Parallel Solution of 2-D Heat Equation Using Laplace Finite Difference

OAK RIDGE NATIONAL LABORATORYU.S. DEPARTMENT OF ENERGY

Parallel Solution of 2-D Heat Equation Using Laplace Finite Difference

Presented by Valerie SpencerMentored by Jim Kohl

Oak Ridge National Laboratory RAM Internship

June 03 – August 16, 2002

This research was performed under the Research Alliance for Minorities Program administered through the Computer Science and Mathematics Division, Oak Ridge

National Laboratory. This Program is sponsored by the Mathematical, Information, and Computational Sciences Division; Office of Advanced Scientific Computing

Research; U.S. Department of Energy. Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-

00OR22725. This work has been authored by a contractor of the U.S. Government under contract DE-AC05-00OR22725. Accordingly, the U.S. Government retains a

nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.


Table of Contents Purpose

Background work

Problem Statement

Linux

Parallel Virtual Machine

Two parts

Using pvm

Two models

Parallel Program Used

Jacobi Iteration Method

Parallel cluster used

Results

Conclusion


Purpose

The research involves solving a multi-dimensional conduction heat

transfer problem on a Linux cluster using Parallel Virtual

Machine (PVM). Temperature distribution in a X*Y size wall can

be determined by solving the Laplace Equation. The larger the

size gets, the more computing power is needed.


Background Work

lab experiment at Alabama A&M University - determine the

temperature distribution inside a medium of size 1.0 m x 1.0 m

using the finite difference method

heat equation (2-D array notation Laplace Eq.)

[(∂2T / ∂x2)+(∂2T / ∂y2)]I,J = 0

T(I,J) = (TE + TW + TN + TS) / 4, where

TE = T(I+1,J)

TW = T(I-1,J)

TN = T(I,J+1)

TS = T(I-1,J)


Background Cont’d

Fortran program -

Microsoft Developer

Studio software

Visual image of the

temperature

distribution -Tecplot

Software

X

Y

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1T

328.041

316.216

304.392

292.568

280.743

268.919

257.095

245.27

233.446

221.622

209.797

197.973

186.149

174.324

162.5

Temperature Distribution


Problem Statement

Decrease experiment cycle time for solving the heat equation

over large problems

Solution:

Parallelize the heat equation simulation program and run using

multiple computers in a cluster


a sophisticated multitasking virtual memory operating system

directly controls the hardware

provides:

true multitasking

virtual memory

shared libraries

demand loading

shared copy-on-write executables

TCP/IP networking

file systems


software that permits a heterogeneous collection of Unix and/or

NT computers connected together by a network to be used as a

single large parallel computer


Two Part PVM System

daemon is a special purpose process that runs on behalf of the

system to handle all the incoming and outgoing messages; it is

represented by “pvmd3” or “pvmd” and any user with a valid

login id can install and execute this on a machine

library of routines that allows the computers to interact “in

parallel” (resource and task management “addhost” and

“spawn”, pack and send messages…)


Using PVM

The problem must be able to be broken down into several tasks

functional parallelism - breaking the application into different

tasks that perform different functions

data parallelism - having several similar tasks that are the

same, solve over different parts of the data


Two Models of PVM Codes

master/worker model - the master task creates all other tasks

that are designed to work on the problem, then coordinates the

input of initial data to each task, and collects the output of

results from each task

hostless model - the initial task spawns off copies of itself as

tasks and then starts working on its portion of the problem while

the created tasks immediately begin working on their portion


Jacobi Iteration Method

Based on solving for every

variable locally with respect

to the other variables

One iteration of the method

corresponds to solving for

every variable once

Simple parallel data structure

Processes exchange

columns with neighbors Process 0 Process 1 Process 2 Process 3

Boundary Point

Interior Point


TORC(Tennessee Oak Ridge Cluster)

Consists of 18 processors

1 Dell PowerEdge 1300

Dual PIII 450MHz, 512MB RAM

4 Dell PowerEdge 6350

Quad PII Xeon 450MHz, 2GB

RAM(node4 – 1GB RAM)

PGI v3.2 components

pgf77 – fortran 77 compiler

pgf90 – fortran 90 compiler

pghpf – high performance

fortran compiler


ResultsTest Run #1

Additional Processor to Speed Ratio vs. Ideal Speedup for Different Medium Sizes Ranging from (1 x 1)m to (10 x 10)m

0

2

4

6

8

10

12

14

16

18

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Number of Processors

Speedup

Ideal

100 x 100

200 x 200

500 x 500

1000 x 1000


ResultsTest Run #2

Additional Processor to Speed Ratio vs. Ideal Speedup for Different Medium Sizes Ranging from (1 x 1)m to (10 x 10)m

0

2

4

6

8

10

12

14

16

18

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Number of Processors

Speedup

Ideal

100 x 100

200 x 200

500 x 500

1000 x 1000


Conclusion

Larger problem, more computational power needed

Clustered personal computers provides adequate computing

power

Parallelization is not needed for small problems

PVM on a Linux system is an excellent tool to solve large

engineering problems


About Myself

Alabama A&M University, Senior

Mechanical Engineering, G.P.A – 3.8

B.S. - May 2003

U.S. Navy- Nupoc Program


Acknowledgements

I appreciate Dr. Z. T. Deng’s decision of selecting me to be a

part of the Summer 2002 RAM Program. I appreciate Dr. Jim

Kohl for guiding my research, dedicating time to discuss my

findings, and facilitating resources from the Computer

Science and Mathematics Division. I would also like to

acknowledge Debbie McCoy for organizing the RAM

Program that allowed me to spend the summer gaining

significant information at ORNL. Other acknowledgements:

Stephen Scott and Cheryl Hamby.

Parallel Solution of 2-D Heat Equation Using Laplace Finite Difference

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