Based on Alternative sources of

wateR: aDvanced INtegrated technologies and managEment

Aims to investigate the viability of artificial recharge in groundwater aquifers, and produce decision support mechanisms

Partners include: Portugal, Germany, Greece, Spain, Belgium, Israel, Palestine

University of Nottingham is developing numerical methods to handle: Phreatic aquifers (with associated moving surfaces) Transport processes Unsaturated zone problems (with nonlinear Governing equations)

RBF Interpolation Methods

xPxxu m

N

jjj 1

1

Apply the above at each of the N test locations:

Niff ii ,...,1

N unknowns…

ijji fx

222)(

)ln()(m

m

crr

rrr

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Kansa’s Method

N

jjj xxu

1

onxgxuB

inxfxuL

)()]([

)()]([ 2

2

.jx

Leg

(Diffusion operator)

i

ij

ijx

ijx

f

g

L

B

][

][Collocating:

Hermitian Method

N

NBjjj

NB

jjj xLxBxu

11

onxgxuB

inxfxuL

)()]([

)()]([

i

ij

ijxijx

ijxijx

f

g

LLBL

LBBB

][][

][][

Operators are also applied to basis functions:

SYMMETRIC SYSTEM

RBF – Features and Drawbacks Partial derivatives are obtained cheaply and accurately by

differentiating the (known) basis functions

Leads to a highly flexible formulation allowing boundary operators to be implemented exactly and directly

Once solution weights are obtained, a continuous solution can be reconstructed over the entire solution domain

A densely populated linear system must be solved to obtain the solution weights

Leads to high computational cost – O(N2), and numerical conditioning issues with large systems, setting a practical limit of ~1000 points

Formulation: LHI method

111

m

N

NBjjj

NB

jjj PxxBxu

inxfxu

onxgxuB

)()(

)()]([

Hermitian Interpolation using solution values, and boundary operator(s) if present

Initial value problem:

Lut

u

LHI method formulation cont…

)()()( kkk dH

Form N systems, based on a local support:

011

1

1

Tm

Tm

mxxx

m

PBP

PBBBB

PB

H

0

)( i

i

xg

u

d

Hence, can reconstruct the solution in the vicinity of local system k via:

.)()( )()( xRxu kk

)(,,)( 1 xpxBxxR mii where:

H ~ 10 x 10 matrix

CV-RBF (Modified CV Scheme) Classic CV approach

Polynomial interpolation to compute the flux

CV-RBF approach

Apply internal operator

Apply Dirichlet operator

Apply Boundary operator

RBF interpolation to compute the flux

L x f x

B x g x

3xon

on

Simulation Workflow

Dataset Generation

Pre-Processing

SimulationPost Processing

GridGenRBF

CVRBF

Triangle

Meshless

TecPlot

CAD

Meshless

RBF specific

Our Code

Convection-Diffusion: Validation

Pe = 50 - LHIM (c = 0.12)

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.75 0.80 0.85 0.90 0.95 1.00

Analytical

Calculated

Pe = 100 - LHIM (c = 0.08)

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.75 0.80 0.85 0.90 0.95 1.00

Analytical

Calculated

k = 120 - c = 0.04

0

50

100

150

200

250

300

0.0 0.2 0.4 0.6 0.8 1.0

Calculated

Analytic

Both methods have been validated against a series of 1D and 3D linear and nonlinear advection-diffusion reaction problems, eg:

Pe = 200 - LHIM (c = 0.03)

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.75 0.80 0.85 0.90 0.95 1.00

Analytical

Calculated

CV-RBF: Infiltration well + Pumping

75 , 31 , 50y zx m L m L mL

Well diameter: 3d m

1.0 0.3 0.2x z yK K m h K m h Soil properties:

Pumping location: 25m from the infiltration well (height y=15m)3

20m

I P qAh

Infiltration-Pumping rate:

CV-RBF: Infiltration well + Pumping 3D model: Mesh (60000 cells) and BC

Boundary conditions31m

0 : 0yn

Everywhere else

CV-RBF: Infiltration well + Pumping

0.2welldh m

Piezometric head contour and streamlines at t=30h

Maximum Displacements: 0.85pumpdh m

plane at z=25m plane at y=29m

Length scale: 100m

LHI: Infiltration model - Setup

Infiltrate over 10m x 10m region at ground surface

Infiltration pressure = 2m

Zero-flux boundary at base (‘solid wall’)Fixed pressure distribution at sides

Initial pressure:(8.0 )

1.0

z m Saturated zone

m Unsaturated zone

Solving Richards’ equation:2

0

1 ( ) ( )( )

'( ) ijp i j

K

t S x x z

sK

LHI: Infiltration model – Soil properties

40 10S

1(1.0, 1.0, 0.2)s mhK

sp SS

)(

00

2

3

2

2

1)(

g

sh

hKK

2

1

2

2

1)()(

g

rsrh

h

Using Van-Genuchten soil representation:

Saturated conductivity:

Storativity:

Relative Hydraultic Conductivity

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Water pressure (m)

Rel

ativ

e co

nd

uct

ivit

y

Water Content

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0

Water pressure (m)

Wat

er C

on

ten

t

0.15, 0.45,r s

LHI: Infiltration model - Results 11,585 points arranged in 17 layers

‘Short’ runs: solution to 48 hours

‘Long’ run: solution to 160 hours

Richards’ equation - First example Using the steady-state example given in Tracy (2006)* for the solution of Richards’ equation, with:

eKK s)( ersr )()(

On a domain:

mz

my

mx

24.150

24.150

24.150

With:

24.15sin

24.15sin)1(ln

1)(

yxee rr hh

x

24.15)( x

Top face

All other faces

* F.T.Tracy, Clean two- and three-dimensional analytical solutions of Richards’ equation for testing numerical solvers

00 pS

Richards’ equation - First example

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 2 4 6 8 10 12 14 16 18

z (m)

Hyd

. H

ead

(m

)

AnalyticalCalculated

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 2 4 6 8 10 12 14 16 18

z (m)

Hy

d.

He

ad

(m

)

Analytical

Calculated

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 2 4 6 8 10 12 14 16 18

z (m)

Hyd

. H

ead

(m

)

AnalyticalCalculated

With: (11 x 11 x 11) and (21 x 21 x 21) uniformly spaced points

α = 0.164N = 11

N = 22

α = 0.328N =11

N = 22

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

0 2 4 6 8 10 12 14 16 18

z (m)

Hyd.

Hea

d (

m)

Analytical

Calculated

Richards’ equation - First example (error analysis)

Alpha = 0.164

11*11*11 1.72E-02 3.16E-02

21*21*21 3.52E-03 7.09E-03 6.62e-2

Alpha = 0.328

11*11*11 1.61E-02 3.20E-02

21*21*21 4.24E-03 8.57E-03 1.11

Alpha = 0.492

11*11*11 1.77E-02 2.43E-02

21*21*21 5.16E-03 7.26E-03 5.13

L2 error norm Max errorFinite Volume – max error

Improvement factor:

706.6

129.5

9.34

Good improvement over finite volume results from Tracy paper, particularly with rapidly decaying K and θ functions

Reasonable convergence rate, with increasing point density

Future work Future work will focus on large-scale problems, including:

Regional scale models of real-world experiments in Portugal and Greece Country-scale models of aquifer pumping and seawater intrusion across Israel and Gaza

The large problem size will require a parallel implementation for efficient solution – hence our interest in HPC and GPUs

To our knowledge, we are the only group working on large-scale hydrology problems using meshless numerical techniques

Practical implementation will require the parallelisation of large, sparsely-populated, iterative matrix solvers

GPU Computing: GPU: Graphics Processing Unit Originally designed to accelerate floating-point heavy calculations in computer games eg.

Pixel shader effects (Lighting effects, reflection/refraction, other effects) Geometry setup (character meshes etc) Solid-body physics (…not yet widely adapted)

Massively parallel architecture – currently up to 128 floating point processing units

Recent hardware (from Nov 2006) and software (Feb 2007) advances have allowed programmable processing units (rather than units specialised for pixel or vertex processing)

Has led to "General Purpose GPUs" - ‘GPGPUs’

GPU Computing: GPUs are extremely efficient at handling add-multiply instructions in small ‘packets’ (usually the main computational cost

in numerical simulations)

FP capacity outstrips CPUs, in both theoretical capacity and efficiency (if properly implemented)

GPU Computing: Modern GPUs effectively work like a shared-memory cluster:

GPUs have an extremely fast (~1000Mhz vs ~400Mhz), dedicated onboard memory

Onboard memory sizes currently range up to 1.5Gb (in addition to system memory)

CUDA - BLAS and FFT libraries The CUDA toolkit is a C language

development environment for CUDA-enabled GPUs.

Two libraries implemented on top of CUDA: Basic Linear Algebra System

(BLAS) Fast Fourier Transform (FFT)

Pre-parallelised routines.

Available Examples/Demos:•Parallel bitonic sort •Matrix multiplication •Matrix transpose •Performance profiling using timers •Parallel prefix sum (scan) of large arrays •Image convolution •1D DWT using Haar wavelet •graphics interoperation examples •CUDA BLAS and FFT examples •CPU-GPU C and C++ code integration •Binomial Option Pricing •Black-Scholes Option Pricing •Monte-Carlo Option Pricing •Parallel Mersenne Twister•Parallel Histogram •Image Denoising •Sobel Edge Detection Filter

TESLA - GPUs for HPC

Deskside (2 GPUs 3GB) and Rackmount (4 GPUs 6GB) options

With ~500Gflops per GPU, that is ~2 Teraflops per rack

Deskside is listed at around $4200

GPU Computing – Some results

Example: Multiplication of densely populated matrices O(N3) algorithm…

GPU specs:

GPU model Processors

GPU Clock speed

Memory size

Memory bus COST

8600GTS 32 675Mhz 512Mb 256bit ~£75

8800GT 112 600Mhz 512Mb 256bit ~£150

8800GTX 128 575Mhz 768Mb 320bit ~£250

Matrices are broken down into vector portions, and sent to the GPU stream processors

Use CUDA – nVidia’s C-based API for GPUs

Various CPUs vs GPU

Note: Performance of dual-core and quad-core CPUs is approximated from an idealised parallelisation (ie. 100% efficiency)

Improvement of 8800GTX over various CPUs

0

20

40

60

80

100

120

140

160

180

200

0 1000 2000 3000 4000 5000 6000

Matrix size ( column or row size )

Imp

rov

em

en

t fa

cto

r o

ve

r C

PU

Pentium 4 3Ghz

E2160 (1.86Ghz dual)- £55E6850 (3.0Ghz dual) -£190QX9650 (3Ghz quad)- £690

More GPU propaganda:

Dr. John Stone, Beckmann Institute of Advanced Technology, NAMD virus simulations: 110 CPU hours on SGI Itanium supercomputer => 47minutes with a single GPU Represents a 240-fold speedup

Dr. Graham Pullan, University of Cambridge, CFD with turbine blades (LES and RANS models) 40x absolute speedup switching from a CPU cluster to ‘a few’ GPUs Use 10 million cells on GPU, up from 500,000 on CPU cluster

Good anecdotal evidence for improvement in real-world simulations is available from those who have switched to GPU computing:

Closing words GPU performance is advancing at a much faster rate than CPUs. This is expected to continue for some time yet

With CUDA and BLAS, exploiting parallelism of the GPUs is in some cases easier than traditional MPI approaches

Later this year: 2-3 times the performance of current hardware (over 1 TFLOP per card) Native 64bit capability

More info: www.nvidia.com/tesla www.nvidia.com/cuda www.GPGPU.org